Subgroups — #6461

Order 224.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₇	C₂²	1	1	yes	characteristic
15	#126	—	7	1	no	maximal, Sylow
16	C₂²×C₇	C₂³	1	1	yes	characteristic
17	C₂×D₇	—	2	1	no	—
18	C₂×D₇	—	2	1	no	—
19	C₂×D₇	—	2	1	no	—
20	C₄×C₇	—	2	1	no	—
21	C₄×C₇	—	2	1	no	—
22	C₄×C₇	—	2	1	no	—
23	Dic₇	—	2	1	no	—
24	Dic₇	—	2	1	no	—
25	Dic₇	—	2	1	no	—
26	C₄⋊C₄	—	7	1	no	—
27	C₄⋊C₄	—	7	1	no	—
28	C₄⋊C₄	D₇	1	1	yes	characteristic, complemented
29	C₂²⋊C₄	—	7	1	no	—
30	C₂²⋊C₄	—	7	1	no	—
31	C₂²⋊C₄	—	7	1	no	—
32	C₄²	—	7	1	no	—
33	C₂×C₇	#41	1	1	yes	characteristic
34	C₂×C₇	#41	1	1	yes	characteristic
35	C₂×C₇	#41	1	1	yes	characteristic
36	D₇	—	4	1	no	—
37	C₂³	—	7	1	no	—
38	C₂×C₄	—	7	1	no	—
39	C₂×C₄	—	7	1	no	—
40	C₂×C₄	—	7	1	no	—
41	C₂×C₄	C₂×D₇	1	1	yes	characteristic
42	C₂×C₄	C₂×D₇	1	1	yes	characteristic
43	C₂×C₄	C₂×D₇	1	1	yes	characteristic
44	C₇	#126	1	1	yes	characteristic, complemented, Sylow
45	C₂²	—	14	1	no	—
46	C₂²	—	14	1	no	—
47	C₂²	—	14	1	no	—
48	C₂²	C₂²×D₇	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	14	1	no	—
53	C₄	—	14	1	no	—
54	C₄	—	14	1	no	—
55	C₂	—	28	1	no	—
56	C₂	#1157	1	1	yes	characteristic
57	C₂	#1159	1	1	yes	characteristic
58	C₂	#1161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, cosocle	C₂²×C₇	C₂³
Fitting	C₇×C₄⋊C₄	C₂
prime radical	C₂²×D₇	C₂²
hypercenter	C₄⋊C₄	D₇
hypocenter	C₇	#126
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₇×C₄⋊C₄	C₂	semidirect
5. C₄⋊Dic₇	C₂	semidirect
6. Dic₇⋊C₄	C₂	semidirect
7. C₄×Dic₇	C₂	semidirect
28. C₄⋊C₄	D₇	semidirect
44. C₇	#126	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊C₄	2
(C₂×D₇)⋊C₄	2
C₄⋊Dic₇	2
Dic₇⋊C₄	2
C₄×Dic₇	2
#126	7