Subgroups — #89979

Order 448.

190 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	C₂×#1136	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₇⋊C₈)	C₂	1	4	yes	maximal, complemented
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	#1136	—	2	4	no	—
11	C₂×C₇⋊C₈	C₂²	1	4	yes	complemented
12	#407	—	7	1	no	maximal, Sylow
13	C₂³×C₇	—	2	2	no	—
14	C₂³×C₇	—	2	4	no	—
15	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	4	no	—
19	C₂×C₄×C₇	—	2	4	no	—
20	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	D₄	1	4	yes	—
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₇⋊C₈	—	2	4	no	—
25	C₂³×C₄	D₇	1	1	yes	characteristic
26	C₂×M₁₆	—	7	2	no	—
27	C₂²⋊C₈	—	7	4	no	—
28	C₂²×C₇	—	2	8	no	—
29	C₂²×C₇	—	2	2	no	—
30	C₂²×C₇	—	2	2	no	—
31	C₂²×C₇	—	2	2	no	—
32	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₇	M₁₆	1	4	yes	complemented
34	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₇	—	2	2	no	—
36	C₄×C₇	C₂²⋊C₄	1	2	yes	—
37	C₄×C₇	C₂×D₄	1	2	yes	—
38	C₂⁴	Dic₇	1	1	yes	characteristic
39	C₂²×C₄	—	2	4	no	—
40	C₂²×C₄	Dic₇	1	1	yes	characteristic
41	C₂²×C₄	Dic₇	1	2	yes	—
42	C₂²×C₄	C₂×D₇	1	2	yes	—
43	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
44	M₁₆	—	14	4	no	—
45	C₂×C₈	—	7	4	no	—
46	C₂×C₇	—	2	2	no	—
47	C₂×C₇	—	2	4	no	—
48	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₇	C₂×M₁₆	1	2	yes	—
50	C₂³	—	2	4	no	—
51	C₂³	—	2	2	no	—
52	C₂³	C₂×Dic₇	1	2	yes	—
53	C₂³	C₂×Dic₇	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₂×Dic₇	1	2	yes	—
58	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₇	1	4	yes	—
60	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
61	C₈	—	14	4	no	—
62	C₇	#407	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	2	8	no	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	#1136	1	4	yes	complemented
68	C₂²	C₂²⋊Dic₇	1	2	yes	—
69	C₂²	C₂²×Dic₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	C₂²⋊Dic₇	1	2	yes	—
72	C₄	C₂×C₂²⋊D₇	1	2	yes	—
73	C₂	—	2	4	no	—
74	C₂	—	2	2	no	—
75	C₂	C₂×#1136	1	2	yes	—
76	C₂	C₂×C₂²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₂³×C₄×C₇	C₂
prime radical	C₂⁴×C₇	C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂³×C₄	D₇
hypocenter	C₇	#407
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#1136	C₂	semidirect
3. C₂²⋊(C₇⋊C₈)	C₂	semidirect
11. C₂×C₇⋊C₈	C₂²	semidirect
33. C₂²×C₇	M₁₆	semidirect
62. C₇	#407	semidirect
67. C₂²	#1136	semidirect

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
C₂×#1136	2
C₂²⋊(C₇⋊C₈)	2
#407	7