Subgroups — C₂×#20288405 #28292006

Order 13568.

9502 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288405	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811300	C₂	1	8	yes	maximal, complemented
3	C₂×#16811264	C₂	1	8	yes	maximal, complemented
4	C₂×#16811213	C₂	1	4	yes	maximal, complemented
5	C₂×#16811185	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16811136	C₂	1	4	yes	maximal, complemented
7	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
8	C₂×#16811096	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
12	C₂²×#11719949	C₂²	1	4	yes	complemented
13	C₂³×D₂₁₂	C₂²	1	4	yes	complemented
14	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
15	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
16	#16811300	C₂²	1	128	yes	complemented
17	#16811264	C₂²	1	128	yes	complemented
18	#16811213	C₂²	1	64	yes	complemented
19	#16811185	C₂²	1	16	yes	complemented
20	C₂×#13133212	C₂²	1	8	yes	complemented
21	C₂×#13133206	C₂²	1	8	yes	complemented
22	#16811136	C₂²	1	64	yes	complemented
23	C₂×#13133202	C₂²	1	8	yes	complemented
24	C₂×#13133202	C₂²	1	8	yes	complemented
25	C₂×#13133201	C₂²	1	4	yes	complemented
26	C₂×#13133193	C₂²	1	4	yes	complemented
27	C₂×#13133192	C₂²	1	8	yes	complemented
28	C₂×#13133192	C₂²	1	4	yes	complemented
29	C₂×#13133191	C₂²	1	4	yes	complemented
30	C₂×#13133187	C₂²	1	8	yes	—
31	C₂×#13133187	C₂²	1	4	yes	complemented
32	C₂×#13133187	C₂²	1	4	yes	complemented
33	C₂×#13133187	C₂²	1	32	yes	complemented
34	C₂×#13133187	C₂²	1	4	yes	complemented
35	#16811096	C₂²	1	32	yes	complemented
36	C₂×#13133179	C₂²	1	8	yes	complemented
37	C₂×C₄×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
38	C₂×#13133177	C₂²	1	8	yes	complemented
39	C₂×#13133175	C₂²	1	4	yes	complemented
40	C₂×#13133173	C₂²	1	4	yes	complemented
41	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
42	C₂²×#11719921	C₂²	1	2	yes	complemented
43	C₂²×#11719921	C₂²	1	4	yes	complemented
44	C₂×C₂²⋊C₄×D₅₃	C₂²	1	8	yes	complemented
45	C₂×C₂²⋊C₄×D₅₃	C₂²	1	16	yes	complemented
46	C₂×C₂²⋊C₄×D₅₃	C₂²	1	4	yes	complemented
47	C₅₃×#543	C₂²	1	16	yes	complemented
48	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
49	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
50	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
51	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
52	C₂⁴×D₅₃	—	2	8	no	—
53	C₂⁴×D₅₃	—	4	8	no	—
54	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
55	C₂⁴×D₅₃	C₂³	1	4	yes	—
56	C₂⁴×D₅₃	C₂³	1	8	yes	—
57	C₂⁴×D₅₃	C₂³	1	2	yes	—
58	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
59	C₂×#11719949	—	4	16	no	—
60	C₂×#11719949	C₂³	1	32	yes	complemented
61	C₂²×D₂₁₂	—	4	16	no	—
62	C₂²×D₂₁₂	C₂³	1	8	yes	—
63	C₂²×D₂₁₂	C₂³	1	8	yes	—
64	C₂²×D₂₁₂	C₂³	1	32	yes	complemented
65	C₂²×D₂₁₂	C₂³	1	4	yes	complemented
66	C₂²×C₂²⋊D₅₃	—	2	16	no	—
67	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
68	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
69	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	—
70	C₂³×Dic₅₃	C₂³	1	2	yes	—
71	C₂³×Dic₅₃	C₂³	1	2	yes	—
72	C₂²×C₄×D₅₃	—	4	8	no	—
73	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
74	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
75	C₂²×C₄×D₅₃	C₂³	1	16	yes	complemented
76	C₂²×C₄×D₅₃	C₂³	1	8	yes	complemented
77	#13133212	C₂³	1	64	yes	complemented
78	#13133206	C₂³	1	64	yes	complemented
79	#13133202	C₂³	1	64	yes	complemented
80	#13133202	C₂³	1	64	yes	complemented
81	#13133201	C₂³	1	32	yes	complemented
82	#13133193	C₂³	1	32	yes	complemented
83	#13133192	C₂³	1	32	yes	complemented
84	#13133192	C₂³	1	64	yes	complemented
85	#13133191	C₂³	1	32	yes	complemented
86	#13133187	C₂³	1	64	yes	—
87	#13133187	C₂³	1	32	yes	complemented
88	#13133187	C₂³	1	32	yes	complemented
89	#13133187	C₂³	1	32	yes	complemented
90	#13133179	C₂³	1	64	yes	complemented
91	C₄×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
92	#13133177	C₂³	1	64	yes	complemented
93	#13133175	C₂³	1	32	yes	complemented
94	#13133173	C₂³	1	32	yes	complemented
95	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	complemented
96	C₂×#11719919	C₂³	1	8	yes	complemented
97	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
98	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
99	C₂×#11719921	C₂³	1	8	yes	—
100	C₂×#11719921	C₂³	1	8	yes	—
101	C₂×#11719921	C₂³	1	8	yes	—
102	C₂×#11719921	C₂³	1	32	yes	complemented
103	C₂×#11719921	C₂³	1	16	yes	complemented
104	C₂×#11719921	C₂³	1	8	yes	complemented
105	C₂×#11719921	C₂³	1	8	yes	complemented
106	C₂²⋊C₄×D₅₃	C₂³	1	64	yes	complemented
107	C₂²⋊C₄×D₅₃	C₂³	1	32	yes	complemented
108	C₂×C₄×Dic₅₃	C₂³	1	8	yes	complemented
109	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
110	C₂²×Q₈×C₅₃	C₂³	1	4	yes	complemented
111	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
112	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
113	C₂²⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
114	C₂²⋊Q₈×C₅₃	C₂³	1	32	yes	complemented
115	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
116	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
117	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
122	C₂⁴×C₅₃	—	4	4	no	—
123	C₂⁴×C₅₃	#41	1	8	yes	complemented
124	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
125	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
127	C₂³×D₅₃	—	2	4	no	—
128	C₂³×D₅₃	—	2	8	no	—
129	C₂³×D₅₃	—	2	16	no	—
130	C₂³×D₅₃	—	2	4	no	—
131	C₂³×D₅₃	—	2	32	no	—
132	C₂³×D₅₃	—	2	8	no	—
133	C₂³×D₅₃	—	2	16	no	—
134	C₂³×D₅₃	—	4	32	no	—
135	C₂³×D₅₃	—	4	8	no	—
136	C₂³×D₅₃	—	4	4	no	—
137	C₂³×D₅₃	—	4	8	no	—
138	C₂³×D₅₃	—	4	64	no	—
139	C₂³×D₅₃	C₂⁴	1	2	yes	—
140	C₂³×D₅₃	C₂⁴	1	4	yes	—
141	C₂³×D₅₃	C₂⁴	1	4	yes	—
142	C₂³×D₅₃	C₂⁴	1	16	yes	—
143	C₂³×D₅₃	C₂⁴	1	2	yes	—
144	C₂³×D₅₃	C₂⁴	1	8	yes	—
145	C₂³×D₅₃	C₂⁴	1	4	yes	—
146	C₂×Q₈×C₅₃	—	4	8	no	—
147	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
148	C₂²×C₄×C₅₃	—	4	4	no	—
149	C₂²×C₄×C₅₃	—	4	8	no	—
150	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
154	C₂×C₂²⋊D₅₃	—	2	16	no	—
155	C₂×C₂²⋊D₅₃	—	2	16	no	—
156	C₂×C₂²⋊D₅₃	—	2	64	no	—
157	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
158	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
159	C₂²×Dic₅₃	—	2	8	no	—
160	C₂²×Dic₅₃	—	4	4	no	—
161	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
162	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
163	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
164	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
165	#11719949	—	4	64	no	—
166	C₂×D₂₁₂	—	2	32	no	—
167	C₂×D₂₁₂	—	4	64	no	—
168	C₂×D₂₁₂	—	4	16	no	—
169	C₂×D₂₁₂	—	4	8	no	—
170	C₂×D₂₁₂	C₂⁴	1	32	yes	—
171	C₂×D₂₁₂	C₂⁴	1	32	yes	—
172	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
173	C₂×C₄×D₅₃	—	2	16	no	—
174	C₂×C₄×D₅₃	—	4	32	no	—
175	C₂×C₄×D₅₃	—	4	8	no	—
176	C₂×C₄×D₅₃	—	4	16	no	—
177	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
178	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
179	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
180	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
181	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
182	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
183	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
184	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
185	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
186	#11719921	C₂⁴	1	32	yes	—
187	#11719921	C₂⁴	1	32	yes	—
188	#11719921	C₂⁴	1	32	yes	—
189	#11719921	C₂⁴	1	32	yes	complemented
190	#11719921	C₂⁴	1	32	yes	complemented
191	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
192	#11719919	C₂⁴	1	32	yes	complemented
193	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
194	C₂³×C₅₃	—	2	4	no	—
195	C₂³×C₅₃	—	2	8	no	—
196	C₂³×C₅₃	—	4	16	no	—
197	C₂³×C₅₃	—	4	8	no	—
198	C₂³×C₅₃	C₂×#41	1	8	yes	—
199	C₂³×C₅₃	C₂×#41	1	4	yes	—
200	C₂³×C₅₃	C₂×#41	1	16	yes	complemented
201	C₂³×C₅₃	C₂⁵	1	2	yes	—
202	C₂³×C₅₃	C₂⁵	1	4	yes	—
203	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
204	C₂²×D₅₃	—	2	32	no	—
205	C₂²×D₅₃	—	2	8	no	—
206	C₂²×D₅₃	—	2	4	no	—
207	C₂²×D₅₃	—	2	4	no	—
208	C₂²×D₅₃	—	2	16	no	—
209	C₂²×D₅₃	—	2	8	no	—
210	C₂²×D₅₃	—	2	32	no	—
211	C₂²×D₅₃	—	2	8	no	—
212	C₂²×D₅₃	—	2	4	no	—
213	C₂²×D₅₃	—	2	16	no	—
214	C₂²×D₅₃	—	2	8	no	—
215	C₂²×D₅₃	—	2	16	no	—
216	C₂²×D₅₃	—	2	32	no	—
217	C₂²×D₅₃	—	4	16	no	—
218	C₂²×D₅₃	—	4	8	no	—
219	C₂²×D₅₃	—	4	16	no	—
220	C₂²×D₅₃	—	4	64	no	—
221	C₂²×D₅₃	—	4	128	no	—
222	C₂²×D₅₃	—	4	16	no	—
223	C₂²×D₅₃	—	8	16	no	—
224	C₂²×D₅₃	H(5,𝔽₂)	1	8	yes	—
225	C₂²×D₅₃	C₂⁵	1	8	yes	—
226	C₂²×D₅₃	C₂⁵	1	4	yes	—
227	C₂²×D₅₃	C₂⁵	1	4	yes	—
228	Q₈×C₅₃	—	4	16	no	—
229	C₂×C₄×C₅₃	—	2	8	no	—
230	C₂×C₄×C₅₃	—	2	16	no	—
231	C₂×C₄×C₅₃	—	4	8	no	—
232	C₂×C₄×C₅₃	—	4	16	no	—
233	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
234	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
235	C₂²⋊D₅₃	—	2	32	no	—
236	C₂²⋊D₅₃	—	2	32	no	—
237	C₂×Dic₅₃	—	2	16	no	—
238	C₂×Dic₅₃	—	2	8	no	—
239	C₂×Dic₅₃	—	2	8	no	—
240	C₂×Dic₅₃	—	4	8	no	—
241	C₂×Dic₅₃	C₂⁵	1	8	yes	—
242	C₂×Dic₅₃	C₂⁵	1	8	yes	—
243	D₂₁₂	—	2	64	no	—
244	D₂₁₂	—	4	32	no	—
245	D₂₁₂	—	4	16	no	—
246	C₄×D₅₃	—	2	32	no	—
247	C₄×D₅₃	—	4	16	no	—
248	C₄×D₅₃	—	4	32	no	—
249	C₂×#3490	—	53	1	no	maximal, Sylow
250	C₂²×C₅₃	—	2	4	no	—
251	C₂²×C₅₃	—	2	8	no	—
252	C₂²×C₅₃	—	2	8	no	—
253	C₂²×C₅₃	—	2	16	no	—
254	C₂²×C₅₃	—	4	16	no	—
255	C₂²×C₅₃	C₂²×#41	1	8	yes	—
256	C₂²×C₅₃	C₂²×#41	1	4	yes	—
257	C₂²×C₅₃	C₂²×#41	1	2	yes	—
258	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
259	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
260	C₂×D₅₃	—	2	32	no	—
261	C₂×D₅₃	—	2	4	no	—
262	C₂×D₅₃	—	2	4	no	—
263	C₂×D₅₃	—	2	16	no	—
264	C₂×D₅₃	—	2	8	no	—
265	C₂×D₅₃	—	2	4	no	—
266	C₂×D₅₃	—	2	16	no	—
267	C₂×D₅₃	—	2	8	no	—
268	C₂×D₅₃	—	4	64	no	—
269	C₂×D₅₃	—	4	16	no	—
270	C₂×D₅₃	—	4	8	no	—
271	C₂×D₅₃	—	4	16	no	—
272	C₂×D₅₃	—	8	16	no	—
273	C₂×D₅₃	C₂×H(5,𝔽₂)	1	8	yes	—
274	C₄×C₅₃	—	2	8	no	—
275	C₄×C₅₃	—	2	16	no	—
276	Dic₅₃	—	2	8	no	—
277	Dic₅₃	—	2	8	no	—
278	#3490	—	53	32	no	—
279	C₂×#551	—	53	8	no	—
280	C₂×#546	—	53	8	no	—
281	C₂×#543	D₅₃	1	1	yes	characteristic, complemented
282	C₂×#538	—	53	4	no	—
283	C₂×#534	—	53	1	no	—
284	C₂×#525	—	53	4	no	—
285	C₂²×#120	—	53	2	no	—
286	C₂×#518	—	53	2	no	—
287	C₂³×C₂²⋊C₄	—	53	1	no	—
288	C₂×C₅₃	—	2	4	no	—
289	C₂×C₅₃	—	2	8	no	—
290	C₂×C₅₃	#3490	1	4	yes	complemented
291	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
292	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
293	D₅₃	—	2	8	no	—
294	D₅₃	—	4	4	no	—
295	D₅₃	—	4	8	no	—
296	C₂⁶	—	53	1	no	—
297	C₂²×#41	—	53	4	no	—
298	C₂³×D₄	—	53	2	no	—
299	C₂³×D₄	—	53	4	no	—
300	C₂⁴×C₄	—	53	2	no	—
301	#551	—	53	128	no	—
302	#546	—	53	128	no	—
303	#543	C₂×D₅₃	1	16	yes	complemented
304	#538	—	53	64	no	—
305	#534	—	53	16	no	—
306	C₂×#126	—	53	8	no	—
307	C₂×#124	—	53	8	no	—
308	#525	—	53	64	no	—
309	C₂×#123	—	53	8	no	—
310	C₂×#123	—	53	8	no	—
311	C₂×#123	—	53	4	no	—
312	C₂×C₂²⋊Q₈	—	53	4	no	—
313	C₂×C₂²⋊Q₈	C₂×D₅₃	1	4	yes	complemented
314	C₂×C₂²⋊Q₈	C₂×D₅₃	1	8	yes	complemented
315	C₂×#121	—	53	4	no	—
316	C₂×#121	—	53	4	no	—
317	C₂×#121	—	53	8	no	—
318	C₂×#120	—	53	8	no	—
319	C₂×#120	—	53	4	no	—
320	C₂×#120	—	53	4	no	—
321	C₂×#120	—	53	32	no	—
322	C₂×#120	—	53	4	no	—
323	#518	—	53	32	no	—
324	C₂×C₄×D₄	—	53	8	no	—
325	C₂×C₄×D₄	—	53	8	no	—
326	C₂×C₄×D₄	—	53	8	no	—
327	C₂×#117	—	53	4	no	—
328	C₂×#117	—	53	4	no	—
329	C₂²×C₂²⋊C₄	—	53	1	no	—
330	C₂²×C₂²⋊C₄	—	53	2	no	—
331	C₂²×C₂²⋊C₄	—	53	4	no	—
332	C₂²×C₂²⋊C₄	—	53	8	no	—
333	C₂²×C₂²⋊C₄	—	53	16	no	—
334	C₂²×C₂²⋊C₄	—	53	4	no	—
335	C₂²×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
336	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
337	C₅₃	C₂×#3490	1	1	yes	characteristic, complemented, Sylow
338	C₂⁵	—	53	1	no	—
339	C₂⁵	—	53	4	no	—
340	C₂⁵	—	53	8	no	—
341	C₂⁵	—	53	1	no	—
342	C₂⁵	—	53	2	no	—
343	C₂⁵	—	106	8	no	—
344	C₂⁵	—	212	8	no	—
345	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
346	C₂×#41	—	53	32	no	—
347	C₂×#41	—	212	16	no	—
348	C₂²×Q₈	C₂²×D₅₃	1	4	yes	complemented
349	C₂²×D₄	—	53	4	no	—
350	C₂²×D₄	—	53	4	no	—
351	C₂²×D₄	—	53	16	no	—
352	C₂²×D₄	—	53	8	no	—
353	C₂²×D₄	—	53	4	no	—
354	C₂²×D₄	—	53	8	no	—
355	C₂²×D₄	—	53	32	no	—
356	C₂²×D₄	—	106	16	no	—
357	C₂²×D₄	—	212	16	no	—
358	C₂³×C₄	—	53	2	no	—
359	C₂³×C₄	—	53	2	no	—
360	C₂³×C₄	—	53	4	no	—
361	C₂³×C₄	—	53	4	no	—
362	C₂³×C₄	—	53	16	no	—
363	C₂³×C₄	—	53	8	no	—
364	C₂³×C₄	—	212	8	no	—
365	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
366	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
367	#126	—	53	64	no	—
368	#124	—	53	64	no	—
369	#123	—	53	64	no	—
370	#123	—	53	64	no	—
371	#123	—	53	32	no	—
372	C₂²⋊Q₈	—	53	32	no	—
373	C₂²⋊Q₈	C₂²×D₅₃	1	32	yes	complemented
374	C₂²⋊Q₈	C₂²×D₅₃	1	64	yes	complemented
375	#121	—	53	32	no	—
376	#121	—	53	32	no	—
377	#121	—	53	64	no	—
378	#120	—	53	32	no	—
379	#120	—	53	32	no	—
380	#120	—	53	64	no	—
381	#120	—	53	32	no	—
382	C₄×D₄	—	53	64	no	—
383	C₄×D₄	—	53	64	no	—
384	C₄×D₄	—	53	64	no	—
385	#117	—	53	32	no	—
386	#117	—	53	32	no	—
387	C₂×C₄⋊C₄	—	53	8	no	—
388	C₂×C₄⋊C₄	—	53	4	no	—
389	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	complemented
390	C₂×C₄⋊C₄	C₂²×D₅₃	1	8	yes	complemented
391	C₂×C₂²⋊C₄	—	53	4	no	—
392	C₂×C₂²⋊C₄	—	53	8	no	—
393	C₂×C₂²⋊C₄	—	53	8	no	—
394	C₂×C₂²⋊C₄	—	53	16	no	—
395	C₂×C₂²⋊C₄	—	53	8	no	—
396	C₂×C₂²⋊C₄	—	53	8	no	—
397	C₂×C₂²⋊C₄	—	53	8	no	—
398	C₂×C₂²⋊C₄	—	53	32	no	—
399	C₂×C₂²⋊C₄	—	53	8	no	—
400	C₂×C₂²⋊C₄	—	53	64	no	—
401	C₂×C₂²⋊C₄	—	53	32	no	—
402	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
404	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
406	C₂×C₂²⋊C₄	C₂²×D₅₃	1	16	yes	—
407	C₂×C₄²	—	53	8	no	—
408	C₂⁴	—	4	4	no	—
409	C₂⁴	—	53	2	no	—
410	C₂⁴	—	53	4	no	—
411	C₂⁴	—	53	2	no	—
412	C₂⁴	—	53	4	no	—
413	C₂⁴	—	53	16	no	—
414	C₂⁴	—	53	4	no	—
415	C₂⁴	—	53	8	no	—
416	C₂⁴	—	106	8	no	—
417	C₂⁴	—	106	16	no	—
418	C₂⁴	—	106	4	no	—
419	C₂⁴	—	106	8	no	—
420	C₂⁴	—	106	16	no	—
421	C₂⁴	—	106	4	no	—
422	C₂⁴	—	106	32	no	—
423	C₂⁴	—	212	8	no	—
424	C₂⁴	—	212	4	no	—
425	C₂⁴	—	212	8	no	—
426	C₂⁴	—	212	32	no	—
427	C₂⁴	—	212	64	no	—
428	C₂⁴	#11719949	1	8	yes	complemented
429	C₂⁴	C₂³×D₅₃	1	2	yes	—
430	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
431	C₂⁴	C₂³×D₅₃	1	4	yes	—
432	#41	—	212	64	no	—
433	C₂×Q₈	—	4	8	no	—
434	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
435	C₂×D₄	—	53	16	no	—
436	C₂×D₄	—	53	16	no	—
437	C₂×D₄	—	53	32	no	—
438	C₂×D₄	—	53	32	no	—
439	C₂×D₄	—	53	16	no	—
440	C₂×D₄	—	106	32	no	—
441	C₂×D₄	—	106	16	no	—
442	C₂×D₄	—	106	16	no	—
443	C₂×D₄	—	106	64	no	—
444	C₂×D₄	—	212	16	no	—
445	C₂×D₄	—	212	8	no	—
446	C₂×D₄	—	212	64	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	4	8	no	—
449	C₂²×C₄	—	53	8	no	—
450	C₂²×C₄	—	53	4	no	—
451	C₂²×C₄	—	53	4	no	—
452	C₂²×C₄	—	53	8	no	—
453	C₂²×C₄	—	53	16	no	—
454	C₂²×C₄	—	53	16	no	—
455	C₂²×C₄	—	53	32	no	—
456	C₂²×C₄	—	106	16	no	—
457	C₂²×C₄	—	106	8	no	—
458	C₂²×C₄	—	212	8	no	—
459	C₂²×C₄	—	212	16	no	—
460	C₂²×C₄	—	212	32	no	—
461	C₂²×C₄	—	212	4	no	—
462	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
463	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
464	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
465	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
466	C₄⋊C₄	—	53	32	no	—
467	C₄⋊C₄	—	53	16	no	—
468	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
469	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
470	C₂²⋊C₄	—	53	16	no	—
471	C₂²⋊C₄	—	53	32	no	—
472	C₂²⋊C₄	—	53	32	no	—
473	C₂²⋊C₄	—	53	32	no	—
474	C₂²⋊C₄	—	53	32	no	—
475	C₂²⋊C₄	—	53	32	no	—
476	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
477	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
478	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
479	C₄²	—	53	32	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	4	16	no	—
483	C₂³	—	4	8	no	—
484	C₂³	—	53	8	no	—
485	C₂³	—	53	4	no	—
486	C₂³	—	53	4	no	—
487	C₂³	—	53	8	no	—
488	C₂³	—	106	16	no	—
489	C₂³	—	106	32	no	—
490	C₂³	—	106	32	no	—
491	C₂³	—	106	8	no	—
492	C₂³	—	106	4	no	—
493	C₂³	—	106	4	no	—
494	C₂³	—	106	16	no	—
495	C₂³	—	106	8	no	—
496	C₂³	—	106	32	no	—
497	C₂³	—	106	8	no	—
498	C₂³	—	106	4	no	—
499	C₂³	—	106	16	no	—
500	C₂³	—	106	8	no	—
501	C₂³	—	424	16	no	—
502	C₂³	—	212	16	no	—
503	C₂³	—	212	8	no	—
504	C₂³	—	212	16	no	—
505	C₂³	—	212	64	no	—
506	C₂³	—	212	16	no	—
507	C₂³	—	212	128	no	—
508	C₂³	C₂×#11719949	1	8	yes	—
509	C₂³	C₂×#11719949	1	4	yes	—
510	C₂³	C₂×#11719949	1	16	yes	complemented
511	C₂³	C₂⁴×D₅₃	1	4	yes	—
512	C₂³	C₂⁴×D₅₃	1	2	yes	—
513	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
514	Q₈	—	4	16	no	—
515	D₄	—	106	64	no	—
516	D₄	—	106	32	no	—
517	D₄	—	106	32	no	—
518	D₄	—	212	32	no	—
519	D₄	—	212	16	no	—
520	C₂×C₄	—	2	8	no	—
521	C₂×C₄	—	2	16	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	4	16	no	—
524	C₂×C₄	—	53	8	no	—
525	C₂×C₄	—	53	8	no	—
526	C₂×C₄	—	106	32	no	—
527	C₂×C₄	—	106	16	no	—
528	C₂×C₄	—	106	8	no	—
529	C₂×C₄	—	106	8	no	—
530	C₂×C₄	—	212	16	no	—
531	C₂×C₄	—	212	32	no	—
532	C₂×C₄	—	212	8	no	—
533	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
534	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	16	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	4	16	no	—
540	C₂²	—	53	8	no	—
541	C₂²	—	106	32	no	—
542	C₂²	—	106	16	no	—
543	C₂²	—	106	4	no	—
544	C₂²	—	106	4	no	—
545	C₂²	—	106	8	no	—
546	C₂²	—	106	4	no	—
547	C₂²	—	106	16	no	—
548	C₂²	—	106	8	no	—
549	C₂²	—	424	16	no	—
550	C₂²	—	212	64	no	—
551	C₂²	—	212	16	no	—
552	C₂²	—	212	16	no	—
553	C₂²	—	212	8	no	—
554	C₂²	C₂²×#11719949	1	8	yes	—
555	C₂²	C₂²×#11719949	1	4	yes	—
556	C₂²	C₂²×#11719949	1	2	yes	—
557	C₂²	H(5,𝔽₂)×D₅₃	1	4	yes	—
558	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
559	C₄	—	2	8	no	—
560	C₄	—	2	16	no	—
561	C₄	—	106	8	no	—
562	C₄	—	106	8	no	—
563	C₂	—	2	4	no	—
564	C₂	—	2	8	no	—
565	C₂	—	106	8	no	—
566	C₂	—	212	8	no	—
567	C₂	—	212	4	no	—
568	C₂	#20288405	1	4	yes	direct product factor
569	C₂	C₂³×#11719949	1	1	yes	characteristic
570	C₂	C₂×H(5,𝔽₂)×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#543	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#543	D₅₃
hypocenter	C₅₃	C₂×#3490
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. #20288405	C₂	direct (N × H)
2. C₂×#16811300	C₂	semidirect
3. C₂×#16811264	C₂	semidirect
4. C₂×#16811213	C₂	semidirect
5. C₂×#16811185	C₂	semidirect
6. C₂×#16811136	C₂	semidirect
7. C₂²×#13133187	C₂	semidirect
8. C₂×#16811096	C₂	semidirect
9. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
10. C₂×C₅₃×#543	C₂	semidirect
12. C₂²×#11719949	C₂²	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
15. C₂³×C₄×D₅₃	C₂²	semidirect
16. #16811300	C₂²	semidirect
17. #16811264	C₂²	semidirect
18. #16811213	C₂²	semidirect
19. #16811185	C₂²	semidirect
20. C₂×#13133212	C₂²	semidirect
21. C₂×#13133206	C₂²	semidirect
22. #16811136	C₂²	semidirect
23. C₂×#13133202	C₂²	semidirect
24. C₂×#13133202	C₂²	semidirect
25. C₂×#13133201	C₂²	semidirect
26. C₂×#13133193	C₂²	semidirect
27. C₂×#13133192	C₂²	semidirect
28. C₂×#13133192	C₂²	semidirect
29. C₂×#13133191	C₂²	semidirect
31. C₂×#13133187	C₂²	semidirect
32. C₂×#13133187	C₂²	semidirect
33. C₂×#13133187	C₂²	semidirect
34. C₂×#13133187	C₂²	semidirect
35. #16811096	C₂²	semidirect
36. C₂×#13133179	C₂²	semidirect
37. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
38. C₂×#13133177	C₂²	semidirect
39. C₂×#13133175	C₂²	semidirect
40. C₂×#13133173	C₂²	semidirect
42. C₂²×#11719921	C₂²	semidirect
43. C₂²×#11719921	C₂²	semidirect
44. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
45. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
46. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
47. C₅₃×#543	C₂²	semidirect
48. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
49. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
60. C₂×#11719949	C₂³	semidirect
64. C₂²×D₂₁₂	C₂³	semidirect
65. C₂²×D₂₁₂	C₂³	semidirect
75. C₂²×C₄×D₅₃	C₂³	semidirect
76. C₂²×C₄×D₅₃	C₂³	semidirect
77. #13133212	C₂³	semidirect
78. #13133206	C₂³	semidirect
79. #13133202	C₂³	semidirect
80. #13133202	C₂³	semidirect
81. #13133201	C₂³	semidirect
82. #13133193	C₂³	semidirect
83. #13133192	C₂³	semidirect
84. #13133192	C₂³	semidirect
85. #13133191	C₂³	semidirect
87. #13133187	C₂³	semidirect
88. #13133187	C₂³	semidirect
89. #13133187	C₂³	semidirect
90. #13133179	C₂³	semidirect
91. C₄×C₂²⋊D₅₃	C₂³	semidirect
92. #13133177	C₂³	semidirect
93. #13133175	C₂³	semidirect
94. #13133173	C₂³	semidirect
95. C₂×C₄⋊Dic₅₃	C₂³	semidirect
96. C₂×#11719919	C₂³	semidirect
102. C₂×#11719921	C₂³	semidirect
103. C₂×#11719921	C₂³	semidirect
104. C₂×#11719921	C₂³	semidirect
105. C₂×#11719921	C₂³	semidirect
106. C₂²⋊C₄×D₅₃	C₂³	semidirect
107. C₂²⋊C₄×D₅₃	C₂³	semidirect
108. C₂×C₄×Dic₅₃	C₂³	semidirect
110. C₂²×Q₈×C₅₃	C₂³	semidirect
113. C₂²⋊Q₈×C₅₃	C₂³	semidirect
114. C₂²⋊Q₈×C₅₃	C₂³	semidirect
115. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
116. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
123. C₂⁴×C₅₃	#41	semidirect
147. C₂×Q₈×C₅₃	C₂⁴	semidirect
172. C₂×D₂₁₂	C₂⁴	semidirect
179. C₂×C₄×D₅₃	C₂⁴	semidirect
180. C₄⋊C₄×C₅₃	C₂⁴	semidirect
181. C₄⋊C₄×C₅₃	C₂⁴	semidirect
189. #11719921	C₂⁴	semidirect
190. #11719921	C₂⁴	semidirect
191. C₄⋊Dic₅₃	C₂⁴	semidirect
192. #11719919	C₂⁴	semidirect
193. C₄×Dic₅₃	C₂⁴	semidirect
200. C₂³×C₅₃	C₂×#41	semidirect
281. C₂×#543	D₅₃	semidirect
290. C₂×C₅₃	#3490	semidirect
303. #543	C₂×D₅₃	semidirect
313. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
314. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
337. C₅₃	C₂×#3490	semidirect
348. C₂²×Q₈	C₂²×D₅₃	semidirect
373. C₂²⋊Q₈	C₂²×D₅₃	semidirect
374. C₂²⋊Q₈	C₂²×D₅₃	semidirect
389. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
390. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
428. C₂⁴	#11719949	semidirect
434. C₂×Q₈	C₂³×D₅₃	semidirect
468. C₄⋊C₄	C₂³×D₅₃	semidirect
469. C₄⋊C₄	C₂³×D₅₃	semidirect
510. C₂³	C₂×#11719949	semidirect
568. C₂	#20288405	direct (N × H)

Maximal subgroups

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

subgroup	index
#20288405	2
C₂×#16811300	2
C₂×#16811264	2
C₂×#16811213	2
C₂×#16811185	2
C₂×#16811136	2
C₂²×#13133187	2
C₂×#16811096	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂×C₅₃×#543	2
C₂×#3490	53

Subgroups — C2×#20288405 #28292006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288405 #28292006