Subgroups — C₂×#6510 #90505

Order 448.

330 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, complemented
2	C₂²×D₂₈	C₂	1	1	yes	maximal, characteristic
3	#6510	C₂	1	8	yes	maximal, direct product factor
4	C₂×(C₂×D₇)⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₇	C₂²	1	2	yes	—
7	C₂×C₇×Q₈	—	2	4	no	—
8	C₂×C₇×Q₈	C₂²	1	4	yes	complemented
9	C₂²×C₄×C₇	C₂²	1	2	yes	—
10	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₇	C₂²	1	2	yes	—
12	C₂×D₂₈	—	2	4	no	—
13	C₂×D₂₈	C₂²	1	4	yes	—
14	(C₂×D₇)⋊C₄	C₂²	1	16	yes	complemented
15	C₄×Dic₇	C₂²	1	4	yes	complemented
16	C₂×#124	—	7	1	no	maximal, Sylow
17	C₂³×C₇	C₂³	1	1	yes	characteristic
18	C₂²×D₇	—	2	4	no	—
19	C₂²×D₇	—	2	8	no	—
20	C₂²×D₇	C₂³	1	4	yes	—
21	C₇×Q₈	—	2	8	no	—
22	C₂×C₄×C₇	—	2	4	no	—
23	C₂×C₄×C₇	D₄	1	4	yes	—
24	C₂×C₄×C₇	C₂³	1	4	yes	—
25	C₂×C₄×C₇	C₂³	1	2	yes	—
26	C₂×Dic₇	—	2	4	no	—
27	C₂×Dic₇	C₂³	1	4	yes	—
28	D₂₈	—	2	8	no	—
29	C₂²×Q₈	D₇	1	1	yes	characteristic, complemented
30	C₂²×D₄	—	7	1	no	—
31	#124	—	7	8	no	—
32	C₂×C₂²⋊C₄	—	7	4	no	—
33	C₂×C₄²	—	7	1	no	—
34	C₂²×C₇	C₂×D₄	1	2	yes	—
35	C₂²×C₇	#41	1	4	yes	—
36	C₂²×C₇	C₂⁴	1	1	yes	characteristic
37	C₂×D₇	—	2	4	no	—
38	C₂×D₇	—	2	8	no	—
39	C₂×D₇	—	4	8	no	—
40	C₄×C₇	—	2	4	no	—
41	C₄×C₇	C₂×D₄	1	4	yes	—
42	Dic₇	—	2	4	no	—
43	C₂⁴	—	7	2	no	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	C₂×D₇	1	4	yes	complemented
46	C₂×D₄	—	7	4	no	—
47	C₂×D₄	—	14	4	no	—
48	C₂²×C₄	—	7	2	no	—
49	C₂²×C₄	C₂×D₇	1	2	yes	—
50	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
51	C₂²⋊C₄	—	7	16	no	—
52	C₄²	—	7	4	no	—
53	C₂×C₇	#124	1	4	yes	complemented
54	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₇	C₂×#41	1	2	yes	—
56	D₇	—	4	4	no	—
57	C₂³	—	7	4	no	—
58	C₂³	—	14	8	no	—
59	C₂³	—	14	4	no	—
60	C₂³	C₂²×D₇	1	1	yes	characteristic
61	Q₈	—	2	8	no	—
62	D₄	—	14	8	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	7	4	no	—
65	C₂×C₄	—	14	4	no	—
66	C₂×C₄	C₂²⋊D₇	1	4	yes	—
67	C₂×C₄	C₂²×D₇	1	4	yes	—
68	C₂×C₄	C₂²×D₇	1	2	yes	—
69	C₇	C₂×#124	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	14	8	no	—
71	C₂²	—	14	4	no	—
72	C₂²	—	28	8	no	—
73	C₂²	#1161	1	4	yes	—
74	C₂²	C₂×C₂²⋊D₇	1	2	yes	—
75	C₂²	C₂³×D₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	14	4	no	—
78	C₄	C₂×C₂²⋊D₇	1	4	yes	—
79	C₂	—	28	4	no	—
80	C₂	#6510	1	4	yes	direct product factor
81	C₂	C₂×#1161	1	2	yes	—
82	C₂	C₂²×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴
Frattini	C₂²	C₂³×D₇
Fitting	C₂²×C₇×Q₈	C₂
socle	C₂³×C₇	C₂³
prime radical	C₂²×D₂₈	C₂
hypercenter	C₂²×Q₈	D₇
hypocenter	C₇	C₂×#124
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×Q₈	C₂	semidirect
3. #6510	C₂	direct (N × H)
4. C₂×(C₂×D₇)⋊C₄	C₂	semidirect
5. C₂×C₄×Dic₇	C₂	semidirect
8. C₂×C₇×Q₈	C₂²	semidirect
14. (C₂×D₇)⋊C₄	C₂²	semidirect
15. C₄×Dic₇	C₂²	semidirect
29. C₂²×Q₈	D₇	semidirect
45. C₂×Q₈	C₂×D₇	semidirect
53. C₂×C₇	#124	semidirect
69. C₇	C₂×#124	semidirect
80. C₂	#6510	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
C₂²×D₂₈	2
#6510	2
C₂×(C₂×D₇)⋊C₄	2
C₂×C₄×Dic₇	2
C₂×#124	7

Subgroups — C2×#6510 #90505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6510 #90505