Group #31897421 — subgroups

Subgroups — #31897421

Order 18176.

406 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22588512	C₂	1	1	yes	maximal, characteristic
2	#22586795	C₂	1	4	yes	maximal
3	C₂×#17985720	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556064	—	2	2	no	—
6	#17986023	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	Q₈⋊(C₇₁⋊C₈)	—	2	8	no	—
9	#17985720	—	2	2	no	—
10	#533×C₇₁	C₂²	1	2	yes	—
11	#500×C₇₁	C₄	1	2	yes	—
12	#499×C₇₁	—	2	2	no	—
13	#498×C₇₁	C₄	1	2	yes	—
14	#427×C₇₁	—	2	2	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#11765849	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	#14556064	—	4	4	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
20	C₄×C₇₁⋊C₈	—	2	2	no	—
21	C₄×C₇₁⋊C₈	D₄	1	4	yes	—
22	C₂²×C₈×C₇₁	D₄	1	2	yes	—
23	C₄⋊Q₈×C₇₁	C₂×C₄	1	4	yes	—
24	#125×C₇₁	—	2	2	no	—
25	C₂²⋊Q₈×C₇₁	—	2	4	no	—
26	C₄×Q₈×C₇₁	—	2	4	no	—
27	#117×C₇₁	—	2	2	no	—
28	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	#107×C₇₁	—	2	4	no	—
30	#106×C₇₁	—	2	4	no	—
31	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	D₄	1	2	yes	—
34	#11765849	—	4	4	no	—
35	C₂×C₇₁⋊C₈	—	2	4	no	—
36	C₂×C₇₁⋊C₈	—	4	4	no	—
37	C₂×C₇₁⋊C₈	—	4	2	no	—
38	C₂×Q₈×C₇₁	—	2	4	no	—
39	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	#41	1	2	yes	—
41	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
44	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
45	C₄⋊C₄×C₇₁	—	2	4	no	—
46	C₄⋊C₄×C₇₁	—	2	4	no	—
47	C₄⋊C₄×C₇₁	—	4	4	no	—
48	C₂²⋊C₄×C₇₁	—	4	2	no	—
49	C₄²×C₇₁	—	4	2	no	—
50	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇₁	C₂×D₄	1	2	yes	—
52	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₇₁	#124	1	1	yes	characteristic
54	Q₈×C₇₁	—	4	8	no	—
55	C₂×C₄×C₇₁	—	2	4	no	—
56	C₂×C₄×C₇₁	—	4	4	no	—
57	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₇₁	#121	1	2	yes	—
60	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
63	C₈×C₇₁	#103	1	4	yes	—
64	C₈×C₇₁	#103	1	4	yes	—
65	C₇₁⋊C₈	—	4	4	no	—
66	C₇₁⋊C₈	—	8	2	no	—
67	C₂²×C₇₁	—	2	2	no	—
68	C₂²×C₇₁	#390	1	1	yes	characteristic
69	C₂²×C₇₁	#487	1	2	yes	—
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	—	8	4	no	—
72	C₄×C₇₁	C₂×#103	1	2	yes	—
73	C₄×C₇₁	#444	1	4	yes	—
74	C₄×C₇₁	#492	1	1	yes	characteristic
75	C₄×C₇₁	#494	1	1	yes	characteristic
76	#9861	—	71	1	no	maximal, Sylow
77	C₂×C₇₁	—	2	2	no	—
78	C₂×C₇₁	#1956	1	1	yes	characteristic
79	C₂×C₇₁	#1975	1	2	yes	—
80	#3160	D₇₁	1	1	yes	characteristic
81	#1848	—	71	1	no	—
82	#1493	—	71	4	no	—
83	C₂×#322	—	71	1	no	—
84	C₇₁	#9861	1	1	yes	characteristic, complemented, Sylow
85	#533	C₂×D₇₁	1	2	yes	—
86	#500	Dic₇₁	1	2	yes	—
87	#499	—	2	2	no	—
88	#498	Dic₇₁	1	2	yes	—
89	C₂×#108	—	142	2	no	—
90	#427	—	2	2	no	—
91	#423	—	71	2	no	—
92	C₂×C₄×C₈	—	71	2	no	—
93	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
94	Q₈⋊C₈	—	142	8	no	—
95	#322	—	142	2	no	—
96	C₂×M₁₆	—	142	2	no	—
97	C₂²×C₈	—	142	2	no	—
98	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
99	C₄⋊Q₈	C₂×Dic₇₁	1	4	yes	—
100	#125	—	2	2	no	—
101	C₂²⋊Q₈	—	2	4	no	—
102	C₄×Q₈	—	2	4	no	—
103	#117	—	2	2	no	—
104	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
105	#108	—	284	4	no	—
106	#107	—	2	4	no	—
107	#106	—	2	4	no	—
108	C₄⋊C₈	—	142	4	no	—
109	C₄×C₈	—	71	4	no	—
110	C₄×C₈	—	142	2	no	—
111	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
114	C₂×Q₈	—	2	4	no	—
115	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂²×C₄	#11765864	1	2	yes	—
117	M₁₆	—	284	4	no	—
118	C₂×C₈	—	284	4	no	—
119	C₂×C₈	—	284	2	no	—
120	C₂×C₈	—	142	4	no	—
121	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
122	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
123	C₂×C₈	#11765852	1	4	yes	—
124	C₂×C₈	#11765855	1	4	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
131	C₄²	D₄×D₇₁	1	2	yes	—
132	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
133	C₂³	#14556115	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
138	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
139	C₂×C₄	#14556099	1	2	yes	—
140	C₂×C₄	#14556124	1	1	yes	characteristic
141	C₂×C₄	C₂×#11765852	1	2	yes	—
142	C₂×C₄	C₂×#11765855	1	2	yes	—
143	C₈	—	284	4	no	—
144	C₈	—	568	2	no	—
145	C₈	#14556052	1	4	yes	—
146	C₈	#14556052	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#17985902	1	1	yes	characteristic
149	C₂²	#17986349	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×#14556052	1	2	yes	—
153	C₄	#17986121	1	4	yes	—
154	C₄	#17986373	1	1	yes	characteristic
155	C₄	#17986378	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#22589143	1	1	yes	characteristic
158	C₂	#22589234	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588512

C₂

yes

maximal, characteristic

#22586795

C₂

yes

maximal

C₂×#17985720

C₂

yes

maximal, characteristic

C₇₁×#3160

C₂

yes

maximal, characteristic

C₂×#14556064

—

#17986023

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

Q₈⋊(C₇₁⋊C₈)

—

#17985720

—

#533×C₇₁

C₂²

yes

—

#500×C₇₁

C₄

yes

—

#499×C₇₁

—

#498×C₇₁

C₄

yes

—

#427×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₂²×C₇₁⋊C₈

—

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

—

#125×C₇₁

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

#117×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

#106×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

#124

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

—

C₈×C₇₁

#103

yes

—

C₈×C₇₁

#103

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#487

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#103

yes

—

C₄×C₇₁

#444

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

#1956

yes

characteristic

C₂×C₇₁

#1975

yes

—

#3160

D₇₁

yes

—

—

—

C₇₁

#9861

yes

characteristic, complemented, Sylow

#533

C₂×D₇₁

yes

—

#500

Dic₇₁

yes

—

#499

—

#498

Dic₇₁

yes

—

C₂×#108

—

142

—

#427

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

Q₈⋊C₈

—

142

—

#322

—

142

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

142

—

C₂²×C₈

C₂²⋊D₇₁

yes

—

C₄⋊Q₈

C₂×Dic₇₁

yes

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂²×D₇₁

yes

characteristic

105

#108

—

284

—

106

#107

—

107

#106

—

108

C₄⋊C₈

—

142

—

109

C₄×C₈

—

110

C₄×C₈

—

142

—

111

C₄×C₈

C₂×Dic₇₁

yes

characteristic

112

C₄×C₈

C₂×Dic₇₁

yes

characteristic

113

C₄×C₈

C₂²⋊D₇₁

yes

—

114

C₂×Q₈

—

115

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

116

C₂²×C₄

#11765864

yes

—

117

M₁₆

—

284

—

118

C₂×C₈

—

284

—

119

C₂×C₈

—

284

—

120

C₂×C₈

—

142

—

121

C₂×C₈

C₂²⋊Dic₇₁

yes

—

122

C₂×C₈

C₂²⋊Dic₇₁

yes

—

123

C₂×C₈

#11765852

yes

—

124

C₂×C₈

#11765855

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

C₂²×Dic₇₁

yes

characteristic

131

C₄²

D₄×D₇₁

yes

—

132

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

133

C₂³

#14556115

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

138

C₂×C₄

D₄×Dic₇₁

yes

—

139

C₂×C₄

#14556099

yes

—

140

C₂×C₄

#14556124

yes

characteristic

141

C₂×C₄

C₂×#11765852

yes

—

142

C₂×C₄

C₂×#11765855

yes

—

143

C₈

—

284

—

144

C₈

—

568

—

145

C₈

#14556052

yes

—

146

C₈

#14556052

yes

—

147

C₂²

—

148

C₂²

#17985902

yes

characteristic

149

C₂²

#17986349

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×#14556052

yes

—

153

C₄

#17986121

yes

—

154

C₄

#17986373

yes

characteristic

155

C₄

#17986378

yes

characteristic

156

C₂

—

157

C₂

#22589143

yes

characteristic

158

C₂

#22589234

yes

—

	subgroup	G/H
center	C₂×C₄	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#3160	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	#124
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#3160	D₇₁
hypocenter	C₇₁	#9861
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#3160

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
84. C₇₁	#9861	semidirect

H ≅ G/N

type

84. C₇₁

#9861

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22588512	2
#22586795	2
C₂×#17985720	2
C₇₁×#3160	2
#9861	71

subgroup

index

C₇₁×#3160

Subgroups — #31897421

Conjugacy classes of subgroups

Complements

Maximal subgroups