Subgroups — #90001

Order 448.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇⋊C₄	C₂	1	3	yes	maximal
3	C₇⋊#95	C₂	1	3	yes	maximal
4	C₂×C₇×Q₈	—	2	6	no	—
5	C₂²×C₄×C₇	C₂²	1	3	yes	—
6	C₂²×Dic₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₇	C₂²	1	3	yes	—
8	Dic₇⋊C₄	—	2	6	no	—
9	#395	—	7	1	no	maximal, Sylow
10	C₂³×C₇	C₂³	1	1	yes	characteristic
11	C₇×Q₈	—	2	8	no	—
12	C₂×C₄×C₇	—	2	6	no	—
13	C₂×C₄×C₇	D₄	1	6	yes	—
14	C₂×Dic₇	—	2	3	no	—
15	C₂×Dic₇	—	2	3	no	—
16	C₂×Dic₇	—	2	3	no	—
17	C₂×Dic₇	Q₈	1	6	yes	complemented
18	C₂²×Q₈	D₇	1	1	yes	characteristic
19	C₂×C₄⋊C₄	—	7	3	no	—
20	#95	—	7	3	no	—
21	C₂²×C₇	C₂×D₄	1	3	yes	—
22	C₂²×C₇	C₂×Q₈	1	3	yes	—
23	C₂²×C₇	#41	1	1	yes	characteristic
24	C₄×C₇	—	2	6	no	—
25	Dic₇	—	2	6	no	—
26	Dic₇	—	4	1	no	—
27	C₂×Q₈	—	2	6	no	—
28	C₂²×C₄	—	7	3	no	—
29	C₂²×C₄	—	7	1	no	—
30	C₂²×C₄	C₂×D₇	1	3	yes	—
31	C₄⋊C₄	—	14	6	no	—
32	C₂×C₇	#120	1	1	yes	characteristic
33	C₂×C₇	C₂²⋊Q₈	1	3	yes	—
34	C₂×C₇	C₄⋊Q₈	1	3	yes	—
35	C₂³	C₂²×D₇	1	1	yes	characteristic
36	Q₈	—	2	8	no	—
37	C₂×C₄	—	2	6	no	—
38	C₂×C₄	—	7	6	no	—
39	C₂×C₄	—	14	3	no	—
40	C₂×C₄	—	14	3	no	—
41	C₂×C₄	—	14	3	no	—
42	C₂×C₄	C₂²⋊D₇	1	6	yes	—
43	C₇	#395	1	1	yes	characteristic, complemented, Sylow
44	C₂²	Q₈×D₇	1	3	yes	—
45	C₂²	#1161	1	1	yes	characteristic
46	C₂²	C₂×C₂²⋊D₇	1	3	yes	—
47	C₄	—	2	6	no	—
48	C₄	—	14	6	no	—
49	C₄	—	28	1	no	—
50	C₂	#6507	1	3	yes	—
51	C₂	#6509	1	3	yes	—
52	C₂	#6516	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂³
Fitting	C₂²×C₇×Q₈	C₂
hypercenter	C₂²×Q₈	D₇
hypocenter	C₇	#395
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

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

N	H ≅ G/N	type
17. C₂×Dic₇	Q₈	semidirect
43. C₇	#395	semidirect

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