Subgroups — C₃×C₂²⋊(C₇⋊C₈) #10620587

Order 672.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₇⋊C₈)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₇	C₄	1	1	yes	characteristic
5	C₂×C₃×C₄×C₇	—	2	2	no	—
6	C₂×C₃×C₄×C₇	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₃×C₇⋊C₈	—	2	2	no	—
9	C₂²×C₄×C₇	C₂×C₃	1	1	yes	characteristic
10	C₂×C₇⋊C₈	C₂×C₃	1	2	yes	complemented
11	C₃×C₂²⋊C₈	—	7	1	no	maximal, Hall
12	C₂²×C₃×C₇	—	2	1	no	—
13	C₂²×C₃×C₇	—	2	1	no	—
14	C₂²×C₃×C₇	C₈	1	2	yes	complemented
15	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
16	C₃×C₄×C₇	—	2	1	no	—
17	C₃×C₄×C₇	D₄	1	2	yes	—
18	C₂³×C₇	C₃×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	C₃×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
22	C₇⋊C₈	—	2	2	no	—
23	C₂²×C₃×C₄	D₇	1	1	yes	characteristic
24	C₂×C₃×C₈	—	7	2	no	—
25	C₂×C₃×C₇	—	2	2	no	—
26	C₂×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₃×C₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₃×C₇	M₁₆	1	1	yes	characteristic
29	C₂²⋊C₈	—	7	1	no	Sylow
30	C₂²×C₇	—	2	1	no	—
31	C₂²×C₇	—	2	1	no	—
32	C₂²×C₇	C₃×C₈	1	2	yes	complemented
33	C₂²×C₇	C₂×C₃×C₄	1	1	yes	characteristic
34	C₄×C₇	—	2	1	no	—
35	C₄×C₇	C₃×D₄	1	2	yes	—
36	C₂³×C₃	Dic₇	1	1	yes	characteristic
37	C₂×C₃×C₄	—	2	2	no	—
38	C₂×C₃×C₄	Dic₇	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
40	C₃×C₈	—	14	2	no	—
41	C₃×C₇	C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₄	C₃×D₇	1	1	yes	characteristic
43	C₂×C₈	—	7	2	no	—
44	C₂×C₇	—	2	2	no	—
45	C₂×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₇	C₂×C₃×C₈	1	1	yes	characteristic
47	C₂×C₇	C₃×M₁₆	1	1	yes	characteristic
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	—	2	1	no	—
50	C₂²×C₃	C₇⋊C₈	1	2	yes	complemented
51	C₂²×C₃	C₂×Dic₇	1	1	yes	characteristic
52	C₃×C₄	—	2	1	no	—
53	C₃×C₄	C₂²⋊D₇	1	2	yes	—
54	C₂³	C₃×Dic₇	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	C₃×Dic₇	1	1	yes	characteristic
57	C₂×C₄	C₂×C₃×D₇	1	1	yes	characteristic
58	C₈	—	14	2	no	—
59	C₇	C₃×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	2	2	no	—
61	C₂×C₃	C₂×C₇⋊C₈	1	1	yes	characteristic
62	C₂×C₃	#1136	1	1	yes	characteristic
63	C₂×C₃	C₂²⋊Dic₇	1	1	yes	characteristic
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₃×C₇⋊C₈	1	2	yes	complemented
67	C₂²	C₂×C₃×Dic₇	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	C₃×C₂²⋊D₇	1	2	yes	—
70	C₃	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic, direct product factor, Sylow
71	C₂	—	2	2	no	—
72	C₂	C₂×C₃×C₇⋊C₈	1	1	yes	characteristic
73	C₂	C₃×#1136	1	1	yes	characteristic
74	C₂	C₃×C₂²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₇
commutator	C₂×C₇	C₂×C₃×C₈
Frattini	C₂×C₄	C₂×C₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂
socle	C₂²×C₃×C₇	C₂×C₄
prime radical	C₂³×C₃×C₇	C₄
cosocle	C₂×C₄×C₇	C₂²×C₃
hypercenter	C₂²×C₃×C₄	D₇
hypocenter	C₇	C₃×C₂²⋊C₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇⋊C₈	C₂	semidirect
3. C₂²⋊(C₇⋊C₈)	C₃	direct (N × H)
10. C₂×C₇⋊C₈	C₂×C₃	semidirect
14. C₂²×C₃×C₇	C₈	semidirect
32. C₂²×C₇	C₃×C₈	semidirect
41. C₃×C₇	C₂²⋊C₈	semidirect
50. C₂²×C₃	C₇⋊C₈	semidirect
59. C₇	C₃×C₂²⋊C₈	semidirect
66. C₂²	C₃×C₇⋊C₈	semidirect
70. C₃	C₂²⋊(C₇⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₇	2
C₂×C₃×C₇⋊C₈	2
C₂²⋊(C₇⋊C₈)	3
C₃×C₂²⋊C₈	7

Subgroups — C3×C22⋊(C7⋊C8) #10620587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊(C₇⋊C₈) #10620587