Subgroups — #27284104

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272556	C₂	1	4	yes	maximal
4	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×C₄×#88322	—	2	2	no	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810378	—	2	2	no	—
9	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#389	C₄	1	2	yes	—
11	C₅₃×#387	—	2	2	no	—
12	C₅₃×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄×#88322	—	2	8	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	Q₈	1	4	yes	—
22	#125×C₅₃	—	2	2	no	—
23	#125×C₅₃	C₈	1	4	yes	complemented
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	—	2	2	no	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#95×C₅₃	—	2	2	no	—
35	#95×C₅₃	—	2	2	no	—
36	#95×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	—	2	8	no	—
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	4	no	—
52	C₄²×C₅₃	—	4	4	no	—
53	C₄²×C₅₃	C₂×C₈	1	2	yes	—
54	C₄²×C₅₃	M₁₆	1	2	yes	—
55	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
56	C₄²×C₅₃	#41	1	2	yes	—
57	C₂×#88322	—	2	8	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	—	4	4	no	—
61	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₅₃	#104	1	4	yes	—
75	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#131	1	2	yes	—
82	C₂×C₄×C₅₃	#135	1	4	yes	—
83	#88322	—	2	8	no	—
84	#88322	—	8	2	no	—
85	#9871	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	#391	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
88	C₂²×C₅₃	#408	1	1	yes	characteristic
89	C₂²×C₅₃	#416	1	2	yes	—
90	C₂²×C₅₃	C₂×#104	1	2	yes	—
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	—	4	4	no	—
95	C₄×C₅₃	—	8	2	no	—
96	C₄×C₅₃	C₈×Q₈	1	2	yes	—
97	C₄×C₅₃	#446	1	2	yes	—
98	C₄×C₅₃	#488	1	2	yes	—
99	C₄×C₅₃	#499	1	2	yes	—
100	C₄×#125	D₅₃	1	1	yes	characteristic
101	#1831	—	53	1	no	—
102	C₄²×C₈	—	53	1	no	—
103	#1279	—	53	4	no	—
104	C₂×C₅₃	#1476	1	2	yes	—
105	C₂×C₅₃	#1494	1	2	yes	—
106	C₂×C₅₃	#1984	1	1	yes	characteristic
107	C₂×C₅₃	#1991	1	1	yes	characteristic
108	C₂×C₅₃	#1994	1	1	yes	characteristic
109	C₂×#125	Dic₅₃	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	53	2	no	—
111	C₂×C₄×C₈	—	53	2	no	—
112	C₂×C₄×C₈	—	106	2	no	—
113	#389	Dic₅₃	1	2	yes	—
114	#387	—	2	2	no	—
115	#382	Dic₅₃	1	1	yes	characteristic
116	C₄×C₄⋊C₄	—	2	2	no	—
117	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
118	C₄³	C₂×D₅₃	1	1	yes	characteristic
119	#336	—	106	2	no	—
120	C₅₃	#9871	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	106	2	no	—
122	C₂²×C₈	—	106	2	no	—
123	#125	—	2	2	no	—
124	#125	#88322	1	4	yes	complemented
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
128	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
134	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	C₄⋊C₈	—	106	4	no	—
136	C₄×C₈	—	53	4	no	—
137	C₄×C₈	—	106	8	no	—
138	C₄×C₈	—	106	2	no	—
139	#95	—	2	2	no	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
145	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂×C₈	—	106	8	no	—
150	C₂×C₈	—	106	4	no	—
151	C₂×C₈	—	212	2	no	—
152	C₂×C₈	—	212	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	C₂×#88322	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	4	4	no	—
162	C₄²	C₂×#88322	1	2	yes	—
163	C₄²	#11719917	1	2	yes	—
164	C₄²	Q₈×D₅₃	1	2	yes	—
165	C₄²	#11719949	1	2	yes	—
166	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂²⋊#88322	1	4	yes	—
179	C₂×C₄	#13133144	1	4	yes	—
180	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
181	C₂×C₄	#13133193	1	2	yes	—
182	C₂×C₄	#13133206	1	1	yes	characteristic
183	C₂×C₄	#13133215	1	1	yes	characteristic
184	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
186	C₂×C₄	#13133224	1	2	yes	—
187	C₂×C₄	#13133237	1	4	yes	—
188	C₈	—	106	8	no	—
189	C₈	—	424	2	no	—
190	C₂²	#16810530	1	1	yes	characteristic
191	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
192	C₂²	#16810590	1	1	yes	characteristic
193	C₂²	#16810619	1	2	yes	—
194	C₂²	C₂×#13133144	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	4	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	4	no	—
199	C₄	—	8	2	no	—
200	C₄	Q₈×#88322	1	2	yes	—
201	C₄	#16810754	1	2	yes	—
202	C₄	#16810980	1	2	yes	—
203	C₄	#16811022	1	2	yes	—
204	C₂	#20273097	1	2	yes	—
205	C₂	#20273175	1	2	yes	—
206	C₂	#20275652	1	1	yes	characteristic
207	C₂	#20275681	1	1	yes	characteristic
208	C₂	#20275693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#9871
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
23. #125×C₅₃	C₈	semidirect
120. C₅₃	#9871	semidirect
124. #125	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
C₄²×#88322	2
#20272556	2
C₄×#125×C₅₃	2
#9871	53