Subgroups — #15688875

Order 3008.

54 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⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₄₇	—	2	1	no	—
5	C₄²×C₄₇	C₂²	1	1	yes	characteristic
6	C₂×C₄₇⋊C₈	—	2	1	no	—
7	C₂×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄₇⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
11	C₂×C₄×C₄₇	Q₈	1	1	yes	characteristic
12	C₈×C₄₇	—	4	1	no	—
13	C₄₇⋊C₈	—	2	1	no	—
14	C₄₇⋊C₈	—	4	1	no	—
15	C₄₇⋊C₈	C₈	1	2	yes	complemented
16	C₂²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₄₇	—	2	1	no	—
18	C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
19	C₄×C₄₇	M₁₆	1	1	yes	characteristic
20	C₄×C₄₇	D₈	1	1	yes	characteristic
21	C₄×C₄₇	Q₁₆	1	1	yes	characteristic
22	C₂×C₄₇	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₄₇	#107	1	1	yes	characteristic
24	C₂×C₄₇	#108	1	1	yes	characteristic
25	#335	—	47	1	no	maximal, Sylow
26	C₄₇	#335	1	1	yes	characteristic, complemented, Sylow
27	C₄⋊C₈	—	47	1	no	—
28	C₄⋊C₈	D₄₇	1	1	yes	characteristic
29	C₄×C₈	—	47	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	47	1	no	—
32	C₂×C₈	—	47	1	no	—
33	C₂×C₈	—	94	1	no	—
34	C₄²	C₂×D₄₇	1	1	yes	characteristic
35	C₂×C₄	Dic₉₄	1	1	yes	characteristic
36	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
37	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	47	2	no	—
40	C₈	—	94	1	no	—
41	C₈	—	188	1	no	—
42	C₂²	Dic₄₇⋊C₄	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₄₇	1	1	yes	characteristic
45	C₄	#10625324	1	1	yes	characteristic
46	C₄	#10625334	1	1	yes	characteristic
47	C₄	#10625337	1	1	yes	characteristic
48	C₂	Dic₄₇⋊C₈	1	1	yes	characteristic
49	C₂	#13093957	1	1	yes	characteristic
50	C₂	#13093959	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂×C₈
Frattini	C₄²	C₂×D₄₇
Fitting	C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₄⋊C₄
cosocle	C₄²×C₄₇	C₂²
hypercenter	C₄⋊C₈	D₄₇
hypocenter	C₄₇	#335
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
15. C₄₇⋊C₈	C₈	semidirect
26. C₄₇	#335	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×C₄₇⋊C₈	2
C₄⋊C₈×C₄₇	2
#335	47