Group #31884274 — subgroups

Subgroups — #31884274

Order 18176.

216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22586424	C₂	1	2	yes	maximal
2	C₇₁×#1985	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556068	—	2	2	no	—
4	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₈×C₇₁	C₄	1	1	yes	characteristic
6	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
7	#336×C₇₁	—	2	2	no	—
8	#14556068	—	2	4	no	—
9	#14556068	—	4	2	no	—
10	C₂×C₇₁⋊C₁₆	—	4	2	no	—
11	C₂²×Q₈×C₇₁	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₇₁	—	2	1	no	—
13	C₂²×C₈×C₇₁	D₄	1	2	yes	—
14	C₄×Q₈×C₇₁	—	4	1	no	—
15	C₄×Q₈×C₇₁	—	4	1	no	—
16	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
17	C₂×C₄⋊C₄×C₇₁	C₈	1	1	yes	characteristic
18	C₂×C₄²×C₇₁	—	2	1	no	—
19	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₈×C₇₁	—	4	1	no	—
21	#97×C₇₁	—	4	1	no	—
22	C₇₁⋊C₁₆	—	4	4	no	—
23	C₂×Q₈×C₇₁	—	4	2	no	—
24	C₂×Q₈×C₇₁	—	4	1	no	—
25	C₂²×C₄×C₇₁	—	2	1	no	—
26	C₂²×C₄×C₇₁	—	2	1	no	—
27	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
30	C₂×C₈×C₇₁	—	2	2	no	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	—	4	1	no	—
33	C₂×C₈×C₇₁	—	4	2	no	—
34	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
35	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
36	C₄⋊C₄×C₇₁	—	4	1	no	—
37	C₄⋊C₄×C₇₁	—	4	2	no	—
38	C₄²×C₇₁	—	2	1	no	—
39	C₄²×C₇₁	—	2	1	no	—
40	C₄²×C₇₁	—	4	1	no	—
41	C₄²×C₇₁	—	4	1	no	—
42	C₂³×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
43	Q₈×C₇₁	—	4	2	no	—
44	Q₈×C₇₁	—	4	2	no	—
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	—	2	1	no	—
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	—	4	1	no	—
52	C₂×C₄×C₇₁	—	4	2	no	—
53	C₂×C₄×C₇₁	—	4	1	no	—
54	C₂×C₄×C₇₁	#99	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#100	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#103	1	2	yes	—
57	C₂×C₄×C₇₁	#104	1	2	yes	—
58	C₈×C₇₁	—	2	2	no	—
59	C₈×C₇₁	—	2	2	no	—
60	C₈×C₇₁	—	8	1	no	—
61	C₂²×C₇₁	—	2	1	no	—
62	C₂²×C₇₁	—	2	1	no	—
63	C₂²×C₇₁	#323	1	1	yes	characteristic
64	C₂²×C₇₁	Q₈⋊C₈	1	2	yes	—
65	C₄×C₇₁	—	4	2	no	—
66	C₄×C₇₁	—	4	1	no	—
67	C₄×C₇₁	—	4	1	no	—
68	C₄×C₇₁	—	8	1	no	—
69	C₄×C₇₁	#328	1	2	yes	—
70	C₄×C₇₁	#329	1	1	yes	characteristic
71	C₄×C₇₁	#333	1	1	yes	characteristic
72	#7133	—	71	1	no	maximal, Sylow
73	C₂×C₇₁	—	2	2	no	—
74	C₂×C₇₁	#1272	1	1	yes	characteristic
75	C₂×C₇₁	#1320	1	1	yes	characteristic
76	C₂×C₇₁	#1321	1	1	yes	characteristic
77	#1985	D₇₁	1	1	yes	characteristic
78	#1377	—	71	2	no	—
79	C₇₁	#7133	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄×Q₈	Dic₇₁	1	1	yes	characteristic
81	C₂×M₃₂	—	142	2	no	—
82	C₂×C₄⋊C₈	Dic₇₁	1	1	yes	characteristic
83	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
84	#336	—	2	2	no	—
85	C₂²×Q₈	C₇₁⋊C₈	1	1	yes	characteristic
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
88	C₄×Q₈	—	4	1	no	—
89	C₄×Q₈	—	4	1	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	C₇₁⋊C₈	1	1	yes	characteristic
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
94	M₃₂	—	284	2	no	—
95	M₃₂	—	142	4	no	—
96	C₂×C₁₆	—	284	2	no	—
97	C₄⋊C₈	—	4	1	no	—
98	#97	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
104	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
105	C₂²×C₄	#11765849	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	#11765852	1	2	yes	—
111	C₂×C₈	#11765855	1	2	yes	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₁₆	—	284	4	no	—
119	C₂³	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#14556043	1	1	yes	characteristic
132	C₂×C₄	#14556045	1	1	yes	characteristic
133	C₂×C₄	#14556052	1	2	yes	—
134	C₂×C₄	#14556055	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#17985723	1	1	yes	characteristic
141	C₂²	Q₈⋊(C₇₁⋊C₈)	1	2	yes	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	#17985736	1	2	yes	—
147	C₄	#17985739	1	1	yes	characteristic
148	C₄	#17985748	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#22586166	1	1	yes	characteristic
151	C₂	#22586276	1	1	yes	characteristic
152	C₂	#22586278	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22586424

C₂

yes

maximal

C₇₁×#1985

C₂

yes

maximal, characteristic

C₂×#14556068

—

C₂×C₄×Q₈×C₇₁

C₄

yes

characteristic

C₂×C₄⋊C₈×C₇₁

C₄

yes

characteristic

C₂×#97×C₇₁

C₂²

yes

characteristic

#336×C₇₁

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂²×Q₈×C₇₁

C₈

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₈

yes

characteristic

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₁

—

#97×C₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₁

M₁₆

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂²⋊C₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#104

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#323

yes

characteristic

C₂²×C₇₁

Q₈⋊C₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#328

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

characteristic

#1377

—

C₇₁

#7133

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

Dic₇₁

yes

characteristic

C₂×M₃₂

—

142

—

C₂×C₄⋊C₈

Dic₇₁

yes

characteristic

C₂×#97

C₂×D₇₁

yes

characteristic

#336

—

C₂²×Q₈

C₇₁⋊C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇₁

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₇₁⋊C₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

M₃₂

—

284

—

M₃₂

—

142

—

C₂×C₁₆

—

284

—

C₄⋊C₈

—

#97

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊Dic₇₁

yes

characteristic

104

C₂²×C₄

C₂×C₇₁⋊C₈

yes

characteristic

105

C₂²×C₄

#11765849

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

#11765852

yes

—

111

C₂×C₈

#11765855

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₁₆

—

284

—

119

C₂³

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#14556043

yes

characteristic

132

C₂×C₄

#14556045

yes

characteristic

133

C₂×C₄

#14556052

yes

—

134

C₂×C₄

#14556055

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

#17985723

yes

characteristic

141

C₂²

Q₈⋊(C₇₁⋊C₈)

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#17985736

yes

—

147

C₄

#17985739

yes

characteristic

148

C₄

#17985748

yes

characteristic

149

C₂

—

150

C₂

#22586166

yes

characteristic

151

C₂

#22586276

yes

characteristic

152

C₂

#22586278

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556043
commutator	C₂²×C₄×C₇₁	C₂×C₈
Frattini	C₂×#97	C₂×D₇₁
Fitting	C₇₁×#1985	C₂
socle	C₂²×C₇₁	#323
prime radical	C₂³×C₇₁	C₂²⋊C₈
cosocle	C₂×#97×C₇₁	C₂²
hypercenter	#1985	D₇₁
hypocenter	C₇₁	#7133
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1985

subgroup

G/H

center

C₂×C₄

#14556043

commutator

C₂²×C₄×C₇₁

C₂×C₈

Frattini

C₂×#97

C₂×D₇₁

Fitting

C₇₁×#1985

C₂

socle

C₂²×C₇₁

#323

prime radical

C₂³×C₇₁

C₂²⋊C₈

cosocle

C₂×#97×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1985

N	H ≅ G/N	type
79. C₇₁	#7133	semidirect

H ≅ G/N

type

79. C₇₁

#7133

semidirect

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

subgroup	index
#22586424	2
C₇₁×#1985	2
#7133	71

subgroup

index

#22586424

C₇₁×#1985

#7133

Subgroups — #31884274

Conjugacy classes of subgroups

Complements

Maximal subgroups