Subgroups — #26104089

Order 12032.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241599	C₂	1	2	yes	maximal, complemented
2	C₂×C₄₇×M₆₄	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊C₆₄	—	2	2	no	—
4	C₂²×C₁₆×C₄₇	C₄	1	1	yes	characteristic
5	C₄₇×M₆₄	—	2	2	no	—
6	C₂×C₃₂×C₄₇	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₄₇	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₄₇	—	2	1	no	—
10	C₂×C₁₆×C₄₇	—	2	1	no	—
11	C₂×C₁₆×C₄₇	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₄₇	—	2	1	no	—
14	C₃₂×C₄₇	D₄	1	2	yes	—
15	C₂²×C₄×C₄₇	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₄₇	—	2	1	no	—
17	C₂×C₈×C₄₇	—	2	1	no	—
18	C₂×C₈×C₄₇	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₄₇	—	2	1	no	—
21	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
23	C₂³×C₄₇	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₄₇	—	2	1	no	—
25	C₂×C₄×C₄₇	—	2	1	no	—
26	C₂×C₄×C₄₇	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₄₇	—	2	1	no	—
29	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₄₇	M₃₂	1	1	yes	characteristic
31	#7513	—	47	1	no	maximal, Sylow
32	C₂²×C₄₇	—	2	1	no	—
33	C₂²×C₄₇	—	4	1	no	—
34	C₂²×C₄₇	C₂×C₃₂	1	1	yes	characteristic
35	C₄×C₄₇	—	2	1	no	—
36	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
37	C₄×C₄₇	M₆₄	1	1	yes	characteristic
38	C₂×M₆₄	D₄₇	1	1	yes	characteristic
39	M₁₂₈	—	47	2	no	—
40	C₂×C₄₇	—	2	1	no	—
41	C₂×C₄₇	—	4	1	no	—
42	C₂×C₄₇	C₂²⋊C₃₂	1	1	yes	characteristic
43	C₂²×C₁₆	Dic₄₇	1	1	yes	characteristic
44	M₆₄	—	2	2	no	—
45	C₂×C₃₂	Dic₄₇	1	1	yes	characteristic
46	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
47	C₆₄	—	94	2	no	—
48	C₄₇	#7513	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	C₄₇⋊C₈	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	C₄₇⋊C₈	1	1	yes	characteristic
53	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
54	C₃₂	—	2	1	no	—
55	C₃₂	C₂²⋊D₄₇	1	2	yes	—
56	C₂²×C₄	C₄₇⋊C₁₆	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	C₄₇⋊C₁₆	1	1	yes	characteristic
60	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	#10625329	1	1	yes	characteristic
63	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
64	C₂³	C₄₇⋊C₃₂	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	C₄₇⋊C₃₂	1	1	yes	characteristic
68	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
71	C₈	#13093964	1	1	yes	characteristic
72	C₂²	—	2	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₄₇⋊C₃₂	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
77	C₄	#15688957	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4	1	no	—
80	C₂	C₂²⋊(C₄₇⋊C₃₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₂×C₄₇×M₆₄	C₂
socle	C₂×C₄₇	C₂²⋊C₃₂
prime radical	C₂³×C₄₇	C₃₂
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	C₂×M₆₄	D₄₇
hypocenter	C₄₇	#7513
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
1. #20241599	C₂	semidirect
48. C₄₇	#7513	semidirect

Maximal subgroups

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

subgroup	index
#20241599	2
C₂×C₄₇×M₆₄	2
#7513	47