Subgroups — C₂×#20274770 #27383755

Order 13568.

1330 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133223	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810639	C₂	1	2	yes	maximal
4	#20274770	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810394	C₂	1	2	yes	maximal, complemented
6	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719945	C₂²	1	1	yes	characteristic
8	C₂×#13133223	—	2	1	no	—
9	C₂×#13133223	—	2	1	no	—
10	C₂×#13133223	—	2	2	no	—
11	C₂×#13133223	—	2	2	no	—
12	C₂×#13133223	C₄	1	2	yes	complemented
13	C₂×#13133223	C₄	1	2	yes	complemented
14	C₂×#13133223	C₄	1	4	yes	complemented
15	C₂×#13133223	C₂²	1	4	yes	—
16	C₂²×#11719917	C₂²	1	1	yes	characteristic
17	C₂²×#11719912	—	2	1	no	—
18	C₂²×C₈×D₅₃	—	2	1	no	—
19	C₂×#13133176	C₂²	1	2	yes	—
20	#16810661	—	2	2	no	—
21	#16810661	C₂²	1	4	yes	complemented
22	C₂×#13133151	—	2	1	no	—
23	C₂×#13133149	—	2	1	no	—
24	#16810639	—	2	4	no	—
25	#16810639	C₂²	1	8	yes	—
26	C₂×#13133143	—	2	4	no	—
27	#16810394	C₂²	1	8	yes	complemented
28	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
29	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#108×C₅₃	—	2	1	no	—
31	C₂×#108×C₅₃	—	2	1	no	—
32	C₂×#108×C₅₃	—	2	2	no	—
33	C₂×#108×C₅₃	C₂²	1	4	yes	complemented
34	C₂×#11719945	—	2	1	no	—
35	C₂×#11719945	—	2	1	no	—
36	C₂×#11719945	—	2	1	no	—
37	C₂×#11719945	—	2	1	no	—
38	C₂×#11719945	—	2	1	no	—
39	C₂×#11719945	—	2	1	no	—
40	C₂×#11719945	C₂×C₄	1	2	yes	—
41	C₂×#11719945	C₂×C₄	1	2	yes	—
42	C₂×#11719945	D₄	1	2	yes	—
43	C₂×#11719945	D₄	1	2	yes	—
44	C₂×#11719945	Q₈	1	2	yes	—
45	C₂×#11719945	C₂³	1	2	yes	—
46	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
47	C₂²×D₂₁₂	Q₈	1	1	yes	characteristic
48	C₂²×C₂²⋊D₅₃	—	2	1	no	—
49	C₂²×C₄×D₅₃	—	2	1	no	—
50	#13133223	—	2	4	no	—
51	#13133223	—	2	4	no	—
52	#13133223	—	2	4	no	—
53	#13133223	—	2	4	no	—
54	#13133223	—	2	4	no	—
55	#13133223	—	2	4	no	—
56	#13133223	C₂×C₄	1	8	yes	complemented
57	#13133223	C₂×C₄	1	8	yes	complemented
58	C₂×#11719917	—	2	1	no	—
59	C₂×#11719917	—	2	1	no	—
60	C₂×#11719917	—	2	1	no	—
61	C₂×#11719917	C₂×C₄	1	2	yes	—
62	C₂×#11719917	C₂×C₄	1	2	yes	—
63	C₂×#11719917	C₂³	1	2	yes	—
64	C₂×#11719912	—	2	2	no	—
65	C₂×#11719912	—	2	2	no	—
66	C₂×#11719912	—	2	4	no	—
67	C₂×#11719912	—	2	4	no	—
68	C₂²×#88322	—	2	1	no	—
69	C₂×C₈×D₅₃	—	2	2	no	—
70	C₂×C₈×D₅₃	—	2	2	no	—
71	C₂×C₈×D₅₃	—	2	4	no	—
72	C₂×C₈×D₅₃	—	2	4	no	—
73	#13133176	—	2	2	no	—
74	#13133176	—	2	2	no	—
75	#13133176	—	2	2	no	—
76	#13133176	C₂³	1	4	yes	—
77	C₂×C₄⋊Dic₅₃	—	2	2	no	—
78	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
79	C₂×C₄×Dic₅₃	—	2	2	no	—
80	#13133151	—	2	4	no	—
81	#13133149	—	2	4	no	—
82	#13133143	—	2	16	no	—
83	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₅₃	—	2	2	no	—
85	C₂×M₁₆×C₅₃	—	2	2	no	—
86	C₂×M₁₆×C₅₃	—	2	2	no	—
87	C₂×M₁₆×C₅₃	—	2	2	no	—
88	C₂×M₁₆×C₅₃	—	2	2	no	—
89	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
90	C₂²×C₈×C₅₃	—	2	2	no	—
91	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
94	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
95	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
96	C₂²×C₈×C₅₃	D₄	1	2	yes	—
97	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
98	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
99	C₂²×C₈×C₅₃	D₄	1	2	yes	—
100	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
101	#108×C₅₃	—	2	4	no	—
102	#108×C₅₃	—	2	4	no	—
103	C₂⁴×C₅₃	#41	1	1	yes	characteristic
104	C₂³×D₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂×C₂²⋊D₅₃	—	2	2	no	—
120	C₂×C₂²⋊D₅₃	—	2	2	no	—
121	C₂×C₂²⋊D₅₃	—	2	2	no	—
122	C₂×C₂²⋊D₅₃	—	2	2	no	—
123	C₂×C₂²⋊D₅₃	—	2	2	no	—
124	C₂×C₂²⋊D₅₃	—	2	2	no	—
125	C₂²×Dic₅₃	—	2	1	no	—
126	C₂²×Dic₅₃	—	4	2	no	—
127	#11719945	—	2	2	no	—
128	#11719945	—	2	2	no	—
129	#11719945	—	2	2	no	—
130	#11719945	—	2	2	no	—
131	#11719945	—	2	2	no	—
132	#11719945	—	2	2	no	—
133	#11719945	—	2	2	no	—
134	#11719945	—	2	2	no	—
135	#11719945	—	2	2	no	—
136	#11719945	—	2	2	no	—
137	#11719945	—	2	2	no	—
138	#11719945	—	2	2	no	—
139	#11719945	C₄⋊C₄	1	4	yes	—
140	#11719945	C₄⋊C₄	1	4	yes	—
141	#11719945	C₂²×C₄	1	4	yes	—
142	#11719945	C₂×D₄	1	4	yes	—
143	C₂×D₂₁₂	—	2	1	no	—
144	C₂×D₂₁₂	—	2	1	no	—
145	C₂×D₂₁₂	—	2	1	no	—
146	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
147	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
148	C₂×D₂₁₂	C₂×Q₈	1	2	yes	—
149	C₂×C₄×D₅₃	—	2	2	no	—
150	C₂×C₄×D₅₃	—	2	2	no	—
151	C₂×C₄×D₅₃	—	2	2	no	—
152	C₂×C₄×D₅₃	—	2	2	no	—
153	C₂×C₄×D₅₃	—	2	2	no	—
154	C₂×C₄×D₅₃	—	2	2	no	—
155	C₂×Dic₁₀₆	—	2	1	no	—
156	C₂×Dic₁₀₆	—	2	1	no	—
157	C₂×Dic₁₀₆	—	2	1	no	—
158	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
159	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
160	C₂×Dic₁₀₆	C₂×Q₈	1	2	yes	—
161	M₁₆×C₅₃	—	2	4	no	—
162	M₁₆×C₅₃	—	2	4	no	—
163	M₁₆×C₅₃	—	4	2	no	—
164	M₁₆×C₅₃	—	4	2	no	—
165	C₂×C₈×C₅₃	—	2	4	no	—
166	C₂×C₈×C₅₃	—	4	2	no	—
167	C₂×C₈×C₅₃	—	4	2	no	—
168	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
171	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
174	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
175	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
176	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
177	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
178	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
179	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
180	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
181	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
183	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
184	C₂²⋊Dic₅₃	—	2	4	no	—
185	C₂²⋊Dic₅₃	—	4	2	no	—
186	C₄⋊Dic₅₃	—	2	4	no	—
187	C₄⋊Dic₅₃	—	4	2	no	—
188	C₄×Dic₅₃	—	2	4	no	—
189	C₄×Dic₅₃	—	4	2	no	—
190	#11719917	—	2	2	no	—
191	#11719917	—	2	2	no	—
192	#11719917	—	2	2	no	—
193	#11719917	C₂²×C₄	1	4	yes	—
194	C₂×#88322	—	2	2	no	—
195	C₂×#88322	—	2	2	no	—
196	C₂×#88322	—	2	2	no	—
197	#11719912	—	2	8	no	—
198	#11719912	—	2	8	no	—
199	C₈×D₅₃	—	2	8	no	—
200	C₈×D₅₃	—	2	8	no	—
201	C₂³×C₅₃	—	4	2	no	—
202	C₂³×C₅₃	#117	1	1	yes	characteristic
203	C₂³×C₅₃	#117	1	1	yes	characteristic
204	C₂³×C₅₃	#121	1	1	yes	characteristic
205	C₂³×C₅₃	#121	1	1	yes	characteristic
206	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
208	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
209	C₂²×D₅₃	—	2	2	no	—
210	C₂²×D₅₃	—	2	2	no	—
211	C₂²×D₅₃	—	2	2	no	—
212	C₂²×D₅₃	—	4	4	no	—
213	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
222	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
223	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
224	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
225	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
226	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
227	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
228	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
231	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
240	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
241	C₂²⋊D₅₃	—	2	4	no	—
242	C₂²⋊D₅₃	—	2	4	no	—
243	C₂²⋊D₅₃	—	2	4	no	—
244	C₂²⋊D₅₃	—	2	4	no	—
245	C₂×Dic₅₃	—	2	2	no	—
246	C₂×Dic₅₃	—	2	2	no	—
247	C₂×Dic₅₃	—	2	2	no	—
248	C₂×Dic₅₃	—	4	4	no	—
249	C₂×Dic₅₃	—	8	4	no	—
250	D₂₁₂	—	2	2	no	—
251	D₂₁₂	—	2	2	no	—
252	D₂₁₂	—	2	2	no	—
253	D₂₁₂	C₂×C₄⋊C₄	1	4	yes	—
254	C₄×D₅₃	—	2	4	no	—
255	C₄×D₅₃	—	2	4	no	—
256	C₄×D₅₃	—	2	4	no	—
257	C₄×D₅₃	—	2	4	no	—
258	Dic₁₀₆	—	2	2	no	—
259	Dic₁₀₆	—	2	2	no	—
260	Dic₁₀₆	—	2	2	no	—
261	Dic₁₀₆	C₂×C₄⋊C₄	1	4	yes	—
262	C₈×C₅₃	—	4	4	no	—
263	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
264	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
265	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
266	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
267	#88322	—	2	4	no	—
268	C₂×#1814	—	53	1	no	maximal, Sylow
269	C₂²×C₅₃	—	2	2	no	—
270	C₂²×C₅₃	—	2	2	no	—
271	C₂²×C₅₃	—	2	2	no	—
272	C₂²×C₅₃	—	4	1	no	—
273	C₂²×C₅₃	—	4	1	no	—
274	C₂²×C₅₃	—	4	1	no	—
275	C₂²×C₅₃	—	4	1	no	—
276	C₂²×C₅₃	#380	1	1	yes	characteristic
277	C₂²×C₅₃	#380	1	1	yes	characteristic
278	C₂²×C₅₃	#380	1	1	yes	characteristic
279	C₂²×C₅₃	#380	1	1	yes	characteristic
280	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
281	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
282	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
283	C₂×D₅₃	—	2	4	no	—
284	C₂×D₅₃	—	4	4	no	—
285	C₂×D₅₃	—	4	4	no	—
286	C₂×D₅₃	—	4	4	no	—
287	C₄×C₅₃	#380	1	1	yes	characteristic
288	C₄×C₅₃	#380	1	1	yes	characteristic
289	C₄×C₅₃	#380	1	1	yes	characteristic
290	C₄×C₅₃	#380	1	1	yes	characteristic
291	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
292	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
293	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
294	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
295	Dic₅₃	—	2	4	no	—
296	Dic₅₃	—	8	4	no	—
297	C₂²×#131	—	53	1	no	—
298	C₂²×#108	D₅₃	1	1	yes	characteristic, complemented
299	C₂×#427	—	53	1	no	—
300	C₂×#421	—	53	2	no	—
301	#1814	—	53	8	no	—
302	C₂×#344	—	53	2	no	—
303	C₂×C₅₃	—	2	1	no	—
304	C₂×C₅₃	—	2	1	no	—
305	C₂×C₅₃	—	2	1	no	—
306	C₂×C₅₃	—	2	1	no	—
307	C₂×C₅₃	—	2	1	no	—
308	C₂×C₅₃	—	2	1	no	—
309	C₂×C₅₃	#1814	1	2	yes	complemented
310	C₂×C₅₃	C₂×#380	1	1	yes	characteristic
311	D₅₃	—	4	4	no	—
312	C₂²×#41	—	53	1	no	—
313	C₂×#131	—	53	2	no	—
314	C₂×#131	—	53	2	no	—
315	C₂×#131	—	53	4	no	—
316	C₂×#131	—	53	4	no	—
317	C₂×#131	—	106	1	no	—
318	C₂×#131	—	106	1	no	—
319	C₂×#131	—	106	2	no	—
320	C₂×#131	—	106	2	no	—
321	C₂²×M₁₆	—	53	1	no	—
322	C₂²×M₁₆	—	106	1	no	—
323	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
324	C₂³×C₈	—	106	1	no	—
325	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
326	C₂×#117	—	53	2	no	—
327	C₂×#108	—	2	1	no	—
328	C₂×#108	—	2	1	no	—
329	C₂×#108	—	2	2	no	—
330	C₂×#108	C₂×D₅₃	1	4	yes	complemented
331	#427	—	53	4	no	—
332	#427	—	106	2	no	—
333	C₂×#107	—	106	1	no	—
334	C₂×#106	—	106	1	no	—
335	#421	—	53	8	no	—
336	#421	—	106	4	no	—
337	C₂×#104	—	106	4	no	—
338	#344	—	53	8	no	—
339	C₅₃	C₂×#1814	1	1	yes	characteristic, complemented, Sylow
340	C₂×#41	—	53	2	no	—
341	C₂×#41	—	53	2	no	—
342	C₂×#41	—	53	2	no	—
343	C₂×#41	—	53	2	no	—
344	C₂×#41	—	53	2	no	—
345	C₂×#41	—	53	2	no	—
346	C₂×#41	—	106	1	no	—
347	C₂×#41	—	106	1	no	—
348	C₂×#41	—	106	1	no	—
349	C₂×#41	—	106	1	no	—
350	C₂×#41	—	106	1	no	—
351	C₂×#41	—	106	1	no	—
352	C₂²×Q₈	—	53	1	no	—
353	C₂²×D₄	—	53	1	no	—
354	C₂²×D₄	—	106	1	no	—
355	C₂³×C₄	—	106	1	no	—
356	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
357	#131	—	53	8	no	—
358	#131	—	53	8	no	—
359	#131	—	106	4	no	—
360	#131	—	106	4	no	—
361	#131	—	106	4	no	—
362	#131	—	106	4	no	—
363	#131	—	106	4	no	—
364	#131	—	106	4	no	—
365	C₂×M₁₆	—	2	2	no	—
366	C₂×M₁₆	—	2	2	no	—
367	C₂×M₁₆	—	2	2	no	—
368	C₂×M₁₆	—	2	2	no	—
369	C₂×M₁₆	—	2	2	no	—
370	C₂×M₁₆	—	53	2	no	—
371	C₂×M₁₆	—	53	2	no	—
372	C₂×M₁₆	—	53	2	no	—
373	C₂×M₁₆	—	106	1	no	—
374	C₂×M₁₆	—	106	1	no	—
375	C₂×M₁₆	—	106	1	no	—
376	C₂×M₁₆	—	106	4	no	—
377	C₂×M₁₆	—	106	2	no	—
378	C₂×M₁₆	—	106	2	no	—
379	C₂×M₁₆	—	106	4	no	—
380	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
381	C₂²×C₈	—	2	2	no	—
382	C₂²×C₈	—	106	1	no	—
383	C₂²×C₈	—	106	4	no	—
384	C₂²×C₈	—	106	2	no	—
385	C₂²×C₈	—	106	2	no	—
386	C₂²×C₈	—	106	4	no	—
387	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
388	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
389	C₂²×C₈	C₄×D₅₃	1	2	yes	—
390	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
391	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
392	C₂²×C₈	D₂₁₂	1	2	yes	—
393	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
394	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
395	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
396	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
397	#117	—	53	4	no	—
398	#117	—	106	2	no	—
399	#117	—	106	2	no	—
400	#117	—	106	2	no	—
401	C₂×C₄⋊C₄	—	106	2	no	—
402	C₂×C₂²⋊C₄	—	106	2	no	—
403	C₂×C₄²	—	106	2	no	—
404	#108	—	2	4	no	—
405	#108	—	2	4	no	—
406	#107	—	106	4	no	—
407	#106	—	106	4	no	—
408	#104	—	106	16	no	—
409	C₂⁴	—	106	1	no	—
410	C₂⁴	#11719949	1	1	yes	characteristic
411	#41	—	53	4	no	—
412	#41	—	53	4	no	—
413	#41	—	53	4	no	—
414	#41	—	53	4	no	—
415	#41	—	106	2	no	—
416	#41	—	106	2	no	—
417	#41	—	106	2	no	—
418	#41	—	106	2	no	—
419	#41	—	106	2	no	—
420	#41	—	106	2	no	—
421	#41	—	106	2	no	—
422	#41	—	106	2	no	—
423	#41	—	106	2	no	—
424	#41	—	106	2	no	—
425	#41	—	106	2	no	—
426	#41	—	106	2	no	—
427	C₂×Q₈	—	53	2	no	—
428	C₂×Q₈	—	53	2	no	—
429	C₂×Q₈	—	53	2	no	—
430	C₂×Q₈	—	106	1	no	—
431	C₂×Q₈	—	106	1	no	—
432	C₂×Q₈	—	106	1	no	—
433	C₂×D₄	—	53	2	no	—
434	C₂×D₄	—	53	2	no	—
435	C₂×D₄	—	53	2	no	—
436	C₂×D₄	—	106	2	no	—
437	C₂×D₄	—	106	2	no	—
438	C₂×D₄	—	106	2	no	—
439	C₂×D₄	—	106	2	no	—
440	C₂×D₄	—	106	2	no	—
441	C₂×D₄	—	106	2	no	—
442	C₂×D₄	—	106	1	no	—
443	C₂×D₄	—	106	1	no	—
444	C₂×D₄	—	106	1	no	—
445	C₂²×C₄	—	106	1	no	—
446	C₂²×C₄	—	106	2	no	—
447	C₂²×C₄	—	106	2	no	—
448	C₂²×C₄	—	106	2	no	—
449	C₂²×C₄	—	106	2	no	—
450	C₂²×C₄	—	106	2	no	—
451	C₂²×C₄	—	106	2	no	—
452	C₂²×C₄	—	212	2	no	—
453	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
454	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
457	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
458	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
459	C₂²×C₄	#11719947	1	1	yes	characteristic
460	C₂²×C₄	#11719947	1	1	yes	characteristic
461	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
462	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
463	C₂²×C₄	#11719949	1	1	yes	characteristic
464	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
465	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
466	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
467	M₁₆	—	2	4	no	—
468	M₁₆	—	2	4	no	—
469	M₁₆	—	4	2	no	—
470	M₁₆	—	4	2	no	—
471	M₁₆	—	53	4	no	—
472	M₁₆	—	106	2	no	—
473	M₁₆	—	106	2	no	—
474	M₁₆	—	106	2	no	—
475	M₁₆	—	106	8	no	—
476	M₁₆	—	106	8	no	—
477	C₂×C₈	—	2	4	no	—
478	C₂×C₈	—	4	2	no	—
479	C₂×C₈	—	4	2	no	—
480	C₂×C₈	—	106	2	no	—
481	C₂×C₈	—	106	2	no	—
482	C₂×C₈	—	106	2	no	—
483	C₂×C₈	—	106	8	no	—
484	C₂×C₈	—	106	8	no	—
485	C₂×C₈	#11719921	1	1	yes	characteristic
486	C₂×C₈	#11719921	1	1	yes	characteristic
487	C₂×C₈	#11719921	1	2	yes	—
488	C₂×C₈	#11719921	1	1	yes	characteristic
489	C₂×C₈	#11719921	1	1	yes	characteristic
490	C₂×C₈	#11719921	1	2	yes	—
491	C₂×C₈	#11719921	1	2	yes	—
492	C₂×C₈	#11719921	1	2	yes	—
493	C₂×C₈	#11719921	1	2	yes	—
494	C₂×C₈	#11719921	1	2	yes	—
495	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
496	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
497	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
498	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
499	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
500	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
501	C₄⋊C₄	—	106	4	no	—
502	C₄⋊C₄	—	212	2	no	—
503	C₂²⋊C₄	—	106	4	no	—
504	C₂²⋊C₄	—	212	2	no	—
505	C₄²	—	106	4	no	—
506	C₄²	—	212	2	no	—
507	C₂³	—	4	2	no	—
508	C₂³	—	106	2	no	—
509	C₂³	—	106	2	no	—
510	C₂³	—	106	2	no	—
511	C₂³	—	212	4	no	—
512	C₂³	#13133175	1	1	yes	characteristic
513	C₂³	#13133175	1	1	yes	characteristic
514	C₂³	#13133192	1	1	yes	characteristic
515	C₂³	#13133192	1	1	yes	characteristic
516	C₂³	#13133193	1	1	yes	characteristic
517	C₂³	#13133193	1	1	yes	characteristic
518	C₂³	C₂×#11719949	1	1	yes	characteristic
519	Q₈	—	53	4	no	—
520	Q₈	—	106	2	no	—
521	Q₈	—	106	2	no	—
522	Q₈	—	106	2	no	—
523	D₄	—	53	4	no	—
524	D₄	—	106	4	no	—
525	D₄	—	106	4	no	—
526	D₄	—	106	4	no	—
527	D₄	—	106	4	no	—
528	D₄	—	106	2	no	—
529	D₄	—	106	2	no	—
530	D₄	—	106	2	no	—
531	C₂×C₄	—	106	2	no	—
532	C₂×C₄	—	106	2	no	—
533	C₂×C₄	—	106	2	no	—
534	C₂×C₄	—	106	4	no	—
535	C₂×C₄	—	106	4	no	—
536	C₂×C₄	—	106	4	no	—
537	C₂×C₄	—	106	4	no	—
538	C₂×C₄	—	424	4	no	—
539	C₂×C₄	—	212	4	no	—
540	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
541	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
542	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
543	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
544	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
545	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
546	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
547	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
548	C₂×C₄	#13133175	1	1	yes	characteristic
549	C₂×C₄	#13133175	1	1	yes	characteristic
550	C₂×C₄	#13133188	1	1	yes	characteristic
551	C₂×C₄	#13133188	1	1	yes	characteristic
552	C₂×C₄	#13133188	1	1	yes	characteristic
553	C₂×C₄	#13133188	1	1	yes	characteristic
554	C₂×C₄	#13133192	1	1	yes	characteristic
555	C₂×C₄	#13133192	1	1	yes	characteristic
556	C₂×C₄	#13133193	1	1	yes	characteristic
557	C₂×C₄	#13133193	1	1	yes	characteristic
558	C₂×C₄	#13133197	1	1	yes	characteristic
559	C₂×C₄	#13133197	1	1	yes	characteristic
560	C₂×C₄	#13133197	1	1	yes	characteristic
561	C₂×C₄	#13133197	1	1	yes	characteristic
562	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
563	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
564	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
566	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
567	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
568	C₈	—	4	4	no	—
569	C₈	—	106	4	no	—
570	C₈	C₂×#11719921	1	2	yes	—
571	C₈	C₂×#11719921	1	2	yes	—
572	C₈	C₂×#11719921	1	2	yes	—
573	C₈	C₂×#11719921	1	2	yes	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	106	4	no	—
582	C₂²	—	212	4	no	—
583	C₂²	—	212	4	no	—
584	C₂²	—	212	4	no	—
585	C₂²	#16810480	1	1	yes	characteristic
586	C₂²	#16810480	1	1	yes	characteristic
587	C₂²	#16810480	1	1	yes	characteristic
588	C₂²	#16810480	1	1	yes	characteristic
589	C₂²	C₂×#13133175	1	1	yes	characteristic
590	C₂²	C₂×#13133192	1	1	yes	characteristic
591	C₂²	C₂×#13133193	1	1	yes	characteristic
592	C₄	—	106	4	no	—
593	C₄	—	424	4	no	—
594	C₄	#16810480	1	1	yes	characteristic
595	C₄	#16810480	1	1	yes	characteristic
596	C₄	#16810480	1	1	yes	characteristic
597	C₄	#16810480	1	1	yes	characteristic
598	C₄	C₂²×#11719921	1	1	yes	characteristic
599	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
600	C₄	C₂×#13133188	1	1	yes	characteristic
601	C₄	C₂×#13133197	1	1	yes	characteristic
602	C₂	—	2	1	no	—
603	C₂	—	2	1	no	—
604	C₂	—	2	1	no	—
605	C₂	—	2	1	no	—
606	C₂	—	2	1	no	—
607	C₂	—	2	1	no	—
608	C₂	—	212	4	no	—
609	C₂	#20274770	1	2	yes	direct product factor
610	C₂	C₂×#16810480	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×#11719945	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂×#1814
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
2. C₂×#16810661	C₂	semidirect
4. #20274770	C₂	direct (N × H)
5. C₂×#16810394	C₂	semidirect
6. C₂²×#108×C₅₃	C₂	semidirect
12. C₂×#13133223	C₄	semidirect
13. C₂×#13133223	C₄	semidirect
14. C₂×#13133223	C₄	semidirect
21. #16810661	C₂²	semidirect
27. #16810394	C₂²	semidirect
33. C₂×#108×C₅₃	C₂²	semidirect
56. #13133223	C₂×C₄	semidirect
57. #13133223	C₂×C₄	semidirect
298. C₂²×#108	D₅₃	semidirect
309. C₂×C₅₃	#1814	semidirect
330. C₂×#108	C₂×D₅₃	semidirect
339. C₅₃	C₂×#1814	semidirect
609. C₂	#20274770	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×#13133223	2
C₂×#16810661	2
C₂×#16810639	2
#20274770	2
C₂×#16810394	2
C₂²×#108×C₅₃	2
C₂×#1814	53

Subgroups — C2×#20274770 #27383755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274770 #27383755