Group #16856242 — subgroups

Subgroups — #16856242

Order 3968.

340 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 165 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	#14521677	C₂	1	2	yes	maximal
2	#14521458	C₂	1	4	yes	maximal, complemented
3	#14521451	C₂	1	2	yes	maximal, complemented
4	#14521447	C₂	1	2	yes	maximal, complemented
5	#14521091	C₂	1	2	yes	maximal
6	#14521087	C₂	1	1	yes	maximal, characteristic
7	C₂×#11758405	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#117	C₂²	1	1	yes	characteristic
12	#11758503	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
14	#11758467	—	2	2	no	—
15	#11758465	—	2	2	no	—
16	#11758455	—	2	4	no	—
17	#11758447	C₂²	1	4	yes	—
18	#11758446	C₂²	1	4	yes	—
19	C₃₁×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	#11758406	C₂²	1	4	yes	—
22	#11758406	C₂²	1	2	yes	—
23	#11758405	C₂²	1	4	yes	—
24	#11758405	C₂²	1	1	yes	characteristic
25	#11758405	C₂²	1	2	yes	—
26	#11758405	C₂²	1	1	yes	characteristic
27	#11758405	C₂²	1	2	yes	—
28	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃₁	—	2	2	no	—
30	C₂×Dic₆₂	—	2	4	no	—
31	M₁₆×C₃₁	Q₈	1	4	yes	complemented
32	C₂×C₈×C₃₁	Q₈	1	4	yes	complemented
33	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₃₁	C₂³	1	2	yes	—
35	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
37	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
38	C₄²×C₃₁	C₂³	1	2	yes	—
39	C₂²⋊Dic₃₁	—	2	2	no	—
40	C₄⋊Dic₃₁	—	2	4	no	—
41	C₄⋊Dic₃₁	—	2	2	no	—
42	C₄⋊Dic₃₁	C₂³	1	2	yes	—
43	C₄⋊Dic₃₁	C₂³	1	4	yes	—
44	C₄⋊Dic₃₁	C₂³	1	2	yes	—
45	Dic₃₁⋊C₄	—	2	4	no	—
46	Dic₃₁⋊C₄	—	2	4	no	—
47	C₄×Dic₃₁	—	2	2	no	—
48	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₃₁	—	2	2	no	—
55	C₂×Dic₃₁	—	2	2	no	—
56	C₂×Dic₃₁	—	2	4	no	—
57	C₂×Dic₃₁	—	8	2	no	—
58	Dic₆₂	—	8	4	no	—
59	C₈×C₃₁	C₂×Q₈	1	2	yes	—
60	C₈×C₃₁	C₂×Q₈	1	2	yes	—
61	C₈×C₃₁	C₂×Q₈	1	4	yes	—
62	#3163	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
65	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₃₁	—	2	4	no	—
67	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
68	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
70	Dic₃₁	—	4	2	no	—
71	Dic₃₁	—	4	4	no	—
72	Dic₃₁	—	4	2	no	—
73	#544	—	31	2	no	—
74	#501	—	31	4	no	—
75	#499	—	31	2	no	—
76	#498	—	31	2	no	—
77	#428	—	31	2	no	—
78	#427	—	31	1	no	—
79	C₂×#106	—	31	1	no	—
80	#405	D₃₁	1	1	yes	characteristic
81	C₂×C₃₁	—	2	2	no	—
82	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₃₁	#577	1	2	yes	—
84	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
86	C₄⋊Q₈	—	31	4	no	—
87	C₄⋊Q₈	—	62	2	no	—
88	#125	—	31	4	no	—
89	#125	—	62	2	no	—
90	C₂²⋊Q₈	—	62	4	no	—
91	#117	—	31	2	no	—
92	#117	C₂×D₃₁	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	31	2	no	—
94	#107	—	31	2	no	—
95	#107	—	31	4	no	—
96	#106	—	31	2	no	—
97	#106	—	31	2	no	—
98	#106	—	31	1	no	—
99	#106	—	31	1	no	—
100	#106	—	31	4	no	—
101	#97	C₂×D₃₁	1	2	yes	—
102	C₄×C₈	C₂×D₃₁	1	2	yes	—
103	C₃₁	#3163	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	62	4	no	—
105	C₂²×C₄	—	62	2	no	—
106	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
107	M₁₆	Dic₆₂	1	4	yes	complemented
108	C₂×C₈	Dic₆₂	1	4	yes	complemented
109	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₃₁	1	2	yes	—
111	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
112	C₄⋊C₄	—	31	4	no	—
113	C₄⋊C₄	—	31	2	no	—
114	C₄⋊C₄	—	31	2	no	—
115	C₄⋊C₄	—	62	4	no	—
116	C₄⋊C₄	—	62	4	no	—
117	C₄⋊C₄	—	62	2	no	—
118	C₄⋊C₄	—	62	4	no	—
119	C₄⋊C₄	D₁₂₄	1	2	yes	—
120	C₂²⋊C₄	—	62	2	no	—
121	C₂²⋊C₄	D₁₂₄	1	2	yes	—
122	C₄²	—	62	2	no	—
123	C₄²	C₂²×D₃₁	1	2	yes	—
124	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
125	Q₈	—	248	4	no	—
126	C₂×C₄	—	62	2	no	—
127	C₂×C₄	—	62	2	no	—
128	C₂×C₄	—	62	4	no	—
129	C₂×C₄	—	248	2	no	—
130	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
131	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
132	C₂×C₄	C₂×D₁₂₄	1	4	yes	—
133	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
135	C₈	C₂×Dic₆₂	1	2	yes	—
136	C₈	C₂×Dic₆₂	1	2	yes	—
137	C₈	C₂×Dic₆₂	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#11758446	1	2	yes	—
140	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	124	4	no	—
143	C₄	—	124	2	no	—
144	C₄	—	124	2	no	—
145	C₄	#11758446	1	2	yes	—
146	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
147	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#11758446	1	1	yes	characteristic
150	C₂	#14521861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521677

C₂

yes

maximal

#14521458

C₂

yes

maximal, complemented

#14521451

C₂

yes

maximal, complemented

#14521447

C₂

yes

maximal, complemented

#14521091

C₂

yes

maximal

#14521087

C₂

yes

maximal, characteristic

C₂×#11758405

C₂

yes

maximal, characteristic

C₃₁×#405

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#117

C₂²

yes

characteristic

#11758503

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

—

—

—

C₂²

yes

—

#11758446

C₂²

yes

—

C₃₁×#97

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

#11758406

C₂²

yes

—

#11758406

C₂²

yes

—

#11758405

C₂²

yes

—

#11758405

C₂²

yes

characteristic

#11758405

C₂²

yes

—

#11758405

C₂²

yes

characteristic

#11758405

C₂²

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

M₁₆×C₃₁

Q₈

yes

complemented

C₂×C₈×C₃₁

Q₈

yes

complemented

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

C₂³

yes

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

D₄

yes

—

C₂²⋊C₄×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

C₄⋊Dic₃₁

C₂³

yes

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

C₈×C₃₁

C₂×Q₈

yes

—

C₈×C₃₁

C₂×Q₈

yes

—

C₈×C₃₁

C₂×Q₈

yes

—

#3163

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄⋊Q₈

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂²×Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

—

—

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

#577

yes

—

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

—

—

—

—

—

—

100

#106

—

101

#97

C₂×D₃₁

yes

—

102

C₄×C₈

C₂×D₃₁

yes

—

103

C₃₁

#3163

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₃₁

yes

characteristic

107

M₁₆

Dic₆₂

yes

complemented

108

C₂×C₈

Dic₆₂

yes

complemented

109

C₂×C₈

C₂²×D₃₁

yes

characteristic

110

C₂×C₈

C₂²×D₃₁

yes

—

111

C₂×C₈

C₂²×D₃₁

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

D₁₂₄

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

D₁₂₄

yes

—

122

C₄²

—

123

C₄²

C₂²×D₃₁

yes

—

124

C₂³

C₂×D₁₂₄

yes

characteristic

125

Q₈

—

248

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

248

—

130

C₂×C₄

C₂×Dic₆₂

yes

—

131

C₂×C₄

C₂×Dic₆₂

yes

—

132

C₂×C₄

C₂×D₁₂₄

yes

—

133

C₂×C₄

C₂×D₁₂₄

yes

characteristic

134

C₂×C₄

C₂³×D₃₁

yes

characteristic

135

C₈

C₂×Dic₆₂

yes

—

136

C₈

C₂×Dic₆₂

yes

—

137

C₈

C₂×Dic₆₂

yes

—

138

C₂²

—

139

C₂²

#11758446

yes

—

140

C₂²

C₂²×D₁₂₄

yes

characteristic

141

C₄

—

142

C₄

—

124

—

143

C₄

—

124

—

144

C₄

—

124

—

145

C₄

#11758446

yes

—

146

C₄

C₂²×Dic₆₂

yes

characteristic

147

C₄

C₂²×Dic₆₂

yes

characteristic

148

C₂

—

149

C₂

C₂×#11758446

yes

characteristic

150

C₂

#14521861

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#405	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#405	D₃₁
hypocenter	C₃₁	#3163
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₁₂₄

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#405

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #14521458	C₂	semidirect
3. #14521451	C₂	semidirect
4. #14521447	C₂	semidirect
31. M₁₆×C₃₁	Q₈	semidirect
32. C₂×C₈×C₃₁	Q₈	semidirect
103. C₃₁	#3163	semidirect
107. M₁₆	Dic₆₂	semidirect
108. C₂×C₈	Dic₆₂	semidirect

H ≅ G/N

type

2. #14521458

C₂

semidirect

3. #14521451

C₂

semidirect

4. #14521447

C₂

semidirect

31. M₁₆×C₃₁

Q₈

semidirect

32. C₂×C₈×C₃₁

Q₈

semidirect

103. C₃₁

#3163

semidirect

107. M₁₆

Dic₆₂

semidirect

108. C₂×C₈

Dic₆₂

semidirect

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

subgroup	index
#14521677	2
#14521458	2
#14521451	2
#14521447	2
#14521091	2
#14521087	2
C₂×#11758405	2
C₃₁×#405	2
#3163	31

subgroup

index

C₃₁×#405

Subgroups — #16856242

Conjugacy classes of subgroups

Complements

Maximal subgroups