Subgroups — C₂×#20272552 #27273584

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20272552	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₄	1	2	yes	—
6	#16810676	—	2	1	no	—
7	#16810676	—	2	1	no	—
8	#16810676	C₄	1	2	yes	—
9	#16810676	C₂²	1	2	yes	—
10	C₂×C₄⋊#88322	C₄	1	2	yes	—
11	C₂×C₂²⋊#88322	—	2	1	no	—
12	C₄×#11719917	—	2	1	no	—
13	C₄×#11719917	—	2	1	no	—
14	C₄×#11719917	C₄	1	2	yes	—
15	C₄×#11719917	C₂²	1	2	yes	—
16	C₂×#13133128	—	2	1	no	—
17	C₂×C₄×#88322	—	2	1	no	—
18	C₂×C₄×#88322	—	2	1	no	—
19	C₂³×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	—	2	1	no	—
22	C₂×C₄×C₈×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	—	2	1	no	—
24	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	4	no	—
29	C₂×#11719917	C₈	1	8	yes	complemented
30	C₂×#11719917	C₂×C₄	1	4	yes	—
31	C₂²×#88322	—	2	2	no	—
32	C₂²×#88322	—	2	2	no	—
33	C₄⋊#88322	—	2	2	no	—
34	C₄⋊#88322	C₂×C₄	1	4	yes	—
35	C₂²⋊#88322	—	2	2	no	—
36	C₂²⋊#88322	—	2	2	no	—
37	#13133128	—	2	2	no	—
38	#13133128	—	2	2	no	—
39	C₄×#88322	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	C₄×#88322	—	2	2	no	—
42	C₄×#88322	—	2	2	no	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	—	2	8	no	—
47	C₂²×C₈×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₃	—	2	2	no	—
63	C₄×C₈×C₅₃	—	2	2	no	—
64	C₄×C₈×C₅₃	—	2	2	no	—
65	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
66	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
85	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₅₃	—	2	8	no	—
90	C₂×C₈×C₅₃	—	2	8	no	—
91	C₂×C₈×C₅₃	—	2	8	no	—
92	C₂×C₈×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	C₄²	1	1	yes	characteristic
94	C₄²×C₅₃	C₄²	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	#11719917	—	2	8	no	—
110	#11719917	—	2	4	no	—
111	#11719917	C₂×C₈	1	8	yes	complemented
112	C₂×#88322	—	2	8	no	—
113	C₂×#88322	—	2	4	no	—
114	C₂×#88322	—	2	4	no	—
115	C₂×#88322	—	4	4	no	—
116	C₂³×C₅₃	—	2	2	no	—
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	#95	1	1	yes	characteristic
119	C₂³×C₅₃	#95	1	1	yes	characteristic
120	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
121	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
139	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
140	C₂×C₄×C₅₃	#97	1	2	yes	—
141	C₂×C₄×C₅₃	#97	1	2	yes	—
142	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
143	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
144	C₂×C₄×C₅₃	#104	1	4	yes	—
145	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
146	C₂×C₄×C₅₃	#108	1	4	yes	—
147	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
154	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
155	C₈×C₅₃	—	2	8	no	—
156	#88322	—	2	8	no	—
157	#88322	—	4	4	no	—
158	C₂×#1278	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	8	no	—
164	C₂²×C₅₃	#336	1	2	yes	—
165	C₂²×C₅₃	#336	1	2	yes	—
166	C₂²×C₅₃	#339	1	2	yes	—
167	C₂²×C₅₃	#341	1	2	yes	—
168	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
169	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	#336	1	2	yes	—
175	C₄×C₅₃	#336	1	2	yes	—
176	C₄×C₅₃	#339	1	2	yes	—
177	C₄×C₅₃	#341	1	2	yes	—
178	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
179	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
180	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
181	C₄×C₅₃	C₂×#104	1	2	yes	—
182	C₄×C₅₃	C₂×#108	1	2	yes	—
183	C₂×#431	—	53	1	no	—
184	C₂×C₄×M₁₆	—	53	1	no	—
185	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
186	#1278	—	53	4	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	#1278	1	4	yes	complemented
190	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
191	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
192	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
193	C₂²×M₁₆	—	53	2	no	—
194	C₂³×C₈	—	2	1	no	—
195	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
196	#431	—	53	2	no	—
197	#431	—	53	2	no	—
198	#431	—	106	1	no	—
199	#431	—	106	1	no	—
200	C₂×C₄⋊C₈	—	53	2	no	—
201	C₂×C₂²⋊C₈	—	106	1	no	—
202	C₄×M₁₆	—	53	2	no	—
203	C₄×M₁₆	—	53	2	no	—
204	C₄×M₁₆	—	106	1	no	—
205	C₄×M₁₆	—	106	1	no	—
206	C₂×#97	—	106	1	no	—
207	C₂×C₄×C₈	—	2	1	no	—
208	C₂×C₄×C₈	—	2	1	no	—
209	C₂×C₄×C₈	—	2	1	no	—
210	C₂×C₄×C₈	—	106	1	no	—
211	C₂×C₄×C₈	—	106	1	no	—
212	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
213	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
214	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
215	C₅₃	C₂×#1278	1	1	yes	characteristic, complemented, Sylow
216	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
217	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
218	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂×M₁₆	—	53	4	no	—
220	C₂×M₁₆	—	53	8	no	—
221	C₂×M₁₆	—	106	4	no	—
222	C₂×M₁₆	—	106	2	no	—
223	C₂²×C₈	—	2	8	no	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	—	2	2	no	—
227	C₂²×C₈	—	106	2	no	—
228	C₂²×C₈	—	106	2	no	—
229	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
230	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
231	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
234	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
235	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
241	C₄⋊C₈	—	53	4	no	—
242	C₄⋊C₈	—	106	2	no	—
243	C₂²⋊C₈	—	106	2	no	—
244	C₂²⋊C₈	—	106	2	no	—
245	#97	—	106	2	no	—
246	#97	—	106	2	no	—
247	C₄×C₈	—	2	2	no	—
248	C₄×C₈	—	2	2	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	—	106	2	no	—
251	C₄×C₈	—	106	2	no	—
252	C₄×C₈	—	106	2	no	—
253	C₄×C₈	—	106	2	no	—
254	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
255	C₂⁴	#11719919	1	1	yes	characteristic
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	C₈×D₅₃	1	4	yes	—
259	C₂²×C₄	#11719912	1	4	yes	—
260	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
262	C₂²×C₄	#11719919	1	1	yes	characteristic
263	C₂²×C₄	#11719919	1	1	yes	characteristic
264	C₂²×C₄	#11719919	1	1	yes	characteristic
265	C₂²×C₄	#11719919	1	1	yes	characteristic
266	C₂²×C₄	#11719919	1	1	yes	characteristic
267	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
269	C₂²×C₄	#11719921	1	1	yes	characteristic
270	C₂²×C₄	#11719921	1	1	yes	characteristic
271	C₂²×C₄	#11719921	1	1	yes	characteristic
272	C₂²×C₄	#11719921	1	1	yes	characteristic
273	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
278	M₁₆	—	53	8	no	—
279	M₁₆	—	106	8	no	—
280	M₁₆	—	106	4	no	—
281	C₂×C₈	—	2	8	no	—
282	C₂×C₈	—	2	8	no	—
283	C₂×C₈	—	2	8	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	106	8	no	—
286	C₂×C₈	—	106	4	no	—
287	C₂×C₈	—	106	4	no	—
288	C₂×C₈	—	212	4	no	—
289	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
290	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
291	C₄²	#11719919	1	1	yes	characteristic
292	C₄²	#11719919	1	1	yes	characteristic
293	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
294	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
295	C₄²	#11719921	1	1	yes	characteristic
296	C₄²	#11719921	1	1	yes	characteristic
297	C₄²	#11719921	1	1	yes	characteristic
298	C₄²	#11719921	1	1	yes	characteristic
299	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
300	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
302	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
303	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
304	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
305	C₂³	—	2	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	#13133124	1	1	yes	characteristic
308	C₂³	#13133124	1	1	yes	characteristic
309	C₂³	#13133146	1	4	yes	—
310	C₂³	C₂×#11719919	1	1	yes	characteristic
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	#13133124	1	1	yes	characteristic
322	C₂×C₄	#13133124	1	1	yes	characteristic
323	C₂×C₄	#13133124	1	1	yes	characteristic
324	C₂×C₄	#13133124	1	1	yes	characteristic
325	C₂×C₄	#13133124	1	1	yes	characteristic
326	C₂×C₄	#13133124	1	1	yes	characteristic
327	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
328	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
329	C₂×C₄	#13133127	1	2	yes	—
330	C₂×C₄	#13133127	1	2	yes	—
331	C₂×C₄	#13133129	1	4	yes	—
332	C₂×C₄	#13133129	1	4	yes	—
333	C₂×C₄	#13133145	1	4	yes	—
334	C₂×C₄	#13133146	1	4	yes	—
335	C₂×C₄	#13133153	1	4	yes	—
336	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
338	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
339	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
343	C₂×C₄	C₂×#11719912	1	2	yes	—
344	C₈	—	2	8	no	—
345	C₈	—	106	8	no	—
346	C₈	—	212	4	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	8	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	#16810377	1	2	yes	—
353	C₂²	#16810377	1	2	yes	—
354	C₂²	#16810383	1	2	yes	—
355	C₂²	#16810388	1	2	yes	—
356	C₂²	C₂×#13133124	1	1	yes	characteristic
357	C₂²	C₂×#13133146	1	2	yes	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	#16810377	1	2	yes	—
363	C₄	#16810377	1	2	yes	—
364	C₄	#16810383	1	2	yes	—
365	C₄	#16810388	1	2	yes	—
366	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
367	C₄	C₂×#13133127	1	1	yes	characteristic
368	C₄	C₂×#13133129	1	2	yes	—
369	C₄	C₂×#13133145	1	2	yes	—
370	C₄	C₂×#13133153	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	2	2	no	—
373	C₂	#20272552	1	4	yes	direct product factor
374	C₂	C₂×#16810377	1	1	yes	characteristic
375	C₂	C₂×#16810383	1	1	yes	characteristic
376	C₂	C₂×#16810388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₂₁₂
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1278
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #20272552	C₂	direct (N × H)
29. C₂×#11719917	C₈	semidirect
111. #11719917	C₂×C₈	semidirect
189. C₂×C₅₃	#1278	semidirect
215. C₅₃	C₂×#1278	semidirect
373. C₂	#20272552	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂×C₄×#11719917	2
#20272552	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1278	53

Subgroups — C2×#20272552 #27273584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272552 #27273584