Subgroups — #90413

Order 448.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#6507	C₂	1	4	yes	maximal
3	#6451	C₂	1	4	yes	maximal
4	#6450	C₂	1	4	yes	maximal
5	C₄×Dic₁₄	C₂	1	2	yes	maximal
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×Dic₁₄	C₂²	1	4	yes	—
8	C₂×Dic₁₄	C₂²	1	2	yes	—
9	C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₇	C₂²	1	2	yes	—
12	Dic₇⋊C₄	C₂²	1	4	yes	—
13	Dic₇⋊C₄	C₂²	1	8	yes	—
14	C₄×Dic₇	C₂²	1	4	yes	—
15	C₄×Dic₇	C₂²	1	4	yes	—
16	Q₈²	—	7	1	no	maximal, Sylow
17	C₇×Q₈	—	4	2	no	—
18	C₂×C₄×C₇	C₂³	1	4	yes	—
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₇	C₂³	1	2	yes	—
21	C₂×Dic₇	C₂³	1	4	yes	—
22	C₂×Dic₇	C₂³	1	4	yes	—
23	Dic₁₄	—	4	4	no	—
24	Dic₁₄	Q₈	1	8	yes	complemented
25	C₄⋊Q₈	—	7	4	no	—
26	C₄⋊Q₈	—	7	4	no	—
27	C₄⋊Q₈	D₇	1	1	yes	characteristic
28	C₄×Q₈	—	7	4	no	—
29	C₄×Q₈	—	7	2	no	—
30	C₂²×C₇	C₂⁴	1	1	yes	characteristic
31	C₄×C₇	—	2	4	no	—
32	C₄×C₇	—	2	1	no	—
33	C₄×C₇	C₂×Q₈	1	4	yes	—
34	Dic₇	—	2	4	no	—
35	Dic₇	C₂×Q₈	1	8	yes	—
36	C₂×Q₈	—	7	2	no	—
37	C₂×Q₈	—	7	4	no	—
38	C₂×Q₈	C₂×D₇	1	2	yes	—
39	C₄⋊C₄	—	7	8	no	—
40	C₄⋊C₄	—	7	4	no	—
41	C₄⋊C₄	—	7	2	no	—
42	C₄⋊C₄	C₂×D₇	1	4	yes	—
43	C₄²	—	7	4	no	—
44	C₄²	—	7	4	no	—
45	C₄²	C₂×D₇	1	1	yes	characteristic
46	C₂×C₇	C₂²×Q₈	1	2	yes	—
47	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	7	8	no	—
50	Q₈	—	28	4	no	—
51	C₂×C₄	—	7	4	no	—
52	C₂×C₄	—	7	4	no	—
53	C₂×C₄	C₂²×D₇	1	4	yes	—
54	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₇	1	2	yes	—
56	C₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
57	C₂²	C₂³×D₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	7	8	no	—
61	C₄	—	14	4	no	—
62	C₄	Q₈×D₇	1	4	yes	—
63	C₂	#6548	1	1	yes	characteristic
64	C₂	C₂×Q₈×D₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	Q₈²
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. Dic₁₄	Q₈	semidirect
56. C₇	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊Q₈	2
#6507	2
#6451	2
#6450	2
C₄×Dic₁₄	2
Q₈²	7