Subgroups — #6500

Order 224.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
4	C₂²⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₇	C₂²	1	1	yes	characteristic
7	C₂³×C₇	C₂²	1	1	yes	characteristic, complemented
8	C₂²×D₇	—	2	2	no	—
9	C₂²×D₇	—	2	4	no	—
10	C₂²×D₇	C₂²	1	2	yes	—
11	C₇×D₄	—	2	2	no	—
12	C₂×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
13	C₂²⋊D₇	—	2	4	no	—
14	C₂×Dic₇	C₂²	1	2	yes	complemented
15	#120	—	7	1	no	maximal, Sylow
16	C₂²×C₇	—	2	2	no	—
17	C₂²×C₇	—	2	2	no	—
18	C₂²×C₇	—	2	1	no	—
19	C₂²×C₇	D₄	1	2	yes	complemented
20	C₂²×C₇	C₂³	1	1	yes	characteristic
21	C₂×D₇	—	2	8	no	—
22	C₂×D₇	—	2	2	no	—
23	C₂×D₇	—	2	2	no	—
24	C₂×D₇	D₄	1	4	yes	complemented
25	C₄×C₇	—	2	1	no	—
26	Dic₇	—	2	2	no	—
27	C₂⁴	—	7	1	no	—
28	C₂×D₄	—	7	2	no	—
29	C₂×D₄	D₇	1	1	yes	characteristic, complemented
30	C₂²⋊C₄	—	7	1	no	—
31	C₂²⋊C₄	—	7	2	no	—
32	C₂×C₇	—	2	2	no	—
33	C₂×C₇	—	4	1	no	—
34	C₂×C₇	C₂×D₄	1	2	yes	—
35	C₂×C₇	C₂×D₄	1	1	yes	characteristic
36	D₇	—	2	4	no	—
37	C₂³	—	7	2	no	—
38	C₂³	—	14	4	no	—
39	C₂³	—	14	2	no	—
40	C₂³	C₂×D₇	1	1	yes	characteristic
41	C₂³	C₂×D₇	1	1	yes	characteristic, complemented
42	D₄	—	2	2	no	—
43	D₄	—	14	4	no	—
44	C₂×C₄	—	7	2	no	—
45	C₂×C₄	C₂×D₇	1	1	yes	characteristic, complemented
46	C₇	#120	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	2	2	no	—
48	C₂²	—	2	2	no	—
49	C₂²	—	2	1	no	—
50	C₂²	—	7	4	no	—
51	C₂²	—	14	8	no	—
52	C₂²	—	14	2	no	—
53	C₂²	—	14	2	no	—
54	C₂²	C₂²⋊D₇	1	2	yes	complemented
55	C₂²	C₂²×D₇	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	14	2	no	—
58	C₂	—	2	2	no	—
59	C₂	—	4	1	no	—
60	C₂	—	14	4	no	—
61	C₂	D₄×D₇	1	2	yes	—
62	C₂	C₂×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, cosocle	C₂²×C₇	C₂³
Fitting	C₂×C₇×D₄	C₂
hypercenter	C₂×D₄	D₇
hypocenter	C₇	#120
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₇	C₂	semidirect
2. C₂×C₇×D₄	C₂	semidirect
3. C₂×C₂²⋊D₇	C₂	semidirect
4. C₂²⋊Dic₇	C₂	semidirect
5. (C₂×D₇)⋊C₄	C₂	semidirect
7. C₂³×C₇	C₂²	semidirect
12. C₂×C₄×C₇	C₂²	semidirect
14. C₂×Dic₇	C₂²	semidirect
19. C₂²×C₇	D₄	semidirect
24. C₂×D₇	D₄	semidirect
29. C₂×D₄	D₇	semidirect
41. C₂³	C₂×D₇	semidirect
45. C₂×C₄	C₂×D₇	semidirect
46. C₇	#120	semidirect
54. C₂²	C₂²⋊D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₇	2
C₂×C₇×D₄	2
C₂×C₂²⋊D₇	2
C₂²⋊Dic₇	2
(C₂×D₇)⋊C₄	2
#120	7