Group #16844396 — subgroups

Subgroups — #16844396

Order 3968.

240 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 101 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	#14521102	C₂	1	1	yes	maximal, characteristic
2	#14521073	C₂	1	1	yes	maximal, characteristic
3	#14520795	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₁	—	2	1	no	—
6	C₃₁×C₄⋊Q₈	—	2	1	no	—
7	C₃₁×C₄⋊Q₈	—	2	1	no	—
8	C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
9	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#92474	—	2	1	no	—
14	C₂²⋊(C₃₁⋊C₈)	—	2	1	no	—
15	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
16	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
17	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
18	C₄×C₃₁⋊C₈	—	2	1	no	—
19	C₂×Q₈×C₃₁	—	2	2	no	—
20	C₂×Q₈×C₃₁	—	2	2	no	—
21	C₂×Q₈×C₃₁	—	4	4	no	—
22	C₂²×C₄×C₃₁	—	2	2	no	—
23	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₃₁	D₄	1	1	yes	characteristic
32	C₄²×C₃₁	D₄	1	1	yes	characteristic
33	C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	#92474	—	4	2	no	—
35	C₂×C₃₁⋊C₈	—	2	2	no	—
36	C₂×C₃₁⋊C₈	—	2	2	no	—
37	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
38	Q₈×C₃₁	—	4	4	no	—
39	Q₈×C₃₁	—	4	4	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	2	2	no	—
42	C₂×C₄×C₃₁	—	2	4	no	—
43	C₂×C₄×C₃₁	—	4	4	no	—
44	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₃₁	Q₁₆	1	2	yes	—
49	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₃₁⋊C₈	—	4	2	no	—
53	C₃₁⋊C₈	—	4	2	no	—
54	#1549	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	—	2	1	no	—
56	C₂²×C₃₁	—	2	1	no	—
57	C₂²×C₃₁	#103	1	2	yes	—
58	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₃₁	—	2	1	no	—
60	C₄×C₃₁	—	2	1	no	—
61	C₄×C₃₁	—	4	4	no	—
62	C₄×C₃₁	#101	1	2	yes	—
63	C₄×C₃₁	#103	1	2	yes	—
64	C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
66	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
67	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
68	#431	—	31	1	no	—
69	#423	—	31	1	no	—
70	#332	—	31	4	no	—
71	C₂×C₃₁	—	2	2	no	—
72	C₂×C₃₁	C₂×#101	1	1	yes	characteristic
73	C₂×C₃₁	C₂×#103	1	1	yes	characteristic
74	C₂×C₃₁	#418	1	1	yes	characteristic
75	C₂²×Q₈	—	2	1	no	—
76	C₂×M₁₆	—	62	1	no	—
77	C₄⋊Q₈	—	2	1	no	—
78	C₄⋊Q₈	—	2	1	no	—
79	C₄⋊Q₈	Dic₃₁	1	2	yes	—
80	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	Dic₃₁	1	2	yes	—
83	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
84	C₄⋊C₈	—	31	2	no	—
85	C₄⋊C₈	—	31	2	no	—
86	C₄⋊C₈	—	62	1	no	—
87	C₂²⋊C₈	—	62	1	no	—
88	C₄×C₈	—	62	1	no	—
89	C₃₁	#1549	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	4	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
97	M₁₆	—	124	2	no	—
98	C₂×C₈	—	62	2	no	—
99	C₂×C₈	—	62	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
104	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
105	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
106	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
107	C₄²	C₂²×D₃₁	1	1	yes	characteristic
108	C₂³	C₂²⋊Dic₃₁	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	Q₈	—	4	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	#92481	1	2	yes	—
116	C₂×C₄	#92482	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
120	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
123	C₈	—	124	2	no	—
124	C₈	—	124	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#11758423	1	2	yes	—
128	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	4	no	—
132	C₄	#11758423	1	2	yes	—
133	C₄	#11758424	1	2	yes	—
134	C₄	#11758509	1	2	yes	—
135	C₄	C₂×#92481	1	1	yes	characteristic
136	C₄	C₂×#92482	1	1	yes	characteristic
137	C₂	—	2	2	no	—
138	C₂	C₂×#11758424	1	1	yes	characteristic
139	C₂	C₂×#11758423	1	1	yes	characteristic
140	C₂	#14521054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521102

C₂

yes

maximal, characteristic

#14521073

C₂

yes

maximal, characteristic

#14520795

C₂

yes

maximal, complemented

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

C₄

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

—

C₂²⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄×C₃₁⋊C₈

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²⋊C₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

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

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1549

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#103

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#101

yes

—

C₄×C₃₁

#103

yes

—

C₄×C₃₁

C₂×SD₁₆

yes

characteristic

C₄×C₃₁

C₂×Q₁₆

yes

characteristic

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₂×C₄⋊Q₈

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₃₁

yes

—

C₄⋊Q₈

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₃₁

#1549

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

M₁₆

—

124

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂×Dic₃₁

yes

—

104

C₄²

C₂×Dic₃₁

yes

characteristic

105

C₄²

C₂²⋊D₃₁

yes

characteristic

106

C₄²

C₂²⋊D₃₁

yes

characteristic

107

C₄²

C₂²×D₃₁

yes

characteristic

108

C₂³

C₂²⋊Dic₃₁

yes

characteristic

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#92481

yes

—

116

C₂×C₄

#92482

yes

—

117

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

118

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

119

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

120

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

123

C₈

—

124

—

124

C₈

—

124

—

125

C₂²

—

126

C₂²

—

127

C₂²

#11758423

yes

—

128

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#11758423

yes

—

133

C₄

#11758424

yes

—

134

C₄

#11758509

yes

—

135

C₄

C₂×#92481

yes

characteristic

136

C₄

C₂×#92482

yes

characteristic

137

C₂

—

138

C₂

C₂×#11758424

yes

characteristic

139

C₂

C₂×#11758423

yes

characteristic

140

C₂

#14521054

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₁	C₂²⋊C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#1549
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₁

C₂²⋊C₄

cosocle

C₄²×C₃₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₃₁

hypocenter

C₃₁

#1549

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #14520795	C₂	semidirect
89. C₃₁	#1549	semidirect

H ≅ G/N

type

3. #14520795

C₂

semidirect

89. C₃₁

#1549

semidirect

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

subgroup	index
#14521102	2
#14521073	2
#14520795	2
C₂×C₃₁×C₄⋊Q₈	2
#1549	31

subgroup

index

C₂×C₃₁×C₄⋊Q₈

Subgroups — #16844396

Conjugacy classes of subgroups

Complements

Maximal subgroups