Subgroups — C₂×#20288403 #28292004

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	#20288403	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811296	C₂	1	8	yes	maximal, complemented
3	C₂×#16811260	C₂	1	8	yes	maximal, complemented
4	C₂×#16811249	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16811210	C₂	1	4	yes	maximal, complemented
6	C₂×#16811133	C₂	1	4	yes	maximal, complemented
7	C₂²×#13133185	C₂	1	2	yes	maximal, complemented
8	C₂×#16811098	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#534	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
12	C₂²×#11719947	C₂²	1	4	yes	complemented
13	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
14	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
15	#16811296	C₂²	1	128	yes	complemented
16	#16811260	C₂²	1	128	yes	complemented
17	#16811249	C₂²	1	16	yes	complemented
18	#16811210	C₂²	1	64	yes	complemented
19	C₂×#13133212	C₂²	1	8	yes	complemented
20	C₂×#13133203	C₂²	1	8	yes	complemented
21	#16811133	C₂²	1	64	yes	complemented
22	C₂×#13133201	C₂²	1	8	yes	complemented
23	C₂×#13133199	C₂²	1	8	yes	complemented
24	C₂×#13133197	C₂²	1	4	yes	complemented
25	C₂×#13133196	C₂²	1	4	yes	complemented
26	C₂×#13133194	C₂²	1	8	yes	complemented
27	C₂×#13133189	C₂²	1	8	yes	complemented
28	C₂×#13133188	C₂²	1	4	yes	complemented
29	C₂×#13133185	C₂²	1	8	yes	—
30	C₂×#13133185	C₂²	1	4	yes	complemented
31	C₂×#13133185	C₂²	1	4	yes	complemented
32	C₂×#13133185	C₂²	1	32	yes	complemented
33	#16811098	C₂²	1	32	yes	complemented
34	C₂×D₄×Dic₅₃	C₂²	1	8	yes	complemented
35	C₂×D₄×Dic₅₃	C₂²	1	8	yes	complemented
36	C₂×#13133177	C₂²	1	8	yes	complemented
37	C₂×#13133176	C₂²	1	4	yes	complemented
38	C₂×#13133175	C₂²	1	4	yes	complemented
39	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
40	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
41	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	complemented
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	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
47	C₅₃×#534	C₂²	1	16	yes	complemented
48	C₂×#123×C₅₃	C₂²	1	4	yes	complemented
49	C₂×#121×C₅₃	C₂²	1	4	yes	complemented
50	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
51	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
58	C₂×#11719947	—	4	16	no	—
59	C₂×#11719947	C₂³	1	32	yes	complemented
60	C₂²×Dic₁₀₆	C₂³	1	4	yes	complemented
61	C₂²×C₂²⋊D₅₃	—	4	16	no	—
62	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
63	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
64	C₂²×C₂²⋊D₅₃	C₂³	1	32	yes	complemented
65	C₂³×Dic₅₃	C₂³	1	2	yes	—
66	C₂³×Dic₅₃	C₂³	1	4	yes	—
67	C₂³×Dic₅₃	C₂³	1	8	yes	complemented
68	C₂²×C₄×D₅₃	—	4	8	no	—
69	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
70	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
71	C₂²×C₄×D₅₃	C₂³	1	16	yes	complemented
72	#13133212	C₂³	1	64	yes	complemented
73	#13133203	C₂³	1	64	yes	complemented
74	#13133201	C₂³	1	64	yes	complemented
75	#13133199	C₂³	1	64	yes	complemented
76	#13133197	C₂³	1	32	yes	complemented
77	#13133196	C₂³	1	32	yes	complemented
78	#13133194	C₂³	1	64	yes	complemented
79	#13133189	C₂³	1	64	yes	complemented
80	#13133188	C₂³	1	32	yes	complemented
81	#13133185	C₂³	1	64	yes	—
82	#13133185	C₂³	1	32	yes	complemented
83	#13133185	C₂³	1	32	yes	complemented
84	D₄×Dic₅₃	C₂³	1	64	yes	complemented
85	D₄×Dic₅₃	C₂³	1	64	yes	complemented
86	#13133177	C₂³	1	64	yes	complemented
87	#13133176	C₂³	1	32	yes	complemented
88	#13133175	C₂³	1	32	yes	complemented
89	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	complemented
90	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	complemented
91	C₂×#11719919	C₂³	1	8	yes	complemented
92	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
93	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
94	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	—
97	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	complemented
98	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	complemented
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	C₂²⋊C₄×D₅₃	C₂³	1	64	yes	complemented
105	C₂×C₄×Dic₅₃	C₂³	1	8	yes	complemented
106	C₂⁵×C₅₃	C₂³	1	2	yes	—
107	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
108	C₂²×D₄×C₅₃	—	2	16	no	—
109	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
110	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
111	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
112	C₂²×D₄×C₅₃	C₂³	1	4	yes	complemented
113	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
114	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
115	#123×C₅₃	C₂³	1	32	yes	complemented
116	#121×C₅₃	C₂³	1	32	yes	complemented
117	#120×C₅₃	C₂³	1	32	yes	complemented
118	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
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	8	yes	complemented
122	C₂⁴×C₅₃	—	2	8	no	—
123	C₂⁴×C₅₃	—	4	4	no	—
124	C₂⁴×C₅₃	—	4	8	no	—
125	C₂⁴×C₅₃	#41	1	8	yes	complemented
126	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
127	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
128	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
129	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
130	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
131	C₂³×D₅₃	—	2	4	no	—
132	C₂³×D₅₃	—	2	8	no	—
133	C₂³×D₅₃	—	2	16	no	—
134	C₂³×D₅₃	—	2	4	no	—
135	C₂³×D₅₃	—	2	32	no	—
136	C₂³×D₅₃	—	2	16	no	—
137	C₂³×D₅₃	—	4	32	no	—
138	C₂³×D₅₃	—	4	8	no	—
139	C₂³×D₅₃	—	4	4	no	—
140	C₂³×D₅₃	—	4	64	no	—
141	C₂³×D₅₃	C₂⁴	1	2	yes	—
142	C₂³×D₅₃	C₂⁴	1	4	yes	—
143	C₂³×D₅₃	C₂⁴	1	16	yes	—
144	C₂³×D₅₃	C₂⁴	1	2	yes	—
145	C₂³×D₅₃	C₂⁴	1	4	yes	—
146	C₂×D₄×C₅₃	—	2	16	no	—
147	C₂×D₄×C₅₃	—	2	16	no	—
148	C₂×D₄×C₅₃	—	2	64	no	—
149	C₂×D₄×C₅₃	—	4	8	no	—
150	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
151	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
152	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
153	C₂²×C₄×C₅₃	—	2	8	no	—
154	C₂²×C₄×C₅₃	—	4	4	no	—
155	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
159	C₂×C₂²⋊D₅₃	—	2	32	no	—
160	C₂×C₂²⋊D₅₃	—	4	16	no	—
161	C₂×C₂²⋊D₅₃	—	4	64	no	—
162	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
163	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
164	C₂²×Dic₅₃	—	4	4	no	—
165	C₂²×Dic₅₃	—	4	8	no	—
166	C₂²×Dic₅₃	—	4	16	no	—
167	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
168	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
169	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
170	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
171	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
172	#11719947	—	4	64	no	—
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	C₂⁴	1	16	yes	—
177	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
178	C₂×Dic₁₀₆	—	4	8	no	—
179	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
180	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
181	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
182	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
183	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
184	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
185	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
186	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
187	#11719921	C₂⁴	1	32	yes	—
188	#11719921	C₂⁴	1	32	yes	—
189	#11719921	C₂⁴	1	32	yes	—
190	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
191	C₄⋊Dic₅₃	C₂⁴	1	32	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	—	2	8	no	—
197	C₂³×C₅₃	—	2	16	no	—
198	C₂³×C₅₃	—	2	16	no	—
199	C₂³×C₅₃	—	4	16	no	—
200	C₂³×C₅₃	—	4	8	no	—
201	C₂³×C₅₃	—	4	16	no	—
202	C₂³×C₅₃	—	8	16	no	—
203	C₂³×C₅₃	C₂×#41	1	8	yes	—
204	C₂³×C₅₃	C₂×#41	1	4	yes	—
205	C₂³×C₅₃	C₂×#41	1	16	yes	complemented
206	C₂³×C₅₃	C₂⁵	1	8	yes	—
207	C₂³×C₅₃	C₂⁵	1	2	yes	—
208	C₂³×C₅₃	C₂⁵	1	4	yes	—
209	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
210	C₂²×D₅₃	—	2	32	no	—
211	C₂²×D₅₃	—	2	8	no	—
212	C₂²×D₅₃	—	2	4	no	—
213	C₂²×D₅₃	—	2	4	no	—
214	C₂²×D₅₃	—	2	16	no	—
215	C₂²×D₅₃	—	2	32	no	—
216	C₂²×D₅₃	—	2	8	no	—
217	C₂²×D₅₃	—	2	4	no	—
218	C₂²×D₅₃	—	2	8	no	—
219	C₂²×D₅₃	—	2	32	no	—
220	C₂²×D₅₃	—	4	16	no	—
221	C₂²×D₅₃	—	4	8	no	—
222	C₂²×D₅₃	—	4	64	no	—
223	C₂²×D₅₃	—	4	128	no	—
224	C₂²×D₅₃	—	4	16	no	—
225	C₂²×D₅₃	H(5,𝔽₂)	1	8	yes	—
226	C₂²×D₅₃	C₂⁵	1	4	yes	—
227	C₂²×D₅₃	C₂⁵	1	4	yes	—
228	D₄×C₅₃	—	2	32	no	—
229	D₄×C₅₃	—	2	32	no	—
230	D₄×C₅₃	—	4	16	no	—
231	C₂×C₄×C₅₃	—	2	16	no	—
232	C₂×C₄×C₅₃	—	2	8	no	—
233	C₂×C₄×C₅₃	—	2	8	no	—
234	C₂×C₄×C₅₃	—	4	8	no	—
235	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
236	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
237	C₂²⋊D₅₃	—	2	64	no	—
238	C₂²⋊D₅₃	—	4	32	no	—
239	C₂×Dic₅₃	—	2	16	no	—
240	C₂×Dic₅₃	—	2	8	no	—
241	C₂×Dic₅₃	—	4	8	no	—
242	C₂×Dic₅₃	—	4	32	no	—
243	C₂×Dic₅₃	—	4	16	no	—
244	C₂×Dic₅₃	C₂⁵	1	16	yes	—
245	C₂×Dic₅₃	C₂⁵	1	8	yes	—
246	C₄×D₅₃	—	2	32	no	—
247	C₄×D₅₃	—	4	16	no	—
248	Dic₁₀₆	—	4	16	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	—	2	8	no	—
255	C₂²×C₅₃	—	2	16	no	—
256	C₂²×C₅₃	—	4	16	no	—
257	C₂²×C₅₃	—	4	16	no	—
258	C₂²×C₅₃	—	8	16	no	—
259	C₂²×C₅₃	C₂²×#41	1	8	yes	—
260	C₂²×C₅₃	C₂²×#41	1	4	yes	—
261	C₂²×C₅₃	C₂²×#41	1	2	yes	—
262	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
263	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
264	C₂×D₅₃	—	2	32	no	—
265	C₂×D₅₃	—	2	4	no	—
266	C₂×D₅₃	—	2	4	no	—
267	C₂×D₅₃	—	2	16	no	—
268	C₂×D₅₃	—	2	4	no	—
269	C₂×D₅₃	—	2	8	no	—
270	C₂×D₅₃	—	4	64	no	—
271	C₂×D₅₃	—	4	16	no	—
272	C₂×D₅₃	—	4	8	no	—
273	C₂×D₅₃	C₂×H(5,𝔽₂)	1	8	yes	—
274	C₄×C₅₃	—	2	8	no	—
275	C₄×C₅₃	—	2	8	no	—
276	Dic₅₃	—	2	16	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	—	53	1	no	—
282	C₂×#538	—	53	4	no	—
283	C₂×#534	D₅₃	1	1	yes	characteristic, complemented
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	—	4	8	no	—
291	C₂×C₅₃	#3490	1	4	yes	complemented
292	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
293	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
294	D₅₃	—	2	8	no	—
295	D₅₃	—	4	4	no	—
296	C₂⁶	—	53	1	no	—
297	C₂²×#41	—	53	4	no	—
298	C₂³×D₄	—	53	4	no	—
299	C₂³×D₄	C₂×D₅₃	1	2	yes	—
300	C₂⁴×C₄	—	53	2	no	—
301	#551	—	53	128	no	—
302	#546	—	53	128	no	—
303	#543	—	53	16	no	—
304	#538	—	53	64	no	—
305	#534	C₂×D₅₃	1	16	yes	complemented
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	C₂×D₅₃	1	4	yes	complemented
312	C₂×C₂²⋊Q₈	—	53	4	no	—
313	C₂×C₂²⋊Q₈	—	53	4	no	—
314	C₂×C₂²⋊Q₈	—	53	8	no	—
315	C₂×#121	—	53	4	no	—
316	C₂×#121	—	53	8	no	—
317	C₂×#121	C₂×D₅₃	1	4	yes	complemented
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	C₂×D₅₃	1	4	yes	complemented
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	2	no	—
333	C₂²×C₂²⋊C₄	—	53	4	no	—
334	C₂²×C₂²⋊C₄	—	53	8	no	—
335	C₂²×C₂²⋊C₄	—	53	16	no	—
336	C₂²×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
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	—	106	8	no	—
343	C₂⁵	—	212	8	no	—
344	C₂⁵	C₂²×D₅₃	1	2	yes	—
345	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
346	C₂×#41	—	53	32	no	—
347	C₂×#41	—	212	16	no	—
348	C₂²×Q₈	—	53	4	no	—
349	C₂²×D₄	—	2	16	no	—
350	C₂²×D₄	—	53	8	no	—
351	C₂²×D₄	—	53	8	no	—
352	C₂²×D₄	—	53	32	no	—
353	C₂²×D₄	—	212	16	no	—
354	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
355	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
356	C₂²×D₄	C₂²×D₅₃	1	16	yes	—
357	C₂²×D₄	C₂²×D₅₃	1	4	yes	complemented
358	C₂³×C₄	—	53	2	no	—
359	C₂³×C₄	—	53	4	no	—
360	C₂³×C₄	—	53	8	no	—
361	C₂³×C₄	—	53	4	no	—
362	C₂³×C₄	—	53	4	no	—
363	C₂³×C₄	—	53	16	no	—
364	C₂³×C₄	—	212	8	no	—
365	C₂³×C₄	C₂²×D₅₃	1	2	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	C₂²×D₅₃	1	32	yes	complemented
372	C₂²⋊Q₈	—	53	32	no	—
373	C₂²⋊Q₈	—	53	32	no	—
374	C₂²⋊Q₈	—	53	64	no	—
375	#121	—	53	32	no	—
376	#121	—	53	64	no	—
377	#121	C₂²×D₅₃	1	32	yes	complemented
378	#120	—	53	32	no	—
379	#120	—	53	32	no	—
380	#120	—	53	64	no	—
381	#120	C₂²×D₅₃	1	32	yes	complemented
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	—	53	8	no	—
390	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	complemented
391	C₂×C₂²⋊C₄	—	53	4	no	—
392	C₂×C₂²⋊C₄	—	53	8	no	—
393	C₂×C₂²⋊C₄	—	53	4	no	—
394	C₂×C₂²⋊C₄	—	53	4	no	—
395	C₂×C₂²⋊C₄	—	53	8	no	—
396	C₂×C₂²⋊C₄	—	53	16	no	—
397	C₂×C₂²⋊C₄	—	53	32	no	—
398	C₂×C₂²⋊C₄	—	53	8	no	—
399	C₂×C₂²⋊C₄	—	53	16	no	—
400	C₂×C₂²⋊C₄	—	53	8	no	—
401	C₂×C₂²⋊C₄	—	53	8	no	—
402	C₂×C₂²⋊C₄	—	53	32	no	—
403	C₂×C₂²⋊C₄	—	53	64	no	—
404	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
406	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	complemented
407	C₂×C₄²	—	53	8	no	—
408	C₂⁴	—	2	8	no	—
409	C₂⁴	—	4	4	no	—
410	C₂⁴	—	4	8	no	—
411	C₂⁴	—	53	2	no	—
412	C₂⁴	—	53	4	no	—
413	C₂⁴	—	53	2	no	—
414	C₂⁴	—	53	16	no	—
415	C₂⁴	—	53	4	no	—
416	C₂⁴	—	106	8	no	—
417	C₂⁴	—	106	16	no	—
418	C₂⁴	—	106	4	no	—
419	C₂⁴	—	106	16	no	—
420	C₂⁴	—	106	4	no	—
421	C₂⁴	—	106	32	no	—
422	C₂⁴	—	212	4	no	—
423	C₂⁴	—	212	8	no	—
424	C₂⁴	—	212	32	no	—
425	C₂⁴	—	212	64	no	—
426	C₂⁴	#11719947	1	8	yes	complemented
427	C₂⁴	C₂³×D₅₃	1	4	yes	—
428	C₂⁴	C₂³×D₅₃	1	2	yes	—
429	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
430	C₂⁴	C₂³×D₅₃	1	8	yes	—
431	C₂⁴	C₂³×D₅₃	1	4	yes	—
432	#41	—	212	64	no	—
433	C₂×Q₈	—	53	16	no	—
434	C₂×Q₈	—	212	8	no	—
435	C₂×D₄	—	2	16	no	—
436	C₂×D₄	—	2	16	no	—
437	C₂×D₄	—	2	64	no	—
438	C₂×D₄	—	4	8	no	—
439	C₂×D₄	—	53	32	no	—
440	C₂×D₄	—	53	32	no	—
441	C₂×D₄	—	106	32	no	—
442	C₂×D₄	—	212	16	no	—
443	C₂×D₄	—	212	64	no	—
444	C₂×D₄	C₂³×D₅₃	1	16	yes	—
445	C₂×D₄	C₂³×D₅₃	1	16	yes	—
446	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
447	C₂²×C₄	—	2	8	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	53	4	no	—
450	C₂²×C₄	—	53	8	no	—
451	C₂²×C₄	—	53	8	no	—
452	C₂²×C₄	—	53	16	no	—
453	C₂²×C₄	—	53	32	no	—
454	C₂²×C₄	—	53	16	no	—
455	C₂²×C₄	—	53	16	no	—
456	C₂²×C₄	—	106	16	no	—
457	C₂²×C₄	—	212	8	no	—
458	C₂²×C₄	—	212	32	no	—
459	C₂²×C₄	—	212	4	no	—
460	C₂²×C₄	—	212	8	no	—
461	C₂²×C₄	—	212	16	no	—
462	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
463	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
464	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
465	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
466	C₄⋊C₄	—	53	32	no	—
467	C₄⋊C₄	—	53	16	no	—
468	C₄⋊C₄	—	53	32	no	—
469	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
470	C₂²⋊C₄	—	53	16	no	—
471	C₂²⋊C₄	—	53	16	no	—
472	C₂²⋊C₄	—	53	16	no	—
473	C₂²⋊C₄	—	53	32	no	—
474	C₂²⋊C₄	—	53	32	no	—
475	C₂²⋊C₄	—	53	32	no	—
476	C₂²⋊C₄	—	53	32	no	—
477	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
478	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
479	C₄²	—	53	32	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	8	no	—
483	C₂³	—	2	16	no	—
484	C₂³	—	2	16	no	—
485	C₂³	—	4	16	no	—
486	C₂³	—	4	8	no	—
487	C₂³	—	4	16	no	—
488	C₂³	—	8	16	no	—
489	C₂³	—	53	8	no	—
490	C₂³	—	53	4	no	—
491	C₂³	—	53	4	no	—
492	C₂³	—	106	16	no	—
493	C₂³	—	106	32	no	—
494	C₂³	—	106	32	no	—
495	C₂³	—	106	8	no	—
496	C₂³	—	106	4	no	—
497	C₂³	—	106	4	no	—
498	C₂³	—	106	32	no	—
499	C₂³	—	106	8	no	—
500	C₂³	—	106	4	no	—
501	C₂³	—	106	8	no	—
502	C₂³	—	212	8	no	—
503	C₂³	—	212	16	no	—
504	C₂³	—	212	64	no	—
505	C₂³	—	212	16	no	—
506	C₂³	—	212	128	no	—
507	C₂³	C₂×#11719947	1	8	yes	—
508	C₂³	C₂×#11719947	1	4	yes	—
509	C₂³	C₂×#11719947	1	16	yes	complemented
510	C₂³	C₂⁴×D₅₃	1	8	yes	—
511	C₂³	C₂⁴×D₅₃	1	4	yes	—
512	C₂³	C₂⁴×D₅₃	1	2	yes	—
513	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
514	Q₈	—	212	16	no	—
515	D₄	—	2	32	no	—
516	D₄	—	2	32	no	—
517	D₄	—	4	16	no	—
518	D₄	—	106	64	no	—
519	D₄	—	212	32	no	—
520	C₂×C₄	—	2	16	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	2	8	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	53	8	no	—
525	C₂×C₄	—	53	16	no	—
526	C₂×C₄	—	106	32	no	—
527	C₂×C₄	—	106	8	no	—
528	C₂×C₄	—	106	16	no	—
529	C₂×C₄	—	212	16	no	—
530	C₂×C₄	—	212	8	no	—
531	C₂×C₄	—	212	16	no	—
532	C₂×C₄	—	212	32	no	—
533	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
534	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
535	C₂²	—	2	16	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	2	8	no	—
540	C₂²	—	2	16	no	—
541	C₂²	—	4	16	no	—
542	C₂²	—	4	16	no	—
543	C₂²	—	8	16	no	—
544	C₂²	—	53	8	no	—
545	C₂²	—	106	32	no	—
546	C₂²	—	106	16	no	—
547	C₂²	—	106	4	no	—
548	C₂²	—	106	4	no	—
549	C₂²	—	106	4	no	—
550	C₂²	—	106	8	no	—
551	C₂²	—	212	64	no	—
552	C₂²	—	212	16	no	—
553	C₂²	—	212	8	no	—
554	C₂²	C₂²×#11719947	1	8	yes	—
555	C₂²	C₂²×#11719947	1	4	yes	—
556	C₂²	C₂²×#11719947	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	8	no	—
561	C₄	—	106	8	no	—
562	C₄	—	106	16	no	—
563	C₂	—	2	8	no	—
564	C₂	—	2	4	no	—
565	C₂	—	4	8	no	—
566	C₂	—	106	8	no	—
567	C₂	—	212	4	no	—
568	C₂	#20288403	1	4	yes	direct product factor
569	C₂	C₂³×#11719947	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×#534	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#534	D₅₃
hypocenter	C₅₃	C₂×#3490
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#534

Complements

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

N	H ≅ G/N	type
1. #20288403	C₂	direct (N × H)
2. C₂×#16811296	C₂	semidirect
3. C₂×#16811260	C₂	semidirect
4. C₂×#16811249	C₂	semidirect
5. C₂×#16811210	C₂	semidirect
6. C₂×#16811133	C₂	semidirect
7. C₂²×#13133185	C₂	semidirect
8. C₂×#16811098	C₂	semidirect
9. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
10. C₂×C₅₃×#534	C₂	semidirect
12. C₂²×#11719947	C₂²	semidirect
13. C₂³×C₂²⋊D₅₃	C₂²	semidirect
14. C₂³×C₄×D₅₃	C₂²	semidirect
15. #16811296	C₂²	semidirect
16. #16811260	C₂²	semidirect
17. #16811249	C₂²	semidirect
18. #16811210	C₂²	semidirect
19. C₂×#13133212	C₂²	semidirect
20. C₂×#13133203	C₂²	semidirect
21. #16811133	C₂²	semidirect
22. C₂×#13133201	C₂²	semidirect
23. C₂×#13133199	C₂²	semidirect
24. C₂×#13133197	C₂²	semidirect
25. C₂×#13133196	C₂²	semidirect
26. C₂×#13133194	C₂²	semidirect
27. C₂×#13133189	C₂²	semidirect
28. C₂×#13133188	C₂²	semidirect
30. C₂×#13133185	C₂²	semidirect
31. C₂×#13133185	C₂²	semidirect
32. C₂×#13133185	C₂²	semidirect
33. #16811098	C₂²	semidirect
34. C₂×D₄×Dic₅₃	C₂²	semidirect
35. C₂×D₄×Dic₅₃	C₂²	semidirect
36. C₂×#13133177	C₂²	semidirect
37. C₂×#13133176	C₂²	semidirect
38. C₂×#13133175	C₂²	semidirect
41. C₂²×C₂²⋊Dic₅₃	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
47. C₅₃×#534	C₂²	semidirect
48. C₂×#123×C₅₃	C₂²	semidirect
49. C₂×#121×C₅₃	C₂²	semidirect
50. C₂×#120×C₅₃	C₂²	semidirect
59. C₂×#11719947	C₂³	semidirect
60. C₂²×Dic₁₀₆	C₂³	semidirect
64. C₂²×C₂²⋊D₅₃	C₂³	semidirect
67. C₂³×Dic₅₃	C₂³	semidirect
71. C₂²×C₄×D₅₃	C₂³	semidirect
72. #13133212	C₂³	semidirect
73. #13133203	C₂³	semidirect
74. #13133201	C₂³	semidirect
75. #13133199	C₂³	semidirect
76. #13133197	C₂³	semidirect
77. #13133196	C₂³	semidirect
78. #13133194	C₂³	semidirect
79. #13133189	C₂³	semidirect
80. #13133188	C₂³	semidirect
82. #13133185	C₂³	semidirect
83. #13133185	C₂³	semidirect
84. D₄×Dic₅₃	C₂³	semidirect
85. D₄×Dic₅₃	C₂³	semidirect
86. #13133177	C₂³	semidirect
87. #13133176	C₂³	semidirect
88. #13133175	C₂³	semidirect
89. C₂×C₄⋊Dic₅₃	C₂³	semidirect
90. C₂×C₄⋊Dic₅₃	C₂³	semidirect
91. C₂×#11719919	C₂³	semidirect
97. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
98. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
102. C₂×#11719921	C₂³	semidirect
103. C₂×#11719921	C₂³	semidirect
104. C₂²⋊C₄×D₅₃	C₂³	semidirect
105. C₂×C₄×Dic₅₃	C₂³	semidirect
112. C₂²×D₄×C₅₃	C₂³	semidirect
115. #123×C₅₃	C₂³	semidirect
116. #121×C₅₃	C₂³	semidirect
117. #120×C₅₃	C₂³	semidirect
118. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
121. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
125. C₂⁴×C₅₃	#41	semidirect
152. C₂×D₄×C₅₃	C₂⁴	semidirect
171. C₂²×Dic₅₃	C₂⁴	semidirect
179. C₂×Dic₁₀₆	C₂⁴	semidirect
180. C₄⋊C₄×C₅₃	C₂⁴	semidirect
182. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
186. C₂²⋊Dic₅₃	C₂⁴	semidirect
190. C₄⋊Dic₅₃	C₂⁴	semidirect
191. C₄⋊Dic₅₃	C₂⁴	semidirect
192. #11719919	C₂⁴	semidirect
193. C₄×Dic₅₃	C₂⁴	semidirect
205. C₂³×C₅₃	C₂×#41	semidirect
283. C₂×#534	D₅₃	semidirect
291. C₂×C₅₃	#3490	semidirect
305. #534	C₂×D₅₃	semidirect
311. C₂×#123	C₂×D₅₃	semidirect
317. C₂×#121	C₂×D₅₃	semidirect
322. C₂×#120	C₂×D₅₃	semidirect
337. C₅₃	C₂×#3490	semidirect
357. C₂²×D₄	C₂²×D₅₃	semidirect
371. #123	C₂²×D₅₃	semidirect
377. #121	C₂²×D₅₃	semidirect
381. #120	C₂²×D₅₃	semidirect
390. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
406. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
426. C₂⁴	#11719947	semidirect
446. C₂×D₄	C₂³×D₅₃	semidirect
469. C₄⋊C₄	C₂³×D₅₃	semidirect
478. C₂²⋊C₄	C₂³×D₅₃	semidirect
509. C₂³	C₂×#11719947	semidirect
568. C₂	#20288403	direct (N × H)

Maximal subgroups

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

subgroup	index
#20288403	2
C₂×#16811296	2
C₂×#16811260	2
C₂×#16811249	2
C₂×#16811210	2
C₂×#16811133	2
C₂²×#13133185	2
C₂×#16811098	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂×C₅₃×#534	2
C₂×#3490	53

Subgroups — C2×#20288403 #28292004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288403 #28292004