Group #16844382 — subgroups

Subgroups — #16844382

Order 3968.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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×#11758446	C₂	1	1	yes	maximal, characteristic
2	#14521073	C₂	1	1	yes	maximal, characteristic
3	#14520794	C₂	1	4	yes	maximal, complemented
4	C₃₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₆₂	—	2	1	no	—
6	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃₁	—	2	1	no	—
8	C₂×C₄⋊Dic₃₁	C₄	1	2	yes	—
9	C₂×#92474	—	2	1	no	—
10	#11758446	—	2	1	no	—
11	#11758446	—	2	1	no	—
12	#11758446	C₄	1	2	yes	—
13	#11758446	C₂²	1	2	yes	—
14	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₃₁×C₂²⋊C₈	—	2	1	no	—
16	C₄×C₈×C₃₁	—	2	1	no	—
17	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
18	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
19	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×Dic₃₁	—	2	2	no	—
23	C₂×Dic₆₂	—	2	2	no	—
24	C₂×Dic₆₂	—	2	2	no	—
25	C₂×Dic₆₂	—	4	2	no	—
26	C₂×Dic₆₂	—	4	2	no	—
27	C₂×C₈×C₃₁	—	2	2	no	—
28	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₃₁	D₄	1	1	yes	characteristic
30	C₄²×C₃₁	D₄	1	1	yes	characteristic
31	C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₃₁	—	2	2	no	—
33	C₄⋊Dic₃₁	—	2	2	no	—
34	C₄⋊Dic₃₁	—	2	2	no	—
35	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
36	#92474	—	4	2	no	—
37	C₂×C₃₁⋊C₈	—	2	2	no	—
38	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	D₈	1	2	yes	—
45	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
46	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₃₁	—	2	4	no	—
50	C₂×Dic₃₁	—	4	4	no	—
51	Dic₆₂	—	4	4	no	—
52	Dic₆₂	—	4	2	no	—
53	Dic₆₂	—	4	2	no	—
54	C₈×C₃₁	—	4	2	no	—
55	C₃₁⋊C₈	—	4	2	no	—
56	#1546	—	31	1	no	maximal, Sylow
57	C₂²×C₃₁	—	2	1	no	—
58	C₂²×C₃₁	—	2	1	no	—
59	C₂²×C₃₁	#102	1	2	yes	—
60	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₃₁	—	2	1	no	—
62	C₄×C₃₁	—	2	1	no	—
63	C₄×C₃₁	#101	1	2	yes	—
64	C₄×C₃₁	#102	1	2	yes	—
65	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
66	C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₃₁	#137	1	2	yes	—
68	Dic₃₁	—	4	4	no	—
69	C₂×C₄⋊Q₈	—	31	1	no	—
70	#431	D₃₁	1	1	yes	characteristic
71	#423	—	31	1	no	—
72	#332	—	31	4	no	—
73	C₂×C₃₁	—	2	2	no	—
74	C₂×C₃₁	C₂×#101	1	1	yes	characteristic
75	C₂×C₃₁	C₂×#102	1	1	yes	characteristic
76	C₂×C₃₁	#419	1	1	yes	characteristic
77	C₂²×Q₈	—	62	1	no	—
78	C₂×M₁₆	—	62	1	no	—
79	C₄⋊Q₈	—	31	2	no	—
80	C₄⋊Q₈	—	31	2	no	—
81	C₄⋊Q₈	—	62	1	no	—
82	C₄⋊Q₈	—	62	1	no	—
83	C₂×C₄⋊C₄	—	31	2	no	—
84	C₂×C₄⋊C₄	—	62	1	no	—
85	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
86	C₄⋊C₈	—	31	2	no	—
87	C₄⋊C₈	—	62	1	no	—
88	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
89	C₂²⋊C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₃₁	#1546	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	62	2	no	—
93	C₂×Q₈	—	62	2	no	—
94	C₂×Q₈	—	124	2	no	—
95	C₂×Q₈	—	124	2	no	—
96	C₂²×C₄	—	62	2	no	—
97	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
98	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
100	M₁₆	—	124	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	62	2	no	—
103	C₄⋊C₄	—	31	4	no	—
104	C₄⋊C₄	—	62	2	no	—
105	C₄⋊C₄	—	62	2	no	—
106	C₄⋊C₄	—	62	2	no	—
107	C₄²	C₄×D₃₁	1	1	yes	characteristic
108	C₄²	D₁₂₄	1	1	yes	characteristic
109	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
110	C₄²	C₂²×D₃₁	1	1	yes	characteristic
111	C₂³	(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
112	Q₈	—	124	4	no	—
113	Q₈	—	124	2	no	—
114	Q₈	—	124	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	62	4	no	—
118	C₂×C₄	—	124	4	no	—
119	C₂×C₄	#92470	1	2	yes	—
120	C₂×C₄	D₂₄₈	1	2	yes	—
121	C₂×C₄	(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	124	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#11758409	1	2	yes	—
132	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	124	4	no	—
136	C₄	#11758409	1	2	yes	—
137	C₄	#11758412	1	2	yes	—
138	C₄	C₂×#92470	1	1	yes	characteristic
139	C₄	C₂×D₂₄₈	1	1	yes	characteristic
140	C₄	#11758528	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#11758412	1	1	yes	characteristic
143	C₂	C₂×#11758409	1	1	yes	characteristic
144	C₂	#14521057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

#14521073

C₂

yes

maximal, characteristic

#14520794

C₂

yes

maximal, complemented

C₃₁×#431

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₄

yes

—

—

—

—

C₄

yes

—

#11758446

C₂²

yes

—

C₃₁×C₄⋊C₈

C₂²

yes

—

C₃₁×C₂²⋊C₈

—

C₄×C₈×C₃₁

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

D₈

yes

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1546

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#102

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#101

yes

—

C₄×C₃₁

#102

yes

—

C₄×C₃₁

C₂×D₈

yes

characteristic

C₄×C₃₁

C₂×SD₁₆

yes

characteristic

C₄×C₃₁

#137

yes

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

#431

D₃₁

yes

characteristic

#423

—

#332

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₃₁

#1546

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₁

yes

characteristic

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

100

M₁₆

—

124

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

C₄×D₃₁

yes

characteristic

108

C₄²

D₁₂₄

yes

characteristic

109

C₄²

C₂²⋊D₃₁

yes

characteristic

110

C₄²

C₂²×D₃₁

yes

characteristic

111

C₂³

(C₂×D₃₁)⋊C₄

yes

characteristic

112

Q₈

—

124

—

113

Q₈

—

124

—

114

Q₈

—

124

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

124

—

119

C₂×C₄

#92470

yes

—

120

C₂×C₄

D₂₄₈

yes

—

121

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

characteristic

123

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

characteristic

124

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

125

C₂×C₄

C₂×D₁₂₄

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

127

C₈

—

128

C₈

—

124

—

129

C₂²

—

130

C₂²

—

131

C₂²

#11758409

yes

—

132

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

124

—

136

C₄

#11758409

yes

—

137

C₄

#11758412

yes

—

138

C₄

C₂×#92470

yes

characteristic

139

C₄

C₂×D₂₄₈

yes

characteristic

140

C₄

#11758528

yes

—

141

C₂

—

142

C₂

C₂×#11758412

yes

characteristic

143

C₂

C₂×#11758409

yes

characteristic

144

C₂

#14521057

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₃₁)⋊C₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#431	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₁	C₂²⋊C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	#431	D₃₁
hypocenter	C₃₁	#1546
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×(C₂×D₃₁)⋊C₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#431

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₁

C₂²⋊C₄

cosocle

C₄²×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #14520794	C₂	semidirect
91. C₃₁	#1546	semidirect

H ≅ G/N

type

3. #14520794

C₂

semidirect

91. C₃₁

#1546

semidirect

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

subgroup	index
C₂×#11758446	2
#14521073	2
#14520794	2
C₃₁×#431	2
#1546	31

subgroup

index

C₃₁×#431

Subgroups — #16844382

Conjugacy classes of subgroups

Complements

Maximal subgroups