Subgroups — #27536611

Order 13568.

2046 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811000	C₂	1	12	yes	maximal, complemented
2	#20283110	C₂	1	2	yes	maximal, complemented
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄₂₄	C₂²	1	16	yes	—
5	C₂×#13133214	—	2	8	no	—
6	C₂×#13133214	C₂²	1	12	yes	—
7	#16811000	—	2	48	no	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×D₂₁₂	—	2	24	no	—
11	C₂²×D₂₁₂	C₂³	1	8	yes	—
12	C₂×D₄₂₄	—	2	96	no	—
13	#13133214	—	2	48	no	—
14	#13133214	—	4	32	no	—
15	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
16	C₂×C₄²×C₅₃	D₄	1	4	yes	—
17	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
18	C₄×C₈×C₅₃	D₄	1	24	yes	complemented
19	C₂³×D₅₃	—	2	8	no	—
20	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
21	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
22	C₂×D₂₁₂	—	2	48	no	—
23	C₂×D₂₁₂	—	4	48	no	—
24	C₂×D₂₁₂	—	4	96	no	—
25	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	—
26	C₄²×C₅₃	D₈	1	16	yes	complemented
27	C₄²×C₅₃	C₂×D₄	1	12	yes	—
28	D₄₂₄	—	4	64	no	—
29	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
30	C₂²×D₅₃	—	4	24	no	—
31	C₂²×D₅₃	—	4	32	no	—
32	C₂×C₄×C₅₃	#127	1	12	yes	—
33	C₂×C₄×C₅₃	C₂×D₈	1	24	yes	—
34	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
35	D₂₁₂	—	4	32	no	—
36	D₂₁₂	—	8	96	no	—
37	C₈×C₅₃	#127	1	16	yes	complemented
38	#33301	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	C₂×#127	1	3	yes	—
40	C₂²×C₅₃	C₂²×D₈	1	4	yes	—
41	C₂×D₅₃	—	8	8	no	—
42	C₂×D₅₃	—	8	48	no	—
43	C₄×C₅₃	#493	1	24	yes	complemented
44	C₄×C₅₃	C₂×#127	1	4	yes	—
45	C₂×#493	—	53	12	no	—
46	#2867	—	53	2	no	—
47	C₄²×C₈	D₅₃	1	1	yes	characteristic, complemented
48	C₂×C₅₃	#2867	1	1	yes	characteristic
49	C₂×C₅₃	C₂×#493	1	6	yes	—
50	D₅₃	—	16	8	no	—
51	C₂²×D₈	—	53	16	no	—
52	C₂×#127	—	53	12	no	—
53	C₂×#127	—	106	8	no	—
54	#493	—	106	48	no	—
55	C₂×C₄×C₈	C₂×D₅₃	1	6	yes	—
56	C₄³	C₂×D₅₃	1	1	yes	characteristic
57	C₅₃	#33301	1	1	yes	characteristic, complemented, Sylow
58	C₂²×D₄	—	53	8	no	—
59	C₂²×D₄	—	106	24	no	—
60	C₂×D₈	—	106	96	no	—
61	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
62	#127	—	106	48	no	—
63	#127	—	212	32	no	—
64	C₂×C₄²	D₂₁₂	1	4	yes	—
65	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
66	C₄×C₈	D₂₁₂	1	24	yes	complemented
67	C₂⁴	—	106	8	no	—
68	C₂×D₄	—	106	48	no	—
69	C₂×D₄	—	212	48	no	—
70	C₂×D₄	—	212	96	no	—
71	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
72	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
73	D₈	—	212	64	no	—
74	C₂×C₈	C₂×D₂₁₂	1	24	yes	—
75	C₄²	D₄₂₄	1	16	yes	complemented
76	C₄²	C₂×D₂₁₂	1	12	yes	—
77	C₂³	—	212	32	no	—
78	C₂³	—	212	24	no	—
79	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
80	D₄	—	424	96	no	—
81	D₄	—	212	32	no	—
82	C₂×C₄	#13133214	1	12	yes	—
83	C₂×C₄	C₂×D₄₂₄	1	24	yes	—
84	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
85	C₈	#13133214	1	16	yes	complemented
86	C₂²	—	424	48	no	—
87	C₂²	—	424	8	no	—
88	C₂²	C₂×#13133214	1	3	yes	—
89	C₂²	C₂²×D₄₂₄	1	4	yes	—
90	C₄	#16811000	1	24	yes	complemented
91	C₄	C₂×#13133214	1	4	yes	—
92	C₂	—	848	8	no	—
93	C₂	#20283110	1	1	yes	characteristic
94	C₂	C₂×#16811000	1	6	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#33301
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16811000	C₂	semidirect
2. #20283110	C₂	semidirect
3. C₄²×C₈×C₅₃	C₂	semidirect
18. C₄×C₈×C₅₃	D₄	semidirect
26. C₄²×C₅₃	D₈	semidirect
37. C₈×C₅₃	#127	semidirect
43. C₄×C₅₃	#493	semidirect
47. C₄²×C₈	D₅₃	semidirect
57. C₅₃	#33301	semidirect
66. C₄×C₈	D₂₁₂	semidirect
75. C₄²	D₄₂₄	semidirect
85. C₈	#13133214	semidirect
90. C₄	#16811000	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811000	2
#20283110	2
C₄²×C₈×C₅₃	2
#33301	53