Subgroups — #15689184

Order 3008.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
3	#13093935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#117×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×C₄₇	C₄	1	2	yes	—
8	C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
9	C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂³×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₄₇	—	2	2	no	—
15	C₂×C₄×C₄₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₄₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₄₇⋊C₈	—	2	2	no	—
20	C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇	—	2	1	no	—
22	C₂²×C₄₇	—	2	2	no	—
23	C₂²×C₄₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₄₇	—	2	1	no	—
25	C₄×C₄₇	—	2	2	no	—
26	C₄×C₄₇	—	2	2	no	—
27	C₄×C₄₇	#41	1	2	yes	—
28	C₂×C₄₇	—	4	1	no	—
29	C₂×C₄₇	#117	1	1	yes	characteristic
30	C₂×C₄₇	#131	1	2	yes	—
31	#433	—	47	1	no	maximal, Sylow
32	C₄₇	#433	1	1	yes	characteristic, complemented, Sylow
33	#117	D₄₇	1	1	yes	characteristic
34	C₄⋊C₈	—	47	2	no	—
35	C₂²⋊C₈	—	47	2	no	—
36	#97	—	47	1	no	—
37	C₄×C₈	—	47	1	no	—
38	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
39	C₂×C₈	—	47	2	no	—
40	C₂×C₈	—	47	2	no	—
41	C₄⋊C₄	Dic₄₇	1	2	yes	—
42	C₂²⋊C₄	Dic₄₇	1	2	yes	—
43	C₄²	C₂×D₄₇	1	1	yes	characteristic
44	C₄²	C₂×D₄₇	1	1	yes	characteristic
45	C₂³	C₂×Dic₄₇	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂×Dic₄₇	1	2	yes	—
48	C₂×C₄	C₂×Dic₄₇	1	2	yes	—
49	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
51	C₈	—	94	2	no	—
52	C₈	—	94	2	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	2	no	—
55	C₂²	C₂²×Dic₄₇	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	2	2	no	—
58	C₄	—	2	2	no	—
59	C₄	#10625350	1	2	yes	—
60	C₂	—	4	1	no	—
61	C₂	#13093983	1	1	yes	characteristic
62	C₂	#13094031	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄
Fitting	#117×C₄₇	C₂
prime radical	C₂³×C₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂³
hypercenter	#117	D₄₇
hypocenter	C₄₇	#433
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. #13093935	C₂	semidirect
4. C₄×C₄₇⋊C₈	C₂	semidirect
32. C₄₇	#433	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
C₂²⋊(C₄₇⋊C₈)	2
#13093935	2
C₄×C₄₇⋊C₈	2
#117×C₄₇	2
#433	47