Group #31891268 — subgroups

Subgroups — #31891268

Order 18176.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22588477	C₂	1	2	yes	maximal
2	#22587098	C₂	1	2	yes	maximal
3	#22587096	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1566	C₂	1	1	yes	maximal, characteristic
6	#17986023	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#17985763	—	2	4	no	—
10	#17985751	—	2	4	no	—
11	#17985750	—	2	2	no	—
12	#17985721	C₄	1	4	yes	—
13	#423×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#334×C₇₁	—	2	2	no	—
16	C₂×#11765849	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
18	C₂²×C₇₁⋊C₈	Q₈	1	2	yes	—
19	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
24	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
25	C₂×M₁₆×C₇₁	—	2	2	no	—
26	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
27	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₇₁	—	2	2	no	—
29	C₄⋊C₈×C₇₁	—	2	2	no	—
30	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₁	D₄	1	2	yes	—
32	#11765849	—	4	4	no	—
33	C₂×C₇₁⋊C₈	—	2	4	no	—
34	C₂×C₇₁⋊C₈	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	M₁₆×C₇₁	—	4	4	no	—
42	C₂×C₈×C₇₁	—	4	2	no	—
43	C₂×C₈×C₇₁	—	4	2	no	—
44	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
45	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
49	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#135	1	4	yes	—
59	C₈×C₇₁	—	2	4	no	—
60	C₈×C₇₁	—	8	2	no	—
61	C₇₁⋊C₈	—	2	4	no	—
62	C₇₁⋊C₈	—	2	4	no	—
63	C₇₁⋊C₈	—	8	2	no	—
64	C₂²×C₇₁	—	2	1	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	#384	1	1	yes	characteristic
67	C₂²×C₇₁	#499	1	2	yes	—
68	C₄×C₇₁	—	2	1	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	#427	1	2	yes	—
71	C₄×C₇₁	#430	1	2	yes	—
72	C₄×C₇₁	#430	1	2	yes	—
73	C₄×C₇₁	#495	1	2	yes	—
74	#8827	—	71	1	no	maximal, Sylow
75	C₂×C₇₁	—	2	2	no	—
76	C₂×C₇₁	#1846	1	1	yes	characteristic
77	C₂×C₇₁	#1849	1	1	yes	characteristic
78	C₂×C₇₁	#1849	1	1	yes	characteristic
79	#1837	—	71	2	no	—
80	#1567	—	71	2	no	—
81	#1566	—	71	1	no	—
82	#1566	D₇₁	1	1	yes	characteristic
83	C₂×#322	—	71	1	no	—
84	C₇₁	#8827	1	1	yes	characteristic, complemented, Sylow
85	#423	—	71	2	no	—
86	#423	C₂×D₇₁	1	2	yes	—
87	C₂×C₄×C₈	—	71	1	no	—
88	C₂×C₄×C₈	—	71	1	no	—
89	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
90	#341	—	142	4	no	—
91	#335	—	142	4	no	—
92	#334	—	2	2	no	—
93	#334	—	142	2	no	—
94	#322	—	71	4	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	142	2	no	—
97	C₂²×C₈	—	71	2	no	—
98	C₂²×C₈	—	71	2	no	—
99	C₂²×C₈	Dic₁₄₂	1	2	yes	—
100	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	142	2	no	—
104	C₄⋊C₈	—	142	2	no	—
105	C₄×C₈	—	71	2	no	—
106	C₄×C₈	—	71	2	no	—
107	C₄×C₈	—	71	2	no	—
108	C₄×C₈	—	71	2	no	—
109	C₄×C₈	C₄×D₇₁	1	2	yes	—
110	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
111	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
112	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
114	M₁₆	—	4	4	no	—
115	M₁₆	—	284	4	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	284	2	no	—
119	C₂×C₈	—	284	2	no	—
120	C₂×C₈	—	71	4	no	—
121	C₂×C₈	—	71	4	no	—
122	C₂×C₈	—	142	4	no	—
123	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
124	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
125	C₄²	D₄×D₇₁	1	1	yes	characteristic
126	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
127	C₄²	Q₈×D₇₁	1	1	yes	characteristic
128	C₂³	#14556127	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
132	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
133	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
134	C₂×C₄	#14556124	1	1	yes	characteristic
135	C₂×C₄	#14556126	1	1	yes	characteristic
136	C₂×C₄	#14556127	1	1	yes	characteristic
137	C₂×C₄	#14556149	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	8	2	no	—
140	C₈	—	568	2	no	—
141	C₈	—	142	4	no	—
142	C₈	—	142	4	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#17985868	1	1	yes	characteristic
146	C₂²	#17986399	1	2	yes	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#17986034	1	2	yes	—
150	C₄	#17986046	1	2	yes	—
151	C₄	#17986047	1	2	yes	—
152	C₄	#17986380	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#22588501	1	1	yes	characteristic
155	C₂	#22588513	1	1	yes	characteristic
156	C₂	#22588514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588477

C₂

yes

maximal

#22587098

C₂

yes

maximal

#22587096

C₂

yes

maximal, characteristic

C₂×#17985721

C₂

yes

maximal, characteristic

C₇₁×#1566

C₂

yes

maximal, characteristic

#17986023

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

—

—

C₄

yes

—

#423×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#334×C₇₁

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

D₄

yes

—

C₂²×C₇₁⋊C₈

Q₈

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₄×C₇₁⋊C₈

Q₈

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

Q₈

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#384

yes

characteristic

C₂²×C₇₁

#499

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#427

yes

—

C₄×C₇₁

#430

yes

—

C₄×C₇₁

#430

yes

—

C₄×C₇₁

#495

yes

—

#8827

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

—

D₇₁

yes

characteristic

C₂×#322

—

C₇₁

#8827

yes

characteristic, complemented, Sylow

#423

—

#423

C₂×D₇₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#341

—

142

—

#335

—

142

—

#334

—

#334

—

142

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₁₄₂

yes

—

100

C₂×C₄²

C₂²×D₇₁

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

142

—

104

C₄⋊C₈

—

142

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₄×D₇₁

yes

—

110

C₄×C₈

C₂²⋊D₇₁

yes

—

111

C₂²×C₄

D₄×D₇₁

yes

characteristic

112

C₂²×C₄

Q₈×D₇₁

yes

characteristic

113

C₂²×C₄

C₂×Dic₁₄₂

yes

characteristic

114

M₁₆

—

115

M₁₆

—

284

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

284

—

119

C₂×C₈

—

284

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

142

—

123

C₂×C₈

Dic₇₁⋊C₄

yes

—

124

C₄²

C₂×C₄×D₇₁

yes

characteristic

125

C₄²

D₄×D₇₁

yes

characteristic

126

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

127

C₄²

Q₈×D₇₁

yes

characteristic

128

C₂³

#14556127

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

132

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

133

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

134

C₂×C₄

#14556124

yes

characteristic

135

C₂×C₄

#14556126

yes

characteristic

136

C₂×C₄

#14556127

yes

characteristic

137

C₂×C₄

#14556149

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

568

—

141

C₈

—

142

—

142

C₈

—

142

—

143

C₂²

—

144

C₂²

—

145

C₂²

#17985868

yes

characteristic

146

C₂²

#17986399

yes

—

147

C₄

—

148

C₄

—

149

C₄

#17986034

yes

—

150

C₄

#17986046

yes

—

151

C₄

#17986047

yes

—

152

C₄

#17986380

yes

—

153

C₂

—

154

C₂

#22588501

yes

characteristic

155

C₂

#22588513

yes

characteristic

156

C₂

#22588514

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1566	C₂
socle	C₂²×C₇₁	#384
prime radical	C₂³×C₇₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1566	D₇₁
hypocenter	C₇₁	#8827
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1566

C₂

socle

C₂²×C₇₁

#384

prime radical

C₂³×C₇₁

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
84. C₇₁	#8827	semidirect

H ≅ G/N

type

84. C₇₁

#8827

semidirect

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

subgroup	index
#22588477	2
#22587098	2
#22587096	2
C₂×#17985721	2
C₇₁×#1566	2
#8827	71

subgroup

index

C₇₁×#1566

Subgroups — #31891268

Conjugacy classes of subgroups

Complements

Maximal subgroups