Subgroups — #89997

Order 448.

150 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	#6489	C₂	1	1	yes	maximal, characteristic
3	C₂×#1136	C₂	1	1	yes	maximal, characteristic
4	#6409	C₂	1	4	yes	maximal, complemented
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	C₂²	1	2	yes	—
13	C₄×Dic₇	—	2	1	no	—
14	#1136	—	2	2	no	—
15	C₂×C₇⋊C₈	C₂²	1	2	yes	—
16	#419	—	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	—
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	C₇⋊C₈	—	2	2	no	—
29	C₂²×Q₈	D₇	1	1	yes	characteristic
30	C₂×M₁₆	—	7	1	no	—
31	#117	—	7	1	no	—
32	#103	—	7	4	no	—
33	C₂²×C₇	—	2	2	no	—
34	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₇	—	2	4	no	—
37	C₄×C₇	C₂²⋊C₄	1	2	yes	—
38	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	Dic₇	—	4	2	no	—
41	C₂×Q₈	—	2	2	no	—
42	C₂×Q₈	—	2	1	no	—
43	C₂×Q₈	Dic₇	1	4	yes	complemented
44	C₂×Q₈	C₂×D₇	1	2	yes	—
45	C₂²×C₄	—	2	1	no	—
46	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
47	M₁₆	—	14	2	no	—
48	C₂×C₈	—	7	2	no	—
49	C₄⋊C₄	—	7	2	no	—
50	C₂²⋊C₄	—	14	1	no	—
51	C₄²	—	14	1	no	—
52	C₂×C₇	—	2	2	no	—
53	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₇	#137	1	2	yes	—
55	C₂³	C₂²⋊D₇	1	1	yes	characteristic
56	Q₈	—	2	4	no	—
57	Q₈	—	2	2	no	—
58	Q₈	C₂×Dic₇	1	4	yes	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	14	2	no	—
62	C₂×C₄	C₂×Dic₇	1	2	yes	—
63	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₇	1	2	yes	—
65	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
66	C₈	—	14	2	no	—
67	C₇	#419	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	2	2	no	—
69	C₂²	C₂²⋊Dic₇	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
71	C₄	—	2	4	no	—
72	C₄	—	28	2	no	—
73	C₄	C₂²⋊Dic₇	1	2	yes	—
74	C₄	C₂²×Dic₇	1	1	yes	characteristic
75	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	#6505	1	2	yes	—
78	C₂	C₂×C₂²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂²×C₇×Q₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²×Q₈	D₇
hypocenter	C₇	#419
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
4. #6409	C₂	semidirect
7. C₂×C₇×Q₈	C₄	semidirect
43. C₂×Q₈	Dic₇	semidirect
67. C₇	#419	semidirect

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
#6489	2
C₂×#1136	2
#6409	2
#419	7