Subgroups — #89333

Order 448.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×Q₈	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂²×Dic₇	—	2	1	no	—
7	C₇×C₄⋊C₄	—	2	1	no	—
8	C₇×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
9	C₇×C₂²⋊C₄	—	2	1	no	—
10	C₂×C₇⋊C₈	—	2	1	no	—
11	#328	—	7	1	no	maximal, Sylow
12	C₂³×C₇	D₄	1	1	yes	characteristic
13	C₇×Q₈	—	4	1	no	—
14	C₂×C₄×C₇	—	2	1	no	—
15	C₂×C₄×C₇	—	2	1	no	—
16	C₂×C₄×C₇	—	4	1	no	—
17	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂×Dic₇	—	2	2	no	—
20	C₂×Dic₇	—	4	1	no	—
21	C₂×Dic₇	—	4	1	no	—
22	C₇⋊C₈	—	4	1	no	—
23	C₂²⋊Q₈	D₇	1	1	yes	characteristic
24	C₂²⋊C₈	—	7	1	no	—
25	#95	—	7	1	no	—
26	C₂²×C₇	—	2	1	no	—
27	C₂²×C₇	—	2	1	no	—
28	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₇	SD₁₆	1	1	yes	characteristic
30	C₂²×C₇	Q₁₆	1	1	yes	characteristic
31	C₄×C₇	—	2	1	no	—
32	C₄×C₇	—	2	1	no	—
33	C₄×C₇	—	4	1	no	—
34	C₄×C₇	—	4	1	no	—
35	Dic₇	—	4	2	no	—
36	C₂×Q₈	Dic₇	1	1	yes	characteristic, complemented
37	C₂²×C₄	—	14	1	no	—
38	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
39	C₂×C₈	—	14	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	Dic₇	1	1	yes	characteristic, complemented
42	C₂²⋊C₄	—	2	1	no	—
43	C₂×C₇	—	2	1	no	—
44	C₂×C₇	—	2	1	no	—
45	C₂×C₇	#99	1	1	yes	characteristic
46	C₂×C₇	#103	1	1	yes	characteristic
47	C₂×C₇	#104	1	1	yes	characteristic
48	C₂³	C₂²⋊D₇	1	1	yes	characteristic
49	Q₈	—	4	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	14	2	no	—
54	C₂×C₄	—	28	1	no	—
55	C₂×C₄	—	28	1	no	—
56	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
58	C₈	—	28	1	no	—
59	C₇	#328	1	1	yes	characteristic, complemented, Sylow
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	#1143	1	1	yes	characteristic
63	C₂²	#1144	1	1	yes	characteristic
64	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	1	no	—
68	C₄	—	4	1	no	—
69	C₄	—	28	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	#6408	1	1	yes	characteristic
73	C₂	#6409	1	1	yes	characteristic
74	C₂	#6411	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂×C₄
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂³×C₇	D₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂²⋊Q₈	D₇
hypocenter	C₇	#328
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
4. C₂×C₇×Q₈	C₄	semidirect
8. C₇×C₄⋊C₄	C₄	semidirect
36. C₂×Q₈	Dic₇	semidirect
41. C₄⋊C₄	Dic₇	semidirect
59. C₇	#328	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₂²⋊Q₈	2
C₇⋊#95	2
C₂²⋊(C₇⋊C₈)	2
#328	7