Subgroups — #15688869

Order 3008.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₄₇	—	2	1	no	—
5	C₄⋊C₄×C₄₇	—	2	1	no	—
6	C₄⋊C₄×C₄₇	C₄	1	2	yes	—
7	C₄²×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄₇⋊C₈	—	2	1	no	—
9	C₂×C₄₇⋊C₈	—	2	1	no	—
10	Q₈×C₄₇	—	4	1	no	—
11	Q₈×C₄₇	—	4	1	no	—
12	C₂×C₄×C₄₇	—	2	2	no	—
13	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
16	C₄₇⋊C₈	—	4	1	no	—
17	C₄₇⋊C₈	—	4	1	no	—
18	C₂²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₄₇	—	2	1	no	—
20	C₄×C₄₇	—	4	2	no	—
21	C₄×C₄₇	D₈	1	1	yes	characteristic
22	C₄×C₄₇	SD₁₆	1	1	yes	characteristic
23	C₄×C₄₇	SD₁₆	1	1	yes	characteristic
24	C₄×C₄₇	Q₁₆	1	1	yes	characteristic
25	C₂×C₄₇	#101	1	1	yes	characteristic
26	C₂×C₄₇	#102	1	1	yes	characteristic
27	C₂×C₄₇	#103	1	1	yes	characteristic
28	#332	—	47	1	no	maximal, Sylow
29	C₄₇	#332	1	1	yes	characteristic, complemented, Sylow
30	C₄⋊Q₈	D₄₇	1	1	yes	characteristic
31	C₄⋊C₈	—	47	1	no	—
32	C₄⋊C₈	—	47	1	no	—
33	C₂×Q₈	—	2	1	no	—
34	C₂×C₈	—	94	1	no	—
35	C₂×C₈	—	94	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	Dic₄₇	1	2	yes	—
38	C₄²	C₂×D₄₇	1	1	yes	characteristic
39	Q₈	—	4	1	no	—
40	Q₈	—	4	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
44	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
45	C₈	—	188	1	no	—
46	C₈	—	188	1	no	—
47	C₂²	C₂²⋊Dic₄₇	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	4	2	no	—
50	C₄	#10625334	1	1	yes	characteristic
51	C₄	#10625335	1	1	yes	characteristic
52	C₄	#10625336	1	1	yes	characteristic
53	C₄	#10625337	1	1	yes	characteristic
54	C₂	#13093943	1	1	yes	characteristic
55	C₂	#13093945	1	1	yes	characteristic
56	C₂	#13093948	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄
Frattini	C₄²	C₂×D₄₇
Fitting	C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₂²⋊C₄
cosocle	C₄²×C₄₇	C₂²
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#332
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
29. C₄₇	#332	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
C₄⋊Q₈×C₄₇	2
#332	47