Subgroups — #6446

Order 224.

78 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
2	C₂×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊Dic₇	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	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂²⋊D₇	—	2	2	no	—
13	C₂×Dic₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₇	C₂²	1	1	yes	characteristic
16	C₄×D₇	—	2	2	no	—
17	#123	—	7	1	no	maximal, Sylow
18	C₂²×C₇	—	2	1	no	—
19	C₂²×C₇	—	2	1	no	—
20	C₂²×C₇	—	2	1	no	—
21	C₂²×C₇	C₂³	1	1	yes	characteristic
22	C₂×D₇	—	2	1	no	—
23	C₂×D₇	—	2	1	no	—
24	C₂×D₇	D₄	1	2	yes	—
25	C₄×C₇	—	2	1	no	—
26	C₄×C₇	—	2	1	no	—
27	Dic₇	—	2	1	no	—
28	Dic₇	—	2	1	no	—
29	Dic₇	—	2	1	no	—
30	C₂×D₄	—	7	1	no	—
31	C₂²×C₄	—	7	1	no	—
32	C₄⋊C₄	—	7	1	no	—
33	C₄⋊C₄	—	7	1	no	—
34	C₂²⋊C₄	—	7	1	no	—
35	C₂²⋊C₄	—	7	1	no	—
36	C₂²⋊C₄	D₇	1	1	yes	characteristic, complemented
37	C₂×C₇	—	4	1	no	—
38	C₂×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₇	#41	1	1	yes	characteristic
40	C₂×C₇	#41	1	1	yes	characteristic
41	D₇	—	2	2	no	—
42	C₂³	—	7	1	no	—
43	C₂³	C₂×D₇	1	1	yes	characteristic
44	D₄	—	14	2	no	—
45	C₂×C₄	—	7	1	no	—
46	C₂×C₄	—	7	1	no	—
47	C₂×C₄	—	7	1	no	—
48	C₂×C₄	—	14	2	no	—
49	C₂×C₄	C₂×D₇	1	1	yes	characteristic
50	C₂×C₄	C₂×D₇	1	1	yes	characteristic
51	C₇	#123	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	7	2	no	—
56	C₂²	—	14	1	no	—
57	C₂²	—	14	1	no	—
58	C₂²	C₂²×D₇	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	14	1	no	—
62	C₄	—	14	1	no	—
63	C₄	—	14	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	14	2	no	—
66	C₂	#1157	1	1	yes	characteristic
67	C₂	D₄×D₇	1	1	yes	characteristic
68	C₂	#1159	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, cosocle	C₂²×C₇	C₂³
Fitting	C₇×C₂²⋊C₄	C₂
prime radical	C₂×C₂²⋊D₇	C₂
hypercenter	C₂²⋊C₄	D₇
hypocenter	C₇	#123
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
2. C₂×C₄×D₇	C₂	semidirect
3. C₇×C₂²⋊C₄	C₂	semidirect
4. C₂²⋊Dic₇	C₂	semidirect
5. (C₂×D₇)⋊C₄	C₂	semidirect
6. C₄⋊Dic₇	C₂	semidirect
7. Dic₇⋊C₄	C₂	semidirect
36. C₂²⋊C₄	D₇	semidirect
51. C₇	#123	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×C₄×D₇	2
C₇×C₂²⋊C₄	2
C₂²⋊Dic₇	2
(C₂×D₇)⋊C₄	2
C₄⋊Dic₇	2
Dic₇⋊C₄	2
#123	7