Group #22589848 — subgroups

Subgroups — #22589848

Order 9088.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#17986042	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556061	C₂	1	1	yes	maximal, characteristic
3	#17986026	C₂	1	1	yes	maximal, characteristic
4	#17985768	C₂	1	1	yes	maximal, characteristic
5	#17985767	C₂	1	1	yes	maximal, characteristic
6	#17985763	C₂	1	1	yes	maximal, characteristic
7	#428×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765849	C₂²	1	1	yes	characteristic
9	C₂×#11765849	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
12	#14556061	—	2	2	no	—
13	#14556061	—	2	1	no	—
14	#14556059	—	2	1	no	—
15	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
16	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#117×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
19	#107×C₇₁	—	2	1	no	—
20	#106×C₇₁	—	2	1	no	—
21	C₂²×Dic₇₁	—	2	1	no	—
22	#11765849	—	2	1	no	—
23	#11765849	—	2	1	no	—
24	#11765849	—	2	1	no	—
25	#11765849	Q₈	1	2	yes	—
26	C₂×C₇₁⋊C₈	—	2	1	no	—
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₂²×C₄×C₇₁	—	2	1	no	—
34	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
35	M₁₆×C₇₁	—	2	1	no	—
36	M₁₆×C₇₁	Q₈	1	2	yes	—
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₄⋊C₄×C₇₁	—	2	1	no	—
40	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
42	C₂²⋊C₄×C₇₁	—	2	1	no	—
43	C₄²×C₇₁	—	2	1	no	—
44	C₂³×C₇₁	#41	1	1	yes	characteristic
45	C₂×C₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	—	2	1	no	—
47	C₂×C₄×C₇₁	—	4	1	no	—
48	C₂×C₄×C₇₁	—	8	1	no	—
49	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
55	C₂×Dic₇₁	—	4	1	no	—
56	C₂×Dic₇₁	—	8	1	no	—
57	C₈×C₇₁	—	2	2	no	—
58	C₇₁⋊C₈	—	2	2	no	—
59	C₇₁⋊C₈	—	2	2	no	—
60	C₇₁⋊C₈	—	4	1	no	—
61	C₂²×C₇₁	—	2	1	no	—
62	C₂²×C₇₁	—	2	1	no	—
63	C₂²×C₇₁	#121	1	1	yes	characteristic
64	C₂²×C₇₁	#125	1	1	yes	characteristic
65	C₂²×C₇₁	#125	1	1	yes	characteristic
66	C₄×C₇₁	—	4	1	no	—
67	C₄×C₇₁	—	4	1	no	—
68	C₄×C₇₁	—	8	1	no	—
69	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₇₁	#123	1	1	yes	characteristic
72	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₇₁	—	8	1	no	—
74	C₂×C₇₁	—	2	1	no	—
75	C₂×C₇₁	—	2	1	no	—
76	C₂×C₇₁	#398	1	1	yes	characteristic
77	C₂×C₇₁	#472	1	1	yes	characteristic
78	C₂×C₇₁	#473	1	1	yes	characteristic
79	#2089	—	71	1	no	maximal, Sylow
80	C₇₁	#2089	1	1	yes	characteristic, complemented, Sylow
81	#428	—	71	1	no	—
82	#428	D₇₁	1	1	yes	characteristic
83	C₂×#107	—	71	1	no	—
84	#424	—	71	1	no	—
85	#343	—	71	1	no	—
86	#343	—	71	1	no	—
87	#341	—	71	1	no	—
88	C₂×M₁₆	—	71	1	no	—
89	C₂×M₁₆	—	71	1	no	—
90	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
91	C₂²×C₈	—	71	1	no	—
92	#117	C₂×D₇₁	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	71	1	no	—
94	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
95	#107	—	2	1	no	—
96	#107	—	142	2	no	—
97	#107	—	142	1	no	—
98	#106	—	2	1	no	—
99	#106	—	142	1	no	—
100	C₄⋊C₈	—	142	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	142	1	no	—
103	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
104	M₁₆	—	2	1	no	—
105	M₁₆	—	71	2	no	—
106	M₁₆	—	142	1	no	—
107	M₁₆	—	142	1	no	—
108	M₁₆	—	142	1	no	—
109	M₁₆	Dic₁₄₂	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	71	2	no	—
112	C₂×C₈	—	142	1	no	—
113	C₂×C₈	—	142	1	no	—
114	C₂×C₈	—	142	1	no	—
115	C₂×C₈	—	142	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	142	1	no	—
119	C₄⋊C₄	—	142	1	no	—
120	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
121	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₂³	#11765864	1	1	yes	characteristic
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	284	1	no	—
130	C₂×C₄	—	568	1	no	—
131	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
135	C₂×C₄	#11765865	1	1	yes	characteristic
136	C₂×C₄	#11765866	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	284	1	no	—
139	C₈	—	142	2	no	—
140	C₈	—	142	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#14556100	1	1	yes	characteristic
144	C₂²	#14556118	1	1	yes	characteristic
145	C₂²	#14556119	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	568	1	no	—
150	C₄	#14556104	1	1	yes	characteristic
151	C₄	#14556107	1	1	yes	characteristic
152	C₄	#14556113	1	1	yes	characteristic
153	C₄	#14556127	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#17985932	1	1	yes	characteristic
157	C₂	#17986267	1	1	yes	characteristic
158	C₂	#17986272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986042

C₂

yes

maximal, characteristic

C₂×#14556061

C₂

yes

maximal, characteristic

#17986026

C₂

yes

maximal, characteristic

#17985768

C₂

yes

maximal, characteristic

#17985767

C₂

yes

maximal, characteristic

#17985763

C₂

yes

maximal, characteristic

#428×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

—

—

C₄⋊(C₇₁⋊C₈)

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

#107×C₇₁

—

#106×C₇₁

—

C₂²×Dic₇₁

—

—

—

—

Q₈

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

Q₈

yes

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

#41

yes

characteristic

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

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#123

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#428

—

#428

D₇₁

yes

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

#117

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

#107

—

#107

—

142

—

#107

—

142

—

#106

—

#106

—

142

—

100

C₄⋊C₈

—

142

—

101

C₂²×C₄

—

102

C₂²×C₄

—

142

—

103

C₂²×C₄

C₂²×D₇₁

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

142

—

107

M₁₆

—

142

—

108

M₁₆

—

142

—

109

M₁₆

Dic₁₄₂

yes

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

142

—

113

C₂×C₈

—

142

—

114

C₂×C₈

—

142

—

115

C₂×C₈

—

142

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

142

—

119

C₄⋊C₄

—

142

—

120

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

121

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

122

C₂²⋊C₄

—

123

C₄²

—

124

C₂³

#11765864

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

284

—

130

C₂×C₄

—

568

—

131

C₂×C₄

D₄×D₇₁

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

133

C₂×C₄

Q₈×D₇₁

yes

characteristic

134

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

135

C₂×C₄

#11765865

yes

characteristic

136

C₂×C₄

#11765866

yes

characteristic

137

C₈

—

138

C₈

—

284

—

139

C₈

—

142

—

140

C₈

—

142

—

141

C₂²

—

142

C₂²

—

143

C₂²

#14556100

yes

characteristic

144

C₂²

#14556118

yes

characteristic

145

C₂²

#14556119

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

568

—

150

C₄

#14556104

yes

characteristic

151

C₄

#14556107

yes

characteristic

152

C₄

#14556113

yes

characteristic

153

C₄

#14556127

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#17985932

yes

characteristic

157

C₂

#17986267

yes

characteristic

158

C₂

#17986272

yes

characteristic

	subgroup	G/H
center	C₂²	#14556100
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#428×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#428	D₇₁
hypocenter	C₇₁	#2089
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

subgroup

G/H

center

C₂²

#14556100

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#428×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

N	H ≅ G/N	type
80. C₇₁	#2089	semidirect

H ≅ G/N

type

80. C₇₁

#2089

semidirect

Subgroups — #22589848

Conjugacy classes of subgroups

Complements