Subgroups — #15689497

Order 3008.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625336	C₂	1	4	yes	maximal, complemented
2	#13094021	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₄₇	C₂²	1	2	yes	—
6	C₂×D₁₈₈	—	2	2	no	—
7	C₂×D₁₈₈	C₂²	1	2	yes	—
8	C₄⋊C₄×C₄₇	—	2	2	no	—
9	C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	#10625336	—	2	8	no	—
11	C₂×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂²×D₄₇	—	2	2	no	—
13	Q₈×C₄₇	—	2	4	no	—
14	C₂×C₄×C₄₇	—	2	2	no	—
15	C₂×C₄×C₄₇	D₄	1	2	yes	—
16	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
17	D₁₈₈	—	2	4	no	—
18	D₁₈₈	—	4	4	no	—
19	C₄₇⋊C₈	D₄	1	4	yes	complemented
20	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
21	C₂×D₄₇	—	4	2	no	—
22	C₂×D₄₇	—	4	4	no	—
23	C₄×C₄₇	—	4	2	no	—
24	C₄×C₄₇	SD₁₆	1	4	yes	complemented
25	C₄×C₄₇	C₂×D₄	1	2	yes	—
26	C₂×C₄₇	#127	1	1	yes	characteristic
27	C₂×C₄₇	C₂×SD₁₆	1	2	yes	—
28	D₄₇	—	8	2	no	—
29	#492	—	47	1	no	maximal, Sylow
30	C₄₇	#492	1	1	yes	characteristic, complemented, Sylow
31	C₂×SD₁₆	—	47	4	no	—
32	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
33	#127	—	47	1	no	—
34	C₄×C₈	—	47	1	no	—
35	C₂×Q₈	C₂×D₄₇	1	2	yes	—
36	C₂×D₄	—	47	2	no	—
37	C₂×D₄	—	94	2	no	—
38	SD₁₆	—	94	8	no	—
39	C₂×C₈	—	47	2	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄²	C₂×D₄₇	1	1	yes	characteristic
42	C₂³	—	94	2	no	—
43	Q₈	—	2	4	no	—
44	D₄	—	94	4	no	—
45	D₄	—	188	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂²⋊D₄₇	1	2	yes	—
48	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
49	C₈	—	47	4	no	—
50	C₂²	—	188	2	no	—
51	C₂²	—	188	4	no	—
52	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	#10625336	1	4	yes	complemented
55	C₄	D₄×D₄₇	1	2	yes	—
56	C₂	—	376	2	no	—
57	C₂	#13094020	1	1	yes	characteristic
58	C₂	C₂×#10625336	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂³
Frattini	C₂×C₄	C₂²×D₄₇
Fitting	C₄⋊Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×D₄
prime radical	#13094021	C₂
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#492
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#10625336	C₂	semidirect
3. C₄×C₄₇⋊C₈	C₂	semidirect
4. C₄⋊Q₈×C₄₇	C₂	semidirect
19. C₄₇⋊C₈	D₄	semidirect
24. C₄×C₄₇	SD₁₆	semidirect
30. C₄₇	#492	semidirect
32. C₄⋊Q₈	D₄₇	semidirect
54. C₄	#10625336	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625336	2
#13094021	2
C₄×C₄₇⋊C₈	2
C₄⋊Q₈×C₄₇	2
#492	47