Group #31891125 — subgroups

Subgroups — #31891125

Order 18176.

322 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22588457	C₂	1	1	yes	maximal, characteristic
2	#22588086	C₂	1	1	yes	maximal, characteristic
3	#22586174	C₂	1	4	yes	maximal
4	C₇₁×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
7	#17985754	—	2	2	no	—
8	#17985754	C₄	1	2	yes	—
9	#17985754	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
11	C₄³×C₇₁	C₂²	1	1	yes	characteristic
12	#336×C₇₁	—	2	2	no	—
13	C₂²×C₇₁⋊C₈	—	2	2	no	—
14	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
15	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
16	C₄×C₇₁⋊C₈	—	2	2	no	—
17	C₄×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₈×C₇₁	—	2	2	no	—
19	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₁	D₄	1	2	yes	—
22	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
23	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	—	2	4	no	—
25	C₂×C₇₁⋊C₈	—	2	4	no	—
26	C₂×C₇₁⋊C₈	—	2	4	no	—
27	C₂×C₇₁⋊C₈	—	2	4	no	—
28	C₂×C₇₁⋊C₈	—	4	2	no	—
29	C₂×C₇₁⋊C₈	—	4	4	no	—
30	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₇₁	—	4	2	no	—
36	C₂×C₈×C₇₁	—	4	4	no	—
37	C₄²×C₇₁	—	2	2	no	—
38	C₄²×C₇₁	—	2	2	no	—
39	C₄²×C₇₁	—	2	4	no	—
40	C₄²×C₇₁	—	4	2	no	—
41	C₄²×C₇₁	#41	1	2	yes	—
42	C₄²×C₇₁	#41	1	2	yes	—
43	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	#117	1	2	yes	—
52	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#125	1	2	yes	—
55	C₂×C₄×C₇₁	#131	1	2	yes	—
56	C₂×C₄×C₇₁	#131	1	2	yes	—
57	C₂×C₄×C₇₁	#135	1	4	yes	—
58	C₈×C₇₁	—	8	2	no	—
59	C₇₁⋊C₈	—	4	4	no	—
60	C₇₁⋊C₈	—	8	2	no	—
61	C₂²×C₇₁	#382	1	1	yes	characteristic
62	C₂²×C₇₁	#424	1	1	yes	characteristic
63	C₂²×C₇₁	#424	1	1	yes	characteristic
64	C₂²×C₇₁	#427	1	2	yes	—
65	C₂²×C₇₁	#430	1	2	yes	—
66	C₄×C₇₁	—	2	2	no	—
67	C₄×C₇₁	—	4	2	no	—
68	C₄×C₇₁	—	4	2	no	—
69	C₄×C₇₁	#433	1	4	yes	—
70	C₄×C₇₁	#488	1	2	yes	—
71	C₄×C₇₁	#488	1	2	yes	—
72	#8788	—	71	1	no	maximal, Sylow
73	C₂×C₇₁	#1572	1	2	yes	—
74	C₂×C₇₁	#1572	1	2	yes	—
75	C₂×C₇₁	#1832	1	1	yes	characteristic
76	C₂×C₇₁	#1835	1	1	yes	characteristic
77	C₂×C₇₁	#1838	1	1	yes	characteristic
78	#1831	—	71	1	no	—
79	#1831	D₇₁	1	1	yes	characteristic
80	#1744	—	71	1	no	—
81	#1277	—	71	4	no	—
82	C₇₁	#8788	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊C₈	—	71	2	no	—
84	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
85	C₂×C₄×C₈	—	71	2	no	—
86	C₄³	C₂×D₇₁	1	1	yes	characteristic
87	#336	—	2	2	no	—
88	#336	—	71	2	no	—
89	#336	—	71	2	no	—
90	#336	—	142	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	71	4	no	—
93	C₂²×C₈	—	142	2	no	—
94	C₂×C₄²	Dic₁₄₂	1	2	yes	—
95	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
96	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
98	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	142	4	no	—
101	C₄×C₈	—	142	2	no	—
102	C₄×C₈	—	142	2	no	—
103	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
104	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
105	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
107	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	4	no	—
110	C₂×C₈	—	284	2	no	—
111	C₂×C₈	—	284	4	no	—
112	C₂×C₈	—	142	4	no	—
113	C₂×C₈	—	142	4	no	—
114	C₂×C₈	—	142	4	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	4	2	no	—
119	C₄²	#11765864	1	2	yes	—
120	C₄²	#11765866	1	2	yes	—
121	C₂³	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#14556086	1	2	yes	—
130	C₂×C₄	#14556115	1	1	yes	characteristic
131	C₂×C₄	#14556117	1	1	yes	characteristic
132	C₂×C₄	#14556119	1	2	yes	—
133	C₂×C₄	#14556134	1	2	yes	—
134	C₂×C₄	#14556136	1	2	yes	—
135	C₂×C₄	#14556148	1	4	yes	—
136	C₈	—	8	2	no	—
137	C₈	—	284	4	no	—
138	C₈	—	568	2	no	—
139	C₂²	#17985864	1	1	yes	characteristic
140	C₂²	#17986025	1	1	yes	characteristic
141	C₂²	#17986027	1	1	yes	characteristic
142	C₂²	#17986035	1	2	yes	—
143	C₂²	#17986047	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	2	no	—
146	C₄	—	4	2	no	—
147	C₄	#17986059	1	4	yes	—
148	C₄	#17986353	1	2	yes	—
149	C₄	#17986356	1	2	yes	—
150	C₂	#22587118	1	2	yes	—
151	C₂	#22587121	1	2	yes	—
152	C₂	#22588459	1	1	yes	characteristic
153	C₂	#22588470	1	1	yes	characteristic
154	C₂	#22588480	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588457

C₂

yes

maximal, characteristic

#22588086

C₂

yes

maximal, characteristic

#22586174

C₂

yes

maximal

C₇₁×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

#17985754

—

#17985754

C₄

yes

—

#17985754

C₄

yes

—

C₂×C₄⋊C₈×C₇₁

C₂²

yes

—

C₄³×C₇₁

C₂²

yes

characteristic

#336×C₇₁

—

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

Q₈

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

#41

yes

—

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#430

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#433

yes

—

C₄×C₇₁

#488

yes

—

C₄×C₇₁

#488

yes

—

#8788

—

maximal, Sylow

C₂×C₇₁

#1572

yes

—

C₂×C₇₁

#1572

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

#1744

—

#1277

—

C₇₁

#8788

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₇₁

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₇₁

yes

—

—

—

—

142

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂×C₄²

Dic₁₄₂

yes

—

C₂×C₄²

C₄×D₇₁

yes

characteristic

C₂×C₄²

C₄×D₇₁

yes

characteristic

C₂×C₄²

C₂²⋊D₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

142

—

101

C₄×C₈

—

142

—

102

C₄×C₈

—

142

—

103

C₂²×C₄

Dic₇₁⋊C₄

yes

—

104

C₂²×C₄

Dic₇₁⋊C₄

yes

—

105

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

106

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

107

C₂²×C₄

C₂×Dic₁₄₂

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

284

—

111

C₂×C₈

—

284

—

112

C₂×C₈

—

142

—

113

C₂×C₈

—

142

—

114

C₂×C₈

—

142

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

#11765864

yes

—

120

C₄²

#11765866

yes

—

121

C₂³

C₂×Dic₇₁⋊C₄

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#14556086

yes

—

130

C₂×C₄

#14556115

yes

characteristic

131

C₂×C₄

#14556117

yes

characteristic

132

C₂×C₄

#14556119

yes

—

133

C₂×C₄

#14556134

yes

—

134

C₂×C₄

#14556136

yes

—

135

C₂×C₄

#14556148

yes

—

136

C₈

—

137

C₈

—

284

—

138

C₈

—

568

—

139

C₂²

#17985864

yes

characteristic

140

C₂²

#17986025

yes

characteristic

141

C₂²

#17986027

yes

characteristic

142

C₂²

#17986035

yes

—

143

C₂²

#17986047

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#17986059

yes

—

148

C₄

#17986353

yes

—

149

C₄

#17986356

yes

—

150

C₂

#22587118

yes

—

151

C₂

#22587121

yes

—

152

C₂

#22588459

yes

characteristic

153

C₂

#22588470

yes

characteristic

154

C₂

#22588480

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1831	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1831	D₇₁
hypocenter	C₇₁	#8788
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₇₁

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1831

C₂

socle, prime radical

C₂³×C₇₁

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
82. C₇₁	#8788	semidirect

H ≅ G/N

type

82. C₇₁

#8788

semidirect

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

subgroup	index
#22588457	2
#22588086	2
#22586174	2
C₇₁×#1831	2
#8788	71

subgroup

index

C₇₁×#1831

Subgroups — #31891125

Conjugacy classes of subgroups

Complements

Maximal subgroups