Group #22587811 — subgroups

Subgroups — #22587811

Order 9088.

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

normal subgroups only

This group has 168 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	#17986371	C₂	1	1	yes	maximal, characteristic
2	#17986285	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986196	C₂	1	1	yes	maximal, characteristic
4	#17986126	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985751	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985746	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765853	—	2	1	no	—
9	C₂×#11765853	—	2	1	no	—
10	C₂×#11765853	—	2	1	no	—
11	#14556128	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
13	#14556060	—	2	1	no	—
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₇₁⋊C₈	—	2	1	no	—
16	#14556053	—	2	1	no	—
17	#14556049	—	2	1	no	—
18	#14556038	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	#127×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	#11765853	—	4	2	no	—
26	#11765853	—	4	2	no	—
27	#11765853	—	4	2	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
34	Dic₇₁⋊C₄	—	2	1	no	—
35	C₄×Dic₇₁	—	2	1	no	—
36	C₂×D₄×C₇₁	—	2	1	no	—
37	C₂×D₄×C₇₁	—	2	1	no	—
38	C₂×D₄×C₇₁	—	2	1	no	—
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₂³×C₇₁	—	4	1	no	—
43	D₄×C₇₁	—	4	1	no	—
44	D₄×C₇₁	—	4	1	no	—
45	D₄×C₇₁	—	4	1	no	—
46	D₄×C₇₁	—	4	1	no	—
47	D₄×C₇₁	—	8	1	no	—
48	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
51	C₂×Dic₇₁	—	2	1	no	—
52	C₂×Dic₇₁	—	2	1	no	—
53	C₂×Dic₇₁	—	4	1	no	—
54	Dic₁₄₂	—	2	1	no	—
55	Dic₁₄₂	—	2	1	no	—
56	Dic₁₄₂	—	4	1	no	—
57	Dic₁₄₂	—	4	1	no	—
58	C₈×C₇₁	—	4	1	no	—
59	C₈×C₇₁	—	4	1	no	—
60	C₇₁⋊C₈	—	2	1	no	—
61	C₇₁⋊C₈	—	4	1	no	—
62	C₇₁⋊C₈	SD₁₆	1	2	yes	complemented
63	C₂²×C₇₁	—	8	1	no	—
64	C₂²×C₇₁	—	8	1	no	—
65	C₂²×C₇₁	—	8	1	no	—
66	C₂²×C₇₁	#121	1	1	yes	characteristic
67	C₄×C₇₁	—	2	1	no	—
68	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₇₁	#137	1	1	yes	characteristic
72	Dic₇₁	—	4	1	no	—
73	Dic₇₁	—	4	1	no	—
74	Dic₇₁	—	8	1	no	—
75	C₂×C₇₁	—	16	1	no	—
76	C₂×C₇₁	#460	1	1	yes	characteristic
77	C₂×C₇₁	#468	1	1	yes	characteristic
78	C₂×C₇₁	#471	1	1	yes	characteristic
79	#1686	—	71	1	no	maximal, Sylow
80	C₇₁	#1686	1	1	yes	characteristic, complemented, Sylow
81	#492	—	71	1	no	—
82	#475	—	71	1	no	—
83	#460	—	71	1	no	—
84	#446	—	71	1	no	—
85	#335	—	71	1	no	—
86	#332	—	71	1	no	—
87	#331	D₇₁	1	1	yes	characteristic
88	C₂×SD₁₆	—	142	1	no	—
89	C₂×SD₁₆	—	142	1	no	—
90	C₂×SD₁₆	—	142	1	no	—
91	C₄⋊Q₈	—	71	1	no	—
92	#127	C₂×D₇₁	1	1	yes	characteristic
93	C₄×Q₈	—	71	1	no	—
94	#106	—	142	1	no	—
95	C₄⋊C₈	—	71	1	no	—
96	C₄⋊C₈	—	142	1	no	—
97	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
98	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
99	#103	—	142	1	no	—
100	#102	—	142	1	no	—
101	#97	—	142	1	no	—
102	C₄×C₈	—	71	1	no	—
103	C₂×Q₈	—	71	1	no	—
104	C₂×Q₈	—	142	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	SD₁₆	—	284	2	no	—
109	SD₁₆	—	284	2	no	—
110	SD₁₆	—	284	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	71	1	no	—
114	C₂×C₈	—	71	1	no	—
115	C₂×C₈	—	142	1	no	—
116	C₄⋊C₄	—	71	1	no	—
117	C₄⋊C₄	—	142	1	no	—
118	C₄⋊C₄	—	142	1	no	—
119	C₄⋊C₄	—	142	1	no	—
120	C₄²	—	142	1	no	—
121	C₄²	C₂²×D₇₁	1	1	yes	characteristic
122	C₂³	—	4	1	no	—
123	Q₈	—	284	1	no	—
124	Q₈	—	284	1	no	—
125	Q₈	—	142	1	no	—
126	Q₈	—	142	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	C₂×C₄	—	284	1	no	—
133	C₂×C₄	—	142	1	no	—
134	C₂×C₄	—	142	1	no	—
135	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
136	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
137	C₂×C₄	#11765865	1	1	yes	characteristic
138	C₈	—	4	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	284	1	no	—
141	C₈	—	71	2	no	—
142	C₈	—	142	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	#14556101	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	284	1	no	—
149	C₄	—	284	1	no	—
150	C₄	—	568	1	no	—
151	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
152	C₄	#14556152	1	1	yes	characteristic
153	C₄	#14556152	1	1	yes	characteristic
154	C₄	#14556157	1	1	yes	characteristic
155	C₂	—	16	1	no	—
156	C₂	#17986198	1	1	yes	characteristic
157	C₂	#17986244	1	1	yes	characteristic
158	C₂	#17986261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986371

C₂

yes

maximal, characteristic

#17986285

C₂

yes

maximal, characteristic, complemented

#17986196

C₂

yes

maximal, characteristic

#17986126

C₂

yes

maximal, characteristic, complemented

#17985751

C₂

yes

maximal, characteristic, complemented

#17985746

C₂

yes

maximal, characteristic, complemented

#331×C₇₁

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₄₂

C₂²

yes

characteristic

#14556060

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

#14556053

—

#14556049

—

#14556038

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

#127×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

—

—

—

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×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×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

SD₁₆

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic

C₂×SD₁₆

—

142

—

C₂×SD₁₆

—

142

—

C₂×SD₁₆

—

142

—

C₄⋊Q₈

—

#127

C₂×D₇₁

yes

characteristic

C₄×Q₈

—

#106

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

142

—

100

#102

—

142

—

101

#97

—

142

—

102

C₄×C₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

142

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

284

—

109

SD₁₆

—

284

—

110

SD₁₆

—

284

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

142

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

142

—

118

C₄⋊C₄

—

142

—

119

C₄⋊C₄

—

142

—

120

C₄²

—

142

—

121

C₄²

C₂²×D₇₁

yes

characteristic

122

C₂³

—

123

Q₈

—

284

—

124

Q₈

—

284

—

125

Q₈

—

142

—

126

Q₈

—

142

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

284

—

133

C₂×C₄

—

142

—

134

C₂×C₄

—

142

—

135

C₂×C₄

D₄×D₇₁

yes

characteristic

136

C₂×C₄

D₄×D₇₁

yes

characteristic

137

C₂×C₄

#11765865

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

284

—

141

C₈

—

142

C₈

—

142

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#14556101

yes

characteristic

147

C₄

—

148

C₄

—

284

—

149

C₄

—

284

—

150

C₄

—

568

—

151

C₄

SD₁₆×D₇₁

yes

characteristic

152

C₄

#14556152

yes

characteristic

153

C₄

#14556152

yes

characteristic

154

C₄

#14556157

yes

characteristic

155

C₂

—

156

C₂

#17986198

yes

characteristic

157

C₂

#17986244

yes

characteristic

158

C₂

#17986261

yes

characteristic

	subgroup	G/H
center	C₂²	#14556101
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#331×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#331	D₇₁
hypocenter	C₇₁	#1686
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14556101

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#331×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #17986285	C₂	semidirect
4. #17986126	C₂	semidirect
5. #17985751	C₂	semidirect
6. #17985746	C₂	semidirect
62. C₇₁⋊C₈	SD₁₆	semidirect
80. C₇₁	#1686	semidirect

H ≅ G/N

type

2. #17986285

C₂

semidirect

4. #17986126

C₂

semidirect

5. #17985751

C₂

semidirect

6. #17985746

C₂

semidirect

62. C₇₁⋊C₈

SD₁₆

semidirect

80. C₇₁

#1686

semidirect

Subgroups — #22587811

Conjugacy classes of subgroups

Complements