Subgroups — #6447

Order 224.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₇	C₂²	1	1	yes	characteristic
9	C₂²×D₇	C₂²	1	1	yes	characteristic
10	C₂²×D₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²⋊D₇	—	2	2	no	—
14	C₂²⋊D₇	—	2	2	no	—
15	C₂×Dic₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₇	C₂²	1	1	yes	characteristic
17	D₂₈	—	2	2	no	—
18	C₄×D₇	—	2	2	no	—
19	#121	—	7	1	no	maximal, Sylow
20	C₂²×C₇	—	2	1	no	—
21	C₂²×C₇	—	2	1	no	—
22	C₂²×C₇	—	2	1	no	—
23	C₂²×C₇	C₂³	1	1	yes	characteristic
24	C₂×D₇	—	2	1	no	—
25	C₂×D₇	—	2	1	no	—
26	C₂×D₇	—	2	1	no	—
27	C₂×D₇	—	2	1	no	—
28	C₂×D₇	—	2	1	no	—
29	C₂×D₇	D₄	1	2	yes	complemented
30	C₄×C₇	—	2	1	no	—
31	C₄×C₇	—	2	1	no	—
32	Dic₇	—	2	1	no	—
33	Dic₇	D₄	1	2	yes	complemented
34	C₂×D₄	—	7	1	no	—
35	C₂×D₄	—	7	1	no	—
36	C₂×D₄	—	7	1	no	—
37	C₂²×C₄	—	7	1	no	—
38	C₄⋊C₄	—	7	1	no	—
39	C₂²⋊C₄	—	7	1	no	—
40	C₂²⋊C₄	D₇	1	1	yes	characteristic, complemented
41	C₂×C₇	—	4	1	no	—
42	C₂×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₇	#41	1	1	yes	characteristic
45	D₇	—	2	2	no	—
46	D₇	—	4	1	no	—
47	C₂³	—	7	1	no	—
48	C₂³	—	7	1	no	—
49	C₂³	C₂×D₇	1	1	yes	characteristic
50	D₄	—	14	2	no	—
51	D₄	—	14	2	no	—
52	D₄	—	14	2	no	—
53	C₂×C₄	—	7	1	no	—
54	C₂×C₄	—	7	1	no	—
55	C₂×C₄	—	14	2	no	—
56	C₂×C₄	C₂×D₇	1	1	yes	characteristic
57	C₂×C₄	C₂×D₇	1	1	yes	characteristic
58	C₇	#121	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	7	2	no	—
63	C₂²	—	14	1	no	—
64	C₂²	—	14	1	no	—
65	C₂²	—	14	1	no	—
66	C₂²	—	14	1	no	—
67	C₂²	—	14	1	no	—
68	C₂²	C₂²×D₇	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	7	2	no	—
72	C₄	—	14	1	no	—
73	C₂	—	4	1	no	—
74	C₂	—	14	2	no	—
75	C₂	—	28	1	no	—
76	C₂	#1157	1	1	yes	characteristic
77	C₂	D₄×D₇	1	1	yes	characteristic
78	C₂	D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, cosocle	C₂²×C₇	C₂³
Fitting	C₇×C₂²⋊C₄	C₂
hypercenter	C₂²⋊C₄	D₇
hypocenter	C₇	#121
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊D₇	C₂	semidirect
2. C₂×C₂²⋊D₇	C₂	semidirect
3. C₂×D₂₈	C₂	semidirect
4. C₂×C₄×D₇	C₂	semidirect
5. C₇×C₂²⋊C₄	C₂	semidirect
6. (C₂×D₇)⋊C₄	C₂	semidirect
7. Dic₇⋊C₄	C₂	semidirect
29. C₂×D₇	D₄	semidirect
33. Dic₇	D₄	semidirect
40. C₂²⋊C₄	D₇	semidirect
58. C₇	#121	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₇	2
C₂×D₂₈	2
C₂×C₄×D₇	2
C₇×C₂²⋊C₄	2
(C₂×D₇)⋊C₄	2
Dic₇⋊C₄	2
#121	7