Subgroups — #90261

Order 448.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₇	C₂	1	2	yes	maximal
3	#6453	C₂	1	6	yes	maximal
4	#6432	C₂	1	3	yes	maximal
5	C₄×Dic₁₄	C₂	1	3	yes	maximal
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₄	C₂²	1	3	yes	—
8	C₇×C₄⋊C₄	C₂²	1	3	yes	—
9	C₄²×C₇	C₂²	1	3	yes	—
10	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₇	C₂²	1	6	yes	—
12	C₄⋊Dic₇	C₂²	1	6	yes	—
13	Dic₇⋊C₄	C₂²	1	6	yes	—
14	C₄×Dic₇	C₂²	1	6	yes	—
15	#557	—	7	1	no	maximal, Sylow
16	C₇×Q₈	Q₈	1	4	yes	complemented
17	C₂×C₄×C₇	C₂³	1	3	yes	—
18	C₂×C₄×C₇	C₂³	1	3	yes	—
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₇	C₂³	1	2	yes	—
21	C₂×Dic₇	C₂³	1	6	yes	—
22	Dic₁₄	—	2	6	no	—
23	C₄⋊Q₈	—	7	3	no	—
24	#125	—	7	6	no	—
25	C₄×Q₈	—	7	2	no	—
26	C₄×Q₈	—	7	3	no	—
27	C₄×Q₈	D₇	1	1	yes	characteristic
28	C₂²×C₇	C₂⁴	1	1	yes	characteristic
29	C₄×C₇	—	2	3	no	—
30	C₄×C₇	C₂×Q₈	1	6	yes	—
31	C₄×C₇	#41	1	2	yes	—
32	Dic₇	—	2	6	no	—
33	Dic₇	—	2	2	no	—
34	C₂×Q₈	—	7	3	no	—
35	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
36	C₄⋊C₄	—	7	6	no	—
37	C₄⋊C₄	—	7	1	no	—
38	C₄⋊C₄	—	7	6	no	—
39	C₄⋊C₄	—	7	6	no	—
40	C₄⋊C₄	C₂×D₇	1	3	yes	—
41	C₄²	—	7	6	no	—
42	C₄²	C₂×D₇	1	3	yes	—
43	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
44	C₂×C₇	C₂×#41	1	1	yes	characteristic
45	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
46	Q₈	—	14	6	no	—
47	Q₈	Dic₁₄	1	4	yes	complemented
48	C₂×C₄	—	7	2	no	—
49	C₂×C₄	—	7	6	no	—
50	C₂×C₄	C₂²×D₇	1	3	yes	—
51	C₂×C₄	C₂²×D₇	1	3	yes	—
52	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
53	C₇	#557	1	1	yes	characteristic, complemented, Sylow
54	C₂²	C₂³×D₇	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	14	6	no	—
57	C₄	—	14	2	no	—
58	C₄	C₂×Dic₁₄	1	6	yes	—
59	C₄	#1161	1	2	yes	—
60	C₂	C₂²×Dic₁₄	1	1	yes	characteristic
61	C₂	C₂×#1161	1	1	yes	characteristic
62	C₂	#6554	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂²×C₇	C₂⁴
Fitting	C₄×C₇×Q₈	C₂
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	#557
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
16. C₇×Q₈	Q₈	semidirect
47. Q₈	Dic₁₄	semidirect
53. C₇	#557	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×Q₈	2
Q₈×Dic₇	2
#6453	2
#6432	2
C₄×Dic₁₄	2
#557	7