Group #16846634 — subgroups

Subgroups — #16846634

Order 3968.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11758398	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₆₂	C₂²	1	2	yes	—
6	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	—	2	1	no	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	—	2	1	no	—
10	C₂×C₄⋊Dic₃₁	—	2	1	no	—
11	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
12	#11758446	—	2	2	no	—
13	#11758446	C₄	1	4	yes	complemented
14	C₃₁×#95	—	2	1	no	—
15	#11758398	—	2	8	no	—
16	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
17	C₂²×C₄×C₃₁	—	2	2	no	—
18	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃₁	—	2	2	no	—
22	C₂×Dic₆₂	—	2	4	no	—
23	C₂×Dic₆₂	—	2	4	no	—
24	C₂×Dic₆₂	—	2	2	no	—
25	C₂×Dic₆₂	C₂×C₄	1	4	yes	—
26	C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₄⋊C₄×C₃₁	D₄	1	4	yes	—
28	C₄²×C₃₁	—	4	2	no	—
29	C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₄²×C₃₁	D₄	1	2	yes	—
31	C₄⋊Dic₃₁	—	2	4	no	—
32	C₄⋊Dic₃₁	—	2	4	no	—
33	C₂×C₃₁⋊C₈	—	2	2	no	—
34	C₂×C₃₁⋊C₈	—	2	4	no	—
35	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	—	2	4	no	—
38	C₂×C₄×C₃₁	—	4	2	no	—
39	C₂×C₄×C₃₁	—	4	2	no	—
40	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
43	C₂×C₄×C₃₁	Q₁₆	1	2	yes	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	#41	1	2	yes	—
48	C₂×Dic₃₁	—	2	4	no	—
49	C₂×Dic₃₁	—	4	4	no	—
50	Dic₆₂	—	2	8	no	—
51	Dic₆₂	—	4	2	no	—
52	Dic₆₂	—	4	2	no	—
53	C₃₁⋊C₈	—	4	2	no	—
54	#1981	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₃₁	#121	1	2	yes	—
57	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×C₃₁	#137	1	2	yes	—
60	C₄×C₃₁	—	2	2	no	—
61	C₄×C₃₁	—	4	4	no	—
62	C₄×C₃₁	#103	1	4	yes	complemented
63	C₄×C₃₁	C₄×D₄	1	2	yes	—
64	C₄×C₃₁	#124	1	1	yes	characteristic
65	C₄×C₃₁	#127	1	1	yes	characteristic
66	Dic₃₁	—	4	4	no	—
67	C₂×C₄⋊Q₈	—	31	1	no	—
68	C₂×C₄⋊C₈	—	31	1	no	—
69	C₂×#103	—	31	4	no	—
70	C₄×C₄⋊C₄	D₃₁	1	1	yes	characteristic
71	C₂×C₃₁	#390	1	1	yes	characteristic
72	C₂×C₃₁	C₂×#103	1	1	yes	characteristic
73	C₂×C₃₁	#419	1	1	yes	characteristic
74	C₂×C₃₁	#461	1	2	yes	—
75	C₂×C₃₁	#462	1	2	yes	—
76	C₂²×Q₈	—	31	2	no	—
77	C₂²×C₈	—	31	2	no	—
78	C₄⋊Q₈	—	31	4	no	—
79	C₄⋊Q₈	—	62	2	no	—
80	C₂×C₄⋊C₄	—	62	1	no	—
81	C₂×C₄⋊C₄	—	62	1	no	—
82	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
85	C₄⋊C₈	—	62	2	no	—
86	#103	—	62	8	no	—
87	#95	—	2	1	no	—
88	C₃₁	#1981	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	31	4	no	—
90	C₂×Q₈	—	62	2	no	—
91	C₂×Q₈	—	62	4	no	—
92	C₂×Q₈	—	62	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	62	2	no	—
95	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
98	C₂×C₈	—	62	4	no	—
99	C₂×C₈	—	62	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	62	4	no	—
102	C₄⋊C₄	—	62	4	no	—
103	C₄⋊C₄	D₁₂₄	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₃₁	1	2	yes	—
106	C₄²	D₁₂₄	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
108	Q₈	—	62	8	no	—
109	Q₈	—	124	2	no	—
110	Q₈	—	124	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	62	4	no	—
116	C₂×C₄	—	124	4	no	—
117	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
118	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
119	C₂×C₄	#92480	1	2	yes	—
120	C₂×C₄	#92482	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
122	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
123	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
124	C₂×C₄	#92495	1	2	yes	—
125	C₈	—	124	2	no	—
126	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
127	C₂²	#11758507	1	2	yes	—
128	C₂²	C₂×#92480	1	1	yes	characteristic
129	C₂²	C₂×#92482	1	1	yes	characteristic
130	C₂²	#11758528	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	124	4	no	—
134	C₄	#11758398	1	4	yes	complemented
135	C₄	C₄×D₁₂₄	1	2	yes	—
136	C₄	#11758451	1	1	yes	characteristic
137	C₄	#11758452	1	1	yes	characteristic
138	C₂	#14520951	1	1	yes	characteristic
139	C₂	C₂×#11758398	1	1	yes	characteristic
140	C₂	#14521057	1	1	yes	characteristic
141	C₂	#14521254	1	2	yes	—
142	C₂	#14521259	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#11758398

C₂

yes

maximal

C₄×C₄⋊C₄×C₃₁

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758446

—

#11758446

C₄

yes

complemented

C₃₁×#95

—

#11758398

—

C₄⋊(C₃₁⋊C₈)

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₃₁⋊C₈

—

#1981

—

maximal, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#121

yes

—

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

C₂×Q₁₆

yes

characteristic

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#103

yes

complemented

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#462

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

C₃₁

#1981

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

D₁₂₄

yes

—

104

C₄²

—

105

C₄²

C₄×D₃₁

yes

—

106

C₄²

D₁₂₄

yes

—

107

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

108

Q₈

—

109

Q₈

—

124

—

110

Q₈

—

124

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

124

—

117

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

118

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

119

C₂×C₄

#92480

yes

—

120

C₂×C₄

#92482

yes

—

121

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

122

C₂×C₄

C₂×D₁₂₄

yes

—

123

C₂×C₄

C₂×D₁₂₄

yes

characteristic

124

C₂×C₄

#92495

yes

—

125

C₈

—

124

—

126

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

127

C₂²

#11758507

yes

—

128

C₂²

C₂×#92480

yes

characteristic

129

C₂²

C₂×#92482

yes

characteristic

130

C₂²

#11758528

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

124

—

134

C₄

#11758398

yes

complemented

135

C₄

C₄×D₁₂₄

yes

—

136

C₄

#11758451

yes

characteristic

137

C₄

#11758452

yes

characteristic

138

C₂

#14520951

yes

characteristic

139

C₂

C₂×#11758398

yes

characteristic

140

C₂

#14521057

yes

characteristic

141

C₂

#14521254

yes

—

142

C₂

#14521259

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₄×C₄⋊C₄×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₃₁
hypocenter	C₃₁	#1981
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₄×C₄⋊C₄×C₃₁

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₄×C₄⋊C₄

D₃₁

hypocenter

C₃₁

#1981

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
13. #11758446	C₄	semidirect
62. C₄×C₃₁	#103	semidirect
88. C₃₁	#1981	semidirect
134. C₄	#11758398	semidirect

H ≅ G/N

type

13. #11758446

C₄

semidirect

62. C₄×C₃₁

#103

semidirect

88. C₃₁

#1981

semidirect

134. C₄

#11758398

semidirect

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

subgroup	index
C₂×#11758446	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×#11758398	2
C₄×C₄⋊C₄×C₃₁	2
#1981	31

subgroup

index

C₂×#11758446

C₂×C₄⋊(C₃₁⋊C₈)

C₂×#11758398

C₄×C₄⋊C₄×C₃₁

#1981

Subgroups — #16846634

Conjugacy classes of subgroups

Complements

Maximal subgroups