Subgroups — #89999

Order 448.

158 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	C₂×#1143	C₂	1	2	yes	maximal, complemented
3	#6493	C₂	1	1	yes	maximal, characteristic
4	#6409	C₂	1	2	yes	maximal, complemented
5	C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
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×C₂²⋊D₇	—	2	1	no	—
12	C₂×D₂₈	C₂²	1	1	yes	characteristic
13	#1143	—	2	4	no	—
14	(C₂×D₇)⋊C₄	—	2	1	no	—
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	C₂×C₇⋊C₈	C₂²	1	2	yes	—
17	#448	—	7	1	no	maximal, Sylow
18	C₂³×C₇	D₄	1	1	yes	characteristic
19	C₂²×D₇	—	2	1	no	—
20	C₇×Q₈	—	2	4	no	—
21	C₇×Q₈	—	2	2	no	—
22	C₇×Q₈	D₄	1	4	yes	complemented
23	C₂×C₄×C₇	—	2	4	no	—
24	C₂×C₄×C₇	—	2	2	no	—
25	C₂×C₄×C₇	—	2	2	no	—
26	C₂×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂²⋊D₇	—	4	2	no	—
29	C₂×Dic₇	—	2	1	no	—
30	D₂₈	—	2	2	no	—
31	C₇⋊C₈	—	2	2	no	—
32	C₂²×Q₈	D₇	1	1	yes	characteristic, complemented
33	C₂×SD₁₆	—	7	2	no	—
34	#121	—	7	1	no	—
35	#103	—	7	2	no	—
36	C₂²⋊C₈	—	7	1	no	—
37	C₂²×C₇	—	2	1	no	—
38	C₂²×C₇	—	2	1	no	—
39	C₂²×C₇	SD₁₆	1	2	yes	complemented
40	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂×D₇	—	4	1	no	—
42	C₂×D₇	—	4	1	no	—
43	C₂×D₇	—	4	1	no	—
44	C₄×C₇	—	2	4	no	—
45	C₄×C₇	—	2	1	no	—
46	C₄×C₇	C₂×D₄	1	2	yes	—
47	Dic₇	—	4	1	no	—
48	C₂×Q₈	—	2	4	no	—
49	C₂×Q₈	—	2	1	no	—
50	C₂×Q₈	C₂×D₇	1	2	yes	—
51	C₂×D₄	—	7	1	no	—
52	C₂×D₄	—	14	1	no	—
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
55	SD₁₆	—	14	4	no	—
56	C₂×C₈	—	7	2	no	—
57	C₄⋊C₄	—	7	1	no	—
58	C₂²⋊C₄	—	14	1	no	—
59	C₂×C₇	—	2	2	no	—
60	C₂×C₇	#120	1	1	yes	characteristic
61	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₇	#137	1	1	yes	characteristic
63	D₇	—	8	1	no	—
64	C₂³	—	14	1	no	—
65	C₂³	C₂²⋊D₇	1	1	yes	characteristic
66	Q₈	—	2	4	no	—
67	Q₈	—	2	2	no	—
68	Q₈	C₂²⋊D₇	1	4	yes	complemented
69	D₄	—	14	2	no	—
70	D₄	—	28	2	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	14	1	no	—
75	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
77	C₈	—	14	2	no	—
78	C₇	#448	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	28	1	no	—
82	C₂²	—	28	1	no	—
83	C₂²	—	28	1	no	—
84	C₂²	#1143	1	2	yes	complemented
85	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	2	1	no	—
88	C₄	—	28	1	no	—
89	C₄	C₂×C₂²⋊D₇	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	—	56	1	no	—
92	C₂	C₂×#1143	1	1	yes	characteristic
93	C₂	#6505	1	1	yes	characteristic
94	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	#6493	C₂
hypercenter	C₂²×Q₈	D₇
hypocenter	C₇	#448
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
1. C₂²×C₇×Q₈	C₂	semidirect
2. C₂×#1143	C₂	semidirect
4. #6409	C₂	semidirect
5. C₂²⋊(C₇⋊C₈)	C₂	semidirect
22. C₇×Q₈	D₄	semidirect
32. C₂²×Q₈	D₇	semidirect
39. C₂²×C₇	SD₁₆	semidirect
68. Q₈	C₂²⋊D₇	semidirect
78. C₇	#448	semidirect
84. C₂²	#1143	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×#1143	2
#6493	2
#6409	2
C₂²⋊(C₇⋊C₈)	2
#448	7