Group #16856087 — subgroups

Subgroups — #16856087

Order 3968.

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

normal subgroups only

This group has 484 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	C₂×#11758451	C₂	1	1	yes	maximal, characteristic
2	#14521415	C₂	1	4	yes	maximal, complemented
3	#14521397	C₂	1	4	yes	maximal, complemented
4	#14521053	C₂	1	4	yes	maximal, complemented
5	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₂₄	—	2	2	no	—
8	C₂²×D₁₂₄	C₂²	1	2	yes	—
9	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
10	C₃₁×#125	C₂²	1	2	yes	complemented
11	C₃₁×C₂²⋊Q₈	—	2	2	no	—
12	C₄×Q₈×C₃₁	—	2	2	no	—
13	C₃₁×#117	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×#92474	C₂²	1	2	yes	—
16	#11758451	—	2	2	no	—
17	#11758451	C₂²	1	2	yes	—
18	#11758451	C₂²	1	2	yes	—
19	#11758397	C₂²	1	16	yes	complemented
20	#11758391	C₂²	1	2	yes	complemented
21	C₄×C₃₁⋊C₈	C₂²	1	2	yes	complemented
22	C₂³×D₃₁	—	2	2	no	—
23	C₂×Q₈×C₃₁	—	2	2	no	—
24	C₂²×C₄×C₃₁	D₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×D₁₂₄	—	2	8	no	—
27	C₂×D₁₂₄	—	2	2	no	—
28	C₂×D₁₂₄	—	2	4	no	—
29	C₂×D₁₂₄	—	2	4	no	—
30	C₂×D₁₂₄	—	4	8	no	—
31	C₂×D₁₂₄	C₂³	1	4	yes	—
32	C₄⋊C₄×C₃₁	—	2	4	no	—
33	C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
35	C₂²⋊C₄×C₃₁	—	2	2	no	—
36	C₄²×C₃₁	—	2	2	no	—
37	C₄²×C₃₁	D₄	1	2	yes	—
38	C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	#92474	—	2	4	no	—
41	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
42	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₃₁	—	2	4	no	—
44	C₂²×D₃₁	—	4	8	no	—
45	C₂²×D₃₁	—	4	4	no	—
46	Q₈×C₃₁	—	4	4	no	—
47	C₂×C₄×C₃₁	—	2	4	no	—
48	C₂×C₄×C₃₁	—	2	4	no	—
49	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₃₁	#41	1	4	yes	—
53	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	D₁₂₄	—	2	8	no	—
55	D₁₂₄	—	4	8	no	—
56	D₁₂₄	—	4	4	no	—
57	D₁₂₄	—	4	8	no	—
58	C₃₁⋊C₈	—	2	4	no	—
59	#3138	—	31	1	no	maximal, Sylow
60	C₂²×C₃₁	—	2	2	no	—
61	C₂²×C₃₁	#124	1	2	yes	—
62	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
63	C₂×D₃₁	—	4	4	no	—
64	C₂×D₃₁	—	4	8	no	—
65	C₂×D₃₁	—	8	8	no	—
66	C₄×C₃₁	—	2	2	no	—
67	C₄×C₃₁	—	4	4	no	—
68	C₄×C₃₁	#124	1	2	yes	—
69	C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
70	C₄×C₃₁	C₂×#41	1	2	yes	—
71	#533	D₃₁	1	1	yes	characteristic, complemented
72	C₂×#127	—	31	1	no	—
73	#490	—	31	4	no	—
74	#486	—	31	4	no	—
75	#418	—	31	4	no	—
76	C₄×M₁₆	—	31	1	no	—
77	C₂×C₃₁	—	2	2	no	—
78	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
79	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
80	D₃₁	—	8	4	no	—
81	C₂²×D₄	—	31	2	no	—
82	C₂²×D₄	—	62	2	no	—
83	C₂×M₁₆	—	31	2	no	—
84	C₄⋊Q₈	C₂×D₃₁	1	2	yes	complemented
85	#127	—	31	2	no	—
86	#127	—	31	2	no	—
87	#127	—	62	2	no	—
88	#125	C₂×D₃₁	1	2	yes	complemented
89	C₂²⋊Q₈	—	2	2	no	—
90	C₄×Q₈	—	2	2	no	—
91	#117	C₂×D₃₁	1	2	yes	—
92	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
93	#102	—	31	16	no	—
94	#97	—	31	2	no	—
95	C₄×C₈	—	31	2	no	—
96	C₃₁	#3138	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	—	62	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	31	4	no	—
100	C₂×D₄	—	62	8	no	—
101	C₂×D₄	—	62	4	no	—
102	C₂×D₄	—	62	4	no	—
103	C₂×D₄	—	62	2	no	—
104	C₂×D₄	—	124	8	no	—
105	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
106	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
107	M₁₆	—	62	4	no	—
108	C₂×C₈	—	31	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₃₁	1	2	yes	—
115	C₄²	C₂²×D₃₁	1	1	yes	characteristic
116	C₄²	C₂²×D₃₁	1	1	yes	characteristic
117	C₂³	—	62	4	no	—
118	C₂³	—	124	8	no	—
119	C₂³	—	124	4	no	—
120	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	62	8	no	—
123	D₄	—	124	8	no	—
124	D₄	—	124	4	no	—
125	D₄	—	124	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	#92499	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
132	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
133	C₈	—	62	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	124	4	no	—
136	C₂²	—	124	8	no	—
137	C₂²	—	248	8	no	—
138	C₂²	#11758524	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#11758524	1	2	yes	—
143	C₄	#11758526	1	4	yes	—
144	C₄	C₂×#92499	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	248	4	no	—
147	C₂	C₂×#11758524	1	1	yes	characteristic
148	C₂	C₂×#11758526	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758451

C₂

yes

maximal, characteristic

#14521415

C₂

yes

maximal, complemented

#14521397

C₂

yes

maximal, complemented

#14521053

C₂

yes

maximal, complemented

C₄×#92474

C₂

yes

maximal, characteristic, complemented

C₃₁×#533

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₂₄

—

C₂²×D₁₂₄

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

complemented

C₃₁×#125

C₂²

yes

complemented

C₃₁×C₂²⋊Q₈

—

C₄×Q₈×C₃₁

—

C₃₁×#117

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

#11758451

—

#11758451

C₂²

yes

—

#11758451

C₂²

yes

—

#11758397

C₂²

yes

complemented

#11758391

C₂²

yes

complemented

C₄×C₃₁⋊C₈

C₂²

yes

complemented

C₂³×D₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

#92474

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

—

Q₈×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

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

C₃₁⋊C₈

—

#3138

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

C₂×#41

yes

—

#533

D₃₁

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#124

yes

characteristic

C₂×C₃₁

C₂×AGL(1,ℤ/8)

yes

—

D₃₁

—

C₂²×D₄

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃₁

yes

—

—

—

C₂×D₃₁

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₃₁

#3138

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

124

—

105

C₂²×C₄

C₂²⋊D₃₁

yes

—

106

C₂²×C₄

C₂²×D₃₁

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₃₁

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₃₁

yes

—

115

C₄²

C₂²×D₃₁

yes

characteristic

116

C₄²

C₂²×D₃₁

yes

characteristic

117

C₂³

—

118

C₂³

—

124

—

119

C₂³

—

124

—

120

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

121

Q₈

—

122

D₄

—

123

D₄

—

124

—

124

D₄

—

124

—

125

D₄

—

124

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#92499

yes

—

129

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

130

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

132

C₂×C₄

C₂³×D₃₁

yes

characteristic

133

C₈

—

134

C₂²

—

135

C₂²

—

124

—

136

C₂²

—

124

—

137

C₂²

—

248

—

138

C₂²

#11758524

yes

—

139

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#11758524

yes

—

143

C₄

#11758526

yes

—

144

C₄

C₂×#92499

yes

—

145

C₂

—

146

C₂

—

248

—

147

C₂

C₂×#11758524

yes

characteristic

148

C₂

C₂×#11758526

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#533	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#11758451	C₂
hypercenter	#533	D₃₁
hypocenter	C₃₁	#3138
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#533

C₂

socle

C₂²×C₃₁

C₂²×D₄

C₂

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14521415	C₂	semidirect
3. #14521397	C₂	semidirect
4. #14521053	C₂	semidirect
5. C₄×#92474	C₂	semidirect
6. C₃₁×#533	C₂	semidirect
9. C₃₁×C₄⋊Q₈	C₂²	semidirect
10. C₃₁×#125	C₂²	semidirect
19. #11758397	C₂²	semidirect
20. #11758391	C₂²	semidirect
21. C₄×C₃₁⋊C₈	C₂²	semidirect
71. #533	D₃₁	semidirect
84. C₄⋊Q₈	C₂×D₃₁	semidirect
88. #125	C₂×D₃₁	semidirect
96. C₃₁	#3138	semidirect

H ≅ G/N

type

2. #14521415

C₂

semidirect

3. #14521397

C₂

semidirect

4. #14521053

C₂

semidirect

5. C₄×#92474

C₂

semidirect

6. C₃₁×#533

C₂

semidirect

9. C₃₁×C₄⋊Q₈

C₂²

semidirect

10. C₃₁×#125

C₂²

semidirect

19. #11758397

C₂²

semidirect

20. #11758391

C₂²

semidirect

21. C₄×C₃₁⋊C₈

C₂²

semidirect

71. #533

D₃₁

semidirect

84. C₄⋊Q₈

C₂×D₃₁

semidirect

88. #125

C₂×D₃₁

semidirect

96. C₃₁

#3138

semidirect

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

subgroup	index
C₂×#11758451	2
#14521415	2
#14521397	2
#14521053	2
C₄×#92474	2
C₃₁×#533	2
#3138	31

subgroup

index

C₃₁×#533

Subgroups — #16856087

Conjugacy classes of subgroups

Complements

Maximal subgroups