Subgroups — C₂×Q₈×Dic₇ #90502

Order 448.

298 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	Q₈×Dic₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₇	C₂	1	3	yes	maximal
4	C₂×C₄×Dic₇	C₂	1	3	yes	maximal
5	C₂×C₇×Q₈	C₄	1	8	yes	complemented
6	C₂×C₇×Q₈	C₂²	1	4	yes	—
7	C₂²×C₄×C₇	C₂²	1	3	yes	—
8	C₂²×Dic₇	C₂²	1	3	yes	—
9	C₂²×Dic₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₇	C₂²	1	12	yes	—
11	C₄×Dic₇	C₂²	1	12	yes	—
12	C₂×C₄×Q₈	—	7	1	no	maximal, Sylow
13	C₂³×C₇	C₂³	1	1	yes	characteristic
14	C₇×Q₈	C₂×C₄	1	16	yes	complemented
15	C₂×C₄×C₇	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₇	C₂³	1	6	yes	—
17	C₂×Dic₇	—	2	6	no	—
18	C₂×Dic₇	Q₈	1	4	yes	direct product factor
19	C₂×Dic₇	C₂³	1	6	yes	—
20	C₂×Dic₇	C₂³	1	2	yes	—
21	C₂²×Q₈	D₇	1	1	yes	characteristic
22	C₄×Q₈	—	7	8	no	—
23	C₂×C₄⋊C₄	—	7	3	no	—
24	C₂×C₄²	—	7	3	no	—
25	C₂²×C₇	C₂²×C₄	1	2	yes	—
26	C₂²×C₇	C₂×Q₈	1	2	yes	—
27	C₂²×C₇	#41	1	2	yes	—
28	C₂²×C₇	C₂⁴	1	1	yes	characteristic
29	C₄×C₇	C₂²×C₄	1	12	yes	—
30	Dic₇	—	2	6	no	—
31	Dic₇	C₂×Q₈	1	4	yes	direct product factor
32	C₂×Q₈	Dic₇	1	8	yes	direct product factor
33	C₂×Q₈	C₂×D₇	1	4	yes	—
34	C₂²×C₄	—	7	1	no	—
35	C₂²×C₄	—	7	3	no	—
36	C₂²×C₄	C₂×D₇	1	3	yes	—
37	C₄⋊C₄	—	7	12	no	—
38	C₄²	—	7	12	no	—
39	C₂×C₇	C₄×Q₈	1	4	yes	complemented
40	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₇	C₂×#41	1	1	yes	characteristic
43	C₂³	C₂²×D₇	1	1	yes	characteristic
44	Q₈	C₂×Dic₇	1	16	yes	direct product factor
45	C₂×C₄	—	7	4	no	—
46	C₂×C₄	—	7	2	no	—
47	C₂×C₄	—	7	6	no	—
48	C₂×C₄	—	14	6	no	—
49	C₂×C₄	C₂×Dic₇	1	12	yes	—
50	C₂×C₄	C₂²×D₇	1	6	yes	—
51	C₇	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
52	C₂²	Q₈×D₇	1	2	yes	—
53	C₂²	#1161	1	2	yes	—
54	C₂²	C₂²×Dic₇	1	2	yes	—
55	C₂²	C₂³×D₇	1	1	yes	characteristic
56	C₄	—	7	4	no	—
57	C₄	—	14	6	no	—
58	C₄	C₂²×Dic₇	1	12	yes	—
59	C₂	Q₈×Dic₇	1	4	yes	direct product factor
60	C₂	C₂×Q₈×D₇	1	1	yes	characteristic
61	C₂	C₂×#1161	1	1	yes	characteristic
62	C₂	C₂³×Dic₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₇	C₂	direct (N × H)
5. C₂×C₇×Q₈	C₄	semidirect
14. C₇×Q₈	C₂×C₄	semidirect
18. C₂×Dic₇	Q₈	direct (N × H)
31. Dic₇	C₂×Q₈	direct (N × H)
32. C₂×Q₈	Dic₇	direct (N × H)
39. C₂×C₇	C₄×Q₈	semidirect
44. Q₈	C₂×Dic₇	direct (N × H)
51. C₇	C₂×C₄×Q₈	semidirect
59. C₂	Q₈×Dic₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×Q₈	2
Q₈×Dic₇	2
C₂×C₄⋊Dic₇	2
C₂×C₄×Dic₇	2
C₂×C₄×Q₈	7

Subgroups — C2×Q8×Dic7 #90502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₇ #90502