Subgroups — #27277033

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274711	C₂	1	2	yes	maximal
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20273471	C₂	1	1	yes	maximal, characteristic
4	#20272590	C₂	1	2	yes	maximal
5	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
6	#16810647	C₂²	1	2	yes	—
7	#16810647	C₂²	1	2	yes	—
8	C₂×#13133136	—	2	2	no	—
9	#16810392	—	2	2	no	—
10	#16810376	—	2	1	no	—
11	#16810376	—	2	1	no	—
12	#16810374	—	2	1	no	—
13	#16810374	—	2	1	no	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#446	—	2	1	no	—
16	C₅₃×#446	—	2	1	no	—
17	C₅₃×#446	—	2	1	no	—
18	C₅₃×#446	—	2	1	no	—
19	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₃	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	#13133136	—	4	4	no	—
30	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	—	2	1	no	—
35	C₄×Q₈×C₅₃	—	2	1	no	—
36	C₄×Q₈×C₅₃	—	2	1	no	—
37	C₄×Q₈×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	—	2	1	no	—
45	C₄⋊C₈×C₅₃	—	2	1	no	—
46	#97×C₅₃	—	2	2	no	—
47	#97×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	—	2	1	no	—
53	C₂×Q₈×C₅₃	—	2	1	no	—
54	C₂×Q₈×C₅₃	—	2	2	no	—
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	4	1	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₄⋊C₄×C₅₃	—	2	1	no	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	#41	1	1	yes	characteristic
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	#11719917	—	4	2	no	—
80	#11719917	—	4	2	no	—
81	#11719917	—	4	2	no	—
82	#11719917	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	Q₈×C₅₃	—	4	2	no	—
87	Q₈×C₅₃	—	4	1	no	—
88	Q₈×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₅₃	#137	1	2	yes	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	4	2	no	—
108	C₈×C₅₃	#101	1	2	yes	complemented
109	#88322	—	8	2	no	—
110	#88322	—	8	2	no	—
111	#8619	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	#380	1	1	yes	characteristic
115	C₂²×C₅₃	#428	1	2	yes	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	4	2	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
122	C₄×C₅₃	#413	1	1	yes	characteristic
123	C₄×C₅₃	#468	1	2	yes	—
124	C₄×C₅₃	#469	1	1	yes	characteristic
125	C₄×C₅₃	#470	1	1	yes	characteristic
126	C₂×#446	D₅₃	1	1	yes	characteristic
127	#1801	—	53	2	no	—
128	#1567	—	53	1	no	—
129	#1566	—	53	1	no	—
130	#1292	—	53	2	no	—
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	#1801	1	1	yes	characteristic
133	C₂×C₅₃	#1809	1	1	yes	characteristic
134	C₂×C₅₃	#1813	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
136	#446	—	2	1	no	—
137	#446	—	2	1	no	—
138	#446	—	2	1	no	—
139	#446	—	2	1	no	—
140	#423	—	53	2	no	—
141	#423	—	53	2	no	—
142	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
143	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
144	C₂×#101	—	106	2	no	—
145	C₂×#97	Dic₅₃	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
147	#343	—	106	2	no	—
148	#335	—	106	1	no	—
149	#335	—	106	1	no	—
150	#334	—	106	1	no	—
151	#334	—	106	1	no	—
152	C₅₃	#8619	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
154	C₂×M₁₆	—	106	2	no	—
155	C₂×M₁₆	—	106	2	no	—
156	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
158	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	—	2	1	no	—
163	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
164	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
166	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	106	2	no	—
172	C₄⋊C₈	—	106	2	no	—
173	C₄⋊C₈	—	106	2	no	—
174	C₄⋊C₈	—	106	2	no	—
175	#101	—	212	4	no	—
176	#97	—	2	2	no	—
177	#97	—	2	2	no	—
178	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
188	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
191	M₁₆	—	212	2	no	—
192	M₁₆	—	212	2	no	—
193	M₁₆	—	212	2	no	—
194	M₁₆	—	212	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	#11719945	1	1	yes	characteristic
212	C₄²	#11719945	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
216	Q₈	—	4	2	no	—
217	Q₈	—	4	1	no	—
218	Q₈	—	4	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	#13133176	1	1	yes	characteristic
228	C₂×C₄	#13133191	1	1	yes	characteristic
229	C₂×C₄	#13133191	1	1	yes	characteristic
230	C₂×C₄	#13133196	1	1	yes	characteristic
231	C₂×C₄	#13133196	1	1	yes	characteristic
232	C₂×C₄	#13133245	1	2	yes	—
233	C₂×C₄	#13133250	1	2	yes	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	—	424	2	no	—
239	C₈	—	424	2	no	—
240	C₈	#13133136	1	2	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#16810483	1	1	yes	characteristic
244	C₂²	#16810665	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂×#13133136	1	1	yes	characteristic
251	C₄	#16810607	1	1	yes	characteristic
252	C₄	#16810869	1	2	yes	—
253	C₄	#16810876	1	1	yes	characteristic
254	C₄	#16810881	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#20274711	1	1	yes	characteristic
257	C₂	#20274748	1	1	yes	characteristic
258	C₂	#20274765	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810483
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#8619
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
108. C₈×C₅₃	#101	semidirect
152. C₅₃	#8619	semidirect
240. C₈	#13133136	semidirect

Maximal subgroups

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

subgroup	index
#20274711	2
#20273474	2
#20273471	2
#20272590	2
C₂×C₅₃×#446	2
#8619	53