Group #22590495 — subgroups

Subgroups — #22590495

Order 9088.

136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986400	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985826	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985825	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985800	C₂	1	1	yes	maximal, characteristic
7	#357×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556119	—	2	1	no	—
9	D₄×Dic₇₁	—	2	1	no	—
10	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14556062	C₂²	1	1	yes	characteristic
12	#14556062	C₂²	1	1	yes	characteristic
13	#14556059	—	2	1	no	—
14	#14556049	—	2	1	no	—
15	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	#107×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×Dic₇₁	—	4	1	no	—
20	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊Dic₇₁	—	2	1	no	—
22	C₄⋊Dic₇₁	—	2	1	no	—
23	C₄⋊Dic₇₁	—	4	1	no	—
24	Dic₇₁⋊C₄	—	4	1	no	—
25	C₂²⋊Dic₇₁	—	4	1	no	—
26	C₄×Dic₇₁	D₄	1	1	yes	characteristic
27	C₇₁⋊C₁₆	—	2	1	no	—
28	C₂×D₄×C₇₁	—	2	1	no	—
29	D₈×C₇₁	—	2	1	no	—
30	D₈×C₇₁	Q₈	1	2	yes	—
31	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₁₆×C₇₁	—	2	1	no	—
34	C₂³×C₇₁	—	4	1	no	—
35	D₄×C₇₁	—	2	2	no	—
36	D₄×C₇₁	—	4	1	no	—
37	C₂×C₄×C₇₁	—	4	1	no	—
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₇₁	—	4	1	no	—
40	C₂×Dic₇₁	—	4	1	no	—
41	C₂×Dic₇₁	—	8	2	no	—
42	C₂×Dic₇₁	D₈	1	1	yes	characteristic
43	C₂×Dic₇₁	D₈	1	1	yes	characteristic
44	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₇₁	#41	1	1	yes	characteristic
46	C₇₁⋊C₈	—	2	1	no	—
47	C₂²×C₇₁	—	4	2	no	—
48	C₂²×C₇₁	—	8	1	no	—
49	C₂²×C₇₁	—	8	1	no	—
50	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
51	C₄×C₇₁	—	8	1	no	—
52	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₇₁	#137	1	1	yes	characteristic
54	Dic₇₁	—	2	1	no	—
55	Dic₇₁	—	2	1	no	—
56	Dic₇₁	—	8	1	no	—
57	Dic₇₁	—	8	1	no	—
58	C₂×C₇₁	—	8	2	no	—
59	C₂×C₇₁	#474	1	1	yes	characteristic
60	C₂×C₇₁	#508	1	1	yes	characteristic
61	C₂×C₇₁	#509	1	1	yes	characteristic
62	#2228	—	71	1	no	maximal, Sylow
63	C₇₁	#2228	1	1	yes	characteristic, complemented, Sylow
64	#499	—	71	1	no	—
65	C₄×D₈	—	71	1	no	—
66	#367	—	71	1	no	—
67	#366	—	71	1	no	—
68	C₄⋊C₁₆	—	71	1	no	—
69	#357	—	71	1	no	—
70	#357	D₇₁	1	1	yes	characteristic
71	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
72	#125	—	142	1	no	—
73	C₄×D₄	—	142	1	no	—
74	C₂×C₁₆	—	71	1	no	—
75	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
76	#107	—	71	1	no	—
77	#107	—	71	1	no	—
78	#107	C₂×D₇₁	1	1	yes	characteristic
79	#106	—	142	1	no	—
80	#102	—	142	1	no	—
81	C₄×C₈	—	71	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂²×C₄	—	284	1	no	—
84	D₈	—	2	1	no	—
85	D₈	Dic₁₄₂	1	2	yes	—
86	C₂×C₈	—	71	1	no	—
87	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	284	1	no	—
90	C₄⋊C₄	—	284	1	no	—
91	C₄⋊C₄	—	142	1	no	—
92	C₄⋊C₄	—	142	1	no	—
93	C₂²⋊C₄	—	284	1	no	—
94	C₄²	—	71	1	no	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	142	1	no	—
97	C₂³	—	4	1	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	284	1	no	—
102	C₂×C₄	—	284	1	no	—
103	C₂×C₄	—	568	2	no	—
104	C₂×C₄	—	71	1	no	—
105	C₂×C₄	—	71	1	no	—
106	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
107	C₈	—	142	1	no	—
108	C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
109	C₈	#11765864	1	1	yes	characteristic
110	C₂²	—	4	2	no	—
111	C₂²	—	8	1	no	—
112	C₂²	—	8	1	no	—
113	C₂²	D₈×D₇₁	1	1	yes	characteristic
114	C₄	—	8	1	no	—
115	C₄	—	568	1	no	—
116	C₄	—	568	1	no	—
117	C₄	—	142	1	no	—
118	C₄	—	142	1	no	—
119	C₄	#14556105	1	1	yes	characteristic
120	C₄	#14556157	1	1	yes	characteristic
121	C₂	—	8	2	no	—
122	C₂	#17986277	1	1	yes	characteristic
123	C₂	#17986428	1	1	yes	characteristic
124	C₂	#17986436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986400

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₁

C₂

yes

maximal, characteristic

#17985826

C₂

yes

maximal, characteristic, complemented

#17985825

C₂

yes

maximal, characteristic, complemented

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#17985800

C₂

yes

maximal, characteristic

#357×C₇₁

C₂

yes

maximal, characteristic

#14556119

—

D₄×Dic₇₁

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556059

—

#14556049

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×D₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

Q₈

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

C₈×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#137

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#499

—

C₄×D₈

—

#367

—

#366

—

C₄⋊C₁₆

—

#357

—

#357

D₇₁

yes

characteristic

C₂×D₈

C₂×D₇₁

yes

characteristic

#125

—

142

—

C₄×D₄

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

—

—

C₂×D₇₁

yes

characteristic

#106

—

142

—

#102

—

142

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

284

—

D₈

—

D₈

Dic₁₄₂

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

C₄²

—

C₁₆

—

C₁₆

—

142

—

C₂³

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

284

—

102

C₂×C₄

—

284

—

103

C₂×C₄

—

568

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

D₄×D₇₁

yes

characteristic

107

C₈

—

142

—

108

C₈

C₂×Dic₁₄₂

yes

characteristic

109

C₈

#11765864

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

D₈×D₇₁

yes

characteristic

114

C₄

—

115

C₄

—

568

—

116

C₄

—

568

—

117

C₄

—

142

—

118

C₄

—

142

—

119

C₄

#14556105

yes

characteristic

120

C₄

#14556157

yes

characteristic

121

C₂

—

122

C₂

#17986277

yes

characteristic

123

C₂

#17986428

yes

characteristic

124

C₂

#17986436

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#357×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#357	D₇₁
hypocenter	C₇₁	#2228
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#357×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #17986400	C₂	semidirect
3. #17985826	C₂	semidirect
4. #17985825	C₂	semidirect
5. Dic₇₁⋊C₁₆	C₂	semidirect
63. C₇₁	#2228	semidirect

H ≅ G/N

type

1. #17986400

C₂

semidirect

3. #17985826

C₂

semidirect

4. #17985825

C₂

semidirect

5. Dic₇₁⋊C₁₆

C₂

semidirect

63. C₇₁

#2228

semidirect

Subgroups — #22590495

Conjugacy classes of subgroups

Complements