Subgroups — C₂×#20288954 #28291327

Order 13568.

8890 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288954	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811363	C₂	1	4	yes	maximal, complemented
3	C₂×#16811319	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
5	C₂×#16811255	C₂	1	4	yes	maximal, complemented
6	C₂×#16811228	C₂	1	2	yes	maximal, complemented
7	C₂×#16811213	C₂	1	4	yes	maximal, complemented
8	C₂×#16811187	C₂	1	2	yes	maximal, complemented
9	C₂×#16811181	C₂	1	4	yes	maximal, complemented
10	C₂²×#13133191	C₂	1	2	yes	maximal, complemented
11	C₂×C₅₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
13	C₂³×D₂₁₂	C₂²	1	2	yes	complemented
14	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
15	C₂²×D₄×D₅₃	C₂²	1	8	yes	complemented
16	#16811363	C₂²	1	64	yes	complemented
17	#16811319	C₂²	1	64	yes	complemented
18	D₄×D₂₁₂	C₂²	1	64	yes	complemented
19	#16811255	C₂²	1	64	yes	complemented
20	#16811228	C₂²	1	32	yes	complemented
21	#16811213	C₂²	1	64	yes	complemented
22	#16811187	C₂²	1	32	yes	complemented
23	#16811181	C₂²	1	64	yes	complemented
24	C₂×#13133214	C₂²	1	2	yes	complemented
25	C₂×#13133213	C₂²	1	4	yes	complemented
26	C₂×#13133212	C₂²	1	8	yes	complemented
27	C₂×#13133209	C₂²	1	2	yes	complemented
28	C₂×#13133206	C₂²	1	4	yes	complemented
29	C₂×#13133202	C₂²	1	8	yes	complemented
30	C₂×#13133199	C₂²	1	8	yes	complemented
31	C₂×#13133192	C₂²	1	8	yes	complemented
32	C₂×#13133191	C₂²	1	4	yes	complemented
33	C₂×#13133191	C₂²	1	4	yes	complemented
34	C₂×#13133191	C₂²	1	4	yes	complemented
35	C₂×#13133191	C₂²	1	4	yes	complemented
36	C₂×#13133191	C₂²	1	4	yes	complemented
37	C₂×#13133191	C₂²	1	32	yes	complemented
38	C₂×#13133190	C₂²	1	8	yes	complemented
39	C₂×#13133189	C₂²	1	8	yes	complemented
40	C₂×#13133187	C₂²	1	8	yes	complemented
41	C₂×#13133187	C₂²	1	8	yes	complemented
42	C₂×#13133185	C₂²	1	4	yes	complemented
43	C₂×#13133185	C₂²	1	4	yes	complemented
44	C₂×C₄×D₂₁₂	C₂²	1	4	yes	complemented
45	C₂×#13133177	C₂²	1	8	yes	complemented
46	C₂×#13133176	C₂²	1	2	yes	complemented
47	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
48	C₂²×#11719921	C₂²	1	4	yes	complemented
49	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
51	C₅₃×#525	C₂²	1	16	yes	complemented
52	C₂×#123×C₅₃	C₂²	1	2	yes	complemented
53	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
54	C₂×#121×C₅₃	C₂²	1	2	yes	complemented
55	C₂×#120×C₅₃	C₂²	1	2	yes	complemented
56	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
57	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₂⁴×D₅₃	C₂³	1	4	yes	—
59	C₂⁴×D₅₃	C₂³	1	4	yes	—
60	C₂²×D₂₁₂	—	4	8	no	—
61	C₂²×D₂₁₂	—	4	8	no	—
62	C₂²×D₂₁₂	C₂³	1	4	yes	—
63	C₂²×D₂₁₂	C₂³	1	8	yes	—
64	C₂²×D₂₁₂	C₂³	1	4	yes	—
65	C₂²×D₂₁₂	C₂³	1	4	yes	—
66	C₂²×D₂₁₂	C₂³	1	16	yes	complemented
67	C₂²×D₂₁₂	C₂³	1	16	yes	complemented
68	C₂²×C₂²⋊D₅₃	—	4	16	no	—
69	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
70	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
71	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
72	C₂²×C₂²⋊D₅₃	C₂³	1	32	yes	complemented
73	C₂×D₄×D₅₃	—	4	32	no	—
74	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
75	C₂³×Dic₅₃	C₂³	1	4	yes	—
76	C₂²×C₄×D₅₃	C₂³	1	8	yes	—
77	#13133214	C₂³	1	16	yes	complemented
78	#13133213	C₂³	1	32	yes	complemented
79	#13133212	C₂³	1	64	yes	complemented
80	#13133209	C₂³	1	16	yes	complemented
81	#13133206	C₂³	1	32	yes	complemented
82	#13133202	C₂³	1	64	yes	complemented
83	#13133199	C₂³	1	64	yes	complemented
84	#13133192	C₂³	1	64	yes	complemented
85	#13133191	C₂³	1	32	yes	complemented
86	#13133191	C₂³	1	32	yes	complemented
87	#13133191	C₂³	1	32	yes	complemented
88	#13133191	C₂³	1	32	yes	complemented
89	#13133191	C₂³	1	32	yes	complemented
90	#13133190	C₂³	1	64	yes	complemented
91	#13133189	C₂³	1	64	yes	complemented
92	#13133187	C₂³	1	64	yes	complemented
93	#13133187	C₂³	1	64	yes	complemented
94	#13133185	C₂³	1	32	yes	complemented
95	#13133185	C₂³	1	32	yes	complemented
96	C₄×D₂₁₂	C₂³	1	32	yes	complemented
97	#13133177	C₂³	1	64	yes	complemented
98	#13133176	C₂³	1	16	yes	complemented
99	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
100	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
101	C₂×C₄⋊Dic₅₃	C₂³	1	16	yes	complemented
102	C₂×#11719919	C₂³	1	8	yes	—
103	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
104	C₂×#11719921	C₂³	1	8	yes	—
105	C₂×#11719921	C₂³	1	8	yes	—
106	C₂×#11719921	C₂³	1	8	yes	—
107	C₂×#11719921	C₂³	1	8	yes	—
108	C₂×#11719921	C₂³	1	32	yes	complemented
109	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
110	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
111	C₂×#41×C₅₃	—	4	4	no	—
112	C₂×#41×C₅₃	C₂³	1	8	yes	complemented
113	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
115	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
116	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
117	C₂³×C₄×C₅₃	—	4	4	no	—
118	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
119	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
120	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
121	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
122	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
123	C₂³×C₄×C₅₃	C₂³	1	8	yes	complemented
124	#123×C₅₃	C₂³	1	16	yes	complemented
125	C₂²⋊Q₈×C₅₃	C₂³	1	16	yes	complemented
126	#121×C₅₃	C₂³	1	16	yes	complemented
127	#120×C₅₃	C₂³	1	16	yes	complemented
128	C₄×D₄×C₅₃	C₂³	1	32	yes	complemented
129	#117×C₅₃	C₂³	1	8	yes	complemented
130	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
132	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
135	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
136	C₂⁴×C₅₃	—	2	8	no	—
137	C₂⁴×C₅₃	—	4	2	no	—
138	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
139	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
140	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
141	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
142	C₂³×D₅₃	—	2	16	no	—
143	C₂³×D₅₃	—	4	16	no	—
144	C₂³×D₅₃	—	4	8	no	—
145	C₂³×D₅₃	—	4	8	no	—
146	C₂³×D₅₃	—	4	8	no	—
147	C₂³×D₅₃	C₂⁴	1	8	yes	—
148	C₂³×D₅₃	C₂⁴	1	16	yes	—
149	C₂³×D₅₃	C₂⁴	1	16	yes	—
150	#41×C₅₃	—	4	16	no	—
151	C₂×Q₈×C₅₃	—	4	2	no	—
152	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
153	C₂×D₄×C₅₃	—	2	16	no	—
154	C₂×D₄×C₅₃	—	4	4	no	—
155	C₂×D₄×C₅₃	—	4	2	no	—
156	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
157	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
158	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₅₃	—	2	4	no	—
160	C₂²×C₄×C₅₃	—	2	8	no	—
161	C₂²×C₄×C₅₃	—	2	8	no	—
162	C₂²×C₄×C₅₃	—	4	16	no	—
163	C₂²×C₄×C₅₃	—	4	2	no	—
164	C₂²×C₄×C₅₃	—	4	4	no	—
165	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
171	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
172	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
173	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
175	C₂×C₂²⋊D₅₃	—	2	32	no	—
176	C₂×C₂²⋊D₅₃	—	4	16	no	—
177	C₂×C₂²⋊D₅₃	—	4	16	no	—
178	C₂×C₂²⋊D₅₃	—	4	64	no	—
179	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
180	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
181	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
182	C₂²×Dic₅₃	—	4	8	no	—
183	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
184	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
185	D₄×D₅₃	—	4	128	no	—
186	C₂×D₂₁₂	—	2	16	no	—
187	C₂×D₂₁₂	—	4	32	no	—
188	C₂×D₂₁₂	—	4	32	no	—
189	C₂×D₂₁₂	—	4	16	no	—
190	C₂×D₂₁₂	—	4	8	no	—
191	C₂×D₂₁₂	—	4	8	no	—
192	C₂×D₂₁₂	C₂⁴	1	16	yes	—
193	C₂×D₂₁₂	C₂⁴	1	32	yes	—
194	C₂×D₂₁₂	C₂⁴	1	16	yes	—
195	C₂×D₂₁₂	C₂⁴	1	16	yes	—
196	C₂×C₄×D₅₃	—	4	16	no	—
197	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
198	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
199	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
200	C₂²⋊C₄×C₅₃	C₂⁴	1	8	yes	—
201	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
202	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
203	C₄²×C₅₃	C₂⁴	1	8	yes	—
204	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
205	#11719921	C₂⁴	1	32	yes	—
206	#11719921	C₂⁴	1	32	yes	—
207	#11719921	C₂⁴	1	32	yes	—
208	#11719921	C₂⁴	1	32	yes	—
209	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
210	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
211	#11719919	C₂⁴	1	32	yes	—
212	C₄×Dic₅₃	C₂⁴	1	16	yes	—
213	C₂³×C₅₃	—	2	4	no	—
214	C₂³×C₅₃	—	2	4	no	—
215	C₂³×C₅₃	—	2	16	no	—
216	C₂³×C₅₃	—	2	4	no	—
217	C₂³×C₅₃	—	2	8	no	—
218	C₂³×C₅₃	—	2	2	no	—
219	C₂³×C₅₃	—	2	4	no	—
220	C₂³×C₅₃	—	4	4	no	—
221	C₂³×C₅₃	—	4	16	no	—
222	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
223	C₂³×C₅₃	C₂⁵	1	2	yes	—
224	C₂³×C₅₃	C₂⁵	1	4	yes	—
225	C₂³×C₅₃	C₂⁵	1	4	yes	—
226	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
227	C₂²×D₅₃	—	2	16	no	—
228	C₂²×D₅₃	—	2	16	no	—
229	C₂²×D₅₃	—	2	16	no	—
230	C₂²×D₅₃	—	2	32	no	—
231	C₂²×D₅₃	—	4	16	no	—
232	C₂²×D₅₃	—	4	16	no	—
233	C₂²×D₅₃	—	4	32	no	—
234	C₂²×D₅₃	—	4	16	no	—
235	C₂²×D₅₃	—	8	32	no	—
236	C₂²×D₅₃	—	8	32	no	—
237	C₂²×D₅₃	C₂⁵	1	16	yes	—
238	Q₈×C₅₃	—	4	4	no	—
239	D₄×C₅₃	—	2	32	no	—
240	D₄×C₅₃	—	4	8	no	—
241	D₄×C₅₃	—	4	4	no	—
242	C₂×C₄×C₅₃	—	2	8	no	—
243	C₂×C₄×C₅₃	—	2	4	no	—
244	C₂×C₄×C₅₃	—	2	4	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	2	16	no	—
247	C₂×C₄×C₅₃	—	2	16	no	—
248	C₂×C₄×C₅₃	—	2	8	no	—
249	C₂×C₄×C₅₃	—	4	4	no	—
250	C₂×C₄×C₅₃	—	4	8	no	—
251	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
252	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
253	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
254	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
255	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
256	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
257	C₂²⋊D₅₃	—	2	64	no	—
258	C₂²⋊D₅₃	—	4	32	no	—
259	C₂²⋊D₅₃	—	4	32	no	—
260	C₂×Dic₅₃	—	2	16	no	—
261	C₂×Dic₅₃	—	4	16	no	—
262	C₂×Dic₅₃	C₂⁵	1	16	yes	—
263	D₂₁₂	—	2	32	no	—
264	D₂₁₂	—	4	16	no	—
265	D₂₁₂	—	4	16	no	—
266	D₂₁₂	—	4	32	no	—
267	C₄×D₅₃	—	4	32	no	—
268	C₂×#3537	—	53	1	no	maximal, Sylow
269	C₂²×C₅₃	—	2	4	no	—
270	C₂²×C₅₃	—	2	4	no	—
271	C₂²×C₅₃	—	2	16	no	—
272	C₂²×C₅₃	—	2	4	no	—
273	C₂²×C₅₃	—	2	8	no	—
274	C₂²×C₅₃	—	2	2	no	—
275	C₂²×C₅₃	—	2	4	no	—
276	C₂²×C₅₃	—	4	16	no	—
277	C₂²×C₅₃	—	4	8	no	—
278	C₂²×C₅₃	—	4	4	no	—
279	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
280	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	8	yes	—
281	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
282	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
283	C₂×D₅₃	—	2	16	no	—
284	C₂×D₅₃	—	2	16	no	—
285	C₂×D₅₃	—	2	16	no	—
286	C₂×D₅₃	—	4	32	no	—
287	C₂×D₅₃	—	8	32	no	—
288	C₂×D₅₃	—	8	32	no	—
289	C₄×C₅₃	—	2	8	no	—
290	C₄×C₅₃	—	2	4	no	—
291	C₄×C₅₃	—	2	4	no	—
292	C₄×C₅₃	—	2	2	no	—
293	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
294	Dic₅₃	—	2	16	no	—
295	#3537	—	53	32	no	—
296	C₂×#560	—	53	4	no	—
297	C₂×#553	—	53	4	no	—
298	C₂×D₄²	—	53	4	no	—
299	C₂×D₄²	—	53	4	no	—
300	C₂×#540	—	53	2	no	—
301	C₂×#538	—	53	4	no	—
302	C₂×#535	—	53	2	no	—
303	C₂×#534	—	53	4	no	—
304	C₂×#525	D₅₃	1	1	yes	characteristic, complemented
305	C₂²×#121	—	53	2	no	—
306	C₂×C₅₃	—	2	8	no	—
307	C₂×C₅₃	—	4	4	no	—
308	C₂×C₅₃	—	4	2	no	—
309	C₂×C₅₃	#3537	1	4	yes	complemented
310	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
311	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
312	D₅₃	—	4	16	no	—
313	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic, complemented
314	C₂³×D₄	—	53	4	no	—
315	C₂³×D₄	—	53	8	no	—
316	C₂³×D₄	—	53	2	no	—
317	C₂³×D₄	—	53	2	no	—
318	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
319	#560	—	53	64	no	—
320	#553	—	53	64	no	—
321	D₄²	—	53	64	no	—
322	D₄²	—	53	64	no	—
323	#540	—	53	32	no	—
324	#538	—	53	64	no	—
325	#535	—	53	32	no	—
326	#534	—	53	64	no	—
327	C₂×#127	—	53	4	no	—
328	C₂×#127	—	53	2	no	—
329	C₂×#126	—	53	8	no	—
330	C₂×#125	—	53	2	no	—
331	C₂×#124	—	53	4	no	—
332	#525	C₂×D₅₃	1	16	yes	complemented
333	C₂×#123	—	53	8	no	—
334	C₂×#123	—	53	8	no	—
335	C₂×#123	C₂×D₅₃	1	2	yes	complemented
336	C₂×C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
337	C₂×#121	—	53	8	no	—
338	C₂×#121	—	53	4	no	—
339	C₂×#121	—	53	4	no	—
340	C₂×#121	—	53	4	no	—
341	C₂×#121	—	53	4	no	—
342	C₂×#121	—	53	4	no	—
343	C₂×#121	—	53	32	no	—
344	C₂×#121	—	53	8	no	—
345	C₂×#121	—	53	8	no	—
346	C₂×#121	C₂×D₅₃	1	2	yes	complemented
347	C₂×#120	—	53	4	no	—
348	C₂×#120	—	53	4	no	—
349	C₂×#120	—	53	8	no	—
350	C₂×#120	—	53	8	no	—
351	C₂×#120	C₂×D₅₃	1	2	yes	complemented
352	C₂×C₄×D₄	—	53	8	no	—
353	C₂×C₄×D₄	—	53	4	no	—
354	C₂×C₄×D₄	C₂×D₅₃	1	4	yes	complemented
355	C₂×#117	—	53	2	no	—
356	C₂×#117	C₂×D₅₃	1	1	yes	characteristic, complemented
357	C₂²×C₄⋊C₄	—	53	2	no	—
358	C₂²×C₂²⋊C₄	—	53	4	no	—
359	C₅₃	C₂×#3537	1	1	yes	characteristic, complemented, Sylow
360	C₂⁵	—	53	4	no	—
361	C₂⁵	—	53	4	no	—
362	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
363	C₂×#41	—	4	4	no	—
364	C₂×#41	C₂²×D₅₃	1	8	yes	complemented
365	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
366	C₂²×D₄	—	53	8	no	—
367	C₂²×D₄	—	53	8	no	—
368	C₂²×D₄	—	53	8	no	—
369	C₂²×D₄	—	53	32	no	—
370	C₂²×D₄	—	53	64	no	—
371	C₂²×D₄	—	53	4	no	—
372	C₂²×D₄	—	53	4	no	—
373	C₂²×D₄	—	53	8	no	—
374	C₂²×D₄	—	53	4	no	—
375	C₂²×D₄	—	53	16	no	—
376	C₂²×D₄	—	53	16	no	—
377	C₂²×D₄	—	212	8	no	—
378	C₂²×D₄	—	212	8	no	—
379	C₂²×D₄	—	212	32	no	—
380	C₂²×D₄	—	212	16	no	—
381	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
382	C₂²×D₄	C₂²×D₅₃	1	1	yes	characteristic
383	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
384	C₂³×C₄	—	4	4	no	—
385	C₂³×C₄	—	53	4	no	—
386	C₂³×C₄	—	53	8	no	—
387	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
388	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
389	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
390	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
391	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
392	C₂³×C₄	C₂²×D₅₃	1	8	yes	complemented
393	#127	—	53	32	no	—
394	#127	—	53	16	no	—
395	#126	—	53	64	no	—
396	#125	—	53	16	no	—
397	#124	—	53	32	no	—
398	#123	—	53	64	no	—
399	#123	—	53	64	no	—
400	#123	C₂²×D₅₃	1	16	yes	complemented
401	C₂²⋊Q₈	C₂²×D₅₃	1	16	yes	complemented
402	#121	—	53	64	no	—
403	#121	—	53	32	no	—
404	#121	—	53	32	no	—
405	#121	—	53	32	no	—
406	#121	—	53	32	no	—
407	#121	—	53	32	no	—
408	#121	—	53	64	no	—
409	#121	—	53	64	no	—
410	#121	C₂²×D₅₃	1	16	yes	complemented
411	#120	—	53	32	no	—
412	#120	—	53	32	no	—
413	#120	—	53	64	no	—
414	#120	—	53	64	no	—
415	#120	C₂²×D₅₃	1	16	yes	complemented
416	C₄×D₄	—	53	64	no	—
417	C₄×D₄	—	53	32	no	—
418	C₄×D₄	C₂²×D₅₃	1	32	yes	complemented
419	#117	—	53	16	no	—
420	#117	C₂²×D₅₃	1	8	yes	complemented
421	C₂×C₄⋊C₄	—	53	4	no	—
422	C₂×C₄⋊C₄	—	53	4	no	—
423	C₂×C₄⋊C₄	—	53	16	no	—
424	C₂×C₄⋊C₄	—	53	8	no	—
425	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
426	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
427	C₂×C₂²⋊C₄	—	53	4	no	—
428	C₂×C₂²⋊C₄	—	53	8	no	—
429	C₂×C₂²⋊C₄	—	53	8	no	—
430	C₂×C₂²⋊C₄	—	53	32	no	—
431	C₂×C₂²⋊C₄	—	53	8	no	—
432	C₂×C₂²⋊C₄	—	53	8	no	—
433	C₂×C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
435	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
436	C₂×C₄²	—	53	4	no	—
437	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
438	C₂⁴	—	2	8	no	—
439	C₂⁴	—	4	2	no	—
440	C₂⁴	—	53	8	no	—
441	C₂⁴	—	53	16	no	—
442	C₂⁴	—	53	16	no	—
443	C₂⁴	—	106	16	no	—
444	C₂⁴	—	212	8	no	—
445	C₂⁴	—	212	8	no	—
446	C₂⁴	—	212	8	no	—
447	C₂⁴	—	212	16	no	—
448	C₂⁴	C₂³×D₅₃	1	2	yes	—
449	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
450	C₂⁴	C₂³×D₅₃	1	4	yes	—
451	C₂⁴	C₂³×D₅₃	1	2	yes	—
452	#41	—	4	16	no	—
453	C₂×Q₈	—	4	2	no	—
454	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
455	C₂×D₄	—	2	16	no	—
456	C₂×D₄	—	4	2	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	53	32	no	—
459	C₂×D₄	—	53	32	no	—
460	C₂×D₄	—	53	32	no	—
461	C₂×D₄	—	53	16	no	—
462	C₂×D₄	—	53	16	no	—
463	C₂×D₄	—	53	32	no	—
464	C₂×D₄	—	53	16	no	—
465	C₂×D₄	—	106	32	no	—
466	C₂×D₄	—	106	16	no	—
467	C₂×D₄	—	212	32	no	—
468	C₂×D₄	—	212	32	no	—
469	C₂×D₄	—	212	8	no	—
470	C₂×D₄	—	212	8	no	—
471	C₂×D₄	—	212	16	no	—
472	C₂×D₄	—	212	16	no	—
473	C₂×D₄	—	212	16	no	—
474	C₂×D₄	—	212	128	no	—
475	C₂×D₄	—	212	64	no	—
476	C₂×D₄	C₂³×D₅₃	1	16	yes	—
477	C₂×D₄	C₂³×D₅₃	1	4	yes	—
478	C₂×D₄	C₂³×D₅₃	1	8	yes	—
479	C₂²×C₄	—	2	8	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	8	no	—
482	C₂²×C₄	—	4	16	no	—
483	C₂²×C₄	—	4	4	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	—	53	8	no	—
486	C₂²×C₄	—	53	16	no	—
487	C₂²×C₄	—	53	32	no	—
488	C₂²×C₄	—	212	16	no	—
489	C₂²×C₄	—	212	8	no	—
490	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
491	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
492	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
493	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
494	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
495	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
496	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
497	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
498	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
499	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
500	C₄⋊C₄	—	53	16	no	—
501	C₄⋊C₄	—	53	16	no	—
502	C₄⋊C₄	—	53	32	no	—
503	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
504	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
505	C₂²⋊C₄	—	53	16	no	—
506	C₂²⋊C₄	—	53	32	no	—
507	C₂²⋊C₄	—	53	32	no	—
508	C₂²⋊C₄	—	53	32	no	—
509	C₂²⋊C₄	—	53	32	no	—
510	C₂²⋊C₄	C₂³×D₅₃	1	8	yes	—
511	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
512	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
513	C₄²	—	53	16	no	—
514	C₄²	C₂³×D₅₃	1	8	yes	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	2	8	no	—
520	C₂³	—	2	16	no	—
521	C₂³	—	2	4	no	—
522	C₂³	—	4	4	no	—
523	C₂³	—	4	16	no	—
524	C₂³	—	53	16	no	—
525	C₂³	—	106	32	no	—
526	C₂³	—	106	16	no	—
527	C₂³	—	106	16	no	—
528	C₂³	—	106	16	no	—
529	C₂³	—	424	32	no	—
530	C₂³	—	424	32	no	—
531	C₂³	—	212	16	no	—
532	C₂³	—	212	16	no	—
533	C₂³	—	212	16	no	—
534	C₂³	—	212	32	no	—
535	C₂³	#13133263	1	8	yes	—
536	C₂³	C₂⁴×D₅₃	1	4	yes	—
537	C₂³	C₂⁴×D₅₃	1	2	yes	—
538	C₂³	C₂⁴×D₅₃	1	4	yes	—
539	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
540	Q₈	—	4	4	no	—
541	D₄	—	2	32	no	—
542	D₄	—	4	4	no	—
543	D₄	—	4	8	no	—
544	D₄	—	106	64	no	—
545	D₄	—	106	32	no	—
546	D₄	—	212	16	no	—
547	D₄	—	212	16	no	—
548	D₄	—	212	32	no	—
549	D₄	—	212	32	no	—
550	D₄	—	212	32	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	16	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	2	16	no	—
558	C₂×C₄	—	4	8	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	53	16	no	—
561	C₂×C₄	—	106	16	no	—
562	C₂×C₄	—	212	32	no	—
563	C₂×C₄	—	212	16	no	—
564	C₂×C₄	#13133262	1	4	yes	—
565	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
566	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
567	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
568	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
569	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
570	C₂²	—	2	16	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	8	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	4	16	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	8	no	—
580	C₂²	—	106	16	no	—
581	C₂²	—	106	16	no	—
582	C₂²	—	106	16	no	—
583	C₂²	—	424	32	no	—
584	C₂²	—	424	32	no	—
585	C₂²	—	212	32	no	—
586	C₂²	C₂×#13133262	1	2	yes	—
587	C₂²	C₂×#13133263	1	8	yes	—
588	C₂²	C₂×#13133263	1	4	yes	—
589	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
590	C₄	—	2	2	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	4	no	—
593	C₄	—	2	8	no	—
594	C₄	—	106	16	no	—
595	C₄	C₂×#13133262	1	4	yes	—
596	C₂	—	2	8	no	—
597	C₂	—	4	2	no	—
598	C₂	—	4	4	no	—
599	C₂	—	212	16	no	—
600	C₂	#20288954	1	4	yes	direct product factor
601	C₂	C₂²×#13133262	1	1	yes	characteristic
602	C₂	C₂²×#13133263	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#525	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#525	D₅₃
hypocenter	C₅₃	C₂×#3537
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#525

Complements

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

N	H ≅ G/N	type
1. #20288954	C₂	direct (N × H)
2. C₂×#16811363	C₂	semidirect
3. C₂×#16811319	C₂	semidirect
4. C₂×D₄×D₂₁₂	C₂	semidirect
5. C₂×#16811255	C₂	semidirect
6. C₂×#16811228	C₂	semidirect
7. C₂×#16811213	C₂	semidirect
8. C₂×#16811187	C₂	semidirect
9. C₂×#16811181	C₂	semidirect
10. C₂²×#13133191	C₂	semidirect
11. C₂×C₅₃×#525	C₂	semidirect
12. C₂³×D₂₁₂	C₂²	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
14. C₂³×C₂²⋊D₅₃	C₂²	semidirect
15. C₂²×D₄×D₅₃	C₂²	semidirect
16. #16811363	C₂²	semidirect
17. #16811319	C₂²	semidirect
18. D₄×D₂₁₂	C₂²	semidirect
19. #16811255	C₂²	semidirect
20. #16811228	C₂²	semidirect
21. #16811213	C₂²	semidirect
22. #16811187	C₂²	semidirect
23. #16811181	C₂²	semidirect
24. C₂×#13133214	C₂²	semidirect
25. C₂×#13133213	C₂²	semidirect
26. C₂×#13133212	C₂²	semidirect
27. C₂×#13133209	C₂²	semidirect
28. C₂×#13133206	C₂²	semidirect
29. C₂×#13133202	C₂²	semidirect
30. C₂×#13133199	C₂²	semidirect
31. C₂×#13133192	C₂²	semidirect
32. C₂×#13133191	C₂²	semidirect
33. C₂×#13133191	C₂²	semidirect
34. C₂×#13133191	C₂²	semidirect
35. C₂×#13133191	C₂²	semidirect
36. C₂×#13133191	C₂²	semidirect
37. C₂×#13133191	C₂²	semidirect
38. C₂×#13133190	C₂²	semidirect
39. C₂×#13133189	C₂²	semidirect
40. C₂×#13133187	C₂²	semidirect
41. C₂×#13133187	C₂²	semidirect
42. C₂×#13133185	C₂²	semidirect
43. C₂×#13133185	C₂²	semidirect
44. C₂×C₄×D₂₁₂	C₂²	semidirect
45. C₂×#13133177	C₂²	semidirect
46. C₂×#13133176	C₂²	semidirect
47. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
48. C₂²×#11719921	C₂²	semidirect
49. C₂²×#41×C₅₃	C₂²	semidirect
50. C₂⁴×C₄×C₅₃	C₂²	semidirect
51. C₅₃×#525	C₂²	semidirect
52. C₂×#123×C₅₃	C₂²	semidirect
53. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
54. C₂×#121×C₅₃	C₂²	semidirect
55. C₂×#120×C₅₃	C₂²	semidirect
56. C₂×C₄×D₄×C₅₃	C₂²	semidirect
57. C₂×#117×C₅₃	C₂²	semidirect
66. C₂²×D₂₁₂	C₂³	semidirect
67. C₂²×D₂₁₂	C₂³	semidirect
72. C₂²×C₂²⋊D₅₃	C₂³	semidirect
74. C₂×D₄×D₅₃	C₂³	semidirect
77. #13133214	C₂³	semidirect
78. #13133213	C₂³	semidirect
79. #13133212	C₂³	semidirect
80. #13133209	C₂³	semidirect
81. #13133206	C₂³	semidirect
82. #13133202	C₂³	semidirect
83. #13133199	C₂³	semidirect
84. #13133192	C₂³	semidirect
85. #13133191	C₂³	semidirect
86. #13133191	C₂³	semidirect
87. #13133191	C₂³	semidirect
88. #13133191	C₂³	semidirect
89. #13133191	C₂³	semidirect
90. #13133190	C₂³	semidirect
91. #13133189	C₂³	semidirect
92. #13133187	C₂³	semidirect
93. #13133187	C₂³	semidirect
94. #13133185	C₂³	semidirect
95. #13133185	C₂³	semidirect
96. C₄×D₂₁₂	C₂³	semidirect
97. #13133177	C₂³	semidirect
98. #13133176	C₂³	semidirect
101. C₂×C₄⋊Dic₅₃	C₂³	semidirect
108. C₂×#11719921	C₂³	semidirect
112. C₂×#41×C₅₃	C₂³	semidirect
123. C₂³×C₄×C₅₃	C₂³	semidirect
124. #123×C₅₃	C₂³	semidirect
125. C₂²⋊Q₈×C₅₃	C₂³	semidirect
126. #121×C₅₃	C₂³	semidirect
127. #120×C₅₃	C₂³	semidirect
128. C₄×D₄×C₅₃	C₂³	semidirect
129. #117×C₅₃	C₂³	semidirect
304. C₂×#525	D₅₃	semidirect
309. C₂×C₅₃	#3537	semidirect
313. C₂²×#41	C₂×D₅₃	semidirect
318. C₂⁴×C₄	C₂×D₅₃	semidirect
332. #525	C₂×D₅₃	semidirect
335. C₂×#123	C₂×D₅₃	semidirect
336. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
346. C₂×#121	C₂×D₅₃	semidirect
351. C₂×#120	C₂×D₅₃	semidirect
354. C₂×C₄×D₄	C₂×D₅₃	semidirect
356. C₂×#117	C₂×D₅₃	semidirect
359. C₅₃	C₂×#3537	semidirect
364. C₂×#41	C₂²×D₅₃	semidirect
392. C₂³×C₄	C₂²×D₅₃	semidirect
400. #123	C₂²×D₅₃	semidirect
401. C₂²⋊Q₈	C₂²×D₅₃	semidirect
410. #121	C₂²×D₅₃	semidirect
415. #120	C₂²×D₅₃	semidirect
418. C₄×D₄	C₂²×D₅₃	semidirect
420. #117	C₂²×D₅₃	semidirect
600. C₂	#20288954	direct (N × H)

Maximal subgroups

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

subgroup	index
#20288954	2
C₂×#16811363	2
C₂×#16811319	2
C₂×D₄×D₂₁₂	2
C₂×#16811255	2
C₂×#16811228	2
C₂×#16811213	2
C₂×#16811187	2
C₂×#16811181	2
C₂²×#13133191	2
C₂×C₅₃×#525	2
C₂×#3537	53

Subgroups — C2×#20288954 #28291327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288954 #28291327