Subgroups — C₂×C₂²⋊(C₇⋊C₈) #89978

Order 448.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₇⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂⁴×C₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₇	—	2	4	no	—
6	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₄	1	2	yes	—
8	C₂²×C₄×C₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇⋊C₈	—	2	4	no	—
11	C₂×C₇⋊C₈	C₂²	1	4	yes	complemented
12	C₂×C₂²⋊C₈	—	7	1	no	maximal, Sylow
13	C₂³×C₇	—	2	2	no	—
14	C₂³×C₇	—	2	2	no	—
15	C₂³×C₇	C₈	1	4	yes	complemented
16	C₂³×C₇	C₂×C₄	1	2	yes	—
17	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	—	2	4	no	—
20	C₂×C₄×C₇	—	2	4	no	—
21	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₇	D₄	1	4	yes	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇⋊C₈	—	2	4	no	—
26	C₂³×C₄	D₇	1	1	yes	characteristic
27	C₂²×C₈	—	7	2	no	—
28	C₂²⋊C₈	—	7	4	no	—
29	C₂²×C₇	—	2	8	no	—
30	C₂²×C₇	—	2	2	no	—
31	C₂²×C₇	—	2	2	no	—
32	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₇	C₂×C₈	1	2	yes	—
34	C₂²×C₇	C₂×C₈	1	4	yes	complemented
35	C₂²×C₇	M₁₆	1	2	yes	—
36	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₇	—	2	2	no	—
38	C₄×C₇	C₂²⋊C₄	1	2	yes	—
39	C₄×C₇	C₂×D₄	1	2	yes	—
40	C₂⁴	Dic₇	1	1	yes	characteristic
41	C₂²×C₄	—	2	4	no	—
42	C₂²×C₄	Dic₇	1	1	yes	characteristic
43	C₂²×C₄	Dic₇	1	2	yes	—
44	C₂²×C₄	C₂×D₇	1	2	yes	—
45	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
46	C₂×C₈	—	7	4	no	—
47	C₂×C₈	—	14	4	no	—
48	C₂×C₇	—	2	4	no	—
49	C₂×C₇	C₂²⋊C₈	1	4	yes	complemented
50	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
53	C₂³	—	2	2	no	—
54	C₂³	—	2	2	no	—
55	C₂³	C₇⋊C₈	1	4	yes	complemented
56	C₂³	C₂×Dic₇	1	2	yes	—
57	C₂³	C₂×Dic₇	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	C₂×Dic₇	1	2	yes	—
62	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₇	1	4	yes	—
64	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
65	C₈	—	14	4	no	—
66	C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	2	8	no	—
68	C₂²	—	2	2	no	—
69	C₂²	—	2	2	no	—
70	C₂²	C₂×C₇⋊C₈	1	2	yes	—
71	C₂²	C₂×C₇⋊C₈	1	4	yes	complemented
72	C₂²	#1136	1	2	yes	—
73	C₂²	C₂²⋊Dic₇	1	2	yes	—
74	C₂²	C₂²×Dic₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	C₂²⋊Dic₇	1	2	yes	—
77	C₄	C₂×C₂²⋊D₇	1	2	yes	—
78	C₂	—	2	4	no	—
79	C₂	C₂²⋊(C₇⋊C₈)	1	4	yes	direct product factor
80	C₂	C₂²×C₇⋊C₈	1	1	yes	characteristic
81	C₂	C₂×#1136	1	1	yes	characteristic
82	C₂	C₂×C₂²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇
commutator	C₂×C₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂³×C₄×C₇	C₂
socle	C₂³×C₇	C₂×C₄
prime radical	C₂⁴×C₇	C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂³×C₄	D₇
hypocenter	C₇	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇⋊C₈	C₂	semidirect
3. C₂²⋊(C₇⋊C₈)	C₂	direct (N × H)
11. C₂×C₇⋊C₈	C₂²	semidirect
15. C₂³×C₇	C₈	semidirect
34. C₂²×C₇	C₂×C₈	semidirect
49. C₂×C₇	C₂²⋊C₈	semidirect
55. C₂³	C₇⋊C₈	semidirect
66. C₇	C₂×C₂²⋊C₈	semidirect
71. C₂²	C₂×C₇⋊C₈	semidirect
79. C₂	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
C₂³×C₄×C₇	2
C₂²×C₇⋊C₈	2
C₂²⋊(C₇⋊C₈)	2
C₂×C₂²⋊C₈	7

Subgroups — C2×C22⋊(C7⋊C8) #89978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊(C₇⋊C₈) #89978