Group #31913834 — subgroups

Subgroups — #31913834

Order 18176.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22588504	C₂	1	1	yes	maximal, characteristic
3	#22586431	C₂	1	4	yes	maximal
4	C₇₁×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556062	C₂²	1	2	yes	—
7	C₂×#14556060	—	2	1	no	—
8	#17985865	—	2	1	no	—
9	C₄×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₇₁	C₂²	1	2	yes	—
11	C₂×#106×C₇₁	—	2	1	no	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#382×C₇₁	—	2	1	no	—
14	C₂×C₄⋊Dic₇₁	—	2	2	no	—
15	C₂×C₇₁⋊C₁₆	—	2	2	no	—
16	C₂×C₇₁⋊C₁₆	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
18	#14556062	—	2	4	no	—
19	#14556060	—	4	2	no	—
20	C₇₁⋊#95	—	4	2	no	—
21	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
24	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
25	#107×C₇₁	—	2	4	no	—
26	#106×C₇₁	—	4	2	no	—
27	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₁	Q₈	1	2	yes	—
29	#95×C₇₁	—	4	2	no	—
30	C₂²×Dic₇₁	—	4	2	no	—
31	C₄⋊Dic₇₁	—	4	2	no	—
32	C₄⋊Dic₇₁	—	4	2	no	—
33	C₇₁⋊C₁₆	—	2	4	no	—
34	C₂²×C₄×C₇₁	—	4	2	no	—
35	C₂²×C₄×C₇₁	D₈	1	2	yes	—
36	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
39	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
40	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₇₁	#41	1	2	yes	—
43	C₂×C₈×C₇₁	#41	1	2	yes	—
44	C₄⋊C₄×C₇₁	—	4	2	no	—
45	C₄⋊C₄×C₇₁	—	4	2	no	—
46	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
47	C₄²×C₇₁	Q₁₆	1	2	yes	—
48	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₁	—	2	4	no	—
50	C₂×C₄×C₇₁	—	8	2	no	—
51	C₂×C₄×C₇₁	—	8	4	no	—
52	C₂×C₄×C₇₁	#107	1	4	yes	—
53	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₇₁	—	8	2	no	—
60	C₂×Dic₇₁	—	8	4	no	—
61	C₈×C₇₁	#106	1	4	yes	—
62	C₈×C₇₁	#117	1	2	yes	—
63	C₈×C₇₁	#125	1	2	yes	—
64	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
65	C₂²×C₇₁	#486	1	1	yes	characteristic
66	C₂²×C₇₁	#486	1	1	yes	characteristic
67	C₂²×C₇₁	#508	1	4	yes	—
68	C₄×C₇₁	—	2	4	no	—
69	C₄×C₇₁	—	16	2	no	—
70	C₄×C₇₁	#382	1	1	yes	characteristic
71	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
72	C₄×C₇₁	#487	1	1	yes	characteristic
73	C₄×C₇₁	#487	1	1	yes	characteristic
74	Dic₇₁	—	16	2	no	—
75	#12889	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	#1834	1	1	yes	characteristic
77	C₂×C₇₁	#2158	1	2	yes	—
78	C₂×C₇₁	#2242	1	2	yes	—
79	C₂×C₇₁	#2242	1	2	yes	—
80	C₂×C₄×C₁₆	—	71	1	no	—
81	#1846	—	71	1	no	—
82	#1846	D₇₁	1	1	yes	characteristic
83	#1380	—	71	4	no	—
84	C₇₁	#12889	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₁₆	—	71	2	no	—
86	C₂×#107	—	71	2	no	—
87	C₂×#107	C₂×D₇₁	1	2	yes	—
88	C₂×#106	—	2	1	no	—
89	C₂×#106	—	142	1	no	—
90	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
91	#382	—	2	1	no	—
92	#382	—	142	1	no	—
93	C₄×C₁₆	—	71	4	no	—
94	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	142	2	no	—
98	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
99	C₂×C₁₆	—	71	4	no	—
100	C₂×C₁₆	—	142	2	no	—
101	C₂×C₁₆	—	142	2	no	—
102	#107	—	2	4	no	—
103	#107	—	142	4	no	—
104	#106	—	4	2	no	—
105	#106	—	284	2	no	—
106	C₄×C₈	Dic₁₄₂	1	2	yes	—
107	C₄×C₈	C₄×D₇₁	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	284	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	284	2	no	—
112	C₂²×C₄	#11765850	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
114	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
115	C₂×C₈	#11765852	1	2	yes	—
116	C₂×C₈	#11765853	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
119	C₂×C₈	#11765864	1	2	yes	—
120	C₂×C₈	#11765866	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	284	2	no	—
124	C₄⋊C₄	—	284	2	no	—
125	C₄²	Dic₇₁⋊C₄	1	2	yes	—
126	C₄²	#11765855	1	2	yes	—
127	C₁₆	—	142	4	no	—
128	C₂³	C₂×#11765850	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	—	568	2	no	—
133	C₂×C₄	—	568	4	no	—
134	C₂×C₄	#14556061	1	4	yes	—
135	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#14556115	1	1	yes	characteristic
137	C₂×C₄	#14556117	1	1	yes	characteristic
138	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
139	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
140	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
141	C₈	#14556059	1	4	yes	—
142	C₈	#14556086	1	2	yes	—
143	C₈	#14556119	1	2	yes	—
144	C₂²	C₂×#14556061	1	1	yes	characteristic
145	C₂²	#17986345	1	1	yes	characteristic
146	C₂²	#17986347	1	1	yes	characteristic
147	C₂²	#17986429	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	1136	2	no	—
151	C₄	#17985864	1	1	yes	characteristic
152	C₄	C₂×#14556059	1	1	yes	characteristic
153	C₄	#17986349	1	1	yes	characteristic
154	C₄	#17986351	1	1	yes	characteristic
155	C₂	#22588467	1	1	yes	characteristic
156	C₂	#22590122	1	2	yes	—
157	C₂	#22590571	1	2	yes	—
158	C₂	#22590574	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic

#22588504

C₂

yes

maximal, characteristic

#22586431

C₂

yes

maximal

C₇₁×#1846

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

C₂×#14556062

C₂²

yes

—

C₂×#14556060

—

#17985865

—

C₄×C₇₁⋊C₁₆

C₄

yes

complemented

C₂×#107×C₇₁

C₂²

yes

—

C₂×#106×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#382×C₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556062

—

#14556060

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#106×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

#95×C₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

Q₁₆

yes

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#106

yes

—

C₈×C₇₁

#117

yes

—

C₈×C₇₁

#125

yes

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2242

yes

—

C₂×C₇₁

#2242

yes

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₇₁

yes

characteristic

#1380

—

C₇₁

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₇₁

yes

—

C₂×#106

—

C₂×#106

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#382

—

#382

—

142

—

C₄×C₁₆

—

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

142

—

101

C₂×C₁₆

—

142

—

102

#107

—

103

#107

—

142

—

104

#106

—

105

#106

—

284

—

106

C₄×C₈

Dic₁₄₂

yes

—

107

C₄×C₈

C₄×D₇₁

yes

—

108

#95

—

109

#95

—

284

—

110

C₂²×C₄

—

111

C₂²×C₄

—

284

—

112

C₂²×C₄

#11765850

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

114

C₂×C₈

Dic₇₁⋊C₄

yes

—

115

C₂×C₈

#11765852

yes

—

116

C₂×C₈

#11765853

yes

—

117

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

118

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

119

C₂×C₈

#11765864

yes

—

120

C₂×C₈

#11765866

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

284

—

124

C₄⋊C₄

—

284

—

125

C₄²

Dic₇₁⋊C₄

yes

—

126

C₄²

#11765855

yes

—

127

C₁₆

—

142

—

128

C₂³

C₂×#11765850

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

568

—

133

C₂×C₄

—

568

—

134

C₂×C₄

#14556061

yes

—

135

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

136

C₂×C₄

#14556115

yes

characteristic

137

C₂×C₄

#14556117

yes

characteristic

138

C₂×C₄

C₂×#11765852

yes

characteristic

139

C₂×C₄

C₂×#11765853

yes

characteristic

140

C₂×C₄

C₂×#11765855

yes

characteristic

141

C₈

#14556059

yes

—

142

C₈

#14556086

yes

—

143

C₈

#14556119

yes

—

144

C₂²

C₂×#14556061

yes

characteristic

145

C₂²

#17986345

yes

characteristic

146

C₂²

#17986347

yes

characteristic

147

C₂²

#17986429

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

1136

—

151

C₄

#17985864

yes

characteristic

152

C₄

C₂×#14556059

yes

characteristic

153

C₄

#17986349

yes

characteristic

154

C₄

#17986351

yes

characteristic

155

C₂

#22588467

yes

characteristic

156

C₂

#22590122

yes

—

157

C₂

#22590571

yes

—

158

C₂

#22590574

yes

—

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1846	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1846	D₇₁
hypocenter	C₇₁	#12889
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#11765850

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1846

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×C₇₁⋊C₁₆	C₄	semidirect
84. C₇₁	#12889	semidirect

H ≅ G/N

type

9. C₄×C₇₁⋊C₁₆

C₄

semidirect

84. C₇₁

#12889

semidirect

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

subgroup	index
C₂×C₄×C₇₁⋊C₁₆	2
#22588504	2
#22586431	2
C₇₁×#1846	2
#12889	71

subgroup

index

C₂×C₄×C₇₁⋊C₁₆

#22588504

#22586431

C₇₁×#1846

#12889

Subgroups — #31913834

Conjugacy classes of subgroups

Complements

Maximal subgroups