Group #22587857 — subgroups

Subgroups — #22587857

Order 9088.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17986402	C₂	1	1	yes	maximal, characteristic
2	#17986210	C₂	1	1	yes	maximal, characteristic
3	#17986206	C₂	1	1	yes	maximal, characteristic
4	#17986126	C₂	1	1	yes	maximal, characteristic
5	#17985749	C₂	1	1	yes	maximal, characteristic
6	#17985744	C₂	1	1	yes	maximal, characteristic
7	#332×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₈₄	—	2	1	no	—
9	C₂×#11765855	—	2	1	no	—
10	#14556128	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
12	#14556061	—	2	1	no	—
13	#14556061	—	2	1	no	—
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₇₁⋊C₈	—	2	1	no	—
16	#14556053	—	2	1	no	—
17	#14556052	—	2	1	no	—
18	#14556038	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
25	Dic₂₈₄	—	4	2	no	—
26	#11765855	—	4	2	no	—
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
33	Dic₇₁⋊C₄	—	2	1	no	—
34	C₄×Dic₇₁	—	2	1	no	—
35	C₂×Q₈×C₇₁	—	2	1	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₄⋊C₄×C₇₁	—	2	1	no	—
40	C₄²×C₇₁	C₂³	1	1	yes	characteristic
41	Q₈×C₇₁	—	4	1	no	—
42	Q₈×C₇₁	—	4	1	no	—
43	C₂×C₄×C₇₁	—	4	1	no	—
44	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
47	C₂×Dic₇₁	—	2	1	no	—
48	C₂×Dic₇₁	—	2	1	no	—
49	C₂×Dic₇₁	—	4	1	no	—
50	Dic₁₄₂	—	2	1	no	—
51	Dic₁₄₂	—	2	1	no	—
52	Dic₁₄₂	—	4	1	no	—
53	Dic₁₄₂	—	4	1	no	—
54	C₈×C₇₁	—	4	1	no	—
55	C₈×C₇₁	—	4	1	no	—
56	C₇₁⋊C₈	—	2	1	no	—
57	C₇₁⋊C₈	—	4	1	no	—
58	C₇₁⋊C₈	Q₁₆	1	2	yes	complemented
59	C₂²×C₇₁	#121	1	1	yes	characteristic
60	C₄×C₇₁	—	2	1	no	—
61	C₄×C₇₁	—	8	1	no	—
62	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄×C₇₁	#137	1	1	yes	characteristic
66	Dic₇₁	—	4	1	no	—
67	Dic₇₁	—	4	1	no	—
68	Dic₇₁	—	8	1	no	—
69	C₂×C₇₁	#462	1	1	yes	characteristic
70	C₂×C₇₁	#469	1	1	yes	characteristic
71	C₂×C₇₁	#473	1	1	yes	characteristic
72	#1694	—	71	1	no	maximal, Sylow
73	C₇₁	#1694	1	1	yes	characteristic, complemented, Sylow
74	#500	—	71	1	no	—
75	#462	—	71	1	no	—
76	#462	—	71	1	no	—
77	#446	—	71	1	no	—
78	#334	—	71	1	no	—
79	#332	—	71	1	no	—
80	#332	D₇₁	1	1	yes	characteristic
81	C₂×Q₁₆	—	142	1	no	—
82	C₂×Q₁₆	—	142	1	no	—
83	C₄⋊Q₈	—	71	1	no	—
84	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
85	C₄×Q₈	—	71	1	no	—
86	#107	—	142	1	no	—
87	#107	—	142	1	no	—
88	C₄⋊C₈	—	71	1	no	—
89	C₄⋊C₈	—	142	1	no	—
90	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
92	#103	—	142	1	no	—
93	#103	—	142	1	no	—
94	#97	—	142	1	no	—
95	C₄×C₈	—	71	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	71	1	no	—
98	C₂×Q₈	—	142	1	no	—
99	Q₁₆	—	284	2	no	—
100	Q₁₆	—	284	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	71	1	no	—
104	C₂×C₈	—	71	1	no	—
105	C₂×C₈	—	142	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	71	1	no	—
109	C₄⋊C₄	—	142	1	no	—
110	C₄⋊C₄	—	142	1	no	—
111	C₄⋊C₄	—	142	1	no	—
112	C₄²	—	142	1	no	—
113	C₄²	C₂²×D₇₁	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	284	1	no	—
117	Q₈	—	284	1	no	—
118	Q₈	—	142	1	no	—
119	Q₈	—	142	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	284	1	no	—
122	C₂×C₄	—	142	1	no	—
123	C₂×C₄	—	142	1	no	—
124	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
125	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
126	C₂×C₄	#11765865	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	284	1	no	—
130	C₈	—	71	2	no	—
131	C₈	—	142	1	no	—
132	C₂²	#14556101	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	284	1	no	—
136	C₄	—	284	1	no	—
137	C₄	—	568	1	no	—
138	C₄	Q₁₆×D₇₁	1	1	yes	characteristic
139	C₄	#14556152	1	1	yes	characteristic
140	C₄	#14556155	1	1	yes	characteristic
141	C₄	#14556157	1	1	yes	characteristic
142	C₂	#17986208	1	1	yes	characteristic
143	C₂	#17986251	1	1	yes	characteristic
144	C₂	#17986273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986402

C₂

yes

maximal, characteristic

#17986210

C₂

yes

maximal, characteristic

#17986206

C₂

yes

maximal, characteristic

#17986126

C₂

yes

maximal, characteristic

#17985749

C₂

yes

maximal, characteristic

#17985744

C₂

yes

maximal, characteristic

#332×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

—

C₂×#11765855

—

#14556128

C₂²

yes

characteristic

C₄×Dic₁₄₂

C₂²

yes

characteristic

#14556061

—

#14556061

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

#14556053

—

#14556052

—

#14556038

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

Dic₂₈₄

—

#11765855

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

Q₁₆

yes

complemented

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄×Q₈

—

#107

—

142

—

#107

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

142

—

#103

—

142

—

#97

—

142

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

142

—

Q₁₆

—

284

—

100

Q₁₆

—

284

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

142

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

142

—

110

C₄⋊C₄

—

142

—

111

C₄⋊C₄

—

142

—

112

C₄²

—

142

—

113

C₄²

C₂²×D₇₁

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

284

—

117

Q₈

—

284

—

118

Q₈

—

142

—

119

Q₈

—

142

—

120

C₂×C₄

—

121

C₂×C₄

—

284

—

122

C₂×C₄

—

142

—

123

C₂×C₄

—

142

—

124

C₂×C₄

D₄×D₇₁

yes

characteristic

125

C₂×C₄

D₄×D₇₁

yes

characteristic

126

C₂×C₄

#11765865

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

284

—

130

C₈

—

131

C₈

—

142

—

132

C₂²

#14556101

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

284

—

136

C₄

—

284

—

137

C₄

—

568

—

138

C₄

Q₁₆×D₇₁

yes

characteristic

139

C₄

#14556152

yes

characteristic

140

C₄

#14556155

yes

characteristic

141

C₄

#14556157

yes

characteristic

142

C₂

#17986208

yes

characteristic

143

C₂

#17986251

yes

characteristic

144

C₂

#17986273

yes

characteristic

	subgroup	G/H
center	C₂²	#14556101
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#332×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#332	D₇₁
hypocenter	C₇₁	#1694
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14556101

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#332×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
58. C₇₁⋊C₈	Q₁₆	semidirect
73. C₇₁	#1694	semidirect

H ≅ G/N

type

58. C₇₁⋊C₈

Q₁₆

semidirect

73. C₇₁

#1694

semidirect

Subgroups — #22587857

Conjugacy classes of subgroups

Complements