Subgroups — C₂×#20272547 #27273527

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 396 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×C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272547	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×Dic₅₃	—	2	1	no	—
6	C₂²×C₄×Dic₅₃	—	2	1	no	—
7	C₂²×C₄×Dic₅₃	C₄	1	2	yes	—
8	#16810676	—	2	1	no	—
9	#16810676	—	2	1	no	—
10	#16810676	C₄	1	2	yes	complemented
11	#16810676	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₄	1	2	yes	complemented
13	C₂×C₂²⋊#88322	—	2	1	no	—
14	C₂×C₄×#88322	—	2	1	no	—
15	C₄²×Dic₅₃	—	2	1	no	—
16	C₄²×Dic₅₃	—	2	1	no	—
17	C₄²×Dic₅₃	C₄	1	2	yes	—
18	C₄²×Dic₅₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#431	—	2	1	no	—
21	C₅₃×#431	—	2	1	no	—
22	C₅₃×#431	C₄	1	2	yes	complemented
23	C₅₃×#431	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
25	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
26	C₂×C₄×C₈×C₅₃	—	2	1	no	—
27	C₂³×Dic₅₃	—	2	2	no	—
28	C₂²×#88322	—	2	2	no	—
29	C₂×C₄×Dic₅₃	—	2	8	no	—
30	C₂×C₄×Dic₅₃	—	2	8	no	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	—	2	2	no	—
34	C₂×C₄×Dic₅₃	—	2	4	no	—
35	C₂×C₄×Dic₅₃	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	2	no	—
37	C₂×C₄×Dic₅₃	C₈	1	8	yes	complemented
38	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
39	C₄⋊#88322	—	2	2	no	—
40	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
41	C₂²⋊#88322	—	2	2	no	—
42	C₂²⋊#88322	—	2	2	no	—
43	C₄×#88322	—	2	2	no	—
44	C₄×#88322	—	2	2	no	—
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
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	D₄	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	Q₈	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	—	2	2	no	—
62	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
63	C₂²⋊C₈×C₅₃	—	2	2	no	—
64	C₂²⋊C₈×C₅₃	—	2	2	no	—
65	C₄×C₈×C₅₃	—	2	2	no	—
66	C₄×C₈×C₅₃	—	2	2	no	—
67	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	2	4	no	—
70	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄²	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	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
86	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₅₃	—	2	8	no	—
91	C₂²×Dic₅₃	—	2	8	no	—
92	C₂²×Dic₅₃	—	2	8	no	—
93	C₂²×Dic₅₃	—	2	4	no	—
94	C₂×C₈×C₅₃	—	2	4	no	—
95	C₂×C₈×C₅₃	—	4	4	no	—
96	C₄²×C₅₃	C₄²	1	1	yes	characteristic
97	C₄²×C₅₃	C₄²	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	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₄×Dic₅₃	—	2	8	no	—
113	C₄×Dic₅₃	—	2	8	no	—
114	C₄×Dic₅₃	—	2	8	no	—
115	C₄×Dic₅₃	—	2	8	no	—
116	C₄×Dic₅₃	—	2	8	no	—
117	C₄×Dic₅₃	—	2	4	no	—
118	C₄×Dic₅₃	C₂×C₈	1	8	yes	complemented
119	C₂×#88322	—	2	4	no	—
120	C₂×#88322	—	4	4	no	—
121	C₂³×C₅₃	—	2	2	no	—
122	C₂³×C₅₃	—	2	2	no	—
123	C₂³×C₅₃	#95	1	1	yes	characteristic
124	C₂³×C₅₃	#95	1	1	yes	characteristic
125	C₂³×C₅₃	C₂²⋊C₈	1	4	yes	—
126	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
144	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
145	C₂×C₄×C₅₃	#97	1	2	yes	—
146	C₂×C₄×C₅₃	#97	1	2	yes	—
147	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
148	C₂×C₄×C₅₃	#104	1	4	yes	—
149	C₂×C₄×C₅₃	#104	1	4	yes	—
150	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
151	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
152	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
159	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
160	C₂×Dic₅₃	—	2	32	no	—
161	C₂×Dic₅₃	—	2	8	no	—
162	C₂×Dic₅₃	—	2	8	no	—
163	C₂×Dic₅₃	—	2	8	no	—
164	C₈×C₅₃	—	4	4	no	—
165	#88322	—	4	4	no	—
166	C₂×#1276	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	8	no	—
172	C₂²×C₅₃	#336	1	2	yes	—
173	C₂²×C₅₃	#336	1	2	yes	—
174	C₂²×C₅₃	#339	1	2	yes	—
175	C₂²×C₅₃	#339	1	2	yes	—
176	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
177	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	#336	1	2	yes	—
183	C₄×C₅₃	#336	1	2	yes	—
184	C₄×C₅₃	#339	1	2	yes	—
185	C₄×C₅₃	#339	1	2	yes	—
186	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
187	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
188	C₄×C₅₃	C₂×#104	1	2	yes	—
189	C₄×C₅₃	C₂×#104	1	2	yes	—
190	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
191	Dic₅₃	—	2	16	no	—
192	C₂×#431	—	53	1	no	—
193	C₂×#431	D₅₃	1	1	yes	characteristic
194	C₂×C₄³	—	53	1	no	—
195	#1276	—	53	4	no	—
196	C₂×C₅₃	—	2	2	no	—
197	C₂×C₅₃	—	2	2	no	—
198	C₂×C₅₃	#1276	1	4	yes	complemented
199	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
200	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
201	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
202	C₂²×C₄²	—	53	2	no	—
203	C₂²×C₄²	—	106	1	no	—
204	C₂²×C₄²	—	106	1	no	—
205	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
206	#431	—	2	1	no	—
207	#431	—	2	1	no	—
208	#431	—	53	2	no	—
209	#431	—	53	2	no	—
210	#431	—	106	1	no	—
211	#431	—	106	1	no	—
212	#431	Dic₅₃	1	2	yes	complemented
213	#431	C₂×D₅₃	1	2	yes	—
214	C₂×C₄⋊C₈	—	53	2	no	—
215	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	complemented
216	C₂×C₂²⋊C₈	—	2	1	no	—
217	C₂×C₂²⋊C₈	—	106	1	no	—
218	C₂×C₄×C₈	—	2	1	no	—
219	C₂×C₄×C₈	—	106	1	no	—
220	C₄³	—	53	2	no	—
221	C₄³	—	53	2	no	—
222	C₄³	—	106	1	no	—
223	C₄³	—	106	1	no	—
224	C₅₃	C₂×#1276	1	1	yes	characteristic, complemented, Sylow
225	C₂³×C₄	—	106	2	no	—
226	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
227	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
228	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	—	106	2	no	—
231	C₂×C₄²	—	53	4	no	—
232	C₂×C₄²	—	53	8	no	—
233	C₂×C₄²	—	106	8	no	—
234	C₂×C₄²	—	106	8	no	—
235	C₂×C₄²	—	106	2	no	—
236	C₂×C₄²	—	106	2	no	—
237	C₂×C₄²	—	106	4	no	—
238	C₂×C₄²	—	106	2	no	—
239	C₂×C₄²	—	106	2	no	—
240	C₂×C₄²	—	106	2	no	—
241	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
242	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
243	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
246	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
247	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
251	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	53	4	no	—
255	C₄⋊C₈	—	106	2	no	—
256	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	complemented
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	—	2	2	no	—
259	C₂²⋊C₈	—	106	2	no	—
260	C₂²⋊C₈	—	106	2	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	106	2	no	—
264	C₄×C₈	—	106	2	no	—
265	C₂⁴	#11719921	1	1	yes	characteristic
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	106	8	no	—
269	C₂²×C₄	—	106	8	no	—
270	C₂²×C₄	—	106	8	no	—
271	C₂²×C₄	—	106	4	no	—
272	C₂²×C₄	C₈×D₅₃	1	4	yes	—
273	C₂²×C₄	#11719912	1	4	yes	—
274	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
276	C₂²×C₄	#11719919	1	1	yes	characteristic
277	C₂²×C₄	#11719919	1	1	yes	characteristic
278	C₂²×C₄	#11719919	1	1	yes	characteristic
279	C₂²×C₄	#11719919	1	1	yes	characteristic
280	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
282	C₂²×C₄	#11719921	1	1	yes	characteristic
283	C₂²×C₄	#11719921	1	1	yes	characteristic
284	C₂²×C₄	#11719921	1	1	yes	characteristic
285	C₂²×C₄	#11719921	1	1	yes	characteristic
286	C₂²×C₄	#11719921	1	1	yes	characteristic
287	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
288	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
291	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	4	4	no	—
294	C₂×C₈	—	106	4	no	—
295	C₂×C₈	—	212	4	no	—
296	C₄²	—	53	8	no	—
297	C₄²	—	106	8	no	—
298	C₄²	—	106	8	no	—
299	C₄²	—	106	8	no	—
300	C₄²	—	106	8	no	—
301	C₄²	—	106	8	no	—
302	C₄²	—	106	4	no	—
303	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
304	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
305	C₄²	#11719919	1	1	yes	characteristic
306	C₄²	#11719919	1	1	yes	characteristic
307	C₄²	#11719919	1	1	yes	characteristic
308	C₄²	#11719919	1	1	yes	characteristic
309	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
310	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
311	C₄²	#11719921	1	1	yes	characteristic
312	C₄²	#11719921	1	1	yes	characteristic
313	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
314	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
315	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
317	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
318	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	#13133124	1	1	yes	characteristic
322	C₂³	#13133124	1	1	yes	characteristic
323	C₂³	#13133129	1	4	yes	—
324	C₂³	C₂×#11719921	1	1	yes	characteristic
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	106	8	no	—
336	C₂×C₄	—	106	8	no	—
337	C₂×C₄	—	106	32	no	—
338	C₂×C₄	—	106	8	no	—
339	C₂×C₄	#13133124	1	1	yes	characteristic
340	C₂×C₄	#13133124	1	1	yes	characteristic
341	C₂×C₄	#13133124	1	1	yes	characteristic
342	C₂×C₄	#13133124	1	1	yes	characteristic
343	C₂×C₄	#13133124	1	1	yes	characteristic
344	C₂×C₄	#13133124	1	1	yes	characteristic
345	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
346	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
347	C₂×C₄	#13133127	1	2	yes	—
348	C₂×C₄	#13133127	1	2	yes	—
349	C₂×C₄	#13133129	1	4	yes	—
350	C₂×C₄	#13133143	1	4	yes	—
351	C₂×C₄	#13133144	1	4	yes	—
352	C₂×C₄	#13133146	1	4	yes	—
353	C₂×C₄	#13133146	1	4	yes	—
354	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
356	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
358	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
359	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
360	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
361	C₂×C₄	C₂×#11719912	1	2	yes	—
362	C₈	—	4	4	no	—
363	C₈	—	212	4	no	—
364	C₂²	—	2	8	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	#16810377	1	2	yes	—
370	C₂²	#16810377	1	2	yes	—
371	C₂²	#16810382	1	2	yes	—
372	C₂²	#16810384	1	2	yes	—
373	C₂²	C₂×#13133124	1	1	yes	characteristic
374	C₂²	C₂×#13133129	1	2	yes	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	106	16	no	—
380	C₄	#16810377	1	2	yes	—
381	C₄	#16810377	1	2	yes	—
382	C₄	#16810382	1	2	yes	—
383	C₄	#16810384	1	2	yes	—
384	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
385	C₄	C₂×#13133127	1	1	yes	characteristic
386	C₄	C₂×#13133143	1	2	yes	—
387	C₄	C₂×#13133144	1	2	yes	—
388	C₄	C₂×#13133146	1	2	yes	—
389	C₂	—	2	2	no	—
390	C₂	—	2	2	no	—
391	C₂	#20272547	1	4	yes	direct product factor
392	C₂	C₂×#16810377	1	1	yes	characteristic
393	C₂	C₂×#16810382	1	1	yes	characteristic
394	C₂	C₂×#16810384	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	C₂×#1276
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20272547	C₂	direct (N × H)
10. #16810676	C₄	semidirect
12. C₂×C₄⋊#88322	C₄	semidirect
22. C₅₃×#431	C₄	semidirect
24. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
37. C₂×C₄×Dic₅₃	C₈	semidirect
40. C₄⋊#88322	C₂×C₄	semidirect
62. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
118. C₄×Dic₅₃	C₂×C₈	semidirect
198. C₂×C₅₃	#1276	semidirect
212. #431	Dic₅₃	semidirect
215. C₂×C₄⋊C₈	Dic₅₃	semidirect
224. C₅₃	C₂×#1276	semidirect
256. C₄⋊C₈	C₂×Dic₅₃	semidirect
391. C₂	#20272547	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×C₄²×Dic₅₃	2
#20272547	2
C₂×C₅₃×#431	2
C₂×#1276	53

Subgroups — C2×#20272547 #27273527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272547 #27273527