Group #16846543 — subgroups

Subgroups — #16846543

Order 3968.

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

normal subgroups only

This group has 320 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	#14521599	C₂	1	1	yes	maximal, characteristic
2	#14521073	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11758393	C₂	1	4	yes	maximal, complemented
4	C₃₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#92495	C₂²	1	2	yes	—
6	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	#11758507	—	2	2	no	—
9	C₂×#92474	C₂²	1	2	yes	—
10	#11758452	C₄	1	2	yes	complemented
11	#11758451	C₄	1	1	yes	characteristic, complemented
12	C₄×D₁₂₄	—	2	2	no	—
13	#11758446	C₄	1	1	yes	characteristic, complemented
14	C₃₁×#95	—	2	2	no	—
15	#11758393	—	2	8	no	—
16	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
17	C₂²×C₄×C₃₁	—	2	2	no	—
18	C₂²×C₄×C₃₁	D₄	1	2	yes	—
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₃₁	—	2	2	no	—
21	#92495	—	2	4	no	—
22	#92495	—	2	4	no	—
23	C₂×D₁₂₄	—	2	2	no	—
24	C₂×D₁₂₄	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₃₁	—	2	2	no	—
26	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
27	C₄²×C₃₁	—	2	2	no	—
28	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₃₁	D₄	1	2	yes	—
31	C₄²×C₃₁	D₄	1	4	yes	complemented
32	(C₂×D₃₁)⋊C₄	—	2	4	no	—
33	C₄⋊Dic₃₁	—	2	2	no	—
34	#92474	—	2	4	no	—
35	C₂×C₃₁⋊C₈	—	2	2	no	—
36	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	4	no	—
39	C₂×C₄×C₃₁	—	4	2	no	—
40	C₂×C₄×C₃₁	—	4	4	no	—
41	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#41	1	2	yes	—
47	C₂²⋊D₃₁	—	4	4	no	—
48	C₂×Dic₃₁	—	2	2	no	—
49	D₁₂₄	—	2	4	no	—
50	D₁₂₄	—	4	4	no	—
51	C₄×D₃₁	—	4	2	no	—
52	C₄×D₃₁	—	4	2	no	—
53	Dic₆₂	—	2	4	no	—
54	C₃₁⋊C₈	—	4	2	no	—
55	#1963	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	—	2	2	no	—
57	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₃₁	#121	1	2	yes	—
59	C₂×D₃₁	—	4	2	no	—
60	C₂×D₃₁	—	4	4	no	—
61	C₄×C₃₁	—	2	2	no	—
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	—	4	4	no	—
64	C₄×C₃₁	C₄×D₄	1	2	yes	—
65	C₄×C₃₁	#124	1	1	yes	characteristic
66	C₄×C₃₁	#127	1	1	yes	characteristic
67	Dic₃₁	—	4	2	no	—
68	#532	—	31	1	no	—
69	#423	—	31	1	no	—
70	C₂×#104	—	31	4	no	—
71	#376	D₃₁	1	1	yes	characteristic, complemented
72	C₂×C₃₁	—	2	2	no	—
73	C₂×C₃₁	#390	1	1	yes	characteristic
74	C₂×C₃₁	#421	1	2	yes	—
75	D₃₁	—	8	2	no	—
76	C₂×#41	—	31	2	no	—
77	C₂×M₁₆	—	31	2	no	—
78	C₄⋊Q₈	—	31	1	no	—
79	#127	—	31	1	no	—
80	#124	—	31	2	no	—
81	#121	—	62	2	no	—
82	C₄×D₄	—	62	2	no	—
83	C₂×C₄²	C₂×D₃₁	1	2	yes	—
84	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
85	C₄⋊C₈	—	62	2	no	—
86	#104	—	62	8	no	—
87	#95	—	2	2	no	—
88	C₃₁	#1963	1	1	yes	characteristic, complemented, Sylow
89	#41	—	62	4	no	—
90	#41	—	62	4	no	—
91	C₂×Q₈	—	31	2	no	—
92	C₂×D₄	—	31	2	no	—
93	C₂×D₄	—	62	2	no	—
94	C₂×D₄	—	62	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	62	2	no	—
97	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
98	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
99	M₁₆	—	62	4	no	—
100	C₂×C₈	—	62	2	no	—
101	C₄⋊C₄	—	62	2	no	—
102	C₂²⋊C₄	—	62	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₃₁	1	1	yes	characteristic
105	C₄²	C₄×D₃₁	1	1	yes	characteristic
106	C₄²	D₁₂₄	1	2	yes	—
107	C₄²	D₁₂₄	1	4	yes	complemented
108	C₂³	—	62	2	no	—
109	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
110	Q₈	—	62	4	no	—
111	D₄	—	62	4	no	—
112	D₄	—	124	4	no	—
113	D₄	—	124	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	62	2	no	—
118	C₂×C₄	—	124	2	no	—
119	C₂×C₄	—	124	2	no	—
120	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
121	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
124	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
125	C₂×C₄	#92495	1	2	yes	—
126	C₈	—	124	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	124	2	no	—
129	C₂²	—	124	4	no	—
130	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
131	C₂²	#11758507	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	124	2	no	—
136	C₄	C₄×D₁₂₄	1	2	yes	—
137	C₄	#11758451	1	1	yes	characteristic
138	C₄	#11758452	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	248	2	no	—
141	C₂	#14520951	1	1	yes	characteristic
142	C₂	#14521067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521599

C₂

yes

maximal, characteristic

#14521073

C₂

yes

maximal, characteristic, complemented

C₂×#11758393

C₂

yes

maximal, complemented

C₃₁×#376

C₂

yes

maximal, characteristic, complemented

C₂×#92495

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758507

—

C₂×#92474

C₂²

yes

—

#11758452

C₄

yes

complemented

#11758451

C₄

yes

characteristic, complemented

C₄×D₁₂₄

—

#11758446

C₄

yes

characteristic, complemented

C₃₁×#95

—

#11758393

—

C₄⋊(C₃₁⋊C₈)

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₂²⋊D₃₁

—

#92495

—

#92495

—

C₂×D₁₂₄

—

C₂×D₁₂₄

C₂×C₄

yes

—

C₂×C₄×D₃₁

—

C₂×Dic₆₂

C₂×C₄

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

D₄

yes

complemented

(C₂×D₃₁)⋊C₄

—

C₄⋊Dic₃₁

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂²×D₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

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

—

C₂²⋊D₃₁

—

C₂×Dic₃₁

—

D₁₂₄

—

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

—

Dic₆₂

—

C₃₁⋊C₈

—

#1963

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#121

yes

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

—

—

—

D₃₁

yes

characteristic, complemented

C₂×C₃₁

—

C₂×C₃₁

#390

yes

characteristic

C₂×C₃₁

#421

yes

—

D₃₁

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#104

—

#95

—

C₃₁

#1963

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₄×D₃₁

yes

characteristic

105

C₄²

C₄×D₃₁

yes

characteristic

106

C₄²

D₁₂₄

yes

—

107

C₄²

D₁₂₄

yes

complemented

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

124

—

113

D₄

—

124

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

124

—

119

C₂×C₄

—

124

—

120

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

121

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

122

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

123

C₂×C₄

C₂×D₁₂₄

yes

—

124

C₂×C₄

C₂×D₁₂₄

yes

characteristic

125

C₂×C₄

#92495

yes

—

126

C₈

—

124

—

127

C₂²

—

128

C₂²

—

124

—

129

C₂²

—

124

—

130

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

131

C₂²

#11758507

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

124

—

136

C₄

C₄×D₁₂₄

yes

—

137

C₄

#11758451

yes

characteristic

138

C₄

#11758452

yes

characteristic

139

C₂

—

140

C₂

—

248

—

141

C₂

#14520951

yes

characteristic

142

C₂

#14521067

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#376	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	#14521599	C₂
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#376	D₃₁
hypocenter	C₃₁	#1963
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#376

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

prime radical

#14521599

C₂

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #14521073	C₂	semidirect
3. C₂×#11758393	C₂	semidirect
4. C₃₁×#376	C₂	semidirect
10. #11758452	C₄	semidirect
11. #11758451	C₄	semidirect
13. #11758446	C₄	semidirect
31. C₄²×C₃₁	D₄	semidirect
71. #376	D₃₁	semidirect
88. C₃₁	#1963	semidirect
107. C₄²	D₁₂₄	semidirect

H ≅ G/N

type

2. #14521073

C₂

semidirect

3. C₂×#11758393

C₂

semidirect

4. C₃₁×#376

C₂

semidirect

10. #11758452

C₄

semidirect

11. #11758451

C₄

semidirect

13. #11758446

C₄

semidirect

31. C₄²×C₃₁

D₄

semidirect

71. #376

D₃₁

semidirect

88. C₃₁

#1963

semidirect

107. C₄²

D₁₂₄

semidirect

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

subgroup	index
#14521599	2
#14521073	2
C₂×#11758393	2
C₃₁×#376	2
#1963	31

subgroup

index

C₃₁×#376

Subgroups — #16846543

Conjugacy classes of subgroups

Complements

Maximal subgroups