Subgroups — #27293316

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal, complemented
2	#20272655	C₂	1	2	yes	maximal, complemented
3	#20272639	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	—	2	2	no	—
6	C₂²⋊#11719908	—	2	2	no	—
7	C₂²⋊#11719908	C₂²	1	4	yes	—
8	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
11	C₈×D₄×C₅₃	—	2	2	no	—
12	C₈×D₄×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#11719908	—	2	4	no	—
18	C₂×#11719908	—	4	4	no	—
19	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	—	2	1	no	—
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	—	4	2	no	—
25	C₂²×C₈×C₅₃	—	4	2	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×D₄×C₅₃	—	2	2	no	—
33	C₄×D₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₂²⋊C₈×C₅₃	—	2	2	no	—
40	C₂²⋊C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₅₃	—	2	2	no	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	C₁₆	1	4	yes	complemented
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	4	1	no	—
51	C₂²×C₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	—	4	1	no	—
53	C₂²×C₄×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	1	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₂²⋊C₄×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	2	1	no	—
79	#11719908	—	4	4	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	4	1	no	—
83	C₂³×C₅₃	—	4	1	no	—
84	C₂³×C₅₃	—	4	1	no	—
85	C₂³×C₅₃	—	4	1	no	—
86	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
87	C₂³×C₅₃	C₂×C₁₆	1	2	yes	—
88	C₂³×C₅₃	C₂×C₁₆	1	2	yes	complemented
89	C₂³×C₅₃	M₃₂	1	2	yes	—
90	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	D₄×C₅₃	—	2	8	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	complemented
109	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	—	4	1	no	—
114	C₈×C₅₃	—	4	1	no	—
115	C₈×C₅₃	#99	1	2	yes	—
116	#11465	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	4	2	no	—
123	C₂²×C₅₃	—	4	2	no	—
124	C₂²×C₅₃	—	4	2	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	C₂²⋊C₁₆	1	4	yes	—
129	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	4	1	no	—
136	C₄×C₅₃	—	4	1	no	—
137	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
138	C₄×C₅₃	#413	1	1	yes	characteristic
139	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
140	C₂×C₂²⋊C₁₆	—	53	2	no	—
141	#1322	—	53	2	no	—
142	#1314	—	53	2	no	—
143	C₂×C₅₃	—	2	1	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	4	no	—
146	C₂×C₅₃	—	4	2	no	—
147	C₂×C₅₃	#1459	1	1	yes	characteristic
148	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
149	C₂×C₅₃	#2115	1	1	yes	characteristic
150	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
151	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
152	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
153	C₂²×C₁₆	—	106	2	no	—
154	C₈×D₄	—	2	2	no	—
155	C₈×D₄	—	2	2	no	—
156	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
159	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
160	C₂²⋊C₁₆	—	53	4	no	—
161	C₂²⋊C₁₆	—	106	2	no	—
162	C₅₃	#11465	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₄	#88322	1	1	yes	characteristic
164	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	4	2	no	—
169	C₂²×C₈	—	4	2	no	—
170	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
174	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
175	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	#88322	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	#88322	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
182	C₂×C₁₆	—	106	4	no	—
183	C₂×C₁₆	—	212	4	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₂²⋊C₈	—	2	2	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₂⁴	C₂×#88322	1	1	yes	characteristic
190	C₂⁴	C₂×#88322	1	1	yes	characteristic
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	#11719908	1	4	yes	complemented
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
201	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
202	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
203	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
204	C₂²×C₄	#11719917	1	2	yes	—
205	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₁₆	—	212	4	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	—	4	1	no	—
232	C₂³	C₂²⋊#88322	1	2	yes	—
233	C₂³	C₂×#11719908	1	2	yes	—
234	C₂³	C₂×#11719908	1	2	yes	complemented
235	C₂³	#13133157	1	2	yes	—
236	C₂³	C₂²×#88322	1	1	yes	characteristic
237	D₄	—	2	8	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₂²⋊#88322	1	2	yes	—
254	C₂×C₄	C₂×#11719908	1	2	yes	complemented
255	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	#13133132	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	C₂²⋊#11719908	1	4	yes	—
274	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
275	C₂²	C₂²×#11719908	1	1	yes	characteristic
276	C₂²	C₂×#13133157	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₂×#13133132	1	1	yes	characteristic
283	C₄	#16810607	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	4	no	—
287	C₂	—	4	2	no	—
288	C₂	#20273033	1	1	yes	characteristic
289	C₂	C₂×C₂²⋊#11719908	1	1	yes	characteristic
290	C₂	#20276336	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#11465
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊#11719908	C₂	semidirect
2. #20272655	C₂	semidirect
3. #20272639	C₂	semidirect
47. C₂×D₄×C₅₃	C₁₆	semidirect
88. C₂³×C₅₃	C₂×C₁₆	semidirect
108. C₂×C₄×C₅₃	C₂×C₁₆	semidirect
162. C₅₃	#11465	semidirect
193. C₂×D₄	#11719908	semidirect
234. C₂³	C₂×#11719908	semidirect
254. C₂×C₄	C₂×#11719908	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11719908	2
#20272655	2
#20272639	2
C₂×C₈×D₄×C₅₃	2
#11465	53