Subgroups — C₂×#20289120 #28291895

Order 13568.

8802 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289120	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811365	C₂	1	4	yes	maximal, complemented
3	C₂×#16811339	C₂	1	2	yes	maximal, complemented
4	C₂×#16811338	C₂	1	4	yes	maximal, complemented
5	C₂×D₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
6	C₂×#16811258	C₂	1	2	yes	maximal, complemented
7	C₂×#16811189	C₂	1	8	yes	maximal, complemented
8	C₂×#16811187	C₂	1	4	yes	maximal, complemented
9	C₂×#16811173	C₂	1	2	yes	maximal, complemented
10	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#11719945	C₂²	1	4	yes	complemented
12	C₂³×D₂₁₂	C₂²	1	4	yes	complemented
13	C₂²×D₄×D₅₃	C₂²	1	8	yes	complemented
14	#16811365	C₂²	1	64	yes	complemented
15	#16811339	C₂²	1	32	yes	complemented
16	#16811338	C₂²	1	64	yes	complemented
17	D₄×D₂₁₂	C₂²	1	64	yes	complemented
18	#16811258	C₂²	1	32	yes	complemented
19	#16811189	C₂²	1	128	yes	complemented
20	#16811187	C₂²	1	64	yes	complemented
21	#16811173	C₂²	1	32	yes	complemented
22	C₂×#13133217	C₂²	1	2	yes	complemented
23	C₂×#13133214	C₂²	1	2	yes	—
24	C₂×#13133213	C₂²	1	8	yes	complemented
25	C₂×#13133206	C₂²	1	8	yes	complemented
26	C₂×#13133205	C₂²	1	4	yes	complemented
27	C₂×#13133203	C₂²	1	16	yes	complemented
28	C₂×#13133192	C₂²	1	8	yes	complemented
29	C₂×#13133191	C₂²	1	4	yes	complemented
30	C₂×#13133191	C₂²	1	8	yes	complemented
31	C₂×#13133191	C₂²	1	4	yes	complemented
32	C₂×#13133188	C₂²	1	8	yes	complemented
33	C₂×#13133187	C₂²	1	16	yes	complemented
34	C₂×#13133185	C₂²	1	8	yes	complemented
35	C₂×#13133185	C₂²	1	8	yes	complemented
36	C₂×C₄×D₂₁₂	C₂²	1	4	yes	complemented
37	C₂×C₄×D₂₁₂	C₂²	1	4	yes	complemented
38	C₂×#13133176	C₂²	1	4	yes	complemented
39	C₂×#13133175	C₂²	1	8	yes	complemented
40	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
41	C₅₃×#532	C₂²	1	16	yes	complemented
42	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#124×C₅₃	C₂²	1	2	yes	complemented
45	C₂×#121×C₅₃	C₂²	1	4	yes	complemented
46	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
47	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
48	C₂⁴×D₅₃	C₂³	1	8	yes	—
49	C₂×#11719945	—	4	16	no	—
50	C₂×#11719945	C₂³	1	32	yes	complemented
51	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
52	C₂²×D₂₁₂	—	4	16	no	—
53	C₂²×D₂₁₂	C₂³	1	4	yes	—
54	C₂²×D₂₁₂	C₂³	1	4	yes	—
55	C₂²×D₂₁₂	C₂³	1	4	yes	—
56	C₂²×D₂₁₂	C₂³	1	8	yes	—
57	C₂²×D₂₁₂	C₂³	1	32	yes	complemented
58	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
59	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	—
60	C₂×D₄×D₅₃	—	4	32	no	—
61	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
62	C₂²×C₄×D₅₃	C₂³	1	8	yes	—
63	#13133217	C₂³	1	16	yes	complemented
64	#13133214	C₂³	1	16	yes	—
65	#13133213	C₂³	1	64	yes	complemented
66	#13133206	C₂³	1	64	yes	complemented
67	#13133205	C₂³	1	32	yes	complemented
68	#13133203	C₂³	1	128	yes	complemented
69	#13133192	C₂³	1	64	yes	complemented
70	#13133191	C₂³	1	64	yes	complemented
71	#13133191	C₂³	1	32	yes	complemented
72	#13133191	C₂³	1	32	yes	complemented
73	#13133188	C₂³	1	64	yes	complemented
74	#13133187	C₂³	1	128	yes	complemented
75	#13133185	C₂³	1	64	yes	complemented
76	#13133185	C₂³	1	64	yes	complemented
77	C₄×D₂₁₂	C₂³	1	32	yes	complemented
78	C₄×D₂₁₂	C₂³	1	32	yes	complemented
79	#13133176	C₂³	1	32	yes	complemented
80	#13133175	C₂³	1	64	yes	complemented
81	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
82	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
83	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
84	C₂×#11719921	C₂³	1	16	yes	—
85	C₂×#11719921	C₂³	1	8	yes	—
86	C₂×#11719921	C₂³	1	8	yes	—
87	C₂×C₄×Dic₅₃	C₂³	1	8	yes	—
88	C₂×#41×C₅₃	—	4	8	no	—
89	C₂×#41×C₅₃	C₂³	1	16	yes	complemented
90	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
91	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
92	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
93	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
94	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
95	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
97	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	complemented
98	#127×C₅₃	C₂³	1	8	yes	—
99	#124×C₅₃	C₂³	1	16	yes	complemented
100	#121×C₅₃	C₂³	1	32	yes	complemented
101	C₄×D₄×C₅₃	C₂³	1	32	yes	complemented
102	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
104	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
105	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₅₃	C₂³	1	8	yes	complemented
108	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
109	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂³×D₅₃	—	4	32	no	—
111	C₂³×D₅₃	—	4	16	no	—
112	C₂³×D₅₃	C₂⁴	1	8	yes	—
113	C₂³×D₅₃	C₂⁴	1	32	yes	—
114	#41×C₅₃	—	4	32	no	—
115	C₂×Q₈×C₅₃	—	4	4	no	—
116	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
117	C₂×D₄×C₅₃	—	2	16	no	—
118	C₂×D₄×C₅₃	—	4	8	no	—
119	C₂×D₄×C₅₃	—	4	4	no	—
120	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
121	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
122	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
123	C₂²×C₄×C₅₃	—	2	4	no	—
124	C₂²×C₄×C₅₃	—	2	16	no	—
125	C₂²×C₄×C₅₃	—	2	8	no	—
126	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
129	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
132	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂×C₂²⋊D₅₃	—	4	32	no	—
135	C₂×C₂²⋊D₅₃	—	4	16	no	—
136	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
137	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
138	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
139	D₄×D₅₃	—	4	128	no	—
140	#11719945	—	4	64	no	—
141	C₂×D₂₁₂	—	2	16	no	—
142	C₂×D₂₁₂	—	2	16	no	—
143	C₂×D₂₁₂	—	4	64	no	—
144	C₂×D₂₁₂	—	4	16	no	—
145	C₂×D₂₁₂	—	4	8	no	—
146	C₂×D₂₁₂	C₂⁴	1	16	yes	—
147	C₂×D₂₁₂	C₂⁴	1	16	yes	—
148	C₂×D₂₁₂	C₂⁴	1	16	yes	—
149	C₂×D₂₁₂	C₂⁴	1	32	yes	—
150	C₂×C₄×D₅₃	—	2	32	no	—
151	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
152	C₂×Dic₁₀₆	—	4	8	no	—
153	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
154	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
155	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
156	C₄²×C₅₃	C₂⁴	1	8	yes	—
157	C₄²×C₅₃	C₂⁴	1	4	yes	—
158	C₄²×C₅₃	C₂⁴	1	4	yes	—
159	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
160	#11719921	C₂⁴	1	64	yes	—
161	#11719921	C₂⁴	1	32	yes	—
162	#11719921	C₂⁴	1	32	yes	—
163	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
164	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
165	C₄×Dic₅₃	C₂⁴	1	32	yes	—
166	C₂³×C₅₃	—	2	4	no	—
167	C₂³×C₅₃	—	2	16	no	—
168	C₂³×C₅₃	—	2	2	no	—
169	C₂³×C₅₃	—	2	8	no	—
170	C₂³×C₅₃	C₂⁵	1	2	yes	—
171	C₂³×C₅₃	C₂⁵	1	8	yes	—
172	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
173	C₂²×D₅₃	—	2	16	no	—
174	C₂²×D₅₃	—	2	16	no	—
175	C₂²×D₅₃	—	2	16	no	—
176	C₂²×D₅₃	—	4	32	no	—
177	C₂²×D₅₃	—	4	64	no	—
178	C₂²×D₅₃	—	8	64	no	—
179	C₂²×D₅₃	C₂⁵	1	16	yes	—
180	Q₈×C₅₃	—	4	8	no	—
181	D₄×C₅₃	—	2	32	no	—
182	D₄×C₅₃	—	4	16	no	—
183	D₄×C₅₃	—	4	8	no	—
184	C₂×C₄×C₅₃	—	2	4	no	—
185	C₂×C₄×C₅₃	—	2	8	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	32	no	—
188	C₂×C₄×C₅₃	—	2	16	no	—
189	C₂×C₄×C₅₃	—	2	8	no	—
190	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
191	C₂×C₄×C₅₃	H(5,𝔽₂)	1	8	yes	—
192	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
193	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
194	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
195	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
196	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
197	C₂²⋊D₅₃	—	4	32	no	—
198	C₂²⋊D₅₃	—	4	64	no	—
199	C₂×Dic₅₃	—	2	16	no	—
200	C₂×Dic₅₃	C₂⁵	1	16	yes	—
201	D₂₁₂	—	2	32	no	—
202	D₂₁₂	—	2	32	no	—
203	D₂₁₂	—	4	32	no	—
204	D₂₁₂	—	4	16	no	—
205	C₄×D₅₃	—	2	64	no	—
206	Dic₁₀₆	—	4	16	no	—
207	C₂×#3549	—	53	1	no	maximal, Sylow
208	C₂²×C₅₃	—	2	4	no	—
209	C₂²×C₅₃	—	2	16	no	—
210	C₂²×C₅₃	—	2	2	no	—
211	C₂²×C₅₃	—	2	8	no	—
212	C₂²×C₅₃	—	4	16	no	—
213	C₂²×C₅₃	—	4	4	no	—
214	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
215	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
216	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
217	C₂×D₅₃	—	2	16	no	—
218	C₂×D₅₃	—	2	16	no	—
219	C₂×D₅₃	—	2	16	no	—
220	C₂×D₅₃	—	4	32	no	—
221	C₂×D₅₃	—	8	64	no	—
222	C₄×C₅₃	—	2	4	no	—
223	C₄×C₅₃	—	2	8	no	—
224	C₄×C₅₃	—	2	2	no	—
225	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
226	C₄×C₅₃	C₂×H(5,𝔽₂)	1	8	yes	—
227	Dic₅₃	—	2	16	no	—
228	#3549	—	53	32	no	—
229	C₂×H(3,𝔽₄)	—	53	4	no	—
230	C₂×#555	—	53	2	no	—
231	C₂×#555	—	53	4	no	—
232	C₂×D₄²	—	53	2	no	—
233	C₂×D₄²	—	53	4	no	—
234	C₂×#535	—	53	4	no	—
235	C₂×#535	—	53	8	no	—
236	C₂×#532	—	53	2	no	—
237	C₂×#532	D₅₃	1	1	yes	characteristic, complemented
238	C₂×C₅₃	—	4	8	no	—
239	C₂×C₅₃	—	4	2	no	—
240	C₂×C₅₃	#3549	1	4	yes	complemented
241	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
243	D₅₃	—	4	16	no	—
244	C₂²×#41	—	53	4	no	—
245	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
246	C₂³×D₄	—	53	8	no	—
247	C₂³×D₄	—	53	4	no	—
248	H(3,𝔽₄)	—	53	64	no	—
249	#555	—	53	32	no	—
250	#555	—	53	64	no	—
251	D₄²	—	53	32	no	—
252	D₄²	—	53	64	no	—
253	#535	—	53	64	no	—
254	#535	—	53	128	no	—
255	#532	—	53	32	no	—
256	#532	C₂×D₅₃	1	16	yes	complemented
257	C₂×C₄⋊Q₈	—	53	2	no	—
258	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
259	C₂×#127	—	53	8	no	—
260	C₂×#127	—	53	2	no	—
261	C₂×#127	C₂×D₅₃	1	1	yes	characteristic
262	C₂×#124	—	53	8	no	—
263	C₂×#124	—	53	16	no	—
264	C₂×#124	—	53	4	no	—
265	C₂×#124	C₂×D₅₃	1	2	yes	complemented
266	C₂×#121	—	53	8	no	—
267	C₂×#121	—	53	4	no	—
268	C₂×#121	—	53	4	no	—
269	C₂×#121	—	53	8	no	—
270	C₂×#121	—	53	8	no	—
271	C₂×#121	C₂×D₅₃	1	4	yes	complemented
272	C₂×#120	—	53	8	no	—
273	C₂×#120	—	53	8	no	—
274	C₂×#120	—	53	16	no	—
275	C₂×C₄×D₄	—	53	4	no	—
276	C₂×C₄×D₄	—	53	4	no	—
277	C₂×C₄×D₄	C₂×D₅₃	1	4	yes	complemented
278	C₂×#117	—	53	4	no	—
279	C₂×#117	—	53	8	no	—
280	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
281	C₅₃	C₂×#3549	1	1	yes	characteristic, complemented, Sylow
282	C₂⁵	—	53	8	no	—
283	C₂×#41	—	4	8	no	—
284	C₂×#41	—	53	32	no	—
285	C₂×#41	—	212	16	no	—
286	C₂×#41	C₂²×D₅₃	1	16	yes	complemented
287	C₂²×Q₈	—	53	4	no	—
288	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
289	C₂²×D₄	—	53	16	no	—
290	C₂²×D₄	—	53	8	no	—
291	C₂²×D₄	—	53	64	no	—
292	C₂²×D₄	—	53	4	no	—
293	C₂²×D₄	—	53	4	no	—
294	C₂²×D₄	—	53	4	no	—
295	C₂²×D₄	—	53	8	no	—
296	C₂²×D₄	—	53	32	no	—
297	C₂²×D₄	—	212	16	no	—
298	C₂²×D₄	—	212	32	no	—
299	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
300	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
301	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
302	C₂³×C₄	—	53	8	no	—
303	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
304	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
305	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
306	C₄⋊Q₈	—	53	16	no	—
307	C₄⋊Q₈	C₂²×D₅₃	1	8	yes	complemented
308	#127	—	53	64	no	—
309	#127	—	53	16	no	—
310	#127	C₂²×D₅₃	1	8	yes	—
311	#124	—	53	64	no	—
312	#124	—	53	128	no	—
313	#124	—	53	32	no	—
314	#124	C₂²×D₅₃	1	16	yes	complemented
315	#121	—	53	64	no	—
316	#121	—	53	32	no	—
317	#121	—	53	32	no	—
318	#121	—	53	64	no	—
319	#121	—	53	64	no	—
320	#121	C₂²×D₅₃	1	32	yes	complemented
321	#120	—	53	64	no	—
322	#120	—	53	64	no	—
323	#120	—	53	128	no	—
324	C₄×D₄	—	53	32	no	—
325	C₄×D₄	—	53	32	no	—
326	C₄×D₄	C₂²×D₅₃	1	32	yes	complemented
327	#117	—	53	32	no	—
328	#117	—	53	64	no	—
329	C₂×C₄⋊C₄	—	53	4	no	—
330	C₂×C₄⋊C₄	—	53	4	no	—
331	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
332	C₂×C₂²⋊C₄	—	53	8	no	—
333	C₂×C₂²⋊C₄	—	53	16	no	—
334	C₂×C₂²⋊C₄	—	53	8	no	—
335	C₂×C₂²⋊C₄	—	53	8	no	—
336	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
337	C₂×C₄²	—	53	8	no	—
338	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
339	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
340	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
341	C₂×C₄²	C₂²×D₅₃	1	8	yes	complemented
342	C₂⁴	—	53	8	no	—
343	C₂⁴	—	53	32	no	—
344	C₂⁴	—	212	16	no	—
345	C₂⁴	—	212	32	no	—
346	C₂⁴	C₂³×D₅₃	1	4	yes	—
347	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
348	#41	—	4	32	no	—
349	#41	—	212	64	no	—
350	C₂×Q₈	—	4	4	no	—
351	C₂×Q₈	—	53	16	no	—
352	C₂×Q₈	—	212	8	no	—
353	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
354	C₂×D₄	—	2	16	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	8	no	—
357	C₂×D₄	—	53	64	no	—
358	C₂×D₄	—	53	32	no	—
359	C₂×D₄	—	53	16	no	—
360	C₂×D₄	—	53	16	no	—
361	C₂×D₄	—	53	16	no	—
362	C₂×D₄	—	53	32	no	—
363	C₂×D₄	—	106	16	no	—
364	C₂×D₄	—	106	16	no	—
365	C₂×D₄	—	212	64	no	—
366	C₂×D₄	—	212	16	no	—
367	C₂×D₄	—	212	8	no	—
368	C₂×D₄	—	212	128	no	—
369	C₂×D₄	—	212	32	no	—
370	C₂×D₄	—	212	16	no	—
371	C₂×D₄	C₂³×D₅₃	1	16	yes	—
372	C₂×D₄	C₂³×D₅₃	1	8	yes	—
373	C₂×D₄	C₂³×D₅₃	1	16	yes	—
374	C₂²×C₄	—	2	16	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	8	no	—
377	C₂²×C₄	—	53	8	no	—
378	C₂²×C₄	—	53	32	no	—
379	C₂²×C₄	—	106	32	no	—
380	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
381	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
382	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
383	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
384	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
385	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
386	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
387	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
388	C₄⋊C₄	—	53	16	no	—
389	C₄⋊C₄	—	53	16	no	—
390	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
391	C₂²⋊C₄	—	53	32	no	—
392	C₂²⋊C₄	—	53	64	no	—
393	C₂²⋊C₄	—	53	32	no	—
394	C₂²⋊C₄	—	53	32	no	—
395	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
396	C₄²	—	53	32	no	—
397	C₄²	C₂³×D₅₃	1	4	yes	—
398	C₄²	C₂³×D₅₃	1	8	yes	—
399	C₄²	C₂³×D₅₃	1	4	yes	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	8	no	—
402	C₂³	—	2	4	no	—
403	C₂³	—	2	16	no	—
404	C₂³	—	53	16	no	—
405	C₂³	—	106	16	no	—
406	C₂³	—	106	16	no	—
407	C₂³	—	106	16	no	—
408	C₂³	—	424	64	no	—
409	C₂³	—	212	32	no	—
410	C₂³	—	212	64	no	—
411	C₂³	C₂⁴×D₅₃	1	8	yes	—
412	C₂³	C₂⁴×D₅₃	1	2	yes	—
413	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
414	Q₈	—	4	8	no	—
415	Q₈	—	212	16	no	—
416	D₄	—	2	32	no	—
417	D₄	—	4	8	no	—
418	D₄	—	4	16	no	—
419	D₄	—	106	32	no	—
420	D₄	—	106	32	no	—
421	D₄	—	212	32	no	—
422	D₄	—	212	16	no	—
423	D₄	—	212	64	no	—
424	D₄	—	212	32	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	32	no	—
429	C₂×C₄	—	2	8	no	—
430	C₂×C₄	—	2	16	no	—
431	C₂×C₄	—	53	16	no	—
432	C₂×C₄	—	106	16	no	—
433	C₂×C₄	—	106	64	no	—
434	C₂×C₄	#13133262	1	4	yes	—
435	C₂×C₄	#13133263	1	8	yes	—
436	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
437	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
438	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
439	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
440	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
441	C₂²	—	2	2	no	—
442	C₂²	—	2	8	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	16	no	—
445	C₂²	—	4	16	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	106	16	no	—
448	C₂²	—	106	16	no	—
449	C₂²	—	106	16	no	—
450	C₂²	—	424	64	no	—
451	C₂²	—	212	32	no	—
452	C₂²	C₂×#13133262	1	2	yes	—
453	C₂²	C₂×#13133263	1	4	yes	—
454	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
455	C₄	—	2	2	no	—
456	C₄	—	2	8	no	—
457	C₄	—	2	4	no	—
458	C₄	—	106	16	no	—
459	C₄	C₂×#13133262	1	4	yes	—
460	C₄	C₂×#13133263	1	8	yes	—
461	C₂	—	4	8	no	—
462	C₂	—	4	2	no	—
463	C₂	—	212	16	no	—
464	C₂	#20289120	1	4	yes	direct product factor
465	C₂	C₂²×#13133262	1	1	yes	characteristic
466	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×#532	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#532	D₅₃
hypocenter	C₅₃	C₂×#3549
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#532

Complements

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

N	H ≅ G/N	type
1. #20289120	C₂	direct (N × H)
2. C₂×#16811365	C₂	semidirect
3. C₂×#16811339	C₂	semidirect
4. C₂×#16811338	C₂	semidirect
5. C₂×D₄×D₂₁₂	C₂	semidirect
6. C₂×#16811258	C₂	semidirect
7. C₂×#16811189	C₂	semidirect
8. C₂×#16811187	C₂	semidirect
9. C₂×#16811173	C₂	semidirect
10. C₂×C₅₃×#532	C₂	semidirect
11. C₂²×#11719945	C₂²	semidirect
12. C₂³×D₂₁₂	C₂²	semidirect
13. C₂²×D₄×D₅₃	C₂²	semidirect
14. #16811365	C₂²	semidirect
15. #16811339	C₂²	semidirect
16. #16811338	C₂²	semidirect
17. D₄×D₂₁₂	C₂²	semidirect
18. #16811258	C₂²	semidirect
19. #16811189	C₂²	semidirect
20. #16811187	C₂²	semidirect
21. #16811173	C₂²	semidirect
22. C₂×#13133217	C₂²	semidirect
24. C₂×#13133213	C₂²	semidirect
25. C₂×#13133206	C₂²	semidirect
26. C₂×#13133205	C₂²	semidirect
27. C₂×#13133203	C₂²	semidirect
28. C₂×#13133192	C₂²	semidirect
29. C₂×#13133191	C₂²	semidirect
30. C₂×#13133191	C₂²	semidirect
31. C₂×#13133191	C₂²	semidirect
32. C₂×#13133188	C₂²	semidirect
33. C₂×#13133187	C₂²	semidirect
34. C₂×#13133185	C₂²	semidirect
35. C₂×#13133185	C₂²	semidirect
36. C₂×C₄×D₂₁₂	C₂²	semidirect
37. C₂×C₄×D₂₁₂	C₂²	semidirect
38. C₂×#13133176	C₂²	semidirect
39. C₂×#13133175	C₂²	semidirect
40. C₂²×#41×C₅₃	C₂²	semidirect
41. C₅₃×#532	C₂²	semidirect
42. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
44. C₂×#124×C₅₃	C₂²	semidirect
45. C₂×#121×C₅₃	C₂²	semidirect
46. C₂×C₄×D₄×C₅₃	C₂²	semidirect
47. C₂²×C₄²×C₅₃	C₂²	semidirect
50. C₂×#11719945	C₂³	semidirect
57. C₂²×D₂₁₂	C₂³	semidirect
61. C₂×D₄×D₅₃	C₂³	semidirect
63. #13133217	C₂³	semidirect
65. #13133213	C₂³	semidirect
66. #13133206	C₂³	semidirect
67. #13133205	C₂³	semidirect
68. #13133203	C₂³	semidirect
69. #13133192	C₂³	semidirect
70. #13133191	C₂³	semidirect
71. #13133191	C₂³	semidirect
72. #13133191	C₂³	semidirect
73. #13133188	C₂³	semidirect
74. #13133187	C₂³	semidirect
75. #13133185	C₂³	semidirect
76. #13133185	C₂³	semidirect
77. C₄×D₂₁₂	C₂³	semidirect
78. C₄×D₂₁₂	C₂³	semidirect
79. #13133176	C₂³	semidirect
80. #13133175	C₂³	semidirect
89. C₂×#41×C₅₃	C₂³	semidirect
97. C₄⋊Q₈×C₅₃	C₂³	semidirect
99. #124×C₅₃	C₂³	semidirect
100. #121×C₅₃	C₂³	semidirect
101. C₄×D₄×C₅₃	C₂³	semidirect
107. C₂×C₄²×C₅₃	C₂³	semidirect
237. C₂×#532	D₅₃	semidirect
240. C₂×C₅₃	#3549	semidirect
245. C₂²×#41	C₂×D₅₃	semidirect
256. #532	C₂×D₅₃	semidirect
258. C₂×C₄⋊Q₈	C₂×D₅₃	semidirect
265. C₂×#124	C₂×D₅₃	semidirect
271. C₂×#121	C₂×D₅₃	semidirect
277. C₂×C₄×D₄	C₂×D₅₃	semidirect
280. C₂²×C₄²	C₂×D₅₃	semidirect
281. C₅₃	C₂×#3549	semidirect
286. C₂×#41	C₂²×D₅₃	semidirect
307. C₄⋊Q₈	C₂²×D₅₃	semidirect
314. #124	C₂²×D₅₃	semidirect
320. #121	C₂²×D₅₃	semidirect
326. C₄×D₄	C₂²×D₅₃	semidirect
341. C₂×C₄²	C₂²×D₅₃	semidirect
464. C₂	#20289120	direct (N × H)

Maximal subgroups

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

subgroup	index
#20289120	2
C₂×#16811365	2
C₂×#16811339	2
C₂×#16811338	2
C₂×D₄×D₂₁₂	2
C₂×#16811258	2
C₂×#16811189	2
C₂×#16811187	2
C₂×#16811173	2
C₂×C₅₃×#532	2
C₂×#3549	53

Subgroups — C2×#20289120 #28291895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289120 #28291895