Subgroups — #6459

Order 224.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₇⋊C₄	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	C₄×D₇	—	2	2	no	—
16	Dic₁₄	—	2	2	no	—
17	C₂²⋊Q₈	—	7	1	no	maximal, Sylow
18	C₂²×C₇	C₂³	1	1	yes	characteristic
19	C₂×D₇	—	2	1	no	—
20	C₂×D₇	—	2	1	no	—
21	C₂×D₇	Q₈	1	2	yes	complemented
22	C₄×C₇	—	2	1	no	—
23	C₄×C₇	—	2	1	no	—
24	C₄×C₇	—	2	1	no	—
25	Dic₇	—	2	1	no	—
26	Dic₇	—	2	1	no	—
27	Dic₇	D₄	1	2	yes	—
28	C₂×Q₈	—	7	1	no	—
29	C₂²×C₄	—	7	1	no	—
30	C₄⋊C₄	—	7	1	no	—
31	C₄⋊C₄	—	7	1	no	—
32	C₄⋊C₄	D₇	1	1	yes	characteristic, complemented
33	C₂²⋊C₄	—	7	1	no	—
34	C₂²⋊C₄	—	7	1	no	—
35	C₂×C₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₇	#41	1	1	yes	characteristic
38	D₇	—	2	2	no	—
39	C₂³	—	7	1	no	—
40	Q₈	—	14	2	no	—
41	C₂×C₄	—	7	1	no	—
42	C₂×C₄	—	7	1	no	—
43	C₂×C₄	—	7	1	no	—
44	C₂×C₄	—	14	2	no	—
45	C₂×C₄	C₂×D₇	1	1	yes	characteristic
46	C₂×C₄	C₂×D₇	1	1	yes	characteristic
47	C₂×C₄	C₂×D₇	1	1	yes	characteristic
48	C₇	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	7	2	no	—
50	C₂²	—	14	1	no	—
51	C₂²	—	14	1	no	—
52	C₂²	C₂²×D₇	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	2	1	no	—
56	C₄	—	7	2	no	—
57	C₄	—	14	1	no	—
58	C₄	—	14	1	no	—
59	C₂	—	14	2	no	—
60	C₂	#1157	1	1	yes	characteristic
61	C₂	D₄×D₇	1	1	yes	characteristic
62	C₂	Q₈×D₇	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	C₂²⋊Q₈
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
2. C₂×Dic₁₄	C₂	semidirect
3. C₇×C₄⋊C₄	C₂	semidirect
6. Dic₇⋊C₄	C₂	semidirect
7. Dic₇⋊C₄	C₂	semidirect
21. C₂×D₇	Q₈	semidirect
32. C₄⋊C₄	D₇	semidirect
48. C₇	C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇	2
C₂×Dic₁₄	2
C₇×C₄⋊C₄	2
(C₂×D₇)⋊C₄	2
Dic₇⋊C₄	2
C₂²⋊Q₈	7