Subgroups — #13050849

Order 1344.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₁₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃⋊(C₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
4	#89316	—	3	1	no	maximal, Hall
5	C₇×Dic₁₂	—	2	1	no	—
6	C₃×D₅₆	—	2	1	no	—
7	C₃×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₇×Q₃₂	—	3	1	no	—
9	D₁₁₂	D₃	1	1	yes	characteristic
10	C₇⋊C₃₂	—	3	1	no	—
11	#4520	—	7	1	no	maximal, Hall
12	C₇×Dic₆	—	4	1	no	—
13	C₃×D₂₈	—	4	1	no	—
14	C₃×C₇×C₈	D₄	1	1	yes	characteristic
15	C₇×Q₁₆	—	6	1	no	—
16	D₅₆	—	2	1	no	—
17	C₇×C₁₆	C₂×D₃	1	1	yes	characteristic
18	C₃×D₁₆	—	7	1	no	—
19	Dic₂₄	D₇	1	1	yes	characteristic, complemented
20	C₃⋊C₃₂	—	7	1	no	—
21	C₂×C₃×D₇	—	8	1	no	—
22	C₃×C₄×C₇	D₈	1	1	yes	characteristic
23	C₇×Dic₃	—	8	1	no	—
24	SD₆₄	—	21	1	no	Sylow
25	C₇×Q₈	—	12	1	no	—
26	D₂₈	—	4	1	no	—
27	C₇×C₈	#67	1	1	yes	characteristic
28	C₃×D₈	—	14	1	no	—
29	Dic₁₂	—	2	1	no	—
30	C₃×C₁₆	C₂×D₇	1	1	yes	characteristic
31	C₂×C₃×C₇	D₁₆	1	1	yes	characteristic
32	C₃×D₇	—	16	1	no	—
33	Q₃₂	—	3	1	no	—
34	D₁₆	—	7	1	no	—
35	C₃₂	—	21	1	no	—
36	C₂×D₇	—	8	1	no	—
37	C₄×C₇	—	24	1	no	—
38	C₄×C₇	#213	1	1	yes	characteristic
39	C₃×D₄	—	28	1	no	—
40	Dic₆	—	4	1	no	—
41	C₃×C₈	C₂²⋊D₇	1	1	yes	characteristic
42	C₃×C₇	SD₆₄	1	1	yes	characteristic, complemented, Hall
43	Q₁₆	—	6	1	no	—
44	D₈	—	14	1	no	—
45	C₁₆	D₃×D₇	1	1	yes	characteristic
46	C₂×C₇	#826	1	1	yes	characteristic
47	D₇	—	16	1	no	—
48	C₂²×C₃	—	56	1	no	—
49	C₃×C₄	#1141	1	1	yes	characteristic
50	Dic₃	—	8	1	no	—
51	Q₈	—	12	1	no	—
52	D₄	—	28	1	no	—
53	C₈	#4245	1	1	yes	characteristic
54	C₇	#4520	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	—	112	1	no	—
56	C₂×C₃	#6400	1	1	yes	characteristic
57	C₂²	—	56	1	no	—
58	C₄	—	24	1	no	—
59	C₄	#66560	1	1	yes	characteristic
60	C₃	#89316	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	112	1	no	—
62	C₂	#10620472	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10620472
commutator, Fitting, cosocle	C₃×C₇×C₁₆	C₂²
Frattini, hypercenter	C₁₆	D₃×D₇
socle	C₂×C₃×C₇	D₁₆
prime radical	C₃×D₁₁₂	C₂
hypocenter	C₃×C₇	SD₆₄
LCS	G ▷ C₃×C₇×C₁₆ ▷ C₃×C₇×C₈ ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
1. C₇×Dic₂₄	C₂	semidirect
3. C₃⋊(C₇⋊C₃₂)	C₂	semidirect
19. Dic₂₄	D₇	semidirect
42. C₃×C₇	SD₆₄	semidirect
54. C₇	#4520	semidirect
60. C₃	#89316	semidirect

Maximal subgroups

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

subgroup	index
C₇×Dic₂₄	2
C₃×D₁₁₂	2
C₃⋊(C₇⋊C₃₂)	2
#89316	3
#4520	7