Group #22587727 — subgroups

Subgroups — #22587727

Order 9088.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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, complemented
2	#17986204	C₂	1	1	yes	maximal, characteristic
3	D₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#17985751	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985746	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985745	C₂	1	1	yes	maximal, characteristic, complemented
7	#461×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765853	—	2	1	no	—
9	C₂²×C₇₁⋊C₈	—	2	1	no	—
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556061	—	2	1	no	—
12	#14556061	—	2	1	no	—
13	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	#14556051	—	2	1	no	—
16	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
17	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₇₁	—	2	1	no	—
19	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	#103×C₇₁	—	2	1	no	—
23	C₂×Dic₁₄₂	—	2	1	no	—
24	#11765853	—	4	2	no	—
25	C₂×C₇₁⋊C₈	—	2	1	no	—
26	C₂×C₇₁⋊C₈	—	2	1	no	—
27	C₂×C₇₁⋊C₈	—	4	2	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	C₂×Q₈×C₇₁	—	2	1	no	—
33	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	—	2	1	no	—
35	SD₁₆×C₇₁	—	4	2	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₄⋊C₄×C₇₁	—	2	1	no	—
38	C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
40	C₂²⋊C₄×C₇₁	—	2	1	no	—
41	C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₂³×C₇₁	—	2	1	no	—
43	Q₈×C₇₁	—	4	1	no	—
44	Q₈×C₇₁	—	4	1	no	—
45	D₄×C₇₁	—	2	1	no	—
46	D₄×C₇₁	D₈	1	2	yes	—
47	C₂×C₄×C₇₁	—	2	1	no	—
48	C₂×C₄×C₇₁	—	4	1	no	—
49	C₂×C₄×C₇₁	—	4	1	no	—
50	C₂×C₄×C₇₁	—	4	1	no	—
51	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₂×Dic₇₁	—	4	1	no	—
55	Dic₁₄₂	—	4	1	no	—
56	Dic₁₄₂	—	4	1	no	—
57	C₈×C₇₁	—	4	1	no	—
58	C₇₁⋊C₈	—	2	1	no	—
59	C₇₁⋊C₈	—	4	1	no	—
60	C₇₁⋊C₈	—	4	1	no	—
61	C₇₁⋊C₈	SD₁₆	1	2	yes	complemented
62	C₂²×C₇₁	—	2	2	no	—
63	C₂²×C₇₁	—	4	1	no	—
64	C₂²×C₇₁	—	4	1	no	—
65	C₂²×C₇₁	#121	1	1	yes	characteristic
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	—	4	1	no	—
68	C₄×C₇₁	—	8	1	no	—
69	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
70	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₇₁	#135	1	1	yes	characteristic
72	C₄×C₇₁	#137	1	1	yes	characteristic
73	Dic₇₁	—	8	1	no	—
74	C₂×C₇₁	—	4	2	no	—
75	C₂×C₇₁	#461	1	1	yes	characteristic
76	C₂×C₇₁	#466	1	1	yes	characteristic
77	C₂×C₇₁	#473	1	1	yes	characteristic
78	#1673	—	71	1	no	maximal, Sylow
79	C₇₁	#1673	1	1	yes	characteristic, complemented, Sylow
80	#500	—	71	1	no	—
81	#461	—	71	1	no	—
82	#461	D₇₁	1	1	yes	characteristic
83	C₈×D₄	—	71	1	no	—
84	#335	—	71	1	no	—
85	#332	—	71	1	no	—
86	#332	—	71	1	no	—
87	C₂×SD₁₆	—	2	1	no	—
88	C₂×SD₁₆	—	142	1	no	—
89	C₂²×C₈	—	142	1	no	—
90	C₄⋊Q₈	—	71	1	no	—
91	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
92	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
93	#107	—	142	1	no	—
94	#107	—	142	1	no	—
95	C₄⋊C₈	—	71	1	no	—
96	C₄⋊C₈	—	71	1	no	—
97	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
98	#103	—	2	1	no	—
99	#103	—	142	1	no	—
100	C₂²⋊C₈	—	142	1	no	—
101	C₄×C₈	—	71	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	142	1	no	—
104	C₂×D₄	C₂²⋊D₇₁	1	1	yes	characteristic
105	C₂²×C₄	—	2	1	no	—
106	SD₁₆	—	4	2	no	—
107	SD₁₆	—	284	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	284	2	no	—
110	C₂×C₈	—	71	1	no	—
111	C₂×C₈	—	71	1	no	—
112	C₂×C₈	—	142	1	no	—
113	C₂×C₈	—	142	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	142	1	no	—
117	C₄⋊C₄	—	142	1	no	—
118	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
119	C₂²⋊C₄	—	2	1	no	—
120	C₄²	C₂²×D₇₁	1	1	yes	characteristic
121	C₂³	—	2	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	284	1	no	—
125	Q₈	—	284	1	no	—
126	D₄	—	2	1	no	—
127	D₄	#11765850	1	2	yes	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	284	1	no	—
133	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
135	C₂×C₄	#11765864	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	284	1	no	—
138	C₈	—	284	1	no	—
139	C₈	—	71	2	no	—
140	C₈	—	142	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	#14556100	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	4	1	no	—
147	C₄	—	8	1	no	—
148	C₄	—	568	1	no	—
149	C₄	C₂×#11765850	1	1	yes	characteristic
150	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
151	C₄	#14556148	1	1	yes	characteristic
152	C₄	#14556157	1	1	yes	characteristic
153	C₂	—	4	2	no	—
154	C₂	#17986202	1	1	yes	characteristic
155	C₂	#17986233	1	1	yes	characteristic
156	C₂	#17986272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986402

C₂

yes

maximal, characteristic, complemented

#17986204

C₂

yes

maximal, characteristic

D₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985751

C₂

yes

maximal, characteristic, complemented

#17985746

C₂

yes

maximal, characteristic, complemented

#17985745

C₂

yes

maximal, characteristic, complemented

#461×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765853

—

C₂²×C₇₁⋊C₈

—

#14556128

C₂²

yes

characteristic

#14556061

—

#14556061

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556051

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₂×Dic₁₄₂

—

#11765853

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

SD₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

D₈

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×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

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

SD₁₆

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#500

—

#461

—

#461

D₇₁

yes

characteristic

C₈×D₄

—

#335

—

#332

—

#332

—

C₂×SD₁₆

—

C₂×SD₁₆

—

142

—

C₂²×C₈

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

C₂×D₇₁

yes

characteristic

#107

—

142

—

#107

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

#103

—

142

—

100

C₂²⋊C₈

—

142

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

142

—

104

C₂×D₄

C₂²⋊D₇₁

yes

characteristic

105

C₂²×C₄

—

106

SD₁₆

—

107

SD₁₆

—

284

—

108

C₂×C₈

—

109

C₂×C₈

—

284

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

142

—

113

C₂×C₈

—

142

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

142

—

117

C₄⋊C₄

—

142

—

118

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

119

C₂²⋊C₄

—

120

C₄²

C₂²×D₇₁

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

284

—

125

Q₈

—

284

—

126

D₄

—

127

D₄

#11765850

yes

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

284

—

133

C₂×C₄

D₄×D₇₁

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

135

C₂×C₄

#11765864

yes

characteristic

136

C₈

—

137

C₈

—

284

—

138

C₈

—

284

—

139

C₈

—

140

C₈

—

142

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#14556100

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

568

—

149

C₄

C₂×#11765850

yes

characteristic

150

C₄

SD₁₆×D₇₁

yes

characteristic

151

C₄

#14556148

yes

characteristic

152

C₄

#14556157

yes

characteristic

153

C₂

—

154

C₂

#17986202

yes

characteristic

155

C₂

#17986233

yes

characteristic

156

C₂

#17986272

yes

characteristic

	subgroup	G/H
center	C₂²	#14556100
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#461×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#461	D₇₁
hypocenter	C₇₁	#1673
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#14556100

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#461×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #17986402	C₂	semidirect
4. #17985751	C₂	semidirect
5. #17985746	C₂	semidirect
6. #17985745	C₂	semidirect
61. C₇₁⋊C₈	SD₁₆	semidirect
79. C₇₁	#1673	semidirect

H ≅ G/N

type

1. #17986402

C₂

semidirect

4. #17985751

C₂

semidirect

5. #17985746

C₂

semidirect

6. #17985745

C₂

semidirect

61. C₇₁⋊C₈

SD₁₆

semidirect

79. C₇₁

#1673

semidirect

Subgroups — #22587727

Conjugacy classes of subgroups

Complements