Subgroups — C₄×C₈×C₇⋊C₈ #13243142

Order 1792.

750 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₇⋊C₈	C₂	1	4	yes	maximal
3	C₄²×C₇⋊C₈	C₂	1	2	yes	maximal
4	C₂×C₄×C₇×C₈	C₄	1	4	yes	—
5	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
6	C₄³×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇⋊C₈	C₄	1	8	yes	—
8	C₂×C₄×C₇⋊C₈	C₂²	1	4	yes	—
9	C₈×C₇⋊C₈	C₄	1	16	yes	direct product factor
10	C₄×C₈²	—	7	1	no	maximal, Sylow
11	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
15	C₂²×C₇⋊C₈	C₂×C₄	1	8	yes	—
16	C₄×C₇×C₈	C₈	1	16	yes	complemented
17	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
18	C₄×C₇⋊C₈	C₈	1	32	yes	direct product factor
19	C₄×C₇⋊C₈	C₂×C₄	1	16	yes	—
20	C₄²×C₈	—	7	2	no	—
21	C₄²×C₈	D₇	1	1	yes	characteristic
22	C₂×C₈²	—	7	4	no	—
23	C₂²×C₄×C₇	C₄²	1	4	yes	—
24	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
25	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₇×C₈	C₄²	1	8	yes	—
27	C₂×C₇×C₈	C₂×C₈	1	16	yes	—
28	C₄²×C₇	C₂×C₈	1	16	yes	—
29	C₄²×C₇	C₂×C₈	1	8	yes	—
30	C₄²×C₇	C₂²×C₄	1	4	yes	—
31	C₂×C₇⋊C₈	C₄²	1	16	yes	—
32	C₂×C₇⋊C₈	C₂×C₈	1	32	yes	—
33	C₂×C₄×C₈	—	7	8	no	—
34	C₂×C₄×C₈	—	7	4	no	—
35	C₂×C₄×C₈	Dic₇	1	4	yes	—
36	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
37	C₄³	C₂×D₇	1	1	yes	characteristic
38	C₈²	—	7	16	no	—
39	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₄×C₇	C₄×C₈	1	8	yes	—
41	C₂×C₄×C₇	C₄×C₈	1	16	yes	—
42	C₂×C₄×C₇	C₂×C₄²	1	4	yes	—
43	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
44	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
45	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
46	C₇×C₈	C₄×C₈	1	16	yes	complemented
47	C₇⋊C₈	C₄×C₈	1	32	yes	direct product factor
48	C₂²×C₈	—	7	8	no	—
49	C₂²×C₈	C₂×Dic₇	1	4	yes	—
50	C₂×C₄²	C₄×D₇	1	4	yes	—
51	C₂×C₄²	C₂×Dic₇	1	2	yes	—
52	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
53	C₄×C₈	—	7	32	no	—
54	C₄×C₈	—	7	16	no	—
55	C₄×C₈	C₇⋊C₈	1	16	yes	direct product factor
56	C₄×C₈	C₄×D₇	1	8	yes	—
57	C₂²×C₇	C₄³	1	1	yes	characteristic
58	C₂²×C₇	C₂×C₄×C₈	1	4	yes	—
59	C₂²×C₇	C₂×C₄×C₈	1	2	yes	—
60	C₄×C₇	C₈²	1	16	yes	complemented
61	C₄×C₇	C₂×C₄×C₈	1	4	yes	—
62	C₄×C₇	C₂×C₄×C₈	1	8	yes	—
63	C₂²×C₄	C₄×Dic₇	1	4	yes	—
64	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
65	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
66	C₂×C₈	—	7	16	no	—
67	C₂×C₈	—	7	32	no	—
68	C₂×C₈	C₂×C₇⋊C₈	1	16	yes	—
69	C₂×C₈	C₄×Dic₇	1	8	yes	—
70	C₄²	C₈×D₇	1	16	yes	—
71	C₄²	C₂×C₇⋊C₈	1	8	yes	—
72	C₄²	C₂×C₄×D₇	1	4	yes	—
73	C₂×C₇	C₂×C₈²	1	4	yes	—
74	C₂×C₇	C₄²×C₈	1	1	yes	characteristic
75	C₂×C₇	C₄²×C₈	1	2	yes	—
76	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
77	C₂×C₄	C₄×C₇⋊C₈	1	8	yes	—
78	C₂×C₄	C₈×Dic₇	1	16	yes	—
79	C₂×C₄	C₄²×D₇	1	4	yes	—
80	C₂×C₄	C₂×C₈×D₇	1	8	yes	—
81	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
82	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
83	C₈	—	7	32	no	—
84	C₈	C₄×C₇⋊C₈	1	16	yes	direct product factor
85	C₇	C₄×C₈²	1	1	yes	characteristic, complemented, Sylow
86	C₂²	C₂×C₄×C₇⋊C₈	1	2	yes	—
87	C₂²	C₄²×Dic₇	1	1	yes	characteristic
88	C₂²	C₂×C₈×Dic₇	1	4	yes	—
89	C₄	C₈×C₇⋊C₈	1	16	yes	direct product factor
90	C₄	C₄×C₈×D₇	1	8	yes	—
91	C₄	C₂×C₄×C₇⋊C₈	1	4	yes	—
92	C₂	C₄²×C₇⋊C₈	1	1	yes	characteristic
93	C₂	C₂×C₈×C₇⋊C₈	1	4	yes	—
94	C₂	C₄×C₈×Dic₇	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₄²×C₈	D₇
commutator, hypocenter	C₇	C₄×C₈²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₄²×C₇×C₈	C₂
socle, prime radical	C₂³×C₇	C₂×C₄²
cosocle	C₂×C₄²×C₇	C₂³
LCS	G ▷ C₇
UCS	C₁ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
9. C₈×C₇⋊C₈	C₄	direct (N × H)
16. C₄×C₇×C₈	C₈	semidirect
18. C₄×C₇⋊C₈	C₈	direct (N × H)
46. C₇×C₈	C₄×C₈	semidirect
47. C₇⋊C₈	C₄×C₈	direct (N × H)
55. C₄×C₈	C₇⋊C₈	direct (N × H)
60. C₄×C₇	C₈²	semidirect
84. C₈	C₄×C₇⋊C₈	direct (N × H)
85. C₇	C₄×C₈²	semidirect
89. C₄	C₈×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
C₄²×C₇×C₈	2
C₂×C₈×C₇⋊C₈	2
C₄²×C₇⋊C₈	2
C₄×C₈²	7

Subgroups — C4×C8×C7⋊C8 #13243142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×C₇⋊C₈ #13243142