Subgroups — #15689176

Order 3008.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
3	C₄×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×C₄₇	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
8	C₄²×C₄₇	C₄	1	1	yes	characteristic
9	C₄²×C₄₇	C₄	1	1	yes	characteristic
10	C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄₇⋊C₈	C₂²	1	4	yes	—
13	C₂³×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₄₇	—	2	2	no	—
15	C₂×C₄×C₄₇	—	2	2	no	—
16	C₂×C₄×C₄₇	C₈	1	4	yes	complemented
17	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇	—	2	1	no	—
25	C₂²×C₄₇	—	2	1	no	—
26	C₂²×C₄₇	C₂×C₈	1	2	yes	—
27	C₂²×C₄₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₄₇	—	2	1	no	—
29	C₄×C₄₇	—	2	1	no	—
30	C₄×C₄₇	C₂×C₈	1	2	yes	—
31	C₄×C₄₇	C₂×C₈	1	2	yes	—
32	C₄×C₄₇	M₁₆	1	2	yes	—
33	C₄×C₄₇	#41	1	2	yes	—
34	C₂×C₄₇	—	2	2	no	—
35	C₂×C₄₇	#117	1	1	yes	characteristic
36	C₂×C₄₇	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₄₇	C₂×M₁₆	1	1	yes	characteristic
38	#431	—	47	1	no	maximal, Sylow
39	C₄₇	#431	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄²	D₄₇	1	1	yes	characteristic
41	C₄⋊C₈	—	47	2	no	—
42	C₂²⋊C₈	—	47	2	no	—
43	C₄×C₈	—	47	2	no	—
44	C₂²×C₄	Dic₄₇	1	1	yes	characteristic
45	C₂²×C₄	Dic₄₇	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
47	C₂×C₈	—	47	4	no	—
48	C₄²	Dic₄₇	1	1	yes	characteristic
49	C₄²	Dic₄₇	1	1	yes	characteristic
50	C₄²	C₂×D₄₇	1	1	yes	characteristic
51	C₄²	C₂×D₄₇	1	1	yes	characteristic
52	C₂³	C₂×Dic₄₇	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₄₇⋊C₈	1	4	yes	complemented
56	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
57	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
58	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
59	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
62	C₈	—	94	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	C₂×C₄₇⋊C₈	1	2	yes	—
66	C₂²	C₂²×Dic₄₇	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	C₂×C₄₇⋊C₈	1	2	yes	—
70	C₄	C₂×C₄₇⋊C₈	1	2	yes	—
71	C₄	#10625329	1	2	yes	—
72	C₄	#10625350	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#13093983	1	1	yes	characteristic
75	C₂	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
76	C₂	C₂×#10625329	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₄₇
commutator	C₂×C₄₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₄₇
Fitting	C₂×C₄²×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄
prime radical	C₂³×C₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂³
hypercenter	C₂×C₄²	D₄₇
hypocenter	C₄₇	#431
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. C₄×C₄₇⋊C₈	C₂	semidirect
16. C₂×C₄×C₄₇	C₈	semidirect
39. C₄₇	#431	semidirect
55. C₂×C₄	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₄₇⋊C₈)	2
C₂²⋊(C₄₇⋊C₈)	2
C₄×C₄₇⋊C₈	2
C₂×C₄²×C₄₇	2
#431	47