Subgroups — #27299221

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287817	C₂	1	1	yes	maximal, characteristic
2	#20276344	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810400	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274561	C₂	1	1	yes	maximal, characteristic
5	#20272801	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272798	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	#16811041	—	2	2	no	—
10	#16811039	C₄	1	4	yes	complemented
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	M₃₂×D₅₃	—	2	2	no	—
14	C₂×#13133143	C₄	1	1	yes	characteristic
15	#16810617	C₄	1	1	yes	characteristic
16	C₂×#13133127	C₂²	1	1	yes	characteristic
17	#16810408	C₄	1	4	yes	complemented
18	#16810400	—	2	2	no	—
19	#16810387	C₄	1	1	yes	characteristic
20	#16810382	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#405	C₂²	1	1	yes	characteristic
24	C₅₃×#347	—	2	2	no	—
25	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
26	#13133223	—	2	1	no	—
27	#13133223	—	2	1	no	—
28	#13133223	C₂×C₄	1	2	yes	—
29	#13133223	C₂×C₄	1	2	yes	—
30	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719912	—	2	1	no	—
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅₃	—	2	1	no	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	#13133157	—	2	1	no	—
36	#13133157	—	2	1	no	—
37	#13133157	—	2	1	no	—
38	#13133157	C₂×C₄	1	2	yes	—
39	#13133156	—	2	4	no	—
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	C₂×C₄	1	2	yes	—
42	C₁₆×D₅₃	—	2	4	no	—
43	#13133143	—	2	2	no	—
44	#13133140	—	2	1	no	—
45	#13133137	—	2	1	no	—
46	#13133127	—	2	1	no	—
47	#13133127	—	2	1	no	—
48	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₅₃	—	2	1	no	—
52	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
53	M₃₂×C₅₃	D₄	1	2	yes	—
54	M₃₂×C₅₃	D₄	1	2	yes	—
55	C₂×C₁₆×C₅₃	—	2	1	no	—
56	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
57	#97×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	—	2	1	no	—
61	C₂²×Dic₅₃	—	2	1	no	—
62	#11719945	—	2	1	no	—
63	#11719945	—	2	1	no	—
64	#11719945	C₄²	1	2	yes	—
65	#11719945	M₁₆	1	2	yes	—
66	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₅₃	—	2	1	no	—
68	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
69	M₁₆×C₅₃	—	2	1	no	—
70	M₁₆×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₂²⋊C₄×C₅₃	—	2	1	no	—
80	C₄²×C₅₃	—	2	1	no	—
81	C₄×Dic₅₃	—	2	1	no	—
82	C₄×Dic₅₃	—	2	1	no	—
83	#11719917	—	2	1	no	—
84	#11719917	C₄²	1	2	yes	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	—	2	1	no	—
87	C₂×#88322	—	2	1	no	—
88	C₂×#88322	—	2	1	no	—
89	#11719912	—	2	2	no	—
90	#11719912	—	2	2	no	—
91	C₈×D₅₃	—	2	2	no	—
92	C₈×D₅₃	—	2	2	no	—
93	C₁₆×C₅₃	—	2	2	no	—
94	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
95	#11719908	—	2	2	no	—
96	#11719908	—	2	2	no	—
97	C₂³×C₅₃	#131	1	1	yes	characteristic
98	C₂²×D₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
106	C₂²⋊D₅₃	—	2	2	no	—
107	C₂²⋊D₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	2	1	no	—
109	C₂×Dic₅₃	—	4	1	no	—
110	C₂×Dic₅₃	—	8	1	no	—
111	D₂₁₂	—	2	1	no	—
112	D₂₁₂	#97	1	2	yes	—
113	C₄×D₅₃	—	2	2	no	—
114	C₄×D₅₃	—	2	2	no	—
115	Dic₁₀₆	—	2	1	no	—
116	Dic₁₀₆	#97	1	2	yes	—
117	C₈×C₅₃	—	4	1	no	—
118	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₈×C₅₃	#117	1	1	yes	characteristic
120	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
122	#88322	—	2	2	no	—
123	#88322	—	4	1	no	—
124	#12618	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	#405	1	1	yes	characteristic
127	C₂²×C₅₃	#435	1	1	yes	characteristic
128	C₂²×C₅₃	#435	1	1	yes	characteristic
129	C₂×D₅₃	—	2	2	no	—
130	C₂×D₅₃	—	8	1	no	—
131	C₄×C₅₃	—	8	1	no	—
132	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
134	C₄×C₅₃	#435	1	1	yes	characteristic
135	C₄×C₅₃	#435	1	1	yes	characteristic
136	Dic₅₃	—	2	2	no	—
137	Dic₅₃	—	8	1	no	—
138	#3407	—	53	1	no	—
139	#2117	—	53	1	no	—
140	#2110	D₅₃	1	1	yes	characteristic, complemented
141	C₂×#347	—	53	1	no	—
142	#1765	—	53	1	no	—
143	#1378	—	53	1	no	—
144	#1377	—	53	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	#1752	1	1	yes	characteristic
149	D₅₃	—	4	2	no	—
150	C₂×#131	—	53	1	no	—
151	#504	—	53	4	no	—
152	#504	—	106	2	no	—
153	C₂×M₃₂	—	53	1	no	—
154	C₂×M₃₂	—	53	1	no	—
155	C₂×M₃₂	—	106	2	no	—
156	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
157	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
158	C₂×#104	—	53	1	no	—
159	#416	—	53	1	no	—
160	#405	C₂×D₅₃	1	1	yes	characteristic
161	C₂×#97	—	53	1	no	—
162	#350	—	53	4	no	—
163	#347	—	2	2	no	—
164	#347	—	106	2	no	—
165	#341	—	53	1	no	—
166	#339	—	53	1	no	—
167	C₅₃	#12618	1	1	yes	characteristic, complemented, Sylow
168	C₂×#41	—	53	1	no	—
169	#131	—	53	2	no	—
170	#131	—	53	2	no	—
171	#131	—	106	1	no	—
172	#131	—	106	1	no	—
173	C₂×M₁₆	—	53	1	no	—
174	C₂×M₁₆	—	106	1	no	—
175	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
176	C₂²×C₈	—	53	1	no	—
177	C₂²×C₈	—	106	1	no	—
178	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
179	#117	C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄²	—	53	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	53	2	no	—
183	M₃₂	—	106	1	no	—
184	M₃₂	—	106	1	no	—
185	M₃₂	—	106	1	no	—
186	M₃₂	—	106	4	no	—
187	M₃₂	C₄×D₅₃	1	2	yes	—
188	M₃₂	D₂₁₂	1	2	yes	—
189	M₃₂	C₂²⋊D₅₃	1	2	yes	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	53	2	no	—
192	C₂×C₁₆	—	106	1	no	—
193	C₂×C₁₆	—	106	4	no	—
194	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
195	#104	—	106	2	no	—
196	#103	—	106	1	no	—
197	#102	—	106	1	no	—
198	#97	—	2	1	no	—
199	#97	—	106	1	no	—
200	#97	—	106	1	no	—
201	C₄×C₈	—	2	1	no	—
202	#41	—	53	2	no	—
203	#41	—	53	2	no	—
204	#41	—	106	1	no	—
205	#41	—	106	1	no	—
206	C₂×Q₈	—	53	1	no	—
207	C₂×D₄	—	53	1	no	—
208	C₂×D₄	—	106	1	no	—
209	C₂²×C₄	—	106	1	no	—
210	C₂²×C₄	—	106	1	no	—
211	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	53	2	no	—
215	M₁₆	—	106	1	no	—
216	M₁₆	—	106	2	no	—
217	M₁₆	—	106	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	106	1	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	106	1	no	—
223	C₂×C₈	—	106	1	no	—
224	C₂×C₈	—	106	1	no	—
225	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
228	C₂×C₈	#11719945	1	1	yes	characteristic
229	C₂×C₈	#11719945	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	106	1	no	—
235	C₄²	—	106	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	106	2	no	—
238	C₁₆	—	106	2	no	—
239	C₁₆	#11719921	1	4	yes	—
240	C₂³	—	106	1	no	—
241	C₂³	#13133225	1	1	yes	characteristic
242	Q₈	—	53	2	no	—
243	Q₈	—	106	1	no	—
244	D₄	—	53	2	no	—
245	D₄	—	106	2	no	—
246	D₄	—	106	2	no	—
247	D₄	—	106	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	106	1	no	—
250	C₂×C₄	—	106	2	no	—
251	C₂×C₄	—	106	2	no	—
252	C₂×C₄	—	424	1	no	—
253	C₂×C₄	—	212	1	no	—
254	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
255	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
257	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
258	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
259	C₂×C₄	#13133225	1	1	yes	characteristic
260	C₈	—	4	1	no	—
261	C₈	—	106	2	no	—
262	C₈	—	212	1	no	—
263	C₈	C₂×#11719921	1	1	yes	characteristic
264	C₈	#13133174	1	1	yes	characteristic
265	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
266	C₈	C₄×D₂₁₂	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	106	2	no	—
269	C₂²	—	424	1	no	—
270	C₂²	#16810580	1	1	yes	characteristic
271	C₂²	#16810692	1	1	yes	characteristic
272	C₂²	#16810696	1	1	yes	characteristic
273	C₄	—	8	1	no	—
274	C₄	—	106	2	no	—
275	C₄	—	424	1	no	—
276	C₄	C₄×#11719921	1	1	yes	characteristic
277	C₄	#97×D₅₃	1	1	yes	characteristic
278	C₄	#16810692	1	1	yes	characteristic
279	C₄	#16810696	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	212	2	no	—
284	C₂	#20274492	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2110	C₂
socle	C₂×C₅₃	#1752
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2110	D₅₃
hypocenter	C₅₃	#12618
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₃₂ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. C₂×#16810400	C₂	semidirect
5. #20272801	C₂	semidirect
6. #20272798	C₂	semidirect
7. C₅₃×#2110	C₂	semidirect
10. #16811039	C₄	semidirect
17. #16810408	C₄	semidirect
140. #2110	D₅₃	semidirect
167. C₅₃	#12618	semidirect

Maximal subgroups

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

subgroup	index
#20287817	2
#20276344	2
C₂×#16810400	2
#20274561	2
#20272801	2
#20272798	2
C₅₃×#2110	2
#12618	53