Subgroups — #15688878

Order 3008.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₇⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₄₇	C₄	1	2	yes	—
4	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
5	C₄²×C₄₇	—	2	2	no	—
6	C₂×C₄₇⋊C₈	—	2	2	no	—
7	C₂×C₄₇⋊C₈	—	2	2	no	—
8	C₂×C₄₇⋊C₈	C₄	1	4	yes	complemented
9	C₂³×C₄₇	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₄₇	—	2	2	no	—
11	C₂×C₄×C₄₇	—	2	2	no	—
12	C₂×C₄×C₄₇	C₈	1	4	yes	complemented
13	C₂×C₄×C₄₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₄₇	D₄	1	2	yes	—
16	C₂×C₄×C₄₇	Q₈	1	1	yes	characteristic
17	C₄₇⋊C₈	—	2	4	no	—
18	C₂²×C₄₇	C₄²	1	1	yes	characteristic
19	C₂²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄₇	C₂×C₈	1	2	yes	—
22	C₂²×C₄₇	M₁₆	1	2	yes	—
23	C₄×C₄₇	—	2	4	no	—
24	C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
25	C₄×C₄₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄₇	#95	1	1	yes	characteristic
27	C₂×C₄₇	C₄×C₈	1	1	yes	characteristic
28	C₂×C₄₇	#97	1	1	yes	characteristic
29	C₂×C₄₇	C₂²⋊C₈	1	2	yes	—
30	C₂×C₄₇	C₄⋊C₈	1	2	yes	—
31	#336	—	47	1	no	maximal, Sylow
32	C₄₇	#336	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₈	—	47	2	no	—
34	C₂×C₄²	D₄₇	1	1	yes	characteristic
35	C₂²×C₄	Dic₄₇	1	2	yes	—
36	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
37	C₂×C₈	—	47	4	no	—
38	C₂×C₈	—	94	2	no	—
39	C₂×C₈	—	94	2	no	—
40	C₄²	—	2	2	no	—
41	C₂³	C₂×Dic₄₇	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	C₄₇⋊C₈	1	4	yes	complemented
45	C₂×C₄	Dic₉₄	1	1	yes	characteristic
46	C₂×C₄	C₄×D₄₇	1	2	yes	—
47	C₂×C₄	D₁₈₈	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₄₇	1	2	yes	—
49	C₈	—	94	4	no	—
50	C₂²	C₂×C₄₇⋊C₈	1	2	yes	—
51	C₂²	#10625329	1	2	yes	—
52	C₂²	C₄×Dic₄₇	1	1	yes	characteristic
53	C₂²	C₄⋊Dic₄₇	1	1	yes	characteristic
54	C₂²	C₂²⋊Dic₄₇	1	1	yes	characteristic
55	C₄	—	2	4	no	—
56	C₄	Dic₄₇⋊C₄	1	2	yes	—
57	C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
58	C₂	C₄₇⋊#95	1	1	yes	characteristic
59	C₂	C₄×C₄₇⋊C₈	1	1	yes	characteristic
60	C₂	#13093935	1	1	yes	characteristic
61	C₂	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
62	C₂	C₄⋊(C₄₇⋊C₈)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₄₇
commutator	C₂×C₄₇	C₄×C₈
Fitting	C₂×C₄²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂²
hypercenter	C₂×C₄²	D₄₇
hypocenter	C₄₇	#336
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
8. C₂×C₄₇⋊C₈	C₄	semidirect
12. C₂×C₄×C₄₇	C₈	semidirect
32. C₄₇	#336	semidirect
44. 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
C₂²×C₄₇⋊C₈	2
C₂×C₄²×C₄₇	2
#336	47