Group #16854363 — subgroups

Subgroups — #16854363

Order 3968.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#92495	C₂	1	1	yes	maximal, characteristic
2	#14521108	C₂	1	4	yes	maximal, complemented
3	#14521104	C₂	1	4	yes	maximal, complemented
4	#14521077	C₂	1	2	yes	maximal, complemented
5	#14521017	C₂	1	2	yes	maximal
6	C₂×#11758391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#92495	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₄×C₂²⋊D₃₁	C₄	1	4	yes	—
12	#11758503	C₂²	1	1	yes	characteristic
13	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
14	C₄×D₁₂₄	C₄	1	2	yes	—
15	#11758449	C₂²	1	2	yes	—
16	C₄²×D₃₁	C₂²	1	2	yes	—
17	C₄×Dic₆₂	C₄	1	2	yes	—
18	C₃₁×#97	C₂²	1	2	yes	—
19	C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	(C₂×D₃₁)⋊C₈	C₂²	1	8	yes	—
21	Dic₃₁⋊C₈	C₂²	1	8	yes	—
22	#11758391	C₂²	1	2	yes	—
23	#11758391	C₂²	1	2	yes	—
24	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
27	#92495	—	2	4	no	—
28	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₃₁	C₂³	1	2	yes	—
31	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
32	M₁₆×C₃₁	—	2	4	no	—
33	C₂×C₈×C₃₁	C₂³	1	4	yes	—
34	C₄²×C₃₁	C₂×C₄	1	2	yes	—
35	C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
38	(C₂×D₃₁)⋊C₄	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
40	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
41	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
42	C₄×Dic₃₁	C₂³	1	2	yes	—
43	C₂×C₃₁⋊C₈	—	2	4	no	—
44	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₃₁	—	4	2	no	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	#41	1	4	yes	—
52	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₃₁	—	2	4	no	—
54	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
55	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
56	D₁₂₄	—	2	2	no	—
57	C₄×D₃₁	—	2	4	no	—
58	C₄×D₃₁	—	4	4	no	—
59	Dic₆₂	—	2	2	no	—
60	C₈×C₃₁	—	2	4	no	—
61	C₃₁⋊C₈	—	2	4	no	—
62	#2923	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	#117	1	2	yes	—
65	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
66	C₂×D₃₁	—	2	2	no	—
67	C₂×D₃₁	—	2	4	no	—
68	C₄×C₃₁	—	2	2	no	—
69	C₄×C₃₁	—	2	2	no	—
70	C₄×C₃₁	#117	1	2	yes	—
71	C₄×C₃₁	C₂×#41	1	2	yes	—
72	Dic₃₁	—	2	2	no	—
73	Dic₃₁	—	2	4	no	—
74	C₄×#41	—	31	1	no	—
75	#433	—	31	4	no	—
76	#432	—	31	4	no	—
77	#424	—	31	2	no	—
78	#408	—	31	2	no	—
79	C₄×M₁₆	D₃₁	1	1	yes	characteristic, complemented
80	C₂×#97	—	31	1	no	—
81	C₂×C₃₁	—	2	2	no	—
82	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
83	C₂×C₃₁	#568	1	2	yes	—
84	D₃₁	—	4	2	no	—
85	C₂×#41	—	31	1	no	—
86	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
87	C₂²×C₈	—	31	2	no	—
88	C₄×Q₈	—	31	2	no	—
89	C₄×D₄	—	31	4	no	—
90	C₄×D₄	—	31	2	no	—
91	#117	—	31	1	no	—
92	#117	—	31	2	no	—
93	C₂×C₄²	—	31	2	no	—
94	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
95	C₄⋊C₈	—	31	8	no	—
96	C₂²⋊C₈	—	31	8	no	—
97	#97	—	31	2	no	—
98	#97	—	31	2	no	—
99	#97	C₂×D₃₁	1	2	yes	—
100	C₄×C₈	C₂×D₃₁	1	2	yes	—
101	C₃₁	#2923	1	1	yes	characteristic, complemented, Sylow
102	#41	—	62	4	no	—
103	C₂×Q₈	—	31	1	no	—
104	C₂×D₄	—	31	2	no	—
105	C₂×D₄	—	31	1	no	—
106	C₂²×C₄	—	31	2	no	—
107	C₂²×C₄	—	31	4	no	—
108	C₂²×C₄	C₄×D₃₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	31	4	no	—
112	C₂×C₈	—	62	4	no	—
113	C₂×C₈	C₂²×D₃₁	1	4	yes	—
114	C₄⋊C₄	—	31	4	no	—
115	C₄⋊C₄	—	31	2	no	—
116	C₂²⋊C₄	—	31	2	no	—
117	C₂²⋊C₄	—	31	4	no	—
118	C₄²	—	31	4	no	—
119	C₄²	—	31	2	no	—
120	C₄²	C₄×D₃₁	1	2	yes	—
121	C₄²	C₂²×D₃₁	1	1	yes	characteristic
122	C₄²	C₂²×D₃₁	1	1	yes	characteristic
123	C₂³	—	31	2	no	—
124	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
125	Q₈	—	62	2	no	—
126	D₄	—	62	4	no	—
127	D₄	—	62	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	31	4	no	—
130	C₂×C₄	—	31	2	no	—
131	C₂×C₄	—	62	4	no	—
132	C₂×C₄	—	124	4	no	—
133	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
136	C₂×C₄	#92495	1	4	yes	—
137	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	62	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	62	2	no	—
142	C₂²	—	62	4	no	—
143	C₂²	#11758449	1	2	yes	—
144	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	62	2	no	—
148	C₄	—	62	4	no	—
149	C₄	#11758449	1	2	yes	—
150	C₄	C₂×#92495	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	124	2	no	—
153	C₂	C₂×#11758449	1	1	yes	characteristic
154	C₂	#14521815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#92495

C₂

yes

maximal, characteristic

#14521108

C₂

yes

maximal, complemented

#14521104

C₂

yes

maximal, complemented

#14521077

C₂

yes

maximal, complemented

#14521017

C₂

yes

maximal

C₂×#11758391

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₃₁

C₂

yes

maximal, characteristic, complemented

C₂×#92495

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₄×C₂²⋊D₃₁

C₄

yes

—

#11758503

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₄×D₁₂₄

C₄

yes

—

#11758449

C₂²

yes

—

C₄²×D₃₁

C₂²

yes

—

C₄×Dic₆₂

C₄

yes

—

C₃₁×#97

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

(C₂×D₃₁)⋊C₈

C₂²

yes

—

Dic₃₁⋊C₈

C₂²

yes

—

#11758391

C₂²

yes

—

#11758391

C₂²

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₂²⋊D₃₁

C₂×C₄

yes

—

#92495

—

C₂×D₁₂₄

C₂×C₄

yes

characteristic

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

C₂³

yes

—

C₂×Dic₆₂

C₂×C₄

yes

characteristic

M₁₆×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

C₂×C₄

yes

—

(C₂×D₃₁)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂²×D₃₁

C₂²×C₄

yes

—

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

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂²⋊D₃₁

—

C₂×Dic₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

C₂²×C₄

yes

—

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#2923

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#117

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#117

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

Dic₃₁

—

C₄×#41

—

—

—

—

—

C₄×M₁₆

D₃₁

yes

characteristic, complemented

C₂×#97

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#117

yes

characteristic

C₂×C₃₁

#568

yes

—

D₃₁

—

C₂×#41

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

C₂×D₃₁

yes

—

100

C₄×C₈

C₂×D₃₁

yes

—

101

C₃₁

#2923

yes

characteristic, complemented, Sylow

102

#41

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₃₁

yes

—

109

C₂²×C₄

C₂²×D₃₁

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₃₁

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₃₁

yes

—

121

C₄²

C₂²×D₃₁

yes

characteristic

122

C₄²

C₂²×D₃₁

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₄×D₃₁

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

124

—

133

C₂×C₄

C₂×C₄×D₃₁

yes

—

134

C₂×C₄

C₂×C₄×D₃₁

yes

—

135

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

136

C₂×C₄

#92495

yes

—

137

C₂×C₄

C₂³×D₃₁

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#11758449

yes

—

144

C₂²

C₂²×C₄×D₃₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#11758449

yes

—

150

C₄

C₂×#92495

yes

—

151

C₂

—

152

C₂

—

124

—

153

C₂

C₂×#11758449

yes

characteristic

154

C₂

#14521815

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₄×M₁₆×C₃₁	C₂
prime radical	C₂×#92495	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×M₁₆	D₃₁
hypocenter	C₃₁	#2923
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₄×M₁₆×C₃₁

C₂

prime radical

C₂×#92495

C₂²

cosocle

C₂×C₄×C₃₁

C₂⁴

hypercenter

C₄×M₁₆

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #14521108	C₂	semidirect
3. #14521104	C₂	semidirect
4. #14521077	C₂	semidirect
6. C₂×#11758391	C₂	semidirect
7. C₄×M₁₆×C₃₁	C₂	semidirect
79. C₄×M₁₆	D₃₁	semidirect
101. C₃₁	#2923	semidirect

H ≅ G/N

type

2. #14521108

C₂

semidirect

3. #14521104

C₂

semidirect

4. #14521077

C₂

semidirect

6. C₂×#11758391

C₂

semidirect

7. C₄×M₁₆×C₃₁

C₂

semidirect

79. C₄×M₁₆

D₃₁

semidirect

101. C₃₁

#2923

semidirect

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

subgroup	index
C₄×#92495	2
#14521108	2
#14521104	2
#14521077	2
#14521017	2
C₂×#11758391	2
C₄×M₁₆×C₃₁	2
#2923	31

subgroup

index

C₄×M₁₆×C₃₁

Subgroups — #16854363

Conjugacy classes of subgroups

Complements

Maximal subgroups