Group #22590570 — subgroups

Subgroups — #22590570

Order 9088.

138 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 47 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	#17986382	C₂	1	1	yes	maximal, characteristic
3	#17985804	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985802	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985800	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
9	C₂×#11765853	—	2	1	no	—
10	#14556119	—	2	1	no	—
11	#14556115	—	2	1	no	—
12	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14556062	C₂²	1	1	yes	characteristic
14	#14556059	—	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	C₂×Dic₁₄₂	—	2	1	no	—
20	Dic₂₈₄	—	2	1	no	—
21	Dic₂₈₄	—	2	1	no	—
22	#11765853	—	4	2	no	—
23	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₇₁	—	2	1	no	—
25	C₄⋊Dic₇₁	—	4	1	no	—
26	Dic₇₁⋊C₄	—	4	1	no	—
27	C₂²⋊Dic₇₁	—	4	1	no	—
28	C₂²⋊Dic₇₁	—	4	1	no	—
29	C₄×Dic₇₁	D₄	1	1	yes	characteristic
30	C₇₁⋊C₁₆	—	2	1	no	—
31	C₂×D₄×C₇₁	—	2	1	no	—
32	D₈×C₇₁	—	2	1	no	—
33	D₈×C₇₁	—	2	1	no	—
34	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₁₆×C₇₁	—	2	1	no	—
37	C₂³×C₇₁	—	4	1	no	—
38	D₄×C₇₁	—	4	1	no	—
39	D₄×C₇₁	—	4	1	no	—
40	C₂×C₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₇₁	—	4	1	no	—
43	C₂×Dic₇₁	—	4	1	no	—
44	C₂×Dic₇₁	D₈	1	1	yes	characteristic
45	C₂×Dic₇₁	D₈	1	1	yes	characteristic
46	Dic₁₄₂	—	4	1	no	—
47	Dic₁₄₂	—	4	1	no	—
48	C₈×C₇₁	#41	1	1	yes	characteristic
49	C₈×C₇₁	#41	1	1	yes	characteristic
50	C₇₁⋊C₈	SD₁₆	1	2	yes	—
51	C₂²×C₇₁	—	8	1	no	—
52	C₂²×C₇₁	—	8	1	no	—
53	C₂²×C₇₁	—	8	1	no	—
54	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₄×C₇₁	—	8	1	no	—
56	C₄×C₇₁	#124	1	1	yes	characteristic
57	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
58	Dic₇₁	—	2	1	no	—
59	Dic₇₁	—	2	1	no	—
60	Dic₇₁	—	8	1	no	—
61	Dic₇₁	—	8	1	no	—
62	C₂×C₇₁	—	16	1	no	—
63	C₂×C₇₁	#486	1	1	yes	characteristic
64	C₂×C₇₁	#508	1	1	yes	characteristic
65	C₂×C₇₁	#508	1	1	yes	characteristic
66	#2242	—	71	1	no	maximal, Sylow
67	C₇₁	#2242	1	1	yes	characteristic, complemented, Sylow
68	#499	—	71	1	no	—
69	#495	—	71	1	no	—
70	#358	—	71	1	no	—
71	#358	—	71	1	no	—
72	#357	—	71	1	no	—
73	#357	D₇₁	1	1	yes	characteristic
74	C₄×C₁₆	—	71	1	no	—
75	C₂×Q₁₆	—	71	1	no	—
76	C₂×SD₁₆	—	142	1	no	—
77	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
78	#125	—	142	1	no	—
79	#124	—	142	1	no	—
80	C₂×C₁₆	—	71	1	no	—
81	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
82	#107	—	71	1	no	—
83	#107	C₂×D₇₁	1	1	yes	characteristic
84	#106	—	142	1	no	—
85	C₄×C₈	—	71	1	no	—
86	C₂×Q₈	—	142	1	no	—
87	C₂×D₄	—	2	1	no	—
88	Q₁₆	—	142	1	no	—
89	Q₁₆	—	142	1	no	—
90	SD₁₆	—	284	2	no	—
91	D₈	—	2	1	no	—
92	D₈	—	2	1	no	—
93	C₂×C₈	—	71	1	no	—
94	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	284	1	no	—
97	C₄⋊C₄	—	284	1	no	—
98	C₄⋊C₄	—	142	1	no	—
99	C₂²⋊C₄	—	284	1	no	—
100	C₂²⋊C₄	—	284	1	no	—
101	C₄²	—	71	1	no	—
102	C₁₆	—	2	1	no	—
103	C₁₆	—	142	1	no	—
104	C₂³	—	4	1	no	—
105	Q₈	—	284	1	no	—
106	Q₈	—	284	1	no	—
107	D₄	—	4	1	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	284	1	no	—
111	C₂×C₄	—	284	1	no	—
112	C₂×C₄	—	71	1	no	—
113	C₂×C₄	—	71	1	no	—
114	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
115	C₈	—	71	2	no	—
116	C₈	#11765864	1	1	yes	characteristic
117	C₈	#11765865	1	1	yes	characteristic
118	C₂²	—	8	1	no	—
119	C₂²	—	8	1	no	—
120	C₂²	—	8	1	no	—
121	C₂²	D₈×D₇₁	1	1	yes	characteristic
122	C₄	—	8	1	no	—
123	C₄	—	568	1	no	—
124	C₄	—	568	1	no	—
125	C₄	—	142	1	no	—
126	C₄	—	142	1	no	—
127	C₄	#14556114	1	1	yes	characteristic
128	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
129	C₂	—	16	1	no	—
130	C₂	#17986344	1	1	yes	characteristic
131	C₂	#17986428	1	1	yes	characteristic
132	C₂	#17986430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986400

C₂

yes

maximal, characteristic, complemented

#17986382

C₂

yes

maximal, characteristic

#17985804

C₂

yes

maximal, characteristic, complemented

#17985802

C₂

yes

maximal, characteristic, complemented

#17985800

C₂

yes

maximal, characteristic

C₁₆×Dic₇₁

C₂

yes

maximal, characteristic, complemented

#357×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

—

—

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556059

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×D₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765853

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

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

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

SD₁₆

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#124

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

142

—

C₂×D₈

C₂×D₇₁

yes

characteristic

#125

—

142

—

#124

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#107

C₂×D₇₁

yes

characteristic

#106

—

142

—

C₄×C₈

—

C₂×Q₈

—

142

—

C₂×D₄

—

Q₁₆

—

142

—

Q₁₆

—

142

—

SD₁₆

—

284

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

100

C₂²⋊C₄

—

284

—

101

C₄²

—

102

C₁₆

—

103

C₁₆

—

142

—

104

C₂³

—

105

Q₈

—

284

—

106

Q₈

—

284

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

284

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

D₄×D₇₁

yes

characteristic

115

C₈

—

116

C₈

#11765864

yes

characteristic

117

C₈

#11765865

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

D₈×D₇₁

yes

characteristic

122

C₄

—

123

C₄

—

568

—

124

C₄

—

568

—

125

C₄

—

142

—

126

C₄

—

142

—

127

C₄

#14556114

yes

characteristic

128

C₄

SD₁₆×D₇₁

yes

characteristic

129

C₂

—

130

C₂

#17986344

yes

characteristic

131

C₂

#17986428

yes

characteristic

132

C₂

#17986430

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	#2242
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. #17985804	C₂	semidirect
4. #17985802	C₂	semidirect
6. C₁₆×Dic₇₁	C₂	semidirect
67. C₇₁	#2242	semidirect

H ≅ G/N

type

1. #17986400

C₂

semidirect

3. #17985804

C₂

semidirect

4. #17985802

C₂

semidirect

6. C₁₆×Dic₇₁

C₂

semidirect

67. C₇₁

#2242

semidirect

Subgroups — #22590570

Conjugacy classes of subgroups

Complements