Subgroups — #27305087

Order 13568.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆₉₆	C₂	1	4	yes	maximal, complemented
2	#20276965	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₄₈	—	2	2	no	—
5	C₂×D₈₄₈	C₂²	1	4	yes	—
6	#16811000	—	2	2	no	—
7	D₁₆₉₆	—	2	8	no	—
8	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
9	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₄₂₄	—	2	4	no	—
11	C₂×D₄₂₄	—	4	2	no	—
12	#13133214	—	4	2	no	—
13	D₈₄₈	—	2	8	no	—
14	D₈₄₈	—	4	4	no	—
15	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	4	yes	complemented
19	C₂×D₂₁₂	—	4	4	no	—
20	C₂×D₂₁₂	—	8	4	no	—
21	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
23	C₄²×C₅₃	D₈	1	1	yes	characteristic
24	D₄₂₄	—	4	4	no	—
25	D₄₂₄	—	4	4	no	—
26	D₄₂₄	—	8	4	no	—
27	C₁₆×C₅₃	D₈	1	2	yes	—
28	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
29	C₂²×D₅₃	—	8	4	no	—
30	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
31	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
32	D₂₁₂	—	8	4	no	—
33	D₂₁₂	—	8	4	no	—
34	D₂₁₂	—	16	8	no	—
35	C₈×C₅₃	D₁₆	1	2	yes	—
36	C₈×C₅₃	#127	1	1	yes	characteristic
37	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
38	#13725	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
40	C₂×D₅₃	—	16	4	no	—
41	C₂×D₅₃	—	16	8	no	—
42	C₄×C₅₃	D₃₂	1	4	yes	complemented
43	C₄×C₅₃	#493	1	1	yes	characteristic
44	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
45	C₂×D₃₂	—	53	4	no	—
46	#2246	—	53	2	no	—
47	C₄×C₃₂	D₅₃	1	1	yes	characteristic, complemented
48	C₂×C₅₃	#2246	1	1	yes	characteristic
49	C₂×C₅₃	C₂×D₃₂	1	2	yes	—
50	D₅₃	—	32	4	no	—
51	C₂×D₁₆	—	53	4	no	—
52	C₂×D₁₆	—	106	2	no	—
53	#493	—	106	2	no	—
54	D₃₂	—	106	8	no	—
55	C₂×C₃₂	C₂×D₅₃	1	2	yes	—
56	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
57	C₅₃	#13725	1	1	yes	characteristic, complemented, Sylow
58	C₂×D₈	—	106	4	no	—
59	C₂×D₈	—	212	2	no	—
60	#127	—	212	2	no	—
61	D₁₆	—	106	8	no	—
62	D₁₆	—	212	4	no	—
63	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
65	C₄×C₈	D₂₁₂	1	1	yes	characteristic
66	C₃₂	D₂₁₂	1	4	yes	complemented
67	C₂×D₄	—	424	4	no	—
68	C₂×D₄	—	212	4	no	—
69	D₈	—	424	4	no	—
70	D₈	—	212	4	no	—
71	D₈	—	212	4	no	—
72	C₂×C₈	D₄₂₄	1	1	yes	characteristic
73	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
74	C₄²	D₄₂₄	1	1	yes	characteristic
75	C₁₆	D₄₂₄	1	2	yes	—
76	C₁₆	C₂×D₂₁₂	1	2	yes	—
77	C₂³	—	424	4	no	—
78	D₄	—	848	8	no	—
79	D₄	—	424	4	no	—
80	D₄	—	424	4	no	—
81	C₂×C₄	D₈₄₈	1	2	yes	—
82	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
83	C₈	D₈₄₈	1	2	yes	—
84	C₈	#13133214	1	1	yes	characteristic
85	C₈	C₂×D₄₂₄	1	1	yes	characteristic
86	C₂²	—	848	8	no	—
87	C₂²	—	848	4	no	—
88	C₂²	C₂×D₈₄₈	1	1	yes	characteristic
89	C₄	D₁₆₉₆	1	4	yes	complemented
90	C₄	#16811000	1	1	yes	characteristic
91	C₄	C₂×D₈₄₈	1	1	yes	characteristic
92	C₂	—	1696	4	no	—
93	C₂	#20276965	1	1	yes	characteristic
94	C₂	C₂×D₁₆₉₆	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₄₈
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₄×C₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#13725
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₆₉₆	C₂	semidirect
2. #20276965	C₂	semidirect
3. C₄×C₃₂×C₅₃	C₂	semidirect
18. C₃₂×C₅₃	D₄	semidirect
42. C₄×C₅₃	D₃₂	semidirect
47. C₄×C₃₂	D₅₃	semidirect
57. C₅₃	#13725	semidirect
66. C₃₂	D₂₁₂	semidirect
89. C₄	D₁₆₉₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆₉₆	2
#20276965	2
C₄×C₃₂×C₅₃	2
#13725	53