Subgroups — C₂×#20288359 #28292136

Order 13568.

11686 conjugacy classes of subgroups.

The subgroup lattice has 532 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	#20288359	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811272	C₂	1	8	yes	maximal, complemented
4	C₂×#16811261	C₂	1	8	yes	maximal, complemented
5	C₂×#16811197	C₂	1	2	yes	maximal, complemented
6	C₂×#16811182	C₂	1	4	yes	maximal, complemented
7	C₂×#16811134	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133200	C₂	1	4	yes	maximal, complemented
9	C₂×#16811098	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133262	—	2	16	no	—
12	C₂×#13133262	C₂²	1	16	yes	complemented
13	C₂²×#11719945	C₂²	1	2	yes	—
14	C₂²×#11719947	C₂²	1	4	yes	complemented
15	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	—
16	C₂²×D₄×D₅₃	C₂²	1	4	yes	complemented
17	#16811272	C₂²	1	128	yes	complemented
18	#16811261	C₂²	1	128	yes	complemented
19	#16811197	C₂²	1	32	yes	complemented
20	#16811182	C₂²	1	64	yes	complemented
21	C₂×#13133215	C₂²	1	4	yes	complemented
22	C₂×#13133206	C₂²	1	4	yes	complemented
23	#16811134	C₂²	1	32	yes	complemented
24	C₂×#13133200	C₂²	1	8	yes	complemented
25	C₂×#13133200	C₂²	1	16	yes	complemented
26	C₂×#13133200	C₂²	1	4	yes	complemented
27	C₂×#13133200	C₂²	1	64	yes	complemented
28	C₂×#13133198	C₂²	1	8	yes	complemented
29	C₂×#13133196	C₂²	1	4	yes	complemented
30	C₂×#13133193	C₂²	1	8	yes	complemented
31	C₂×#13133191	C₂²	1	4	yes	complemented
32	C₂×#13133189	C₂²	1	16	yes	complemented
33	C₂×#13133186	C₂²	1	4	yes	complemented
34	C₂×#13133185	C₂²	1	8	yes	complemented
35	#16811098	C₂²	1	16	yes	complemented
36	C₂×D₄×Dic₅₃	C₂²	1	8	yes	complemented
37	C₂×C₄×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
38	C₂×#13133176	C₂²	1	2	yes	complemented
39	C₂²×#11719919	C₂²	1	8	yes	complemented
40	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	complemented
41	C₂²×#11719921	C₂²	1	8	yes	complemented
42	C₂²×#41×C₅₃	—	2	16	no	—
43	C₂²×#41×C₅₃	C₂²	1	8	yes	complemented
44	C₂²×#41×C₅₃	C₂²	1	16	yes	complemented
45	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
47	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
48	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
49	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂⁴×D₅₃	C₂³	1	4	yes	—
51	#13133262	—	2	128	no	—
52	C₂×#11719945	—	2	16	no	—
53	C₂×#11719945	C₂³	1	16	yes	—
54	C₂×#11719947	—	2	32	no	—
55	C₂×#11719947	C₂³	1	32	yes	complemented
56	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
57	C₂²×D₂₁₂	C₂³	1	2	yes	—
58	C₂²×C₂²⋊D₅₃	—	2	32	no	—
59	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
60	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
61	C₂²×C₂²⋊D₅₃	C₂³	1	32	yes	—
62	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
63	C₂×D₄×D₅₃	—	2	32	no	—
64	C₂×D₄×D₅₃	C₂³	1	32	yes	complemented
65	C₂³×Dic₅₃	C₂³	1	4	yes	—
66	C₂³×Dic₅₃	C₂³	1	8	yes	—
67	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
68	#13133215	C₂³	1	32	yes	complemented
69	#13133206	C₂³	1	32	yes	complemented
70	#13133200	C₂³	1	128	yes	complemented
71	#13133200	C₂³	1	32	yes	complemented
72	#13133200	C₂³	1	64	yes	complemented
73	#13133198	C₂³	1	64	yes	complemented
74	#13133196	C₂³	1	32	yes	complemented
75	#13133193	C₂³	1	64	yes	complemented
76	#13133191	C₂³	1	32	yes	complemented
77	#13133189	C₂³	1	128	yes	complemented
78	#13133186	C₂³	1	32	yes	complemented
79	#13133185	C₂³	1	64	yes	complemented
80	D₄×Dic₅₃	C₂³	1	64	yes	complemented
81	C₄×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
82	#13133176	C₂³	1	16	yes	complemented
83	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
84	C₂×#11719919	C₂³	1	8	yes	—
85	C₂×#11719919	C₂³	1	16	yes	—
86	C₂×#11719919	C₂³	1	64	yes	complemented
87	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
88	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
89	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
90	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	complemented
91	C₂×#11719921	C₂³	1	8	yes	—
92	C₂×#11719921	C₂³	1	16	yes	—
93	C₂×#11719921	C₂³	1	64	yes	complemented
94	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
95	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
96	C₂⁵×C₅₃	C₂³	1	2	yes	—
97	C₂×#41×C₅₃	—	2	64	no	—
98	C₂×#41×C₅₃	—	2	128	no	—
99	C₂×#41×C₅₃	C₂³	1	64	yes	complemented
100	C₂²×Q₈×C₅₃	—	2	8	no	—
101	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
102	C₂²×Q₈×C₅₃	C₂³	1	8	yes	complemented
103	C₂²×D₄×C₅₃	—	2	16	no	—
104	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
105	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
106	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
107	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
108	C₂²×D₄×C₅₃	C₂³	1	16	yes	complemented
109	C₂³×C₄×C₅₃	—	2	16	no	—
110	C₂³×C₄×C₅₃	—	2	8	no	—
111	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
112	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
113	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
114	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
115	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
116	C₂³×C₄×C₅₃	C₂³	1	16	yes	complemented
117	C₂³×C₄×C₅₃	C₂³	1	8	yes	complemented
118	C₂⁴×C₅₃	—	2	4	no	—
119	C₂⁴×C₅₃	—	4	8	no	—
120	C₂⁴×C₅₃	—	4	4	no	—
121	C₂⁴×C₅₃	C₂×D₄	1	16	yes	—
122	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
123	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
124	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
125	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
126	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
127	C₂³×D₅₃	—	2	16	no	—
128	C₂³×D₅₃	—	4	16	no	—
129	C₂³×D₅₃	C₂⁴	1	4	yes	—
130	C₂³×D₅₃	C₂⁴	1	16	yes	—
131	#41×C₅₃	—	2	256	no	—
132	C₂×Q₈×C₅₃	—	2	16	no	—
133	C₂×Q₈×C₅₃	—	2	32	no	—
134	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
135	C₂×D₄×C₅₃	—	2	32	no	—
136	C₂×D₄×C₅₃	—	2	64	no	—
137	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	—
138	C₂×D₄×C₅₃	C₂×D₄	1	64	yes	complemented
139	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
140	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
141	C₂²×C₄×C₅₃	—	2	4	no	—
142	C₂²×C₄×C₅₃	—	2	8	no	—
143	C₂²×C₄×C₅₃	—	2	32	no	—
144	C₂²×C₄×C₅₃	—	2	64	no	—
145	C₂²×C₄×C₅₃	—	2	16	no	—
146	C₂²×C₄×C₅₃	—	2	32	no	—
147	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
148	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
156	C₂×C₂²⋊D₅₃	—	2	32	no	—
157	C₂×C₂²⋊D₅₃	—	2	16	no	—
158	C₂×C₂²⋊D₅₃	—	2	128	no	—
159	C₂×C₂²⋊D₅₃	—	4	16	no	—
160	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
161	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
162	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
163	C₂²×Dic₅₃	—	2	16	no	—
164	C₂²×Dic₅₃	—	4	16	no	—
165	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
166	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
167	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
168	C₂²×Dic₅₃	C₂⁴	1	32	yes	—
169	#11719947	—	2	128	no	—
170	D₄×D₅₃	—	2	128	no	—
171	#11719945	—	2	64	no	—
172	C₂×D₂₁₂	—	2	8	no	—
173	C₂×D₂₁₂	C₂⁴	1	8	yes	—
174	C₂×C₄×D₅₃	—	2	16	no	—
175	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
176	C₂×Dic₁₀₆	—	2	8	no	—
177	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
178	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
179	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
180	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
181	#11719921	C₂⁴	1	32	yes	—
182	#11719921	C₂⁴	1	64	yes	—
183	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
184	#11719919	C₂⁴	1	32	yes	—
185	#11719919	C₂⁴	1	64	yes	—
186	C₄×Dic₅₃	C₂⁴	1	16	yes	—
187	C₂³×C₅₃	—	2	16	no	—
188	C₂³×C₅₃	—	2	4	no	—
189	C₂³×C₅₃	—	2	8	no	—
190	C₂³×C₅₃	—	2	4	no	—
191	C₂³×C₅₃	—	2	8	no	—
192	C₂³×C₅₃	—	4	32	no	—
193	C₂³×C₅₃	—	4	16	no	—
194	C₂³×C₅₃	—	4	8	no	—
195	C₂³×C₅₃	—	4	16	no	—
196	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
197	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
198	C₂³×C₅₃	C₂²×D₄	1	32	yes	—
199	C₂³×C₅₃	C₂⁵	1	4	yes	—
200	C₂³×C₅₃	C₂⁵	1	8	yes	—
201	C₂³×C₅₃	C₂⁵	1	2	yes	—
202	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
203	C₂²×D₅₃	—	2	8	no	—
204	C₂²×D₅₃	—	2	16	no	—
205	C₂²×D₅₃	—	2	32	no	—
206	C₂²×D₅₃	—	4	32	no	—
207	C₂²×D₅₃	—	8	32	no	—
208	C₂²×D₅₃	C₂⁵	1	8	yes	—
209	Q₈×C₅₃	—	2	32	no	—
210	D₄×C₅₃	—	2	64	no	—
211	D₄×C₅₃	C₂²×D₄	1	64	yes	—
212	C₂×C₄×C₅₃	—	2	8	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	16	no	—
215	C₂×C₄×C₅₃	—	2	8	no	—
216	C₂×C₄×C₅₃	—	2	64	no	—
217	C₂×C₄×C₅₃	—	2	32	no	—
218	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
219	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
220	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
221	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
222	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
223	C₂²⋊D₅₃	—	2	32	no	—
224	C₂²⋊D₅₃	—	2	64	no	—
225	C₂²⋊D₅₃	—	4	32	no	—
226	C₂×Dic₅₃	—	2	8	no	—
227	C₂×Dic₅₃	—	2	32	no	—
228	C₂×Dic₅₃	—	2	16	no	—
229	C₂×Dic₅₃	—	4	32	no	—
230	C₂×Dic₅₃	C₂⁵	1	8	yes	—
231	C₂×Dic₅₃	C₂⁵	1	16	yes	—
232	D₂₁₂	—	2	16	no	—
233	C₄×D₅₃	—	2	32	no	—
234	Dic₁₀₆	—	2	16	no	—
235	C₂×#3485	—	53	1	no	maximal, Sylow
236	C₂²×C₅₃	—	2	16	no	—
237	C₂²×C₅₃	—	2	4	no	—
238	C₂²×C₅₃	—	2	32	no	—
239	C₂²×C₅₃	—	2	8	no	—
240	C₂²×C₅₃	—	2	4	no	—
241	C₂²×C₅₃	—	2	8	no	—
242	C₂²×C₅₃	—	4	32	no	—
243	C₂²×C₅₃	—	4	16	no	—
244	C₂²×C₅₃	—	4	8	no	—
245	C₂²×C₅₃	C₂³×D₄	1	16	yes	—
246	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
247	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
248	C₂²×C₅₃	#585	1	4	yes	—
249	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
250	C₂×D₅₃	—	2	8	no	—
251	C₂×D₅₃	—	2	16	no	—
252	C₂×D₅₃	—	4	16	no	—
253	C₂×D₅₃	—	8	32	no	—
254	C₄×C₅₃	—	2	4	no	—
255	C₄×C₅₃	—	2	8	no	—
256	C₄×C₅₃	C₂³×D₄	1	8	yes	—
257	Dic₅₃	—	2	8	no	—
258	Dic₅₃	—	2	16	no	—
259	C₂²×H(5,𝔽₂)	—	53	1	no	—
260	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
261	#3485	—	53	32	no	—
262	C₂×#547	—	53	8	no	—
263	C₂×#546	—	53	8	no	—
264	C₂×#536	—	53	2	no	—
265	C₂×#534	—	53	4	no	—
266	C₂×#525	—	53	2	no	—
267	C₂²×#123	—	53	4	no	—
268	C₂×#518	—	53	1	no	—
269	C₂×C₅₃	—	2	16	no	—
270	C₂×C₅₃	—	2	4	no	—
271	C₂×C₅₃	—	4	4	no	—
272	C₂×C₅₃	#3485	1	4	yes	complemented
273	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
274	C₂×C₅₃	C₂×#585	1	2	yes	—
275	D₅₃	—	4	8	no	—
276	C₂×H(5,𝔽₂)	—	53	16	no	—
277	C₂×H(5,𝔽₂)	—	106	16	no	—
278	C₂²×#41	—	2	16	no	—
279	C₂²×#41	—	53	4	no	—
280	C₂²×#41	—	53	2	no	—
281	C₂²×#41	C₂×D₅₃	1	8	yes	complemented
282	C₂²×#41	C₂×D₅₃	1	16	yes	complemented
283	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
284	C₂³×D₄	—	53	4	no	—
285	C₂³×D₄	—	53	4	no	—
286	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
287	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
288	C₂⁴×C₄	C₂×D₅₃	1	2	yes	complemented
289	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
290	#547	—	53	128	no	—
291	#546	—	53	128	no	—
292	#536	—	53	32	no	—
293	#534	—	53	64	no	—
294	C₂×C₄⋊Q₈	—	53	4	no	—
295	C₂×#124	—	53	4	no	—
296	#525	—	53	32	no	—
297	C₂×#123	—	53	8	no	—
298	C₂×#123	—	53	4	no	—
299	C₂×#123	—	53	8	no	—
300	C₂×#123	—	53	16	no	—
301	C₂×#123	—	53	64	no	—
302	C₂×C₂²⋊Q₈	—	53	8	no	—
303	C₂×C₂²⋊Q₈	—	53	4	no	—
304	C₂×#121	—	53	16	no	—
305	C₂×#121	—	53	4	no	—
306	C₂×#120	—	53	4	no	—
307	C₂×#120	—	53	8	no	—
308	#518	—	53	16	no	—
309	C₂×C₄×D₄	—	53	8	no	—
310	C₂×C₄×D₄	—	53	8	no	—
311	C₂×#117	—	53	2	no	—
312	C₂²×C₄⋊C₄	—	53	8	no	—
313	C₂²×C₂²⋊C₄	—	53	4	no	—
314	C₂²×C₂²⋊C₄	—	53	8	no	—
315	C₅₃	C₂×#3485	1	1	yes	characteristic, complemented, Sylow
316	C₂⁵	—	53	4	no	—
317	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
318	C₂⁵	C₂²×D₅₃	1	2	yes	—
319	H(5,𝔽₂)	—	106	128	no	—
320	C₂×#41	—	2	64	no	—
321	C₂×#41	—	2	128	no	—
322	C₂×#41	—	53	32	no	—
323	C₂×#41	—	53	16	no	—
324	C₂×#41	—	106	32	no	—
325	C₂×#41	—	106	16	no	—
326	C₂×#41	C₂²×D₅₃	1	64	yes	complemented
327	C₂²×Q₈	—	2	8	no	—
328	C₂²×Q₈	—	53	2	no	—
329	C₂²×Q₈	C₂²×D₅₃	1	4	yes	—
330	C₂²×Q₈	C₂²×D₅₃	1	8	yes	complemented
331	C₂²×D₄	—	2	16	no	—
332	C₂²×D₄	—	53	4	no	—
333	C₂²×D₄	—	53	8	no	—
334	C₂²×D₄	—	53	32	no	—
335	C₂²×D₄	—	53	8	no	—
336	C₂²×D₄	—	53	32	no	—
337	C₂²×D₄	—	53	2	no	—
338	C₂²×D₄	—	106	32	no	—
339	C₂²×D₄	—	106	32	no	—
340	C₂²×D₄	C₂²⋊D₅₃	1	16	yes	complemented
341	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
342	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
343	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
344	C₂²×D₄	C₂²×D₅₃	1	16	yes	complemented
345	C₂³×C₄	—	2	16	no	—
346	C₂³×C₄	—	2	8	no	—
347	C₂³×C₄	—	53	4	no	—
348	C₂³×C₄	—	53	8	no	—
349	C₂³×C₄	—	53	4	no	—
350	C₂³×C₄	C₂²×D₅₃	1	8	yes	—
351	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
352	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
353	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
354	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
355	C₂³×C₄	C₂²×D₅₃	1	16	yes	complemented
356	C₂³×C₄	C₂²×D₅₃	1	8	yes	complemented
357	C₄⋊Q₈	—	53	32	no	—
358	#124	—	53	32	no	—
359	#123	—	53	64	no	—
360	#123	—	53	32	no	—
361	#123	—	53	64	no	—
362	#123	—	53	128	no	—
363	C₂²⋊Q₈	—	53	64	no	—
364	C₂²⋊Q₈	—	53	32	no	—
365	#121	—	53	128	no	—
366	#121	—	53	32	no	—
367	#120	—	53	32	no	—
368	#120	—	53	64	no	—
369	C₄×D₄	—	53	64	no	—
370	C₄×D₄	—	53	64	no	—
371	#117	—	53	16	no	—
372	C₂×C₄⋊C₄	—	53	8	no	—
373	C₂×C₄⋊C₄	—	53	16	no	—
374	C₂×C₄⋊C₄	—	53	64	no	—
375	C₂×C₄⋊C₄	—	53	4	no	—
376	C₂×C₂²⋊C₄	—	53	4	no	—
377	C₂×C₂²⋊C₄	—	53	8	no	—
378	C₂×C₂²⋊C₄	—	53	32	no	—
379	C₂×C₂²⋊C₄	—	53	8	no	—
380	C₂×C₂²⋊C₄	—	53	8	no	—
381	C₂×C₂²⋊C₄	—	53	16	no	—
382	C₂×C₂²⋊C₄	—	53	64	no	—
383	C₂×C₄²	—	53	4	no	—
384	C₂⁴	—	2	4	no	—
385	C₂⁴	—	4	8	no	—
386	C₂⁴	—	4	4	no	—
387	C₂⁴	—	53	4	no	—
388	C₂⁴	—	53	16	no	—
389	C₂⁴	—	106	16	no	—
390	C₂⁴	—	212	16	no	—
391	C₂⁴	C₂×C₂²⋊D₅₃	1	16	yes	—
392	C₂⁴	C₂³×D₅₃	1	2	yes	—
393	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
394	C₂⁴	C₂³×D₅₃	1	4	yes	—
395	C₂⁴	C₂³×D₅₃	1	4	yes	—
396	C₂⁴	C₂³×D₅₃	1	8	yes	—
397	#41	—	2	256	no	—
398	#41	—	106	128	no	—
399	#41	—	106	64	no	—
400	C₂×Q₈	—	2	16	no	—
401	C₂×Q₈	—	2	32	no	—
402	C₂×Q₈	—	53	8	no	—
403	C₂×Q₈	—	106	8	no	—
404	C₂×Q₈	C₂³×D₅₃	1	16	yes	—
405	C₂×D₄	—	2	32	no	—
406	C₂×D₄	—	2	64	no	—
407	C₂×D₄	—	53	16	no	—
408	C₂×D₄	—	53	32	no	—
409	C₂×D₄	—	53	32	no	—
410	C₂×D₄	—	53	8	no	—
411	C₂×D₄	—	106	16	no	—
412	C₂×D₄	—	106	32	no	—
413	C₂×D₄	—	106	128	no	—
414	C₂×D₄	—	106	128	no	—
415	C₂×D₄	—	106	8	no	—
416	C₂×D₄	—	212	16	no	—
417	C₂×D₄	C₂×C₂²⋊D₅₃	1	32	yes	—
418	C₂×D₄	C₂×C₂²⋊D₅₃	1	64	yes	complemented
419	C₂×D₄	C₂³×D₅₃	1	32	yes	—
420	C₂×D₄	C₂³×D₅₃	1	16	yes	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	32	no	—
424	C₂²×C₄	—	2	64	no	—
425	C₂²×C₄	—	2	16	no	—
426	C₂²×C₄	—	2	32	no	—
427	C₂²×C₄	—	53	16	no	—
428	C₂²×C₄	—	53	8	no	—
429	C₂²×C₄	—	53	4	no	—
430	C₂²×C₄	—	53	32	no	—
431	C₂²×C₄	—	53	16	no	—
432	C₂²×C₄	—	106	16	no	—
433	C₂²×C₄	—	106	16	no	—
434	C₂²×C₄	—	212	16	no	—
435	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	—
436	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
437	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
438	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
439	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
440	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
441	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
442	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
443	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
444	C₄⋊C₄	—	53	32	no	—
445	C₄⋊C₄	—	53	64	no	—
446	C₄⋊C₄	—	53	16	no	—
447	C₂²⋊C₄	—	53	16	no	—
448	C₂²⋊C₄	—	53	32	no	—
449	C₂²⋊C₄	—	53	32	no	—
450	C₂²⋊C₄	—	53	32	no	—
451	C₂²⋊C₄	—	53	64	no	—
452	C₄²	—	53	16	no	—
453	C₂³	—	2	16	no	—
454	C₂³	—	2	4	no	—
455	C₂³	—	2	8	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	8	no	—
458	C₂³	—	4	32	no	—
459	C₂³	—	4	16	no	—
460	C₂³	—	4	8	no	—
461	C₂³	—	4	16	no	—
462	C₂³	—	53	8	no	—
463	C₂³	—	106	32	no	—
464	C₂³	—	106	8	no	—
465	C₂³	—	106	16	no	—
466	C₂³	—	424	32	no	—
467	C₂³	—	212	32	no	—
468	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
469	C₂³	C₂²×C₂²⋊D₅₃	1	16	yes	—
470	C₂³	C₂²×C₂²⋊D₅₃	1	32	yes	—
471	C₂³	C₂⁴×D₅₃	1	4	yes	—
472	C₂³	C₂⁴×D₅₃	1	2	yes	—
473	C₂³	C₂⁴×D₅₃	1	8	yes	—
474	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
475	Q₈	—	2	32	no	—
476	Q₈	—	106	16	no	—
477	D₄	—	2	64	no	—
478	D₄	—	106	32	no	—
479	D₄	—	106	64	no	—
480	D₄	—	106	16	no	—
481	D₄	—	212	32	no	—
482	D₄	C₂²×C₂²⋊D₅₃	1	64	yes	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	16	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	64	no	—
488	C₂×C₄	—	2	32	no	—
489	C₂×C₄	—	53	16	no	—
490	C₂×C₄	—	53	8	no	—
491	C₂×C₄	—	106	8	no	—
492	C₂×C₄	—	106	32	no	—
493	C₂×C₄	—	106	16	no	—
494	C₂×C₄	—	106	32	no	—
495	C₂×C₄	—	212	32	no	—
496	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
497	C₂×C₄	C₂²×C₂²⋊D₅₃	1	16	yes	—
498	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
499	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
500	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
501	C₂²	—	2	8	no	—
502	C₂²	—	2	32	no	—
503	C₂²	—	2	16	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	8	no	—
507	C₂²	—	4	32	no	—
508	C₂²	—	4	16	no	—
509	C₂²	—	4	8	no	—
510	C₂²	—	106	8	no	—
511	C₂²	—	106	16	no	—
512	C₂²	—	424	32	no	—
513	C₂²	—	212	16	no	—
514	C₂²	C₂³×C₂²⋊D₅₃	1	4	yes	—
515	C₂²	C₂³×C₂²⋊D₅₃	1	16	yes	—
516	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
517	C₂²	#16811459	1	4	yes	—
518	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
519	C₄	—	2	4	no	—
520	C₄	—	2	8	no	—
521	C₄	—	106	8	no	—
522	C₄	—	106	16	no	—
523	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
524	C₂	—	2	4	no	—
525	C₂	—	2	16	no	—
526	C₂	—	4	4	no	—
527	C₂	—	212	8	no	—
528	C₂	#20288359	1	4	yes	direct product factor
529	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
530	C₂	C₂×#16811459	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂×#3485
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133262	C₂	semidirect
2. #20288359	C₂	direct (N × H)
3. C₂×#16811272	C₂	semidirect
4. C₂×#16811261	C₂	semidirect
5. C₂×#16811197	C₂	semidirect
6. C₂×#16811182	C₂	semidirect
7. C₂×#16811134	C₂	semidirect
8. C₂²×#13133200	C₂	semidirect
9. C₂×#16811098	C₂	semidirect
10. C₂³×#41×C₅₃	C₂	semidirect
12. C₂×#13133262	C₂²	semidirect
14. C₂²×#11719947	C₂²	semidirect
16. C₂²×D₄×D₅₃	C₂²	semidirect
17. #16811272	C₂²	semidirect
18. #16811261	C₂²	semidirect
19. #16811197	C₂²	semidirect
20. #16811182	C₂²	semidirect
21. C₂×#13133215	C₂²	semidirect
22. C₂×#13133206	C₂²	semidirect
23. #16811134	C₂²	semidirect
24. C₂×#13133200	C₂²	semidirect
25. C₂×#13133200	C₂²	semidirect
26. C₂×#13133200	C₂²	semidirect
27. C₂×#13133200	C₂²	semidirect
28. C₂×#13133198	C₂²	semidirect
29. C₂×#13133196	C₂²	semidirect
30. C₂×#13133193	C₂²	semidirect
31. C₂×#13133191	C₂²	semidirect
32. C₂×#13133189	C₂²	semidirect
33. C₂×#13133186	C₂²	semidirect
34. C₂×#13133185	C₂²	semidirect
35. #16811098	C₂²	semidirect
36. C₂×D₄×Dic₅₃	C₂²	semidirect
37. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
38. C₂×#13133176	C₂²	semidirect
39. C₂²×#11719919	C₂²	semidirect
40. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
41. C₂²×#11719921	C₂²	semidirect
43. C₂²×#41×C₅₃	C₂²	semidirect
44. C₂²×#41×C₅₃	C₂²	semidirect
45. C₂³×Q₈×C₅₃	C₂²	semidirect
47. C₂³×D₄×C₅₃	C₂²	semidirect
48. C₂⁴×C₄×C₅₃	C₂²	semidirect
49. C₂⁴×C₄×C₅₃	C₂²	semidirect
55. C₂×#11719947	C₂³	semidirect
64. C₂×D₄×D₅₃	C₂³	semidirect
68. #13133215	C₂³	semidirect
69. #13133206	C₂³	semidirect
70. #13133200	C₂³	semidirect
71. #13133200	C₂³	semidirect
72. #13133200	C₂³	semidirect
73. #13133198	C₂³	semidirect
74. #13133196	C₂³	semidirect
75. #13133193	C₂³	semidirect
76. #13133191	C₂³	semidirect
77. #13133189	C₂³	semidirect
78. #13133186	C₂³	semidirect
79. #13133185	C₂³	semidirect
80. D₄×Dic₅₃	C₂³	semidirect
81. C₄×C₂²⋊D₅₃	C₂³	semidirect
82. #13133176	C₂³	semidirect
86. C₂×#11719919	C₂³	semidirect
90. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
93. C₂×#11719921	C₂³	semidirect
99. C₂×#41×C₅₃	C₂³	semidirect
102. C₂²×Q₈×C₅₃	C₂³	semidirect
104. C₂²×D₄×C₅₃	D₄	semidirect
108. C₂²×D₄×C₅₃	C₂³	semidirect
116. C₂³×C₄×C₅₃	C₂³	semidirect
117. C₂³×C₄×C₅₃	C₂³	semidirect
138. C₂×D₄×C₅₃	C₂×D₄	semidirect
260. C₂³×#41	D₅₃	semidirect
272. C₂×C₅₃	#3485	semidirect
281. C₂²×#41	C₂×D₅₃	semidirect
282. C₂²×#41	C₂×D₅₃	semidirect
283. C₂³×Q₈	C₂×D₅₃	semidirect
287. C₂³×D₄	C₂×D₅₃	semidirect
288. C₂⁴×C₄	C₂×D₅₃	semidirect
289. C₂⁴×C₄	C₂×D₅₃	semidirect
315. C₅₃	C₂×#3485	semidirect
326. C₂×#41	C₂²×D₅₃	semidirect
330. C₂²×Q₈	C₂²×D₅₃	semidirect
340. C₂²×D₄	C₂²⋊D₅₃	semidirect
344. C₂²×D₄	C₂²×D₅₃	semidirect
355. C₂³×C₄	C₂²×D₅₃	semidirect
356. C₂³×C₄	C₂²×D₅₃	semidirect
418. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
528. C₂	#20288359	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133262	2
#20288359	2
C₂×#16811272	2
C₂×#16811261	2
C₂×#16811197	2
C₂×#16811182	2
C₂×#16811134	2
C₂²×#13133200	2
C₂×#16811098	2
C₂³×#41×C₅₃	2
C₂×#3485	53

Subgroups — C2×#20288359 #28292136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288359 #28292136