Group #31909442 — subgroups

Subgroups — #31909442

Order 18176.

210 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₄×Dic₅₆₈	C₂	1	1	yes	maximal, characteristic
2	#22587236	C₂	1	2	yes	maximal
3	#22586303	C₂	1	2	yes	maximal
4	C₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₈	C₄	1	1	yes	characteristic
7	#17986126	—	2	2	no	—
8	C₄×Dic₂₈₄	C₂²	1	2	yes	—
9	#17985825	C₄	1	1	yes	characteristic
10	#17985804	C₄	1	2	yes	—
11	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#17985729	—	2	2	no	—
13	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	#335×C₇₁	C₂²	1	2	yes	—
16	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₄₂	—	2	2	no	—
18	Dic₅₆₈	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	#14556062	C₂×C₄	1	2	yes	—
21	Dic₇₁⋊C₈	—	4	2	no	—
22	#14556053	—	2	2	no	—
23	#14556038	—	4	2	no	—
24	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
25	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₄₂	—	2	2	no	—
31	Dic₂₈₄	—	2	2	no	—
32	Dic₂₈₄	—	2	2	no	—
33	C₂×C₇₁⋊C₈	—	2	1	no	—
34	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
35	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
36	C₄⋊Dic₇₁	—	2	2	no	—
37	Dic₇₁⋊C₄	—	4	2	no	—
38	C₄×Dic₇₁	—	4	2	no	—
39	C₇₁⋊C₁₆	—	2	2	no	—
40	C₂×C₈×C₇₁	—	4	2	no	—
41	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
43	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₁	—	2	1	no	—
45	C₁₆×C₇₁	M₁₆	1	2	yes	—
46	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
49	C₂×Dic₇₁	—	4	2	no	—
50	C₂×Dic₇₁	—	4	2	no	—
51	Dic₁₄₂	—	4	2	no	—
52	Dic₁₄₂	—	4	2	no	—
53	C₈×C₇₁	—	2	1	no	—
54	C₈×C₇₁	—	8	2	no	—
55	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₇₁	#131	1	1	yes	characteristic
57	C₇₁⋊C₈	—	2	2	no	—
58	C₇₁⋊C₈	—	2	1	no	—
59	C₇₁⋊C₈	Q₃₂	1	2	yes	complemented
60	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	#436	1	1	yes	characteristic
63	C₄×C₇₁	#443	1	1	yes	characteristic
64	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
65	C₄×C₇₁	#508	1	1	yes	characteristic
66	Dic₇₁	—	8	2	no	—
67	Dic₇₁	—	8	2	no	—
68	#12083	—	71	1	no	maximal, Sylow
69	C₂×C₇₁	#1589	1	1	yes	characteristic
70	C₂×C₇₁	C₄×Q₃₂	1	1	yes	characteristic
71	C₂×C₇₁	#2178	1	1	yes	characteristic
72	C₄×Q₃₂	—	71	1	no	—
73	#1591	—	71	2	no	—
74	#1371	D₇₁	1	1	yes	characteristic
75	#1332	—	71	2	no	—
76	C₈×C₁₆	—	71	1	no	—
77	C₇₁	#12083	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₃₂	—	71	1	no	—
79	#446	—	142	2	no	—
80	C₄×Q₁₆	—	71	2	no	—
81	#366	—	71	1	no	—
82	#358	—	71	2	no	—
83	C₄×C₁₆	—	71	1	no	—
84	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
85	#335	C₂×D₇₁	1	2	yes	—
86	Q₈⋊C₈	—	142	2	no	—
87	C₈²	—	71	1	no	—
88	C₂×Q₁₆	—	71	2	no	—
89	C₄×Q₈	—	142	2	no	—
90	Q₃₂	—	142	2	no	—
91	C₂×C₁₆	—	142	1	no	—
92	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
94	#107	—	71	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	284	2	no	—
97	#103	—	142	2	no	—
98	#97	—	284	2	no	—
99	C₄×C₈	—	71	1	no	—
100	C₄×C₈	—	71	1	no	—
101	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
102	C₂×Q₈	—	142	2	no	—
103	Q₁₆	—	142	2	no	—
104	Q₁₆	—	142	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	71	1	no	—
107	C₂×C₈	—	71	1	no	—
108	C₂×C₈	—	142	1	no	—
109	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
110	C₂×C₈	#11765866	1	1	yes	characteristic
111	C₄⋊C₄	—	284	2	no	—
112	C₄⋊C₄	—	142	2	no	—
113	C₄²	—	284	2	no	—
114	C₄²	D₄×D₇₁	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	142	2	no	—
117	C₁₆	#11765848	1	2	yes	—
118	Q₈	—	284	2	no	—
119	Q₈	—	284	2	no	—
120	C₂×C₄	—	284	2	no	—
121	C₂×C₄	—	284	2	no	—
122	C₂×C₄	#14556090	1	1	yes	characteristic
123	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
124	C₂×C₄	#14556151	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	71	2	no	—
128	C₈	—	142	2	no	—
129	C₈	—	142	1	no	—
130	C₈	C₂×#11765848	1	1	yes	characteristic
131	C₈	#14556136	1	1	yes	characteristic
132	C₂²	#17986081	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	568	2	no	—
135	C₄	—	568	2	no	—
136	C₄	#17986076	1	1	yes	characteristic
137	C₄	#17986115	1	1	yes	characteristic
138	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
139	C₄	#17986428	1	1	yes	characteristic
140	C₂	#22587224	1	1	yes	characteristic
141	C₂	#22590198	1	1	yes	characteristic
142	C₂	#22590220	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₅₆₈

C₂

yes

maximal, characteristic

#22587236

C₂

yes

maximal

#22586303

C₂

yes

maximal

C₁₆×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₇₁×#1371

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

C₄

yes

characteristic

#17986126

—

C₄×Dic₂₈₄

C₂²

yes

—

#17985825

C₄

yes

characteristic

#17985804

C₄

yes

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985729

—

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#335×C₇₁

C₂²

yes

—

C₂×Dic₂₈₄

C₂×C₄

yes

—

C₄×Dic₁₄₂

—

Dic₅₆₈

—

C₂×C₇₁⋊C₁₆

—

#14556062

C₂×C₄

yes

—

Dic₇₁⋊C₈

—

#14556053

—

#14556038

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

Q₃₂

yes

complemented

C₂²×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

C₄×Q₃₂

yes

characteristic

C₂×C₇₁

#2178

yes

characteristic

C₄×Q₃₂

—

#1591

—

#1371

D₇₁

yes

characteristic

#1332

—

C₈×C₁₆

—

C₇₁

#12083

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#446

—

142

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

C₂×D₇₁

yes

—

Q₈⋊C₈

—

142

—

C₈²

—

C₂×Q₁₆

—

C₄×Q₈

—

142

—

Q₃₂

—

142

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₄×D₇₁

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

284

—

#103

—

142

—

#97

—

284

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₇₁

yes

characteristic

102

C₂×Q₈

—

142

—

103

Q₁₆

—

142

—

104

Q₁₆

—

142

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

142

—

109

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

110

C₂×C₈

#11765866

yes

characteristic

111

C₄⋊C₄

—

284

—

112

C₄⋊C₄

—

142

—

113

C₄²

—

284

—

114

C₄²

D₄×D₇₁

yes

characteristic

115

C₁₆

—

116

C₁₆

—

142

—

117

C₁₆

#11765848

yes

—

118

Q₈

—

284

—

119

Q₈

—

284

—

120

C₂×C₄

—

284

—

121

C₂×C₄

—

284

—

122

C₂×C₄

#14556090

yes

characteristic

123

C₂×C₄

D₈×D₇₁

yes

characteristic

124

C₂×C₄

#14556151

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

142

—

129

C₈

—

142

—

130

C₈

C₂×#11765848

yes

characteristic

131

C₈

#14556136

yes

characteristic

132

C₂²

#17986081

yes

characteristic

133

C₄

—

134

C₄

—

568

—

135

C₄

—

568

—

136

C₄

#17986076

yes

characteristic

137

C₄

#17986115

yes

characteristic

138

C₄

Q₃₂×D₇₁

yes

characteristic

139

C₄

#17986428

yes

characteristic

140

C₂

#22587224

yes

characteristic

141

C₂

#22590198

yes

characteristic

142

C₂

#22590220

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1371	C₂
socle, prime radical	C₂²×C₇₁	C₄×D₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1371	D₇₁
hypocenter	C₇₁	#12083
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1371

C₂

socle, prime radical

C₂²×C₇₁

C₄×D₈

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
59. C₇₁⋊C₈	Q₃₂	semidirect
77. C₇₁	#12083	semidirect

H ≅ G/N

type

59. C₇₁⋊C₈

Q₃₂

semidirect

77. C₇₁

#12083

semidirect

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

subgroup	index
C₄×Dic₅₆₈	2
#22587236	2
#22586303	2
C₁₆×C₇₁⋊C₈	2
C₇₁×#1371	2
#12083	71

subgroup

index

C₄×Dic₅₆₈

#22587236

#22586303

C₁₆×C₇₁⋊C₈

C₇₁×#1371

#12083

Subgroups — #31909442

Conjugacy classes of subgroups

Complements

Maximal subgroups