Group #31909292 — subgroups

Subgroups — #31909292

Order 18176.

212 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₁₆×Dic₁₄₂	C₂	1	2	yes	maximal
2	C₈×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
3	#22586404	C₂	1	1	yes	maximal, characteristic
4	#22586320	C₂	1	2	yes	maximal
5	C₇₁×#1312	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₄₂	C₂²	1	2	yes	—
7	C₄×Dic₂₈₄	C₄	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
9	Dic₇₁⋊C₁₆	—	2	2	no	—
10	C₁₆×Dic₇₁	—	2	2	no	—
11	#17985752	C₄	1	1	yes	characteristic
12	#17985731	C₄	1	2	yes	—
13	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₈²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₈₄	C₈	1	1	yes	characteristic
17	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	#14556062	C₈	1	1	yes	characteristic
20	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
21	Dic₇₁⋊C₈	—	2	2	no	—
22	#14556053	C₈	1	2	yes	—
23	C₈×Dic₇₁	—	2	2	no	—
24	C₂×C₁₆×C₇₁	—	2	1	no	—
25	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
26	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₂	C₂×C₈	1	2	yes	—
30	Dic₂₈₄	C₁₆	1	4	yes	complemented
31	C₂×C₇₁⋊C₈	—	2	2	no	—
32	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
33	Dic₇₁⋊C₄	—	2	2	no	—
34	C₄×Dic₇₁	—	2	2	no	—
35	C₇₁⋊C₁₆	—	4	2	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
41	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	2	no	—
43	C₁₆×C₇₁	—	4	1	no	—
44	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
47	C₂×Dic₇₁	—	2	2	no	—
48	C₂×Dic₇₁	—	2	2	no	—
49	Dic₁₄₂	—	2	2	no	—
50	Dic₁₄₂	C₂×C₁₆	1	4	yes	—
51	C₈×C₇₁	—	2	1	no	—
52	C₈×C₇₁	—	2	1	no	—
53	C₈×C₇₁	—	4	1	no	—
54	C₈×C₇₁	C₂×C₁₆	1	2	yes	—
55	C₈×C₇₁	#137	1	2	yes	—
56	C₇₁⋊C₈	—	4	2	no	—
57	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	#441	1	1	yes	characteristic
60	C₄×C₇₁	#444	1	1	yes	characteristic
61	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
62	C₄×C₇₁	#504	1	1	yes	characteristic
63	Dic₇₁	—	2	4	no	—
64	Dic₇₁	—	4	2	no	—
65	#12059	—	71	1	no	maximal, Sylow
66	C₂×C₇₁	#1579	1	1	yes	characteristic
67	C₂×C₇₁	D₄×C₁₆	1	1	yes	characteristic
68	C₂×C₇₁	#2171	1	1	yes	characteristic
69	Q₈×C₁₆	—	71	2	no	—
70	C₈×Q₁₆	—	71	1	no	—
71	#1367	—	71	1	no	—
72	Q₈⋊C₁₆	—	71	2	no	—
73	#1312	D₇₁	1	1	yes	characteristic
74	C₇₁	#12059	1	1	yes	characteristic, complemented, Sylow
75	C₈×Q₈	—	71	2	no	—
76	C₄×Q₁₆	—	71	1	no	—
77	C₄⋊C₁₆	—	71	2	no	—
78	C₄⋊C₁₆	—	142	2	no	—
79	C₄×C₁₆	—	142	2	no	—
80	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
82	#335	—	71	1	no	—
83	Q₈⋊C₈	—	71	2	no	—
84	C₈²	C₂×D₇₁	1	1	yes	characteristic
85	C₂×Q₁₆	—	71	1	no	—
86	C₄×Q₈	—	71	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	142	2	no	—
89	C₂×C₁₆	D₂₈₄	1	2	yes	—
90	#107	—	71	1	no	—
91	C₄⋊C₈	—	71	2	no	—
92	C₄⋊C₈	—	142	2	no	—
93	#103	—	71	2	no	—
94	C₄×C₈	—	142	2	no	—
95	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
96	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
98	C₂×Q₈	—	71	2	no	—
99	Q₁₆	—	71	4	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	142	2	no	—
102	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
103	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
104	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
105	C₂×C₈	#11765865	1	1	yes	characteristic
106	C₄⋊C₄	—	71	2	no	—
107	C₄⋊C₄	—	142	2	no	—
108	C₄²	—	142	2	no	—
109	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	284	2	no	—
113	Q₈	—	71	4	no	—
114	Q₈	—	142	2	no	—
115	C₂×C₄	—	142	2	no	—
116	C₂×C₄	—	142	2	no	—
117	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
119	C₂×C₄	#14556134	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	284	2	no	—
124	C₈	C₁₆×D₇₁	1	2	yes	—
125	C₈	#14556161	1	2	yes	—
126	C₂²	C₈×D₂₈₄	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	284	2	no	—
129	C₄	—	142	4	no	—
130	C₄	#17986107	1	1	yes	characteristic
131	C₄	#17986122	1	1	yes	characteristic
132	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
133	C₄	#17986415	1	1	yes	characteristic
134	C₂	#22587160	1	1	yes	characteristic
135	C₂	C₁₆×D₂₈₄	1	1	yes	characteristic
136	C₂	#22590183	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄₂

C₂

yes

maximal

C₈×Dic₂₈₄

C₂

yes

maximal, characteristic

#22586404

C₂

yes

maximal, characteristic

#22586320

C₂

yes

maximal

C₇₁×#1312

C₂

yes

maximal, characteristic

C₈×Dic₁₄₂

C₂²

yes

—

C₄×Dic₂₈₄

C₄

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

—

Dic₇₁⋊C₁₆

—

C₁₆×Dic₇₁

—

#17985752

C₄

yes

characteristic

#17985731

C₄

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

C₈

yes

characteristic

C₄×Dic₁₄₂

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

—

#14556062

C₈

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂×C₄

yes

—

Dic₇₁⋊C₈

—

#14556053

C₈

yes

—

C₈×Dic₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

C₂×C₈

yes

—

Dic₂₈₄

C₁₆

yes

complemented

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

C₂×C₈

yes

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

#131

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

C₂×C₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₁₆

yes

—

C₈×C₇₁

#137

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₈×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂²×C₁₆

yes

characteristic

C₄×C₇₁

#504

yes

characteristic

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

D₄×C₁₆

yes

characteristic

C₂×C₇₁

#2171

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

D₇₁

yes

characteristic

C₇₁

#12059

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

C₄×C₁₆

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

D₂₈₄

yes

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

#103

—

C₄×C₈

—

142

—

C₄×C₈

C₄×D₇₁

yes

characteristic

C₄×C₈

C₄×D₇₁

yes

characteristic

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

142

—

102

C₂×C₈

C₈×D₇₁

yes

characteristic

103

C₂×C₈

C₈×D₇₁

yes

characteristic

104

C₂×C₈

C₂×D₂₈₄

yes

characteristic

105

C₂×C₈

#11765865

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

142

—

108

C₄²

—

142

—

109

C₄²

C₂×C₄×D₇₁

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

284

—

113

Q₈

—

114

Q₈

—

142

—

115

C₂×C₄

—

142

—

116

C₂×C₄

—

142

—

117

C₂×C₄

C₄×D₂₈₄

yes

characteristic

118

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

119

C₂×C₄

#14556134

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

284

—

124

C₈

C₁₆×D₇₁

yes

—

125

C₈

#14556161

yes

—

126

C₂²

C₈×D₂₈₄

yes

characteristic

127

C₄

—

128

C₄

—

284

—

129

C₄

—

142

—

130

C₄

#17986107

yes

characteristic

131

C₄

#17986122

yes

characteristic

132

C₄

C₂×C₁₆×D₇₁

yes

characteristic

133

C₄

#17986415

yes

characteristic

134

C₂

#22587160

yes

characteristic

135

C₂

C₁₆×D₂₈₄

yes

characteristic

136

C₂

#22590183

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈₄
commutator	C₄×C₇₁	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1312	C₂
socle, prime radical	C₂²×C₇₁	C₈×D₄
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1312	D₇₁
hypocenter	C₇₁	#12059
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₂₈₄

commutator

C₄×C₇₁

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1312

C₂

socle, prime radical

C₂²×C₇₁

C₈×D₄

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
30. Dic₂₈₄	C₁₆	semidirect
74. C₇₁	#12059	semidirect

H ≅ G/N

type

30. Dic₂₈₄

C₁₆

semidirect

74. C₇₁

#12059

semidirect

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

subgroup	index
C₁₆×Dic₁₄₂	2
C₈×Dic₂₈₄	2
#22586404	2
#22586320	2
C₇₁×#1312	2
#12059	71

subgroup

index

C₁₆×Dic₁₄₂

C₈×Dic₂₈₄

#22586404

#22586320

C₇₁×#1312

#12059

Subgroups — #31909292

Conjugacy classes of subgroups

Complements

Maximal subgroups