Subgroups — #72254211

Order 2352.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₃⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
3	#11716031	—	3	1	no	maximal, Hall
4	C₂×C₃×C₇⋊D₇	C₄	1	1	yes	characteristic
5	C₃×C₄×C₇²	C₂²	1	1	yes	characteristic
6	C₃×C₇⋊Dic₇	C₄	1	1	yes	characteristic
7	C₄×C₇⋊D₇	D₃	1	1	yes	characteristic
8	C₇×C₇⋊C₈	—	3	2	no	—
9	#66557	—	7	2	no	maximal
10	C₂×C₃×C₇²	C₂×C₄	1	1	yes	characteristic
11	C₃×C₇⋊D₇	—	2	1	no	—
12	C₂×C₇⋊D₇	Dic₃	1	1	yes	characteristic
13	C₄×C₇²	C₂×D₃	1	1	yes	characteristic
14	C₇⋊Dic₇	Dic₃	1	1	yes	characteristic
15	C₃×C₄×D₇	—	7	2	no	—
16	C₃×C₄×D₇	—	14	3	no	—
17	C₃⋊(C₇⋊C₈)	D₇	1	2	yes	complemented
18	C₇×C₃⋊C₈	—	7	2	no	—
19	C₃×C₇²	M₁₆	1	1	yes	characteristic, complemented, Hall
20	#1131	—	21	2	no	—
21	C₂×C₇²	C₂×Dic₃	1	1	yes	characteristic
22	C₇⋊D₇	—	2	1	no	—
23	C₂×C₃×D₇	—	7	2	no	—
24	C₂×C₃×D₇	—	14	3	no	—
25	C₃×C₄×C₇	—	2	3	no	—
26	C₃×C₄×C₇	C₂×D₇	1	2	yes	—
27	C₃×Dic₇	—	7	2	no	—
28	C₃×Dic₇	—	14	3	no	—
29	C₄×D₇	—	7	2	no	—
30	C₄×D₇	—	14	3	no	—
31	C₇×C₈	—	21	2	no	—
32	C₇⋊C₈	—	3	2	no	—
33	C₇²	#208	1	1	yes	characteristic, complemented, Sylow
34	#208	—	49	1	no	Hall
35	C₂×C₃×C₇	—	2	3	no	—
36	C₂×C₃×C₇	C₄×D₇	1	2	yes	—
37	C₃×D₇	—	14	6	no	—
38	C₃×D₇	—	14	2	no	—
39	C₂×D₇	—	7	2	no	—
40	C₂×D₇	—	14	3	no	—
41	C₄×C₇	—	2	3	no	—
42	C₄×C₇	D₃×D₇	1	2	yes	—
43	Dic₇	—	7	2	no	—
44	Dic₇	—	14	3	no	—
45	C₂×C₃×C₄	—	49	1	no	—
46	C₃⋊C₈	—	7	2	no	—
47	C₃×C₇	—	2	3	no	—
48	C₃×C₇	#1131	1	2	yes	complemented
49	M₁₆	—	147	1	no	Sylow
50	C₂×C₇	—	2	3	no	—
51	C₂×C₇	Dic₃×D₇	1	2	yes	—
52	D₇	—	14	6	no	—
53	D₇	—	14	2	no	—
54	C₂²×C₃	—	49	1	no	—
55	C₃×C₄	—	49	1	no	—
56	C₃×C₄	D₇²	1	1	yes	characteristic
57	C₂×C₄	—	49	1	no	—
58	C₈	—	21	2	no	—
59	C₇	—	2	3	no	—
60	C₇	#66557	1	2	yes	complemented
61	C₂×C₃	—	98	1	no	—
62	C₂×C₃	#87639	1	1	yes	characteristic
63	C₂²	—	49	1	no	—
64	C₄	—	49	1	no	—
65	C₄	D₂₁⋊D₇	1	1	yes	characteristic
66	C₃	#11716031	1	1	yes	characteristic, complemented, Sylow
67	C₂	—	98	1	no	—
68	C₂	#11924252	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₂₁⋊D₇
commutator, socle	C₂×C₃×C₇²	C₂×C₄
Fitting, cosocle	C₃×C₄×C₇²	C₂²
prime radical	C₂×C₃×C₇⋊D₇	C₄
hypocenter	C₃×C₇²	M₁₆
LCS	G ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₇×C₃⋊(C₇⋊C₈)	C₂	semidirect
17. C₃⋊(C₇⋊C₈)	D₇	semidirect
19. C₃×C₇²	M₁₆	semidirect
33. C₇²	#208	semidirect
48. C₃×C₇	#1131	semidirect
60. C₇	#66557	semidirect
66. C₃	#11716031	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₇⋊D₇	2
C₇×C₃⋊(C₇⋊C₈)	2
#11716031	3
#66557	7