Subgroups — #90000

Order 448.

148 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×#1144	C₂	1	2	yes	maximal, complemented
3	#6487	C₂	1	1	yes	maximal, characteristic
4	#6409	C₂	1	2	yes	maximal, complemented
5	C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	—	2	4	no	—
7	C₂×C₇×Q₈	—	2	1	no	—
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	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₇	—	2	1	no	—
13	#1144	—	2	4	no	—
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	Dic₇⋊C₄	—	2	1	no	—
16	C₂×C₇⋊C₈	C₂²	1	2	yes	—
17	#451	—	7	1	no	maximal, Sylow
18	C₂³×C₇	D₄	1	1	yes	characteristic
19	C₇×Q₈	—	2	4	no	—
20	C₇×Q₈	—	2	2	no	—
21	C₇×Q₈	D₄	1	4	yes	—
22	C₂×C₄×C₇	—	2	4	no	—
23	C₂×C₄×C₇	—	2	2	no	—
24	C₂×C₄×C₇	—	2	2	no	—
25	C₂×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇	—	2	1	no	—
28	C₂×Dic₇	—	2	1	no	—
29	Dic₁₄	—	2	2	no	—
30	C₇⋊C₈	—	2	2	no	—
31	C₂²×Q₈	D₇	1	1	yes	characteristic
32	C₂×Q₁₆	—	7	2	no	—
33	C₂²⋊Q₈	—	7	1	no	—
34	#103	—	7	2	no	—
35	C₂²⋊C₈	—	7	1	no	—
36	C₂²×C₇	—	2	1	no	—
37	C₂²×C₇	—	2	1	no	—
38	C₂²×C₇	Q₁₆	1	2	yes	complemented
39	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₇	—	2	4	no	—
41	C₄×C₇	—	2	1	no	—
42	C₄×C₇	C₂×D₄	1	2	yes	—
43	Dic₇	—	4	1	no	—
44	Dic₇	—	4	1	no	—
45	C₂×Q₈	—	2	4	no	—
46	C₂×Q₈	—	2	1	no	—
47	C₂×Q₈	—	7	1	no	—
48	C₂×Q₈	C₂×D₇	1	2	yes	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
51	Q₁₆	—	14	4	no	—
52	C₂×C₈	—	7	2	no	—
53	C₄⋊C₄	—	7	1	no	—
54	C₄⋊C₄	—	14	1	no	—
55	C₂²⋊C₄	—	14	1	no	—
56	C₂×C₇	—	2	2	no	—
57	C₂×C₇	#120	1	1	yes	characteristic
58	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×C₇	#137	1	1	yes	characteristic
60	C₂³	C₂²⋊D₇	1	1	yes	characteristic
61	Q₈	—	2	4	no	—
62	Q₈	—	2	2	no	—
63	Q₈	—	14	2	no	—
64	Q₈	C₂²⋊D₇	1	4	yes	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	14	1	no	—
69	C₂×C₄	—	14	1	no	—
70	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
72	C₈	—	14	2	no	—
73	C₇	#451	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	#1144	1	2	yes	complemented
77	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	2	1	no	—
80	C₄	—	28	1	no	—
81	C₄	—	28	1	no	—
82	C₄	C₂×C₂²⋊D₇	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#6505	1	1	yes	characteristic
85	C₂	C₂×#1144	1	1	yes	characteristic
86	C₂	#6516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂²×C₇×Q₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
hypercenter	C₂²×Q₈	D₇
hypocenter	C₇	#451
LCS	G ▷ C₂×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
2. C₂×#1144	C₂	semidirect
4. #6409	C₂	semidirect
38. C₂²×C₇	Q₁₆	semidirect
73. C₇	#451	semidirect
76. C₂²	#1144	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
C₂×#1144	2
#6487	2
#6409	2
C₂²⋊(C₇⋊C₈)	2
#451	7