Group #31912863 — subgroups

Subgroups — #31912863

Order 18176.

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

normal subgroups only

This group has 314 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	C₂×#17985828	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₇₁	C₂	1	2	yes	maximal
4	#22586434	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556064	C₂²	1	2	yes	—
8	M₁₆×Dic₇₁	—	2	2	no	—
9	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985828	C₄	1	8	yes	complemented
11	C₁₆×Dic₇₁	C₄	1	4	yes	—
12	#17985763	—	2	2	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₁	C₂²	1	2	yes	—
15	C₂×#11765849	—	2	2	no	—
16	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
20	#14556064	—	2	2	no	—
21	#14556064	C₂×C₄	1	4	yes	—
22	#14556038	—	4	2	no	—
23	C₈×Dic₇₁	—	4	2	no	—
24	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
25	C₈×Dic₇₁	Q₈	1	2	yes	—
26	C₂×M₁₆×C₇₁	—	2	2	no	—
27	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₇₁	—	2	2	no	—
29	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
30	#108×C₇₁	—	2	2	no	—
31	#108×C₇₁	C₂×C₄	1	4	yes	—
32	C₂²×Dic₇₁	Q₁₆	1	2	yes	—
33	#11765849	—	2	4	no	—
34	#11765849	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	2	2	no	—
36	C₂×C₇₁⋊C₈	—	4	2	no	—
37	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
38	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
39	C₄×Dic₇₁	D₈	1	1	yes	characteristic
40	C₄×Dic₇₁	D₈	1	1	yes	characteristic
41	C₇₁⋊C₁₆	C₄²	1	4	yes	—
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₇₁	—	2	4	no	—
44	M₁₆×C₇₁	—	4	2	no	—
45	C₂×C₈×C₇₁	—	4	2	no	—
46	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₇₁	#41	1	2	yes	—
50	C₁₆×C₇₁	—	2	2	no	—
51	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#135	1	2	yes	—
56	C₂×Dic₇₁	—	2	4	no	—
57	C₂×Dic₇₁	#107	1	2	yes	—
58	C₂×Dic₇₁	#107	1	2	yes	—
59	C₈×C₇₁	—	4	4	no	—
60	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
61	C₈×C₇₁	#117	1	1	yes	characteristic
62	C₈×C₇₁	C₄×Q₈	1	2	yes	—
63	C₇₁⋊C₈	—	2	2	no	—
64	C₇₁⋊C₈	—	4	4	no	—
65	C₂²×C₇₁	—	2	2	no	—
66	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
67	C₂²×C₇₁	C₄×Q₁₆	1	2	yes	—
68	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₇₁	#427	1	1	yes	characteristic
70	C₄×C₇₁	C₄×D₈	1	2	yes	—
71	Dic₇₁	—	2	2	no	—
72	Dic₇₁	#368	1	4	yes	complemented
73	C₄×#368	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	—	2	2	no	—
75	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
76	C₂×C₇₁	C₂×#368	1	2	yes	—
77	C₂×#368	—	71	2	no	—
78	C₂×C₄×C₁₆	—	71	1	no	—
79	C₄×#108	—	71	2	no	—
80	#1381	—	71	1	no	—
81	#1381	D₇₁	1	1	yes	characteristic
82	C₇₁	C₄×#368	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	—	71	1	no	—
84	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
85	C₂×#108	—	71	2	no	—
86	C₂×#108	C₂×D₇₁	1	2	yes	—
87	C₄×M₁₆	—	142	2	no	—
88	C₂×C₄×C₈	—	71	1	no	—
89	#368	—	71	8	no	—
90	C₄×C₁₆	—	71	4	no	—
91	#341	—	142	2	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	142	2	no	—
94	C₂²×C₈	—	71	1	no	—
95	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
96	C₂×C₄²	—	71	1	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	71	2	no	—
99	C₂×C₁₆	—	71	4	no	—
100	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
101	#108	—	2	2	no	—
102	#108	—	71	4	no	—
103	#108	—	142	2	no	—
104	#108	C₄×D₇₁	1	4	yes	—
105	#97	—	284	2	no	—
106	C₄×C₈	—	284	2	no	—
107	C₄×C₈	—	71	2	no	—
108	C₄×C₈	—	71	2	no	—
109	C₂²×C₄	—	71	2	no	—
110	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
111	M₁₆	—	2	4	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	284	2	no	—
114	M₁₆	—	142	4	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	284	2	no	—
117	C₂×C₈	—	71	2	no	—
118	C₂×C₈	—	142	2	no	—
119	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
121	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
122	C₂×C₈	#11765864	1	2	yes	—
123	C₄²	—	71	2	no	—
124	C₄²	—	71	1	no	—
125	C₄²	—	71	1	no	—
126	C₁₆	—	2	2	no	—
127	C₁₆	—	71	4	no	—
128	C₂³	Q₁₆×D₇₁	1	1	yes	characteristic
129	C₂×C₄	—	71	2	no	—
130	C₂×C₄	—	71	2	no	—
131	C₂×C₄	—	142	4	no	—
132	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
133	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
134	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
135	C₂×C₄	#14556149	1	2	yes	—
136	C₈	—	4	4	no	—
137	C₈	—	284	4	no	—
138	C₈	—	142	2	no	—
139	C₈	C₄²×D₇₁	1	1	yes	characteristic
140	C₈	#14556084	1	1	yes	characteristic
141	C₈	#14556093	1	2	yes	—
142	C₂²	—	2	2	no	—
143	C₂²	#107×D₇₁	1	1	yes	characteristic
144	C₂²	#17986091	1	2	yes	—
145	C₄	—	71	4	no	—
146	C₄	—	142	2	no	—
147	C₄	#17985850	1	1	yes	characteristic
148	C₄	#17986034	1	1	yes	characteristic
149	C₄	#17986079	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#22588232	1	1	yes	characteristic
152	C₂	#368×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985828

C₂

yes

maximal

C₂×C₁₆×Dic₇₁

C₂

yes

maximal, characteristic

#108×Dic₇₁

C₂

yes

maximal

#22586434

C₂

yes

maximal, characteristic

C₇₁×#1381

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

M₁₆×Dic₇₁

—

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

#17985828

C₄

yes

complemented

C₁₆×Dic₇₁

C₄

yes

—

#17985763

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556064

—

#14556064

C₂×C₄

yes

—

#14556038

—

C₈×Dic₇₁

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₈×Dic₇₁

Q₈

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#108×C₇₁

—

#108×C₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

Q₁₆

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₄×Dic₇₁

C₄⋊C₄

yes

—

C₄×Dic₇₁

D₈

yes

characteristic

C₄×Dic₇₁

D₈

yes

characteristic

C₇₁⋊C₁₆

C₄²

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

#107

yes

—

C₂×Dic₇₁

#107

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₄²

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

C₄×Q₁₆

yes

—

C₄×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₁

#427

yes

characteristic

C₄×C₇₁

C₄×D₈

yes

—

Dic₇₁

—

Dic₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

C₄×#107

yes

characteristic

C₂×C₇₁

C₂×#368

yes

—

C₂×#368

—

C₂×C₄×C₁₆

—

C₄×#108

—

#1381

—

#1381

D₇₁

yes

characteristic

C₇₁

C₄×#368

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₇₁

yes

—

C₄×M₁₆

—

142

—

C₂×C₄×C₈

—

#368

—

C₄×C₁₆

—

#341

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

C₄×D₇₁

yes

—

101

#108

—

102

#108

—

103

#108

—

142

—

104

#108

C₄×D₇₁

yes

—

105

#97

—

284

—

106

C₄×C₈

—

284

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₂²×C₄

—

110

C₂²×C₄

D₄×D₇₁

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

284

—

114

M₁₆

—

142

—

115

C₂×C₈

—

116

C₂×C₈

—

284

—

117

C₂×C₈

—

118

C₂×C₈

—

142

—

119

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

120

C₂×C₈

C₂×C₄×D₇₁

yes

—

121

C₂×C₈

Q₈×D₇₁

yes

characteristic

122

C₂×C₈

#11765864

yes

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₁₆

—

127

C₁₆

—

128

C₂³

Q₁₆×D₇₁

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

142

—

132

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

133

C₂×C₄

(C₂²⋊D₇₁)⋊C₄

yes

—

134

C₂×C₄

D₈×D₇₁

yes

characteristic

135

C₂×C₄

#14556149

yes

—

136

C₈

—

137

C₈

—

284

—

138

C₈

—

142

—

139

C₈

C₄²×D₇₁

yes

characteristic

140

C₈

#14556084

yes

characteristic

141

C₈

#14556093

yes

—

142

C₂²

—

143

C₂²

#107×D₇₁

yes

characteristic

144

C₂²

#17986091

yes

—

145

C₄

—

146

C₄

—

142

—

147

C₄

#17985850

yes

characteristic

148

C₄

#17986034

yes

characteristic

149

C₄

#17986079

yes

—

150

C₂

—

151

C₂

#22588232

yes

characteristic

152

C₂

#368×D₇₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1381	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1381	D₇₁
hypocenter	C₇₁	C₄×#368
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₈×C₇₁

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1381

C₂

socle

C₂²×C₇₁

C₂×#107

prime radical

C₂³×C₇₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
10. #17985828	C₄	semidirect
72. Dic₇₁	#368	semidirect
82. C₇₁	C₄×#368	semidirect

H ≅ G/N

type

10. #17985828

C₄

semidirect

72. Dic₇₁

#368

semidirect

82. C₇₁

C₄×#368

semidirect

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

subgroup	index
C₂×#17985828	2
C₂×C₁₆×Dic₇₁	2
#108×Dic₇₁	2
#22586434	2
C₇₁×#1381	2
C₄×#368	71

subgroup

index

C₂×#17985828

C₂×C₁₆×Dic₇₁

#108×Dic₇₁

#22586434

C₇₁×#1381

C₄×#368

Subgroups — #31912863

Conjugacy classes of subgroups

Complements

Maximal subgroups