Subgroups — C₂×#6409 #89996

Order 448.

162 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×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#6409	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×Q₈	—	2	2	no	—
6	C₂×C₇×Q₈	—	2	1	no	—
7	C₂×C₇×Q₈	C₄	1	4	yes	complemented
8	C₂×C₇×Q₈	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	—	2	1	no	—
10	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₇	—	2	1	no	—
12	C₄⋊Dic₇	—	2	1	no	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	C₂×C₇⋊C₈	—	2	2	no	—
15	C₂×C₇⋊C₈	C₂²	1	2	yes	—
16	C₂×#103	—	7	1	no	maximal, Sylow
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₇×Q₈	—	2	4	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₇×Q₈	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₇	—	2	4	no	—
22	C₂×C₄×C₇	—	2	2	no	—
23	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₇	D₄	1	2	yes	—
26	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇	—	2	2	no	—
28	C₂×Dic₇	—	4	2	no	—
29	C₇⋊C₈	—	2	2	no	—
30	C₂²×Q₈	D₇	1	1	yes	characteristic
31	C₂²×C₈	—	7	1	no	—
32	C₂×C₄⋊C₄	—	7	1	no	—
33	#103	—	7	4	no	—
34	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₇	SD₁₆	1	2	yes	—
36	C₂²×C₇	Q₁₆	1	2	yes	—
37	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₇	—	2	4	no	—
39	C₄×C₇	C₂²⋊C₄	1	2	yes	—
40	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	Dic₇	—	4	2	no	—
43	C₂×Q₈	—	2	2	no	—
44	C₂×Q₈	—	2	1	no	—
45	C₂×Q₈	Dic₇	1	4	yes	complemented
46	C₂×Q₈	C₂×D₇	1	2	yes	—
47	C₂²×C₄	—	2	1	no	—
48	C₂²×C₄	—	14	1	no	—
49	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
50	C₂×C₈	—	7	2	no	—
51	C₂×C₈	—	14	2	no	—
52	C₄⋊C₄	—	7	2	no	—
53	C₄⋊C₄	—	14	1	no	—
54	C₂×C₇	#103	1	4	yes	complemented
55	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂³	C₂²⋊D₇	1	1	yes	characteristic
59	Q₈	—	2	4	no	—
60	Q₈	—	2	2	no	—
61	Q₈	C₂×Dic₇	1	4	yes	complemented
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	14	2	no	—
65	C₂×C₄	—	28	2	no	—
66	C₂×C₄	C₂×Dic₇	1	2	yes	—
67	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₇	1	2	yes	—
69	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
70	C₈	—	14	2	no	—
71	C₇	C₂×#103	1	1	yes	characteristic, complemented, Sylow
72	C₂²	#1143	1	2	yes	—
73	C₂²	#1144	1	2	yes	—
74	C₂²	C₂²⋊Dic₇	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	28	2	no	—
78	C₄	C₂²⋊Dic₇	1	2	yes	—
79	C₄	C₂²×Dic₇	1	1	yes	characteristic
80	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
81	C₂	#6409	1	4	yes	direct product factor
82	C₂	C₂×#1143	1	1	yes	characteristic
83	C₂	C₂×#1144	1	1	yes	characteristic
84	C₂	C₂×C₂²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂²×C₇×Q₈	C₂
socle, prime radical	C₂³×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²×Q₈	D₇
hypocenter	C₇	C₂×#103
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
4. #6409	C₂	direct (N × H)
7. C₂×C₇×Q₈	C₄	semidirect
20. C₇×Q₈	C₂×C₄	semidirect
45. C₂×Q₈	Dic₇	semidirect
54. C₂×C₇	#103	semidirect
61. Q₈	C₂×Dic₇	semidirect
71. C₇	C₂×#103	semidirect
81. C₂	#6409	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
C₂×C₄⋊Dic₇	2
C₂²×C₇⋊C₈	2
#6409	2
C₂×#103	7

Subgroups — C2×#6409 #89996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6409 #89996