Subgroups — #89704

Order 448.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₇	C₂	1	2	yes	maximal
3	C₇⋊#95	C₂	1	4	yes	maximal
4	C₂²×C₄×C₇	C₂²	1	2	yes	—
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂²×Dic₇	C₂²	1	4	yes	—
7	C₄²×C₇	C₄	1	4	yes	complemented
8	C₄×Dic₇	C₄	1	8	yes	complemented
9	#376	—	7	1	no	maximal, Sylow
10	C₂³×C₇	C₂³	1	1	yes	characteristic
11	C₂×C₄×C₇	—	2	4	no	—
12	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
13	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
14	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
15	C₂×Dic₇	—	2	8	no	—
16	C₂×Dic₇	C₂×C₄	1	8	yes	—
17	C₂×C₄²	—	7	2	no	—
18	C₂×C₄²	D₇	1	1	yes	characteristic
19	#95	—	7	4	no	—
20	C₂²×C₇	C₂²×C₄	1	2	yes	—
21	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₇	#41	1	4	yes	—
23	C₄×C₇	—	2	4	no	—
24	C₄×C₇	C₄²	1	4	yes	—
25	Dic₇	—	2	8	no	—
26	C₂²×C₄	—	7	4	no	—
27	C₂²×C₄	C₂×D₇	1	2	yes	—
28	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
29	C₄²	—	7	8	no	—
30	C₄²	Dic₇	1	4	yes	complemented
31	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
32	C₂×C₇	#117	1	4	yes	—
33	C₂×C₇	#117	1	2	yes	—
34	C₂³	C₂²×D₇	1	1	yes	characteristic
35	C₂×C₄	—	2	4	no	—
36	C₂×C₄	—	7	8	no	—
37	C₂×C₄	—	14	8	no	—
38	C₂×C₄	C₄×D₇	1	4	yes	—
39	C₂×C₄	C₂×Dic₇	1	4	yes	—
40	C₂×C₄	C₂×Dic₇	1	2	yes	—
41	C₇	#376	1	1	yes	characteristic, complemented, Sylow
42	C₂²	C₂×C₄×D₇	1	2	yes	—
43	C₂²	#1157	1	4	yes	—
44	C₂²	C₂²×Dic₇	1	1	yes	characteristic
45	C₄	—	2	4	no	—
46	C₄	—	14	8	no	—
47	C₄	C₄×Dic₇	1	4	yes	—
48	C₂	#6435	1	4	yes	—
49	C₂	C₂×C₄×Dic₇	1	1	yes	characteristic
50	C₂	#6489	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₇
commutator	C₂×C₇	C₂×C₄²
Frattini	C₂³	C₂²×D₇
Fitting	C₂×C₄²×C₇	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂³
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	#376
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
7. C₄²×C₇	C₄	semidirect
8. C₄×Dic₇	C₄	semidirect
30. C₄²	Dic₇	semidirect
41. C₇	#376	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₇	2
C₂×C₄×Dic₇	2
C₇⋊#95	2
#376	7