Group #16847254 — subgroups

Subgroups — #16847254

Order 3968.

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

normal subgroups only

This group has 254 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×#11758396	C₂	1	2	yes	maximal
2	C₂×#11758423	C₂	1	2	yes	maximal
3	#14520919	C₂	1	1	yes	maximal, characteristic
4	#14520804	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	—	2	2	no	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
12	C₃₁×#95	—	2	2	no	—
13	#11758423	—	2	4	no	—
14	#11758396	—	2	4	no	—
15	C₂×Q₈×C₃₁	—	2	1	no	—
16	C₂×Q₈×C₃₁	—	2	2	no	—
17	C₂×Q₈×C₃₁	—	2	2	no	—
18	C₂×Q₈×C₃₁	Q₈	1	2	yes	—
19	C₂²×C₄×C₃₁	—	2	1	no	—
20	C₂²×C₄×C₃₁	—	2	1	no	—
21	C₂²×C₄×C₃₁	D₄	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₁	—	2	2	no	—
24	C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₄²×C₃₁	—	2	2	no	—
26	C₄⋊Dic₃₁	—	2	2	no	—
27	C₄⋊Dic₃₁	—	2	2	no	—
28	C₄⋊Dic₃₁	—	4	4	no	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
32	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₃₁	—	2	4	no	—
34	Q₈×C₃₁	—	4	2	no	—
35	C₂×C₄×C₃₁	—	2	2	no	—
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	4	1	no	—
39	C₂×C₄×C₃₁	—	4	1	no	—
40	C₂×C₄×C₃₁	—	4	1	no	—
41	C₂×C₄×C₃₁	—	4	2	no	—
42	C₂×C₄×C₃₁	SD₁₆	1	4	yes	—
43	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#41	1	2	yes	—
47	C₂×Dic₃₁	—	4	2	no	—
48	C₂×Dic₃₁	—	4	4	no	—
49	C₃₁⋊C₈	—	2	4	no	—
50	#2096	—	31	1	no	maximal, Sylow
51	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₃₁	#123	1	1	yes	characteristic
53	C₂²×C₃₁	#127	1	1	yes	characteristic
54	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
55	C₂²×C₃₁	#137	1	2	yes	—
56	C₄×C₃₁	—	2	4	no	—
57	C₄×C₃₁	—	4	2	no	—
58	C₄×C₃₁	—	8	1	no	—
59	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₃₁	#123	1	2	yes	—
61	Dic₃₁	—	8	2	no	—
62	C₂×#106	—	31	2	no	—
63	C₂×#103	—	31	2	no	—
64	#391	D₃₁	1	1	yes	characteristic
65	#384	—	31	1	no	—
66	#336	—	31	1	no	—
67	C₂×C₃₁	#399	1	1	yes	characteristic
68	C₂×C₃₁	#475	1	2	yes	—
69	C₂×C₃₁	#483	1	2	yes	—
70	C₂×C₃₁	#492	1	1	yes	characteristic
71	C₂×C₃₁	#497	1	1	yes	characteristic
72	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
73	C₂²×C₈	—	31	2	no	—
74	C₂×C₄⋊C₄	—	31	2	no	—
75	C₂×C₄⋊C₄	—	62	2	no	—
76	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
78	#106	—	62	4	no	—
79	#103	—	62	4	no	—
80	#95	—	2	2	no	—
81	C₃₁	#2096	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	Dic₆₂	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	62	2	no	—
89	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
91	C₂×C₈	—	31	4	no	—
92	C₂×C₈	—	62	2	no	—
93	C₂×C₈	—	62	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	62	2	no	—
96	C₄⋊C₄	—	62	2	no	—
97	C₄⋊C₄	—	124	4	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	124	2	no	—
110	C₂×C₄	—	124	4	no	—
111	C₂×C₄	#92481	1	4	yes	—
112	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
113	C₂×C₄	#92495	1	1	yes	characteristic
114	C₂×C₄	D₄×D₃₁	1	2	yes	—
115	C₂×C₄	#92497	1	2	yes	—
116	C₈	—	62	4	no	—
117	C₂²	#11758501	1	1	yes	characteristic
118	C₂²	#11758512	1	1	yes	characteristic
119	C₂²	#11758517	1	1	yes	characteristic
120	C₂²	C₂×#92481	1	2	yes	—
121	C₂²	#11758528	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	248	2	no	—
126	C₄	#11758455	1	2	yes	—
127	C₄	#11758460	1	2	yes	—
128	C₂	#14520985	1	1	yes	characteristic
129	C₂	#14521334	1	2	yes	—
130	C₂	#14521377	1	2	yes	—
131	C₂	#14521423	1	1	yes	characteristic
132	C₂	#14521444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758396

C₂

yes

maximal

C₂×#11758423

C₂

yes

maximal

#14520919

C₂

yes

maximal, characteristic

#14520804

C₂

yes

maximal, characteristic

C₃₁×#391

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁×#95

—

#11758423

—

#11758396

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#2096

—

maximal, Sylow

C₂²×C₃₁

C₂²⋊Q₈

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

C₂×SD₁₆

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

#123

yes

—

Dic₃₁

—

C₂×#106

—

C₂×#103

—

#391

D₃₁

yes

characteristic

#384

—

#336

—

C₂×C₃₁

#399

yes

characteristic

C₂×C₃₁

#475

yes

—

C₂×C₃₁

#483

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₃₁

#2096

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₆₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄²

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

124

—

110

C₂×C₄

—

124

—

111

C₂×C₄

#92481

yes

—

112

C₂×C₄

C₂×Dic₆₂

yes

characteristic

113

C₂×C₄

#92495

yes

characteristic

114

C₂×C₄

D₄×D₃₁

yes

—

115

C₂×C₄

#92497

yes

—

116

C₈

—

117

C₂²

#11758501

yes

characteristic

118

C₂²

#11758512

yes

characteristic

119

C₂²

#11758517

yes

characteristic

120

C₂²

C₂×#92481

yes

—

121

C₂²

#11758528

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

248

—

126

C₄

#11758455

yes

—

127

C₄

#11758460

yes

—

128

C₂

#14520985

yes

characteristic

129

C₂

#14521334

yes

—

130

C₂

#14521377

yes

—

131

C₂

#14521423

yes

characteristic

132

C₂

#14521444

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#391	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#391	D₃₁
hypocenter	C₃₁	#2096
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#391

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
81. C₃₁	#2096	semidirect

H ≅ G/N

type

81. C₃₁

#2096

semidirect

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

subgroup	index
C₂×#11758396	2
C₂×#11758423	2
#14520919	2
#14520804	2
C₃₁×#391	2
#2096	31

subgroup

index

C₃₁×#391

Subgroups — #16847254

Conjugacy classes of subgroups

Complements

Maximal subgroups