Subgroups — #28293105

Order 13568.

11770 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133262	C₂	1	1	yes	maximal, characteristic, complemented
2	#20288474	C₂	1	16	yes	maximal, complemented
3	#20288359	C₂	1	8	yes	maximal, complemented
4	#20288355	C₂	1	8	yes	maximal, complemented
5	#20288350	C₂	1	8	yes	maximal, complemented
6	#20288346	C₂	1	8	yes	maximal, complemented
7	#20288044	C₂	1	4	yes	maximal, complemented
8	C₂×#16811182	C₂	1	8	yes	maximal, complemented
9	C₂×#16811098	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133262	—	2	16	no	—
12	C₂×#13133262	C₂²	1	8	yes	complemented
13	C₂×#13133262	C₂²	1	8	yes	complemented
14	C₂²×#11719945	C₂²	1	2	yes	complemented
15	C₂²×#11719947	C₂²	1	4	yes	complemented
16	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
17	C₂²×D₄×D₅₃	C₂²	1	4	yes	complemented
18	#16811272	C₂²	1	64	yes	complemented
19	#16811267	C₂²	1	64	yes	complemented
20	#16811261	C₂²	1	64	yes	complemented
21	D₄×C₂²⋊D₅₃	C₂²	1	64	yes	complemented
22	#16811201	C₂²	1	16	yes	complemented
23	#16811197	C₂²	1	8	yes	complemented
24	#16811194	C₂²	1	8	yes	complemented
25	#16811190	C₂²	1	8	yes	complemented
26	#16811187	C₂²	1	8	yes	complemented
27	#16811182	C₂²	1	16	yes	complemented
28	#16811182	C₂²	1	32	yes	complemented
29	#16811182	C₂²	1	32	yes	complemented
30	#16811182	C₂²	1	32	yes	complemented
31	#16811169	C₂²	1	16	yes	complemented
32	#16811134	C₂²	1	16	yes	complemented
33	#16811130	C₂²	1	16	yes	complemented
34	C₂×#13133200	C₂²	1	16	yes	complemented
35	C₂×#13133198	C₂²	1	16	yes	complemented
36	C₂×#13133189	C₂²	1	32	yes	complemented
37	C₂×#13133186	C₂²	1	16	yes	complemented
38	C₂×#13133185	C₂²	1	16	yes	complemented
39	#16811098	C₂²	1	8	yes	complemented
40	#16811098	C₂²	1	8	yes	complemented
41	C₂×D₄×Dic₅₃	C₂²	1	8	yes	complemented
42	C₂×#13133176	C₂²	1	2	yes	complemented
43	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	complemented
44	C₂×H(5,𝔽₂)×C₅₃	—	2	16	no	—
45	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	8	yes	complemented
46	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	8	yes	complemented
47	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
48	C₂²×#41×C₅₃	C₂²	1	4	yes	complemented
49	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
50	C₂³×D₄×C₅₃	C₂²	1	4	yes	complemented
51	C₂³×D₄×C₅₃	C₂²	1	4	yes	complemented
52	C₂⁴×D₅₃	C₂³	1	4	yes	complemented
53	#13133262	—	2	64	no	—
54	#13133262	—	2	64	no	—
55	C₂×#11719945	—	2	16	no	—
56	C₂×#11719945	C₂³	1	8	yes	—
57	C₂×#11719945	C₂³	1	8	yes	complemented
58	C₂×#11719947	—	2	32	no	—
59	C₂×#11719947	C₂³	1	32	yes	complemented
60	C₂²×Dic₁₀₆	C₂³	1	2	yes	complemented
61	C₂²×D₂₁₂	C₂³	1	2	yes	complemented
62	C₂²×C₂²⋊D₅₃	—	2	32	no	—
63	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	—
64	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
65	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	complemented
66	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	complemented
67	C₂²×C₂²⋊D₅₃	C₂³	1	32	yes	complemented
68	C₂×D₄×D₅₃	—	2	32	no	—
69	C₂×D₄×D₅₃	C₂³	1	32	yes	complemented
70	C₂³×Dic₅₃	C₂³	1	4	yes	complemented
71	C₂³×Dic₅₃	C₂³	1	8	yes	complemented
72	C₂²×C₄×D₅₃	C₂³	1	4	yes	complemented
73	#13133215	C₂³	1	16	yes	complemented
74	#13133213	C₂³	1	16	yes	complemented
75	#13133206	C₂³	1	16	yes	complemented
76	#13133204	C₂³	1	16	yes	complemented
77	#13133200	C₂³	1	64	yes	complemented
78	#13133200	C₂³	1	32	yes	complemented
79	#13133198	C₂³	1	64	yes	complemented
80	#13133198	C₂³	1	32	yes	complemented
81	#13133196	C₂³	1	32	yes	complemented
82	#13133193	C₂³	1	32	yes	complemented
83	#13133191	C₂³	1	32	yes	complemented
84	#13133189	C₂³	1	128	yes	complemented
85	#13133189	C₂³	1	64	yes	complemented
86	#13133188	C₂³	1	32	yes	complemented
87	#13133186	C₂³	1	64	yes	complemented
88	#13133186	C₂³	1	32	yes	complemented
89	#13133185	C₂³	1	32	yes	complemented
90	#13133185	C₂³	1	64	yes	complemented
91	D₄×Dic₅₃	C₂³	1	64	yes	complemented
92	C₄×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
93	#13133176	C₂³	1	8	yes	complemented
94	#13133176	C₂³	1	8	yes	complemented
95	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	complemented
96	C₂×#11719919	C₂³	1	32	yes	complemented
97	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
98	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	complemented
99	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	complemented
100	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	complemented
101	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	complemented
102	C₂×#11719921	C₂³	1	32	yes	complemented
103	C₂×C₄×Dic₅₃	C₂³	1	4	yes	complemented
104	C₂⁵×C₅₃	C₂³	1	2	yes	—
105	C₂⁵×C₅₃	C₂³	1	4	yes	complemented
106	H(5,𝔽₂)×C₅₃	—	2	64	no	—
107	H(5,𝔽₂)×C₅₃	—	2	64	no	—
108	C₂×#41×C₅₃	—	2	32	no	—
109	C₂×#41×C₅₃	—	2	16	no	—
110	C₂×#41×C₅₃	C₂³	1	8	yes	—
111	C₂×#41×C₅₃	C₂³	1	32	yes	complemented
112	C₂×#41×C₅₃	C₂³	1	8	yes	complemented
113	C₂²×Q₈×C₅₃	C₂³	1	2	yes	complemented
114	C₂²×D₄×C₅₃	—	2	32	no	—
115	C₂²×D₄×C₅₃	—	2	32	no	—
116	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
117	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
118	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
119	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
120	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
121	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
122	C₂²×D₄×C₅₃	C₂³	1	32	yes	complemented
123	C₂²×D₄×C₅₃	C₂³	1	8	yes	complemented
124	C₂²×D₄×C₅₃	C₂³	1	2	yes	complemented
125	C₂²×D₄×C₅₃	C₂³	1	16	yes	complemented
126	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂³×C₄×C₅₃	C₂³	1	4	yes	complemented
128	C₂³×C₄×C₅₃	C₂³	1	4	yes	complemented
129	C₂⁴×C₅₃	—	2	16	no	—
130	C₂⁴×C₅₃	—	8	4	no	—
131	C₂⁴×C₅₃	—	8	8	no	—
132	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
133	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
134	C₂⁴×C₅₃	C₂⁴	1	16	yes	—
135	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
137	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
138	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
139	C₂³×D₅₃	—	2	16	no	—
140	C₂³×D₅₃	—	8	8	no	—
141	C₂³×D₅₃	C₂⁴	1	16	yes	—
142	C₂³×D₅₃	C₂⁴	1	4	yes	—
143	#41×C₅₃	—	2	128	no	—
144	#41×C₅₃	—	2	32	no	—
145	#41×C₅₃	—	2	32	no	—
146	C₂×Q₈×C₅₃	—	2	8	no	—
147	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
148	C₂×D₄×C₅₃	—	2	16	no	—
149	C₂×D₄×C₅₃	—	2	32	no	—
150	C₂×D₄×C₅₃	—	2	64	no	—
151	C₂×D₄×C₅₃	—	2	64	no	—
152	C₂×D₄×C₅₃	—	2	128	no	—
153	C₂×D₄×C₅₃	—	2	8	no	—
154	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
155	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
156	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
157	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
158	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
159	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
160	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
161	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
162	C₂²×C₄×C₅₃	—	2	16	no	—
163	C₂²×C₄×C₅₃	—	2	16	no	—
164	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
165	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
171	C₂×C₂²⋊D₅₃	—	2	32	no	—
172	C₂×C₂²⋊D₅₃	—	2	16	no	—
173	C₂×C₂²⋊D₅₃	—	2	64	no	—
174	C₂×C₂²⋊D₅₃	—	2	64	no	—
175	C₂×C₂²⋊D₅₃	—	8	32	no	—
176	C₂×C₂²⋊D₅₃	C₂⁴	1	8	yes	—
177	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
178	C₂×C₂²⋊D₅₃	C₂⁴	1	8	yes	—
179	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
180	C₂²×Dic₅₃	—	2	16	no	—
181	C₂²×Dic₅₃	—	8	8	no	—
182	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
183	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
184	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
185	C₂²×Dic₅₃	C₂⁴	1	32	yes	—
186	#11719947	—	2	128	no	—
187	D₄×D₅₃	—	2	128	no	—
188	#11719945	—	2	32	no	—
189	#11719945	—	2	32	no	—
190	C₂×D₂₁₂	—	2	8	no	—
191	C₂×D₂₁₂	C₂⁴	1	8	yes	—
192	C₂×C₄×D₅₃	—	2	16	no	—
193	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
194	C₂×Dic₁₀₆	—	2	8	no	—
195	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
196	C₂²⋊Dic₅₃	C₂⁴	1	8	yes	—
197	C₂²⋊Dic₅₃	C₂⁴	1	8	yes	—
198	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
199	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
200	#11719921	C₂⁴	1	64	yes	—
201	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
202	#11719919	C₂⁴	1	64	yes	—
203	C₄×Dic₅₃	C₂⁴	1	16	yes	—
204	C₂³×C₅₃	—	2	8	no	—
205	C₂³×C₅₃	—	2	32	no	—
206	C₂³×C₅₃	—	4	8	no	—
207	C₂³×C₅₃	—	4	8	no	—
208	C₂³×C₅₃	—	4	2	no	—
209	C₂³×C₅₃	—	4	8	no	—
210	C₂³×C₅₃	—	4	8	no	—
211	C₂³×C₅₃	—	4	32	no	—
212	C₂³×C₅₃	—	8	16	no	—
213	C₂³×C₅₃	—	8	32	no	—
214	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
215	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
216	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
217	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
218	C₂³×C₅₃	C₂⁵	1	8	yes	—
219	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
220	C₂³×C₅₃	C₂⁵	1	8	yes	—
221	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
222	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
223	C₂²×D₅₃	—	2	8	no	—
224	C₂²×D₅₃	—	2	32	no	—
225	C₂²×D₅₃	—	4	8	no	—
226	C₂²×D₅₃	—	4	32	no	—
227	C₂²×D₅₃	—	8	32	no	—
228	C₂²×D₅₃	C₂⁵	1	8	yes	—
229	Q₈×C₅₃	—	2	16	no	—
230	D₄×C₅₃	—	2	16	no	—
231	D₄×C₅₃	—	2	64	no	—
232	D₄×C₅₃	—	2	16	no	—
233	D₄×C₅₃	—	2	16	no	—
234	D₄×C₅₃	C₂²×D₄	1	64	yes	—
235	C₂×C₄×C₅₃	—	2	8	no	—
236	C₂×C₄×C₅₃	—	2	32	no	—
237	C₂×C₄×C₅₃	—	2	32	no	—
238	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
239	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
240	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
241	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
242	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
243	C₂²⋊D₅₃	—	2	16	no	—
244	C₂²⋊D₅₃	—	2	16	no	—
245	C₂²⋊D₅₃	—	2	64	no	—
246	C₂²⋊D₅₃	—	4	64	no	—
247	C₂×Dic₅₃	—	2	8	no	—
248	C₂×Dic₅₃	—	2	32	no	—
249	C₂×Dic₅₃	—	4	8	no	—
250	C₂×Dic₅₃	—	4	32	no	—
251	C₂×Dic₅₃	C₂⁵	1	8	yes	—
252	C₂×Dic₅₃	C₂⁵	1	16	yes	—
253	D₂₁₂	—	2	16	no	—
254	C₄×D₅₃	—	2	32	no	—
255	Dic₁₀₆	—	2	16	no	—
256	#62842	—	53	1	no	maximal, Sylow
257	C₂²×C₅₃	—	2	2	no	—
258	C₂²×C₅₃	—	2	16	no	—
259	C₂²×C₅₃	—	2	2	no	—
260	C₂²×C₅₃	—	2	2	no	—
261	C₂²×C₅₃	—	2	8	no	—
262	C₂²×C₅₃	—	2	16	no	—
263	C₂²×C₅₃	—	4	16	no	—
264	C₂²×C₅₃	—	4	4	no	—
265	C₂²×C₅₃	—	4	16	no	—
266	C₂²×C₅₃	—	4	16	no	—
267	C₂²×C₅₃	—	4	16	no	—
268	C₂²×C₅₃	—	8	32	no	—
269	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
270	C₂²×C₅₃	C₂³×D₄	1	16	yes	—
271	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
272	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
273	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
274	C₂×D₅₃	—	2	8	no	—
275	C₂×D₅₃	—	2	16	no	—
276	C₂×D₅₃	—	4	16	no	—
277	C₂×D₅₃	—	8	32	no	—
278	C₄×C₅₃	—	2	8	no	—
279	C₄×C₅₃	C₂³×D₄	1	8	yes	—
280	Dic₅₃	—	2	8	no	—
281	Dic₅₃	—	2	16	no	—
282	C₂²×H(5,𝔽₂)	—	53	1	no	—
283	C₂²×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic, complemented
284	#3496	—	53	16	no	—
285	#3485	—	53	8	no	—
286	#3485	—	53	8	no	—
287	#3484	—	53	8	no	—
288	#3484	—	53	8	no	—
289	#3449	—	53	4	no	—
290	C₂×#534	—	53	8	no	—
291	C₂×#518	—	53	1	no	—
292	C₂×C₅₃	—	2	2	no	—
293	C₂×C₅₃	—	2	16	no	—
294	C₂×C₅₃	—	2	2	no	—
295	C₂×C₅₃	—	2	2	no	—
296	C₂×C₅₃	—	4	8	no	—
297	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
298	C₂×C₅₃	H(7,𝔽₂)	1	2	yes	—
299	D₅₃	—	4	8	no	—
300	C₂×H(5,𝔽₂)	—	2	16	no	—
301	C₂×H(5,𝔽₂)	—	53	8	no	—
302	C₂×H(5,𝔽₂)	—	53	8	no	—
303	C₂×H(5,𝔽₂)	—	106	16	no	—
304	C₂×H(5,𝔽₂)	C₂×D₅₃	1	8	yes	complemented
305	C₂×H(5,𝔽₂)	C₂×D₅₃	1	8	yes	complemented
306	C₂²×#41	—	53	4	no	—
307	C₂²×#41	—	53	2	no	—
308	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
309	C₂²×#41	C₂×D₅₃	1	4	yes	complemented
310	C₂³×D₄	—	53	4	no	—
311	C₂³×D₄	—	53	4	no	—
312	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
313	C₂³×D₄	C₂×D₅₃	1	4	yes	complemented
314	C₂³×D₄	C₂×D₅₃	1	4	yes	complemented
315	#547	—	53	64	no	—
316	#546	—	53	64	no	—
317	#546	—	53	64	no	—
318	D₄²	—	53	64	no	—
319	#537	—	53	16	no	—
320	#536	—	53	8	no	—
321	#536	—	53	8	no	—
322	#535	—	53	8	no	—
323	#535	—	53	8	no	—
324	#534	—	53	16	no	—
325	#534	—	53	32	no	—
326	#534	—	53	32	no	—
327	#534	—	53	32	no	—
328	#532	—	53	16	no	—
329	#525	—	53	16	no	—
330	#525	—	53	16	no	—
331	C₂×#123	—	53	16	no	—
332	C₂×#123	—	53	16	no	—
333	C₂×#121	—	53	32	no	—
334	C₂×#120	—	53	16	no	—
335	C₂×#120	—	53	16	no	—
336	#518	—	53	8	no	—
337	#518	—	53	8	no	—
338	C₂×C₄×D₄	—	53	8	no	—
339	C₂×#117	—	53	2	no	—
340	C₂²×C₂²⋊C₄	—	53	4	no	—
341	C₅₃	#62842	1	1	yes	characteristic, complemented, Sylow
342	C₂⁵	—	53	4	no	—
343	C₂⁵	C₂²×D₅₃	1	2	yes	—
344	C₂⁵	C₂²×D₅₃	1	4	yes	complemented
345	H(5,𝔽₂)	—	2	64	no	—
346	H(5,𝔽₂)	—	2	64	no	—
347	H(5,𝔽₂)	—	106	64	no	—
348	H(5,𝔽₂)	—	106	64	no	—
349	C₂×#41	—	2	32	no	—
350	C₂×#41	—	2	16	no	—
351	C₂×#41	—	53	32	no	—
352	C₂×#41	—	53	8	no	—
353	C₂×#41	—	53	8	no	—
354	C₂×#41	—	106	32	no	—
355	C₂×#41	—	106	16	no	—
356	C₂×#41	C₂²×D₅₃	1	8	yes	—
357	C₂×#41	C₂²×D₅₃	1	8	yes	complemented
358	C₂×#41	C₂²×D₅₃	1	32	yes	complemented
359	C₂²×Q₈	—	53	2	no	—
360	C₂²×Q₈	C₂²×D₅₃	1	2	yes	complemented
361	C₂²×D₄	—	2	32	no	—
362	C₂²×D₄	—	2	32	no	—
363	C₂²×D₄	—	53	16	no	—
364	C₂²×D₄	—	53	16	no	—
365	C₂²×D₄	—	53	4	no	—
366	C₂²×D₄	—	53	8	no	—
367	C₂²×D₄	—	53	32	no	—
368	C₂²×D₄	—	53	32	no	—
369	C₂²×D₄	—	53	2	no	—
370	C₂²×D₄	—	106	32	no	—
371	C₂²×D₄	—	106	32	no	—
372	C₂²×D₄	C₂²⋊D₅₃	1	16	yes	complemented
373	C₂²×D₄	C₂²×D₅₃	1	16	yes	—
374	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
375	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
376	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
377	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
378	C₂²×D₄	C₂²×D₅₃	1	16	yes	complemented
379	C₂²×D₄	C₂²×D₅₃	1	8	yes	complemented
380	C₂²×D₄	C₂²×D₅₃	1	2	yes	complemented
381	C₂²×D₄	C₂²×D₅₃	1	32	yes	complemented
382	C₂³×C₄	—	53	4	no	—
383	C₂³×C₄	—	53	8	no	—
384	C₂³×C₄	—	53	4	no	—
385	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
386	C₂³×C₄	C₂²×D₅₃	1	4	yes	complemented
387	C₂³×C₄	C₂²×D₅₃	1	4	yes	complemented
388	C₄⋊Q₈	—	53	16	no	—
389	#127	—	53	16	no	—
390	#124	—	53	16	no	—
391	#124	—	53	16	no	—
392	#123	—	53	32	no	—
393	#123	—	53	64	no	—
394	#123	—	53	32	no	—
395	#123	—	53	64	no	—
396	C₂²⋊Q₈	—	53	32	no	—
397	C₂²⋊Q₈	—	53	32	no	—
398	#121	—	53	64	no	—
399	#121	—	53	128	no	—
400	#121	—	53	32	no	—
401	#121	—	53	32	no	—
402	#120	—	53	32	no	—
403	#120	—	53	64	no	—
404	#120	—	53	32	no	—
405	#120	—	53	64	no	—
406	C₄×D₄	—	53	64	no	—
407	C₄×D₄	—	53	64	no	—
408	#117	—	53	8	no	—
409	#117	—	53	8	no	—
410	C₂×C₄⋊C₄	—	53	4	no	—
411	C₂×C₄⋊C₄	—	53	32	no	—
412	C₂×C₂²⋊C₄	—	53	4	no	—
413	C₂×C₂²⋊C₄	—	53	8	no	—
414	C₂×C₂²⋊C₄	—	53	16	no	—
415	C₂×C₂²⋊C₄	—	53	16	no	—
416	C₂×C₂²⋊C₄	—	53	32	no	—
417	C₂×C₂²⋊C₄	—	53	32	no	—
418	C₂×C₄²	—	53	4	no	—
419	C₂⁴	—	2	16	no	—
420	C₂⁴	—	8	4	no	—
421	C₂⁴	—	8	8	no	—
422	C₂⁴	—	53	16	no	—
423	C₂⁴	—	53	4	no	—
424	C₂⁴	—	106	16	no	—
425	C₂⁴	—	424	8	no	—
426	C₂⁴	C₂×C₂²⋊D₅₃	1	16	yes	complemented
427	C₂⁴	C₂³×D₅₃	1	16	yes	—
428	C₂⁴	C₂³×D₅₃	1	4	yes	—
429	C₂⁴	C₂³×D₅₃	1	4	yes	—
430	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
431	C₂⁴	C₂³×D₅₃	1	4	yes	—
432	C₂⁴	C₂³×D₅₃	1	4	yes	—
433	#41	—	2	128	no	—
434	#41	—	2	32	no	—
435	#41	—	2	32	no	—
436	#41	—	106	128	no	—
437	#41	—	106	32	no	—
438	#41	—	106	32	no	—
439	C₂×Q₈	—	2	8	no	—
440	C₂×Q₈	—	53	8	no	—
441	C₂×Q₈	—	106	8	no	—
442	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
443	C₂×D₄	—	2	16	no	—
444	C₂×D₄	—	2	32	no	—
445	C₂×D₄	—	2	64	no	—
446	C₂×D₄	—	2	64	no	—
447	C₂×D₄	—	2	128	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	53	8	no	—
450	C₂×D₄	—	53	32	no	—
451	C₂×D₄	—	53	8	no	—
452	C₂×D₄	—	53	64	no	—
453	C₂×D₄	—	53	8	no	—
454	C₂×D₄	—	106	32	no	—
455	C₂×D₄	—	106	16	no	—
456	C₂×D₄	—	106	64	no	—
457	C₂×D₄	—	106	64	no	—
458	C₂×D₄	—	106	128	no	—
459	C₂×D₄	—	106	8	no	—
460	C₂×D₄	—	424	32	no	—
461	C₂×D₄	C₂×C₂²⋊D₅₃	1	32	yes	complemented
462	C₂×D₄	C₂×C₂²⋊D₅₃	1	32	yes	complemented
463	C₂×D₄	C₂×C₂²⋊D₅₃	1	32	yes	complemented
464	C₂×D₄	C₂³×D₅₃	1	8	yes	—
465	C₂×D₄	C₂³×D₅₃	1	32	yes	—
466	C₂×D₄	C₂³×D₅₃	1	8	yes	—
467	C₂×D₄	C₂³×D₅₃	1	8	yes	—
468	C₂×D₄	C₂³×D₅₃	1	16	yes	—
469	C₂²×C₄	—	2	16	no	—
470	C₂²×C₄	—	2	16	no	—
471	C₂²×C₄	—	53	16	no	—
472	C₂²×C₄	—	53	4	no	—
473	C₂²×C₄	—	53	8	no	—
474	C₂²×C₄	—	53	32	no	—
475	C₂²×C₄	—	53	16	no	—
476	C₂²×C₄	—	106	16	no	—
477	C₂²×C₄	—	106	16	no	—
478	C₂²×C₄	—	424	8	no	—
479	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	complemented
480	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
481	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
482	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
483	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
484	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
485	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
486	C₄⋊C₄	—	53	64	no	—
487	C₄⋊C₄	—	53	16	no	—
488	C₂²⋊C₄	—	53	8	no	—
489	C₂²⋊C₄	—	53	8	no	—
490	C₂²⋊C₄	—	53	32	no	—
491	C₂²⋊C₄	—	53	64	no	—
492	C₂²⋊C₄	—	53	64	no	—
493	C₄²	—	53	16	no	—
494	C₂³	—	2	8	no	—
495	C₂³	—	2	32	no	—
496	C₂³	—	4	8	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	8	no	—
499	C₂³	—	4	8	no	—
500	C₂³	—	4	8	no	—
501	C₂³	—	4	32	no	—
502	C₂³	—	8	16	no	—
503	C₂³	—	8	32	no	—
504	C₂³	—	53	8	no	—
505	C₂³	—	106	32	no	—
506	C₂³	—	106	8	no	—
507	C₂³	—	424	32	no	—
508	C₂³	—	212	32	no	—
509	C₂³	—	212	8	no	—
510	C₂³	C₂²×C₂²⋊D₅₃	1	16	yes	—
511	C₂³	C₂²×C₂²⋊D₅₃	1	16	yes	—
512	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
513	C₂³	C₂²×C₂²⋊D₅₃	1	16	yes	—
514	C₂³	C₂⁴×D₅₃	1	8	yes	—
515	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
516	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
517	C₂³	C₂⁴×D₅₃	1	8	yes	—
518	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
519	Q₈	—	2	16	no	—
520	Q₈	—	106	16	no	—
521	D₄	—	2	64	no	—
522	D₄	—	2	16	no	—
523	D₄	—	2	16	no	—
524	D₄	—	2	16	no	—
525	D₄	—	106	16	no	—
526	D₄	—	106	64	no	—
527	D₄	—	106	16	no	—
528	D₄	—	106	16	no	—
529	D₄	—	212	64	no	—
530	D₄	C₂²×C₂²⋊D₅₃	1	64	yes	—
531	C₂×C₄	—	2	8	no	—
532	C₂×C₄	—	2	32	no	—
533	C₂×C₄	—	2	32	no	—
534	C₂×C₄	—	53	16	no	—
535	C₂×C₄	—	53	8	no	—
536	C₂×C₄	—	106	8	no	—
537	C₂×C₄	—	106	32	no	—
538	C₂×C₄	—	106	32	no	—
539	C₂×C₄	—	212	8	no	—
540	C₂×C₄	—	212	32	no	—
541	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
542	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
543	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
544	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
545	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	16	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	2	8	no	—
551	C₂²	—	2	16	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	16	no	—
554	C₂²	—	4	16	no	—
555	C₂²	—	4	16	no	—
556	C₂²	—	4	16	no	—
557	C₂²	—	8	32	no	—
558	C₂²	—	106	8	no	—
559	C₂²	—	106	16	no	—
560	C₂²	—	424	32	no	—
561	C₂²	—	212	16	no	—
562	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
563	C₂²	C₂³×C₂²⋊D₅₃	1	16	yes	—
564	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
565	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
566	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
567	C₄	—	2	8	no	—
568	C₄	—	106	8	no	—
569	C₄	—	106	16	no	—
570	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
571	C₂	—	2	2	no	—
572	C₂	—	2	2	no	—
573	C₂	—	2	2	no	—
574	C₂	—	2	16	no	—
575	C₂	—	4	8	no	—
576	C₂	—	212	8	no	—
577	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
578	C₂	#20289525	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₂²×H(5,𝔽₂)×C₅₃	C₂
hypercenter	C₂²×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	#62842
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133262	C₂	semidirect
2. #20288474	C₂	semidirect
3. #20288359	C₂	semidirect
4. #20288355	C₂	semidirect
5. #20288350	C₂	semidirect
6. #20288346	C₂	semidirect
7. #20288044	C₂	semidirect
8. C₂×#16811182	C₂	semidirect
9. C₂×#16811098	C₂	semidirect
10. C₂²×H(5,𝔽₂)×C₅₃	C₂	semidirect
12. C₂×#13133262	C₂²	semidirect
13. C₂×#13133262	C₂²	semidirect
14. C₂²×#11719945	C₂²	semidirect
15. C₂²×#11719947	C₂²	semidirect
16. C₂³×C₂²⋊D₅₃	C₂²	semidirect
17. C₂²×D₄×D₅₃	C₂²	semidirect
18. #16811272	C₂²	semidirect
19. #16811267	C₂²	semidirect
20. #16811261	C₂²	semidirect
21. D₄×C₂²⋊D₅₃	C₂²	semidirect
22. #16811201	C₂²	semidirect
23. #16811197	C₂²	semidirect
24. #16811194	C₂²	semidirect
25. #16811190	C₂²	semidirect
26. #16811187	C₂²	semidirect
27. #16811182	C₂²	semidirect
28. #16811182	C₂²	semidirect
29. #16811182	C₂²	semidirect
30. #16811182	C₂²	semidirect
31. #16811169	C₂²	semidirect
32. #16811134	C₂²	semidirect
33. #16811130	C₂²	semidirect
34. C₂×#13133200	C₂²	semidirect
35. C₂×#13133198	C₂²	semidirect
36. C₂×#13133189	C₂²	semidirect
37. C₂×#13133186	C₂²	semidirect
38. C₂×#13133185	C₂²	semidirect
39. #16811098	C₂²	semidirect
40. #16811098	C₂²	semidirect
41. C₂×D₄×Dic₅₃	C₂²	semidirect
42. C₂×#13133176	C₂²	semidirect
43. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
45. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
46. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
47. C₂²×#41×C₅₃	C₂²	semidirect
48. C₂²×#41×C₅₃	C₂²	semidirect
50. C₂³×D₄×C₅₃	C₂²	semidirect
51. C₂³×D₄×C₅₃	C₂²	semidirect
52. C₂⁴×D₅₃	C₂³	semidirect
57. C₂×#11719945	C₂³	semidirect
59. C₂×#11719947	C₂³	semidirect
60. C₂²×Dic₁₀₆	C₂³	semidirect
61. C₂²×D₂₁₂	C₂³	semidirect
65. C₂²×C₂²⋊D₅₃	C₂³	semidirect
66. C₂²×C₂²⋊D₅₃	C₂³	semidirect
67. C₂²×C₂²⋊D₅₃	C₂³	semidirect
69. C₂×D₄×D₅₃	C₂³	semidirect
70. C₂³×Dic₅₃	C₂³	semidirect
71. C₂³×Dic₅₃	C₂³	semidirect
72. C₂²×C₄×D₅₃	C₂³	semidirect
73. #13133215	C₂³	semidirect
74. #13133213	C₂³	semidirect
75. #13133206	C₂³	semidirect
76. #13133204	C₂³	semidirect
77. #13133200	C₂³	semidirect
78. #13133200	C₂³	semidirect
79. #13133198	C₂³	semidirect
80. #13133198	C₂³	semidirect
81. #13133196	C₂³	semidirect
82. #13133193	C₂³	semidirect
83. #13133191	C₂³	semidirect
84. #13133189	C₂³	semidirect
85. #13133189	C₂³	semidirect
86. #13133188	C₂³	semidirect
87. #13133186	C₂³	semidirect
88. #13133186	C₂³	semidirect
89. #13133185	C₂³	semidirect
90. #13133185	C₂³	semidirect
91. D₄×Dic₅₃	C₂³	semidirect
92. C₄×C₂²⋊D₅₃	C₂³	semidirect
93. #13133176	C₂³	semidirect
94. #13133176	C₂³	semidirect
95. C₂×C₄⋊Dic₅₃	C₂³	semidirect
96. C₂×#11719919	C₂³	semidirect
98. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
99. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
100. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
101. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
102. C₂×#11719921	C₂³	semidirect
103. C₂×C₄×Dic₅₃	C₂³	semidirect
105. C₂⁵×C₅₃	C₂³	semidirect
111. C₂×#41×C₅₃	C₂³	semidirect
112. C₂×#41×C₅₃	C₂³	semidirect
113. C₂²×Q₈×C₅₃	C₂³	semidirect
116. C₂²×D₄×C₅₃	D₄	semidirect
122. C₂²×D₄×C₅₃	C₂³	semidirect
123. C₂²×D₄×C₅₃	C₂³	semidirect
124. C₂²×D₄×C₅₃	C₂³	semidirect
125. C₂²×D₄×C₅₃	C₂³	semidirect
127. C₂³×C₄×C₅₃	C₂³	semidirect
128. C₂³×C₄×C₅₃	C₂³	semidirect
132. C₂⁴×C₅₃	C₂×D₄	semidirect
154. C₂×D₄×C₅₃	C₂×D₄	semidirect
155. C₂×D₄×C₅₃	C₂×D₄	semidirect
156. C₂×D₄×C₅₃	C₂×D₄	semidirect
164. C₂²×C₄×C₅₃	C₂×D₄	semidirect
283. C₂²×H(5,𝔽₂)	D₅₃	semidirect
304. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
305. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
308. C₂²×#41	C₂×D₅₃	semidirect
309. C₂²×#41	C₂×D₅₃	semidirect
313. C₂³×D₄	C₂×D₅₃	semidirect
314. C₂³×D₄	C₂×D₅₃	semidirect
341. C₅₃	#62842	semidirect
344. C₂⁵	C₂²×D₅₃	semidirect
357. C₂×#41	C₂²×D₅₃	semidirect
358. C₂×#41	C₂²×D₅₃	semidirect
360. C₂²×Q₈	C₂²×D₅₃	semidirect
372. C₂²×D₄	C₂²⋊D₅₃	semidirect
378. C₂²×D₄	C₂²×D₅₃	semidirect
379. C₂²×D₄	C₂²×D₅₃	semidirect
380. C₂²×D₄	C₂²×D₅₃	semidirect
381. C₂²×D₄	C₂²×D₅₃	semidirect
386. C₂³×C₄	C₂²×D₅₃	semidirect
387. C₂³×C₄	C₂²×D₅₃	semidirect
426. C₂⁴	C₂×C₂²⋊D₅₃	semidirect
461. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
462. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
463. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
479. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133262	2
#20288474	2
#20288359	2
#20288355	2
#20288350	2
#20288346	2
#20288044	2
C₂×#16811182	2
C₂×#16811098	2
C₂²×H(5,𝔽₂)×C₅₃	2
#62842	53