Subgroups — #89708

Order 448.

154 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	—	2	4	no	—
9	#382	—	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	2	yes	—
14	C₂×C₄×C₇	D₄	1	2	yes	—
15	C₂×C₄×C₇	Q₈	1	2	yes	—
16	C₂×Dic₇	—	2	4	no	—
17	C₂×Dic₇	—	2	8	no	—
18	C₂×C₄⋊C₄	—	7	2	no	—
19	C₂×C₄²	D₇	1	1	yes	characteristic
20	#95	—	7	4	no	—
21	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₇	#41	1	4	yes	—
25	C₄×C₇	—	2	4	no	—
26	C₄×C₇	C₄⋊C₄	1	4	yes	—
27	Dic₇	—	4	4	no	—
28	C₂²×C₄	—	7	4	no	—
29	C₂²×C₄	C₂×D₇	1	2	yes	—
30	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
31	C₄⋊C₄	—	14	4	no	—
32	C₄²	Dic₇	1	4	yes	complemented
33	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₇	#117	1	2	yes	—
35	C₂×C₇	#124	1	2	yes	—
36	C₂×C₇	#125	1	2	yes	—
37	C₂³	C₂²×D₇	1	1	yes	characteristic
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	—	14	4	no	—
40	C₂×C₄	—	14	8	no	—
41	C₂×C₄	Dic₁₄	1	2	yes	—
42	C₂×C₄	D₂₈	1	2	yes	—
43	C₂×C₄	C₂×Dic₇	1	4	yes	—
44	C₂×C₄	C₂×Dic₇	1	2	yes	—
45	C₇	#382	1	1	yes	characteristic, complemented, Sylow
46	C₂²	C₂×Dic₁₄	1	1	yes	characteristic
47	C₂²	C₂×D₂₈	1	1	yes	characteristic
48	C₂²	#1157	1	4	yes	—
49	C₂²	C₂²×Dic₇	1	1	yes	characteristic
50	C₄	—	2	4	no	—
51	C₄	—	28	4	no	—
52	C₄	C₄⋊Dic₇	1	4	yes	—
53	C₂	#6433	1	2	yes	—
54	C₂	#6438	1	2	yes	—
55	C₂	C₂×C₄⋊Dic₇	1	1	yes	characteristic
56	C₂	#6489	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₄
Fitting	C₂×C₄²×C₇	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂³
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	#382
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
7. C₄²×C₇	C₄	semidirect
32. C₄²	Dic₇	semidirect
45. C₇	#382	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
#382	7