Subgroups — #15689410

Order 3008.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#13093955	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#13093944	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₇	—	2	1	no	—
7	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄₇	—	2	1	no	—
9	C₄⋊C₄×C₄₇	—	2	1	no	—
10	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
11	C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	(C₂×D₄₇)⋊C₄	—	2	1	no	—
13	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×D₄₇	—	2	1	no	—
16	Q₈×C₄₇	—	4	2	no	—
17	C₂×C₄×C₄₇	—	2	2	no	—
18	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₇	—	2	1	no	—
22	D₁₈₈	—	2	1	no	—
23	D₁₈₈	Q₈	1	2	yes	complemented
24	C₄×D₄₇	—	4	2	no	—
25	C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₄₇	—	2	2	no	—
28	C₂×D₄₇	—	4	1	no	—
29	C₂×D₄₇	—	4	1	no	—
30	C₄×C₄₇	—	2	1	no	—
31	C₄×C₄₇	—	4	2	no	—
32	C₄×C₄₇	SD₁₆	1	2	yes	—
33	C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₄₇	#41	1	1	yes	characteristic
35	Dic₄₇	—	4	1	no	—
36	C₂×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₄₇	—	4	2	no	—
40	#476	—	47	1	no	maximal, Sylow
41	C₄₇	#476	1	1	yes	characteristic, complemented, Sylow
42	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
43	C₄×D₄	—	47	1	no	—
44	#106	—	47	2	no	—
45	C₄⋊C₈	—	47	1	no	—
46	#102	—	47	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	47	1	no	—
49	C₂²×C₄	—	94	1	no	—
50	C₂×C₈	—	47	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	47	1	no	—
53	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
54	C₂²⋊C₄	—	94	1	no	—
55	C₄²	C₂×D₄₇	1	1	yes	characteristic
56	C₂³	—	94	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	47	2	no	—
59	D₄	—	94	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	94	1	no	—
62	C₂×C₄	—	188	2	no	—
63	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
66	C₈	—	94	2	no	—
67	C₂²	—	94	2	no	—
68	C₂²	—	188	1	no	—
69	C₂²	—	188	1	no	—
70	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	188	1	no	—
74	C₄	#10625336	1	2	yes	—
75	C₄	Q₈×D₄₇	1	1	yes	characteristic
76	C₄	#10625354	1	1	yes	characteristic
77	C₂	—	188	2	no	—
78	C₂	#13094000	1	1	yes	characteristic
79	C₂	C₂×#10625336	1	1	yes	characteristic
80	C₂	#13094050	1	1	yes	characteristic

	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	C₂×D₁₈₈	C₂²
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#476
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
2. #13093955	C₂	semidirect
3. C₄⋊(C₄₇⋊C₈)	C₂	semidirect
5. C₄⋊Q₈×C₄₇	C₂	semidirect
23. D₁₈₈	Q₈	semidirect
41. C₄₇	#476	semidirect
42. C₄⋊Q₈	D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₈₈	2
#13093955	2
C₄⋊(C₄₇⋊C₈)	2
#13093944	2
C₄⋊Q₈×C₄₇	2
#476	47