Subgroups — C₂×#20288569 #28292048

Order 13568.

7994 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288569	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811358	C₂	1	4	yes	maximal, complemented
3	C₂×#16811346	C₂	1	2	yes	maximal, complemented
4	C₂×#16811339	C₂	1	2	yes	maximal, complemented
5	C₂×#16811326	C₂	1	4	yes	maximal, complemented
6	C₂×#16811295	C₂	1	2	yes	maximal, complemented
7	C₂×#16811276	C₂	1	2	yes	maximal, complemented
8	C₂×#16811213	C₂	1	4	yes	maximal, complemented
9	C₂×#16811192	C₂	1	2	yes	maximal, complemented
10	C₂×#16811173	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#16811136	C₂	1	4	yes	maximal, complemented
12	C₂×#16811103	C₂	1	2	yes	maximal, complemented
13	C₂²×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#11719949	C₂²	1	4	yes	complemented
16	C₂²×#11719945	C₂²	1	2	yes	—
17	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
18	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
19	#16811358	C₂²	1	64	yes	complemented
20	#16811346	C₂²	1	32	yes	complemented
21	#16811339	C₂²	1	32	yes	complemented
22	#16811326	C₂²	1	64	yes	complemented
23	#16811295	C₂²	1	32	yes	complemented
24	#16811276	C₂²	1	32	yes	complemented
25	#16811213	C₂²	1	64	yes	complemented
26	#16811192	C₂²	1	32	yes	complemented
27	#16811173	C₂²	1	16	yes	complemented
28	C₂×#13133217	C₂²	1	1	yes	characteristic, complemented
29	C₂×#13133214	C₂²	1	4	yes	—
30	C₂×#13133214	C₂²	1	1	yes	characteristic, complemented
31	C₂×#13133212	C₂²	1	8	yes	complemented
32	C₂×#13133206	C₂²	1	8	yes	complemented
33	C₂×#13133205	C₂²	1	4	yes	—
34	C₂×#13133205	C₂²	1	2	yes	complemented
35	#16811136	C₂²	1	64	yes	complemented
36	C₂×#13133202	C₂²	1	8	yes	complemented
37	C₂×#13133202	C₂²	1	8	yes	complemented
38	C₂×#13133193	C₂²	1	4	yes	complemented
39	C₂×#13133192	C₂²	1	4	yes	—
40	C₂×#13133192	C₂²	1	4	yes	—
41	C₂×#13133192	C₂²	1	4	yes	complemented
42	C₂×#13133192	C₂²	1	8	yes	complemented
43	C₂×#13133191	C₂²	1	2	yes	—
44	C₂×#13133191	C₂²	1	2	yes	—
45	C₂×#13133187	C₂²	1	8	yes	—
46	#16811103	C₂²	1	32	yes	complemented
47	C₂×C₄×Dic₁₀₆	C₂²	1	4	yes	complemented
48	C₂×C₄×D₂₁₂	C₂²	1	2	yes	—
49	C₂×C₄×D₂₁₂	C₂²	1	2	yes	—
50	C₂×C₄×D₂₁₂	C₂²	1	2	yes	—
51	C₂×C₄×D₂₁₂	C₂²	1	2	yes	—
52	C₂×C₄×D₂₁₂	C₂²	1	16	yes	complemented
53	C₂×C₄×D₂₁₂	C₂²	1	2	yes	complemented
54	C₂×C₄×D₂₁₂	C₂²	1	4	yes	complemented
55	C₂×C₄×D₂₁₂	C₂²	1	2	yes	complemented
56	C₂×#13133179	C₂²	1	8	yes	complemented
57	C₂×#13133177	C₂²	1	8	yes	—
58	C₂×#13133175	C₂²	1	4	yes	complemented
59	C₂×#13133175	C₂²	1	4	yes	complemented
60	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
61	C₂×C₄⋊C₄×D₅₃	C₂²	1	4	yes	complemented
62	C₂²×#11719921	C₂²	1	2	yes	—
63	C₅₃×#533	C₂²	1	16	yes	complemented
64	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
65	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
66	C₂×#125×C₅₃	C₂²	1	2	yes	complemented
67	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
68	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
69	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
70	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
71	C₂×#117×C₅₃	C₂²	1	2	yes	—
72	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
73	C₂⁴×D₅₃	C₂³	1	2	yes	—
74	C₂×#11719949	—	4	16	no	—
75	C₂×#11719949	C₂³	1	32	yes	complemented
76	C₂×#11719945	—	2	16	no	—
77	C₂×#11719945	C₂³	1	16	yes	—
78	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
79	C₂²×D₂₁₂	—	2	8	no	—
80	C₂²×D₂₁₂	C₂³	1	2	yes	—
81	C₂²×D₂₁₂	C₂³	1	2	yes	—
82	C₂²×D₂₁₂	C₂³	1	2	yes	—
83	C₂²×D₂₁₂	C₂³	1	8	yes	—
84	C₂²×D₂₁₂	C₂³	1	2	yes	—
85	C₂²×D₂₁₂	C₂³	1	8	yes	—
86	C₂²×D₂₁₂	C₂³	1	2	yes	—
87	C₂²×D₂₁₂	C₂³	1	8	yes	—
88	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
89	C₂³×Dic₅₃	C₂³	1	2	yes	—
90	C₂²×C₄×D₅₃	—	4	8	no	—
91	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
92	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
93	C₂²×C₄×D₅₃	C₂³	1	8	yes	—
94	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
95	C₂²×C₄×D₅₃	C₂³	1	16	yes	—
96	#13133217	C₂³	1	8	yes	complemented
97	#13133214	C₂³	1	32	yes	—
98	#13133214	C₂³	1	8	yes	complemented
99	#13133212	C₂³	1	64	yes	complemented
100	#13133206	C₂³	1	64	yes	complemented
101	#13133205	C₂³	1	32	yes	—
102	#13133205	C₂³	1	16	yes	complemented
103	#13133202	C₂³	1	64	yes	complemented
104	#13133202	C₂³	1	64	yes	complemented
105	#13133193	C₂³	1	32	yes	complemented
106	#13133192	C₂³	1	32	yes	—
107	#13133192	C₂³	1	32	yes	—
108	#13133192	C₂³	1	64	yes	complemented
109	#13133192	C₂³	1	32	yes	complemented
110	#13133191	C₂³	1	16	yes	—
111	#13133191	C₂³	1	16	yes	—
112	#13133187	C₂³	1	64	yes	—
113	C₄×Dic₁₀₆	C₂³	1	32	yes	complemented
114	C₄×D₂₁₂	C₂³	1	16	yes	—
115	C₄×D₂₁₂	C₂³	1	16	yes	—
116	C₄×D₂₁₂	C₂³	1	16	yes	—
117	C₄×D₂₁₂	C₂³	1	16	yes	—
118	C₄×D₂₁₂	C₂³	1	16	yes	complemented
119	C₄×D₂₁₂	C₂³	1	32	yes	complemented
120	C₄×D₂₁₂	C₂³	1	16	yes	complemented
121	#13133179	C₂³	1	64	yes	complemented
122	#13133177	C₂³	1	64	yes	—
123	#13133175	C₂³	1	32	yes	complemented
124	#13133175	C₂³	1	32	yes	complemented
125	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
126	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
127	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
128	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
129	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
130	C₂×#11719919	C₂³	1	8	yes	—
131	C₄⋊C₄×D₅₃	C₂³	1	32	yes	complemented
132	C₂×#11719921	C₂³	1	4	yes	—
133	C₂×#11719921	C₂³	1	4	yes	—
134	C₂×#11719921	C₂³	1	16	yes	—
135	C₂×#11719921	C₂³	1	8	yes	—
136	C₂×#11719921	C₂³	1	8	yes	—
137	C₂×#11719921	C₂³	1	4	yes	—
138	C₂×#11719921	C₂³	1	8	yes	—
139	C₂×#11719921	C₂³	1	4	yes	—
140	C₂×C₄×Dic₅₃	C₂³	1	8	yes	—
141	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
142	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
143	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
144	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
145	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
146	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	complemented
147	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	complemented
148	#125×C₅₃	C₂³	1	16	yes	complemented
149	C₂²⋊Q₈×C₅₃	C₂³	1	16	yes	—
150	C₂²⋊Q₈×C₅₃	C₂³	1	16	yes	—
151	C₄×Q₈×C₅₃	C₂³	1	16	yes	complemented
152	C₄×Q₈×C₅₃	C₂³	1	16	yes	complemented
153	#117×C₅₃	C₂³	1	16	yes	—
154	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
155	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
156	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
157	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
158	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
160	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
161	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
162	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
163	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
164	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
165	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
166	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
167	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
168	C₂³×D₅₃	—	2	8	no	—
169	C₂³×D₅₃	—	4	4	no	—
170	C₂³×D₅₃	—	4	4	no	—
171	C₂³×D₅₃	C₂⁴	1	4	yes	—
172	C₂³×D₅₃	C₂⁴	1	4	yes	—
173	C₂³×D₅₃	C₂⁴	1	8	yes	—
174	C₂×Q₈×C₅₃	—	2	8	no	—
175	C₂×Q₈×C₅₃	—	4	4	no	—
176	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
177	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
178	C₂²×C₄×C₅₃	—	2	4	no	—
179	C₂²×C₄×C₅₃	—	4	2	no	—
180	C₂²×C₄×C₅₃	#41	1	8	yes	complemented
181	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
191	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
192	C₂×C₂²⋊D₅₃	—	2	16	no	—
193	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
194	C₂²×Dic₅₃	—	4	4	no	—
195	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
196	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
197	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
198	#11719949	—	4	64	no	—
199	#11719945	—	2	64	no	—
200	C₂×D₂₁₂	—	2	8	no	—
201	C₂×D₂₁₂	—	2	8	no	—
202	C₂×D₂₁₂	—	2	8	no	—
203	C₂×D₂₁₂	—	2	32	no	—
204	C₂×D₂₁₂	—	2	8	no	—
205	C₂×D₂₁₂	—	4	4	no	—
206	C₂×D₂₁₂	—	4	16	no	—
207	C₂×D₂₁₂	—	4	16	no	—
208	C₂×D₂₁₂	C₂⁴	1	8	yes	—
209	C₂×D₂₁₂	C₂⁴	1	8	yes	—
210	C₂×D₂₁₂	C₂⁴	1	8	yes	—
211	C₂×D₂₁₂	C₂⁴	1	8	yes	—
212	C₂×D₂₁₂	C₂⁴	1	32	yes	—
213	C₂×D₂₁₂	C₂⁴	1	8	yes	—
214	C₂×D₂₁₂	C₂⁴	1	32	yes	—
215	C₂×C₄×D₅₃	—	2	16	no	—
216	C₂×C₄×D₅₃	—	2	16	no	—
217	C₂×C₄×D₅₃	—	4	32	no	—
218	C₂×C₄×D₅₃	—	4	8	no	—
219	C₂×C₄×D₅₃	—	4	16	no	—
220	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
221	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
222	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
223	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
224	C₂×Dic₁₀₆	—	2	8	no	—
225	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
226	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
227	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
228	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
229	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
230	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
231	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
232	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
233	C₄²×C₅₃	C₂⁴	1	4	yes	—
234	C₄²×C₅₃	C₂⁴	1	4	yes	—
235	C₄²×C₅₃	C₂⁴	1	4	yes	—
236	C₄²×C₅₃	C₂⁴	1	4	yes	—
237	C₄²×C₅₃	C₂⁴	1	16	yes	—
238	#11719921	C₂⁴	1	16	yes	—
239	#11719921	C₂⁴	1	16	yes	—
240	#11719921	C₂⁴	1	32	yes	—
241	#11719921	C₂⁴	1	32	yes	—
242	#11719921	C₂⁴	1	16	yes	—
243	#11719921	C₂⁴	1	32	yes	—
244	#11719921	C₂⁴	1	16	yes	—
245	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
246	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
247	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
248	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
249	#11719919	C₂⁴	1	32	yes	—
250	C₄×Dic₅₃	C₂⁴	1	32	yes	—
251	C₂³×C₅₃	—	2	2	no	—
252	C₂³×C₅₃	—	2	2	no	—
253	C₂³×C₅₃	C₂×#41	1	4	yes	—
254	C₂³×C₅₃	C₂⁵	1	2	yes	—
255	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
256	C₂²×D₅₃	—	2	8	no	—
257	C₂²×D₅₃	—	2	16	no	—
258	C₂²×D₅₃	—	2	8	no	—
259	C₂²×D₅₃	—	2	8	no	—
260	C₂²×D₅₃	—	2	8	no	—
261	C₂²×D₅₃	—	2	8	no	—
262	C₂²×D₅₃	—	2	8	no	—
263	C₂²×D₅₃	—	4	8	no	—
264	C₂²×D₅₃	—	4	8	no	—
265	C₂²×D₅₃	—	4	32	no	—
266	C₂²×D₅₃	C₂⁵	1	8	yes	—
267	C₂²×D₅₃	C₂⁵	1	8	yes	—
268	Q₈×C₅₃	—	2	16	no	—
269	Q₈×C₅₃	—	4	8	no	—
270	C₂×C₄×C₅₃	—	2	2	no	—
271	C₂×C₄×C₅₃	—	2	8	no	—
272	C₂×C₄×C₅₃	—	2	2	no	—
273	C₂×C₄×C₅₃	—	2	8	no	—
274	C₂×C₄×C₅₃	—	2	2	no	—
275	C₂×C₄×C₅₃	—	2	8	no	—
276	C₂×C₄×C₅₃	—	4	4	no	—
277	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
278	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
279	C₂×C₄×C₅₃	C₂×#41	1	16	yes	complemented
280	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
281	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
282	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
283	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
284	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
285	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
286	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
287	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
288	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
289	C₂²⋊D₅₃	—	2	32	no	—
290	C₂×Dic₅₃	—	2	8	no	—
291	C₂×Dic₅₃	—	2	8	no	—
292	C₂×Dic₅₃	—	4	8	no	—
293	C₂×Dic₅₃	C₂⁵	1	8	yes	—
294	C₂×Dic₅₃	C₂⁵	1	8	yes	—
295	D₂₁₂	—	2	16	no	—
296	D₂₁₂	—	2	16	no	—
297	D₂₁₂	—	2	16	no	—
298	D₂₁₂	—	2	16	no	—
299	D₂₁₂	—	4	8	no	—
300	D₂₁₂	—	4	32	no	—
301	D₂₁₂	—	4	32	no	—
302	C₄×D₅₃	—	2	32	no	—
303	C₄×D₅₃	—	2	32	no	—
304	C₄×D₅₃	—	4	16	no	—
305	C₄×D₅₃	—	4	32	no	—
306	Dic₁₀₆	—	2	16	no	—
307	C₂×#3506	—	53	1	no	maximal, Sylow
308	C₂²×C₅₃	—	2	2	no	—
309	C₂²×C₅₃	—	2	2	no	—
310	C₂²×C₅₃	—	2	8	no	—
311	C₂²×C₅₃	C₂²×#41	1	4	yes	—
312	C₂²×C₅₃	C₂²×#41	1	2	yes	—
313	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
314	C₂²×C₅₃	#585	1	2	yes	—
315	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
316	C₂×D₅₃	—	2	8	no	—
317	C₂×D₅₃	—	2	8	no	—
318	C₂×D₅₃	—	2	8	no	—
319	C₂×D₅₃	—	2	8	no	—
320	C₂×D₅₃	—	2	8	no	—
321	C₂×D₅₃	—	2	8	no	—
322	C₂×D₅₃	—	4	32	no	—
323	C₂×D₅₃	—	4	16	no	—
324	C₂×D₅₃	—	4	16	no	—
325	C₄×C₅₃	—	2	2	no	—
326	C₄×C₅₃	—	2	8	no	—
327	C₄×C₅₃	—	2	2	no	—
328	C₄×C₅₃	—	2	8	no	—
329	C₄×C₅₃	—	2	2	no	—
330	C₄×C₅₃	C₂²×#41	1	4	yes	—
331	C₄×C₅₃	C₂²×#41	1	4	yes	—
332	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
333	Dic₅₃	—	2	8	no	—
334	Dic₅₃	—	2	8	no	—
335	#3506	—	53	32	no	—
336	C₂×#559	—	53	4	no	—
337	C₂×#556	—	53	2	no	—
338	C₂×#555	—	53	2	no	—
339	C₂×#553	—	53	4	no	—
340	C₂×#550	—	53	2	no	—
341	C₂×#547	—	53	2	no	—
342	C₂×#538	—	53	4	no	—
343	C₂×#535	—	53	2	no	—
344	C₂×#533	D₅₃	1	1	yes	characteristic, complemented
345	C₂×#532	—	53	1	no	—
346	C₂×#525	—	53	4	no	—
347	C₂×#520	—	53	2	no	—
348	C₂²×C₄×D₄	—	53	1	no	—
349	C₂×C₅₃	—	2	4	no	—
350	C₂×C₅₃	#3506	1	4	yes	complemented
351	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
352	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
353	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
354	D₅₃	—	4	8	no	—
355	D₅₃	—	4	8	no	—
356	C₂²×#41	—	53	4	no	—
357	C₂²×#41	—	53	2	no	—
358	C₂³×D₄	—	53	1	no	—
359	C₂⁴×C₄	—	53	2	no	—
360	#559	—	53	64	no	—
361	#556	—	53	32	no	—
362	#555	—	53	32	no	—
363	#553	—	53	64	no	—
364	#550	—	53	32	no	—
365	#547	—	53	32	no	—
366	#538	—	53	64	no	—
367	#535	—	53	32	no	—
368	#533	C₂×D₅₃	1	16	yes	complemented
369	#532	—	53	16	no	—
370	C₂×C₄⋊Q₈	—	53	1	no	—
371	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
372	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
373	C₂×#127	—	53	1	no	—
374	C₂×#127	—	53	4	no	—
375	C₂×#126	—	53	8	no	—
376	C₂×#125	C₂×D₅₃	1	2	yes	complemented
377	C₂×#124	—	53	8	no	—
378	C₂×#124	—	53	2	no	—
379	C₂×#124	—	53	4	no	—
380	#525	—	53	64	no	—
381	C₂×#123	—	53	8	no	—
382	C₂×#123	—	53	8	no	—
383	C₂×C₂²⋊Q₈	—	53	4	no	—
384	C₂×C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
385	C₂×C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
386	C₂×#121	—	53	2	no	—
387	C₂×#121	—	53	2	no	—
388	C₂×#121	—	53	4	no	—
389	C₂×#121	—	53	4	no	—
390	C₂×#121	—	53	4	no	—
391	C₂×#121	—	53	8	no	—
392	C₂×#120	—	53	8	no	—
393	#520	—	53	32	no	—
394	C₂×C₄×Q₈	—	53	4	no	—
395	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
396	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
397	C₂×C₄×D₄	—	53	8	no	—
398	C₂×C₄×D₄	—	53	8	no	—
399	C₂×C₄×D₄	—	53	2	no	—
400	C₂×C₄×D₄	—	53	2	no	—
401	C₂×C₄×D₄	—	53	2	no	—
402	C₂×C₄×D₄	—	53	2	no	—
403	C₂×C₄×D₄	—	53	16	no	—
404	C₂×C₄×D₄	—	53	4	no	—
405	C₂×C₄×D₄	—	53	2	no	—
406	C₂×C₄×D₄	—	53	2	no	—
407	C₂×#117	—	53	4	no	—
408	C₂×#117	—	53	4	no	—
409	C₂×#117	C₂×D₅₃	1	2	yes	—
410	C₂²×C₄⋊C₄	—	53	1	no	—
411	C₂²×C₄⋊C₄	—	53	4	no	—
412	C₂²×C₂²⋊C₄	—	53	2	no	—
413	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
414	C₅₃	C₂×#3506	1	1	yes	characteristic, complemented, Sylow
415	C₂⁵	—	53	2	no	—
416	C₂×#41	—	53	32	no	—
417	C₂×#41	—	53	16	no	—
418	C₂×#41	—	106	16	no	—
419	C₂×#41	—	212	16	no	—
420	C₂²×Q₈	—	53	2	no	—
421	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
422	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
423	C₂²×D₄	—	53	4	no	—
424	C₂²×D₄	—	53	2	no	—
425	C₂²×D₄	—	53	2	no	—
426	C₂²×D₄	—	53	2	no	—
427	C₂²×D₄	—	53	8	no	—
428	C₂²×D₄	—	53	2	no	—
429	C₂²×D₄	—	53	8	no	—
430	C₂²×D₄	—	53	2	no	—
431	C₂²×D₄	—	53	8	no	—
432	C₂²×D₄	—	106	8	no	—
433	C₂³×C₄	—	53	2	no	—
434	C₂³×C₄	—	53	4	no	—
435	C₂³×C₄	—	53	4	no	—
436	C₂³×C₄	—	53	8	no	—
437	C₂³×C₄	—	53	4	no	—
438	C₂³×C₄	—	53	16	no	—
439	C₂³×C₄	—	212	8	no	—
440	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
441	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
442	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
443	C₄⋊Q₈	—	53	8	no	—
444	C₄⋊Q₈	C₂²×D₅₃	1	8	yes	complemented
445	C₄⋊Q₈	C₂²×D₅₃	1	8	yes	complemented
446	#127	—	53	8	no	—
447	#127	—	53	32	no	—
448	#126	—	53	64	no	—
449	#125	C₂²×D₅₃	1	16	yes	complemented
450	#124	—	53	64	no	—
451	#124	—	53	16	no	—
452	#124	—	53	32	no	—
453	#123	—	53	64	no	—
454	#123	—	53	64	no	—
455	C₂²⋊Q₈	—	53	32	no	—
456	C₂²⋊Q₈	C₂²×D₅₃	1	16	yes	—
457	C₂²⋊Q₈	C₂²×D₅₃	1	16	yes	—
458	#121	—	53	16	no	—
459	#121	—	53	16	no	—
460	#121	—	53	32	no	—
461	#121	—	53	32	no	—
462	#121	—	53	32	no	—
463	#121	—	53	64	no	—
464	#120	—	53	64	no	—
465	C₄×Q₈	—	53	32	no	—
466	C₄×Q₈	C₂²×D₅₃	1	16	yes	complemented
467	C₄×Q₈	C₂²×D₅₃	1	16	yes	complemented
468	C₄×D₄	—	53	64	no	—
469	C₄×D₄	—	53	64	no	—
470	C₄×D₄	—	53	16	no	—
471	C₄×D₄	—	53	16	no	—
472	C₄×D₄	—	53	16	no	—
473	C₄×D₄	—	53	16	no	—
474	C₄×D₄	—	53	32	no	—
475	C₄×D₄	—	53	16	no	—
476	C₄×D₄	—	53	16	no	—
477	#117	—	53	32	no	—
478	#117	—	53	32	no	—
479	#117	C₂²×D₅₃	1	16	yes	—
480	C₂×C₄⋊C₄	—	53	2	no	—
481	C₂×C₄⋊C₄	—	53	2	no	—
482	C₂×C₄⋊C₄	—	53	8	no	—
483	C₂×C₄⋊C₄	—	53	8	no	—
484	C₂×C₄⋊C₄	—	53	2	no	—
485	C₂×C₄⋊C₄	—	53	2	no	—
486	C₂×C₄⋊C₄	—	53	32	no	—
487	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
488	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
489	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
490	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
491	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
492	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
493	C₂×C₂²⋊C₄	—	53	4	no	—
494	C₂×C₂²⋊C₄	—	53	4	no	—
495	C₂×C₂²⋊C₄	—	53	16	no	—
496	C₂×C₂²⋊C₄	—	53	8	no	—
497	C₂×C₂²⋊C₄	—	53	8	no	—
498	C₂×C₂²⋊C₄	—	53	4	no	—
499	C₂×C₂²⋊C₄	—	53	8	no	—
500	C₂×C₂²⋊C₄	—	53	4	no	—
501	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
502	C₂×C₄²	—	53	8	no	—
503	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
504	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
505	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
506	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
507	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
508	C₂×C₄²	C₂²×D₅₃	1	8	yes	—
509	C₂⁴	—	53	4	no	—
510	C₂⁴	—	53	4	no	—
511	C₂⁴	—	53	8	no	—
512	C₂⁴	—	106	8	no	—
513	C₂⁴	—	212	4	no	—
514	C₂⁴	—	212	4	no	—
515	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
516	#41	—	106	64	no	—
517	#41	—	212	64	no	—
518	C₂×Q₈	—	2	8	no	—
519	C₂×Q₈	—	4	4	no	—
520	C₂×Q₈	—	53	8	no	—
521	C₂×Q₈	—	106	8	no	—
522	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
523	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
524	C₂×D₄	—	53	16	no	—
525	C₂×D₄	—	53	8	no	—
526	C₂×D₄	—	53	8	no	—
527	C₂×D₄	—	53	8	no	—
528	C₂×D₄	—	53	8	no	—
529	C₂×D₄	—	53	32	no	—
530	C₂×D₄	—	53	8	no	—
531	C₂×D₄	—	53	32	no	—
532	C₂×D₄	—	106	16	no	—
533	C₂×D₄	—	106	8	no	—
534	C₂×D₄	—	106	8	no	—
535	C₂×D₄	—	106	8	no	—
536	C₂×D₄	—	106	32	no	—
537	C₂×D₄	—	106	8	no	—
538	C₂×D₄	—	212	4	no	—
539	C₂×D₄	—	212	16	no	—
540	C₂×D₄	—	212	16	no	—
541	C₂²×C₄	—	2	4	no	—
542	C₂²×C₄	—	4	2	no	—
543	C₂²×C₄	—	53	4	no	—
544	C₂²×C₄	—	53	4	no	—
545	C₂²×C₄	—	53	8	no	—
546	C₂²×C₄	—	53	16	no	—
547	C₂²×C₄	—	53	16	no	—
548	C₂²×C₄	—	53	32	no	—
549	C₂²×C₄	—	53	16	no	—
550	C₂²×C₄	—	106	16	no	—
551	C₂²×C₄	—	106	16	no	—
552	C₂²×C₄	—	212	32	no	—
553	C₂²×C₄	—	212	8	no	—
554	C₂²×C₄	—	212	16	no	—
555	C₂²×C₄	—	212	4	no	—
556	C₂²×C₄	#11719949	1	8	yes	complemented
557	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
558	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
559	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
560	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
561	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
562	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
563	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
564	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
565	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
566	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
567	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
568	C₄⋊C₄	—	53	8	no	—
569	C₄⋊C₄	—	53	8	no	—
570	C₄⋊C₄	—	53	32	no	—
571	C₄⋊C₄	—	53	8	no	—
572	C₄⋊C₄	—	53	8	no	—
573	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
574	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
575	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
576	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
577	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
578	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
579	C₂²⋊C₄	—	53	16	no	—
580	C₂²⋊C₄	—	53	16	no	—
581	C₂²⋊C₄	—	53	32	no	—
582	C₂²⋊C₄	—	53	32	no	—
583	C₂²⋊C₄	—	53	16	no	—
584	C₂²⋊C₄	—	53	32	no	—
585	C₂²⋊C₄	—	53	16	no	—
586	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
587	C₄²	—	53	32	no	—
588	C₄²	C₂³×D₅₃	1	4	yes	—
589	C₄²	C₂³×D₅₃	1	4	yes	—
590	C₄²	C₂³×D₅₃	1	4	yes	—
591	C₄²	C₂³×D₅₃	1	4	yes	—
592	C₄²	C₂³×D₅₃	1	16	yes	—
593	C₂³	—	2	2	no	—
594	C₂³	—	2	2	no	—
595	C₂³	—	53	8	no	—
596	C₂³	—	53	8	no	—
597	C₂³	—	106	8	no	—
598	C₂³	—	106	16	no	—
599	C₂³	—	106	8	no	—
600	C₂³	—	106	8	no	—
601	C₂³	—	106	8	no	—
602	C₂³	—	106	8	no	—
603	C₂³	—	106	8	no	—
604	C₂³	—	212	8	no	—
605	C₂³	—	212	8	no	—
606	C₂³	—	212	32	no	—
607	C₂³	C₂×#11719949	1	4	yes	—
608	C₂³	C₂⁴×D₅₃	1	2	yes	—
609	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
610	Q₈	—	2	16	no	—
611	Q₈	—	4	8	no	—
612	Q₈	—	106	16	no	—
613	D₄	—	106	32	no	—
614	D₄	—	106	16	no	—
615	D₄	—	106	16	no	—
616	D₄	—	106	16	no	—
617	D₄	—	106	16	no	—
618	D₄	—	212	8	no	—
619	D₄	—	212	32	no	—
620	D₄	—	212	32	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	8	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	8	no	—
626	C₂×C₄	—	2	8	no	—
627	C₂×C₄	—	4	4	no	—
628	C₂×C₄	—	53	8	no	—
629	C₂×C₄	—	53	8	no	—
630	C₂×C₄	—	106	8	no	—
631	C₂×C₄	—	106	8	no	—
632	C₂×C₄	—	106	32	no	—
633	C₂×C₄	—	106	32	no	—
634	C₂×C₄	—	212	16	no	—
635	C₂×C₄	—	212	32	no	—
636	C₂×C₄	—	212	8	no	—
637	C₂×C₄	C₂×#11719949	1	4	yes	—
638	C₂×C₄	C₂×#11719949	1	4	yes	—
639	C₂×C₄	C₂×#11719949	1	16	yes	complemented
640	C₂×C₄	#13133263	1	4	yes	—
641	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
642	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
643	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
644	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
645	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
646	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
647	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
648	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
649	C₂²	—	2	8	no	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	106	8	no	—
653	C₂²	—	106	8	no	—
654	C₂²	—	106	8	no	—
655	C₂²	—	106	8	no	—
656	C₂²	—	106	8	no	—
657	C₂²	—	106	8	no	—
658	C₂²	—	212	32	no	—
659	C₂²	—	212	16	no	—
660	C₂²	—	212	16	no	—
661	C₂²	C₂²×#11719949	1	4	yes	—
662	C₂²	C₂²×#11719949	1	2	yes	—
663	C₂²	C₂×#13133263	1	2	yes	—
664	C₂²	#16811459	1	2	yes	—
665	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
666	C₄	—	2	2	no	—
667	C₄	—	2	8	no	—
668	C₄	—	2	2	no	—
669	C₄	—	2	2	no	—
670	C₄	—	2	8	no	—
671	C₄	—	106	8	no	—
672	C₄	—	106	8	no	—
673	C₄	C₂²×#11719949	1	4	yes	—
674	C₄	C₂²×#11719949	1	4	yes	—
675	C₄	C₂×#13133263	1	4	yes	—
676	C₂	—	2	4	no	—
677	C₂	—	212	8	no	—
678	C₂	—	212	8	no	—
679	C₂	#20288569	1	4	yes	direct product factor
680	C₂	C₂³×#11719949	1	1	yes	characteristic
681	C₂	C₂²×#13133263	1	1	yes	characteristic
682	C₂	C₂×#16811459	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#533	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#533	D₅₃
hypocenter	C₅₃	C₂×#3506
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#533

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20288569	C₂	direct (N × H)
2. C₂×#16811358	C₂	semidirect
3. C₂×#16811346	C₂	semidirect
4. C₂×#16811339	C₂	semidirect
5. C₂×#16811326	C₂	semidirect
6. C₂×#16811295	C₂	semidirect
7. C₂×#16811276	C₂	semidirect
8. C₂×#16811213	C₂	semidirect
9. C₂×#16811192	C₂	semidirect
10. C₂×#16811173	C₂	semidirect
11. C₂×#16811136	C₂	semidirect
12. C₂×#16811103	C₂	semidirect
13. C₂²×C₄×D₂₁₂	C₂	semidirect
14. C₂×C₅₃×#533	C₂	semidirect
15. C₂²×#11719949	C₂²	semidirect
19. #16811358	C₂²	semidirect
20. #16811346	C₂²	semidirect
21. #16811339	C₂²	semidirect
22. #16811326	C₂²	semidirect
23. #16811295	C₂²	semidirect
24. #16811276	C₂²	semidirect
25. #16811213	C₂²	semidirect
26. #16811192	C₂²	semidirect
27. #16811173	C₂²	semidirect
28. C₂×#13133217	C₂²	semidirect
30. C₂×#13133214	C₂²	semidirect
31. C₂×#13133212	C₂²	semidirect
32. C₂×#13133206	C₂²	semidirect
34. C₂×#13133205	C₂²	semidirect
35. #16811136	C₂²	semidirect
36. C₂×#13133202	C₂²	semidirect
37. C₂×#13133202	C₂²	semidirect
38. C₂×#13133193	C₂²	semidirect
41. C₂×#13133192	C₂²	semidirect
42. C₂×#13133192	C₂²	semidirect
46. #16811103	C₂²	semidirect
47. C₂×C₄×Dic₁₀₆	C₂²	semidirect
52. C₂×C₄×D₂₁₂	C₂²	semidirect
53. C₂×C₄×D₂₁₂	C₂²	semidirect
54. C₂×C₄×D₂₁₂	C₂²	semidirect
55. C₂×C₄×D₂₁₂	C₂²	semidirect
56. C₂×#13133179	C₂²	semidirect
58. C₂×#13133175	C₂²	semidirect
59. C₂×#13133175	C₂²	semidirect
61. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
63. C₅₃×#533	C₂²	semidirect
64. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
65. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
66. C₂×#125×C₅₃	C₂²	semidirect
69. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
70. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
75. C₂×#11719949	C₂³	semidirect
96. #13133217	C₂³	semidirect
98. #13133214	C₂³	semidirect
99. #13133212	C₂³	semidirect
100. #13133206	C₂³	semidirect
102. #13133205	C₂³	semidirect
103. #13133202	C₂³	semidirect
104. #13133202	C₂³	semidirect
105. #13133193	C₂³	semidirect
108. #13133192	C₂³	semidirect
109. #13133192	C₂³	semidirect
113. C₄×Dic₁₀₆	C₂³	semidirect
118. C₄×D₂₁₂	C₂³	semidirect
119. C₄×D₂₁₂	C₂³	semidirect
120. C₄×D₂₁₂	C₂³	semidirect
121. #13133179	C₂³	semidirect
123. #13133175	C₂³	semidirect
124. #13133175	C₂³	semidirect
131. C₄⋊C₄×D₅₃	C₂³	semidirect
146. C₄⋊Q₈×C₅₃	C₂³	semidirect
147. C₄⋊Q₈×C₅₃	C₂³	semidirect
148. #125×C₅₃	C₂³	semidirect
151. C₄×Q₈×C₅₃	C₂³	semidirect
152. C₄×Q₈×C₅₃	C₂³	semidirect
180. C₂²×C₄×C₅₃	#41	semidirect
279. C₂×C₄×C₅₃	C₂×#41	semidirect
344. C₂×#533	D₅₃	semidirect
350. C₂×C₅₃	#3506	semidirect
368. #533	C₂×D₅₃	semidirect
371. C₂×C₄⋊Q₈	C₂×D₅₃	semidirect
372. C₂×C₄⋊Q₈	C₂×D₅₃	semidirect
376. C₂×#125	C₂×D₅₃	semidirect
395. C₂×C₄×Q₈	C₂×D₅₃	semidirect
396. C₂×C₄×Q₈	C₂×D₅₃	semidirect
414. C₅₃	C₂×#3506	semidirect
444. C₄⋊Q₈	C₂²×D₅₃	semidirect
445. C₄⋊Q₈	C₂²×D₅₃	semidirect
449. #125	C₂²×D₅₃	semidirect
466. C₄×Q₈	C₂²×D₅₃	semidirect
467. C₄×Q₈	C₂²×D₅₃	semidirect
556. C₂²×C₄	#11719949	semidirect
639. C₂×C₄	C₂×#11719949	semidirect
679. C₂	#20288569	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20288569	2
C₂×#16811358	2
C₂×#16811346	2
C₂×#16811339	2
C₂×#16811326	2
C₂×#16811295	2
C₂×#16811276	2
C₂×#16811213	2
C₂×#16811192	2
C₂×#16811173	2
C₂×#16811136	2
C₂×#16811103	2
C₂²×C₄×D₂₁₂	2
C₂×C₅₃×#533	2
C₂×#3506	53

Subgroups — C2×#20288569 #28292048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288569 #28292048