Group #16856091 — subgroups

Subgroups — #16856091

Order 3968.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14521606	C₂	1	1	yes	maximal, characteristic
2	#14521417	C₂	1	4	yes	maximal, complemented
3	#14521399	C₂	1	4	yes	maximal, complemented
4	#14521058	C₂	1	4	yes	maximal
5	C₄×#92474	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₃₁×C₄⋊Q₈	—	2	2	no	—
9	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	#11758503	C₂²	1	2	yes	—
14	#11758501	—	2	2	no	—
15	C₂×#92474	C₂²	1	2	yes	—
16	#11758447	C₂²	1	2	yes	—
17	#11758446	C₂²	1	2	yes	—
18	C₄×Dic₆₂	—	2	2	no	—
19	#11758423	C₂²	1	16	yes	—
20	#11758391	C₂²	1	2	yes	—
21	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
22	C₂×Q₈×C₃₁	—	2	2	no	—
23	C₂×Q₈×C₃₁	—	2	8	no	—
24	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₃₁	—	2	2	no	—
26	C₂²×C₄×C₃₁	D₄	1	2	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₆₂	—	2	2	no	—
29	C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₄⋊C₄×C₃₁	—	2	4	no	—
31	C₄²×C₃₁	D₄	1	2	yes	—
32	C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₃₁	—	2	2	no	—
35	C₄⋊Dic₃₁	—	2	2	no	—
36	C₄⋊Dic₃₁	C₂³	1	4	yes	—
37	Dic₃₁⋊C₄	—	2	4	no	—
38	C₄×Dic₃₁	—	2	2	no	—
39	#92474	—	2	4	no	—
40	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
41	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₃₁	—	2	8	no	—
43	Q₈×C₃₁	—	4	4	no	—
44	C₂×C₄×C₃₁	—	2	4	no	—
45	C₂×C₄×C₃₁	—	2	4	no	—
46	C₂×C₄×C₃₁	—	4	4	no	—
47	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₃₁	#41	1	4	yes	—
51	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₃₁	—	2	4	no	—
53	Dic₆₂	—	4	4	no	—
54	C₃₁⋊C₈	—	2	4	no	—
55	#3139	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	—	2	2	no	—
57	C₂²×C₃₁	#124	1	2	yes	—
58	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₃₁	—	2	2	no	—
60	C₄×C₃₁	—	4	4	no	—
61	C₄×C₃₁	#124	1	2	yes	—
62	C₄×C₃₁	#137	1	4	yes	—
63	C₄×C₃₁	C₂×#41	1	2	yes	—
64	Dic₃₁	—	4	4	no	—
65	#533	—	31	1	no	—
66	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
67	#491	—	31	4	no	—
68	#487	—	31	4	no	—
69	#419	—	31	4	no	—
70	C₄×M₁₆	—	31	1	no	—
71	C₂×C₃₁	—	2	2	no	—
72	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
73	C₂×C₃₁	C₂×#137	1	2	yes	—
74	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
75	C₂×M₁₆	—	31	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	—	31	2	no	—
78	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
79	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
80	#125	—	31	2	no	—
81	C₂²⋊Q₈	—	62	2	no	—
82	C₄×Q₈	—	62	2	no	—
83	#117	—	31	2	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
86	#103	—	31	16	no	—
87	#97	—	31	2	no	—
88	C₄×C₈	—	31	2	no	—
89	C₃₁	#3139	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	62	2	no	—
93	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
96	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	M₁₆	—	62	4	no	—
98	C₂×C₈	—	31	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	31	4	no	—
102	C₄⋊C₄	—	62	4	no	—
103	C₄⋊C₄	—	62	2	no	—
104	C₂²⋊C₄	—	62	2	no	—
105	C₄²	—	62	2	no	—
106	C₄²	C₂²⋊D₃₁	1	2	yes	—
107	C₄²	C₂²×D₃₁	1	1	yes	characteristic
108	C₄²	C₂²×D₃₁	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	124	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	62	4	no	—
117	C₂×C₄	#92497	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
121	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
122	C₈	—	62	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#11758513	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	124	4	no	—
129	C₄	#11758513	1	2	yes	—
130	C₄	#11758519	1	4	yes	—
131	C₄	C₂×#92497	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#11758513	1	1	yes	characteristic
134	C₂	C₂×#11758519	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

#14521417

C₂

yes

maximal, complemented

#14521399

C₂

yes

maximal, complemented

#14521058

C₂

yes

maximal

C₄×#92474

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758501

—

C₂×#92474

C₂²

yes

—

#11758447

C₂²

yes

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

—

#11758423

C₂²

yes

—

#11758391

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

#92474

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

Dic₆₂

—

C₃₁⋊C₈

—

#3139

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

#137

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

#533

—

C₂×C₄⋊Q₈

D₃₁

yes

—

—

—

C₄×M₁₆

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#124

yes

characteristic

C₂×C₃₁

C₂×#137

yes

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

—

C₄⋊Q₈

C₂×D₃₁

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₃₁

#3139

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂²⋊D₃₁

yes

—

107

C₄²

C₂²×D₃₁

yes

characteristic

108

C₄²

C₂²×D₃₁

yes

characteristic

109

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

124

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#92497

yes

—

118

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

121

C₂×C₄

C₂³×D₃₁

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

#11758513

yes

—

125

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

124

—

129

C₄

#11758513

yes

—

130

C₄

#11758519

yes

—

131

C₄

C₂×#92497

yes

—

132

C₂

—

133

C₂

C₂×#11758513

yes

characteristic

134

C₂

C₂×#11758519

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#3139
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂³×C₃₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₃₁

hypocenter

C₃₁

#3139

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #14521417	C₂	semidirect
3. #14521399	C₂	semidirect
89. C₃₁	#3139	semidirect

H ≅ G/N

type

2. #14521417

C₂

semidirect

3. #14521399

C₂

semidirect

89. C₃₁

#3139

semidirect

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

subgroup	index
#14521606	2
#14521417	2
#14521399	2
#14521058	2
C₄×#92474	2
C₂×C₃₁×C₄⋊Q₈	2
#3139	31

subgroup

index

C₂×C₃₁×C₄⋊Q₈

Subgroups — #16856091

Conjugacy classes of subgroups

Complements

Maximal subgroups