Group #16854273 — subgroups

Subgroups — #16854273

Order 3968.

484 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 213 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×M₁₆×D₃₁	C₂	1	2	yes	maximal
2	C₂×#11758449	C₂	1	1	yes	maximal, characteristic
3	#14521074	C₂	1	8	yes	maximal, complemented
4	#14521073	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521072	C₂	1	2	yes	maximal, complemented
6	C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₃₁)⋊C₄	C₄	1	2	yes	—
11	C₂×Dic₃₁⋊C₄	C₄	1	2	yes	—
12	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
13	C₂×#92474	C₂²	1	2	yes	—
14	M₁₆×D₃₁	—	2	8	no	—
15	C₂×#92469	C₂²	1	4	yes	—
16	C₂×C₈×D₃₁	C₂²	1	4	yes	—
17	#11758449	C₄	1	4	yes	—
18	#11758449	C₂²	1	4	yes	—
19	C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
20	Dic₃₁⋊C₈	C₂²	1	8	yes	complemented
21	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
22	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
26	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₃₁	D₄	1	4	yes	—
29	C₂×C₄×D₃₁	Q₈	1	4	yes	—
30	C₂×C₄×D₃₁	C₂³	1	2	yes	—
31	M₁₆×C₃₁	—	2	4	no	—
32	C₂×C₈×C₃₁	C₂³	1	4	yes	—
33	C₄²×C₃₁	C₂×C₄	1	2	yes	—
34	C₄²×C₃₁	C₂³	1	2	yes	—
35	(C₂×D₃₁)⋊C₄	C₂×C₄	1	8	yes	—
36	Dic₃₁⋊C₄	C₂×C₄	1	8	yes	—
37	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
38	C₄×Dic₃₁	C₂³	1	2	yes	—
39	#92474	—	2	4	no	—
40	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
41	#92469	—	2	8	no	—
42	C₈×D₃₁	—	2	8	no	—
43	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×D₃₁	—	4	2	no	—
45	C₂²×D₃₁	C₄⋊C₄	1	4	yes	—
46	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₃₁	—	4	2	no	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₃₁	—	4	2	no	—
54	C₂×Dic₃₁	C₄⋊C₄	1	4	yes	—
55	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
56	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
57	C₄×D₃₁	C₄⋊C₄	1	8	yes	—
58	C₄×D₃₁	C₂×D₄	1	4	yes	—
59	C₄×D₃₁	C₂×Q₈	1	4	yes	—
60	C₈×C₃₁	—	2	4	no	—
61	C₃₁⋊C₈	—	2	4	no	—
62	#2905	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
65	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
66	C₂×D₃₁	—	2	4	no	—
67	C₂×D₃₁	—	4	4	no	—
68	C₂×D₃₁	C₂×C₄⋊C₄	1	4	yes	—
69	C₄×C₃₁	—	2	4	no	—
70	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
71	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
72	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
73	Dic₃₁	—	2	4	no	—
74	Dic₃₁	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×M₁₆	—	31	2	no	—
76	C₂×#117	—	31	1	no	—
77	#424	—	31	8	no	—
78	#423	—	31	2	no	—
79	#423	—	31	1	no	—
80	#423	D₃₁	1	1	yes	characteristic, complemented
81	C₂×C₃₁	—	2	2	no	—
82	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₃₁	#568	1	2	yes	—
84	D₃₁	—	2	4	no	—
85	C₂³×C₄	—	31	1	no	—
86	C₂×M₁₆	—	31	2	no	—
87	C₂×M₁₆	—	31	4	no	—
88	C₂×M₁₆	—	62	8	no	—
89	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
90	C₂²×C₈	—	31	4	no	—
91	#117	—	31	4	no	—
92	#117	—	31	4	no	—
93	C₂×C₄⋊C₄	—	31	2	no	—
94	C₂×C₂²⋊C₄	—	31	2	no	—
95	C₂×C₄²	—	31	1	no	—
96	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
97	C₄⋊C₈	—	31	8	no	—
98	C₄⋊C₈	—	31	4	no	—
99	C₄⋊C₈	C₂×D₃₁	1	4	yes	complemented
100	C₃₁	#2905	1	1	yes	characteristic, complemented, Sylow
101	C₂⁴	—	31	1	no	—
102	C₂²×C₄	—	31	1	no	—
103	C₂²×C₄	—	31	2	no	—
104	C₂²×C₄	—	31	4	no	—
105	C₂²×C₄	—	31	4	no	—
106	C₂²×C₄	—	31	2	no	—
107	C₂²×C₄	—	31	2	no	—
108	C₂²×C₄	C₄×D₃₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	62	8	no	—
112	M₁₆	—	62	4	no	—
113	C₂×C₈	—	31	4	no	—
114	C₂×C₈	—	62	8	no	—
115	C₂×C₈	C₂²×D₃₁	1	4	yes	—
116	C₄⋊C₄	—	31	8	no	—
117	C₂²⋊C₄	—	31	8	no	—
118	C₄²	—	31	2	no	—
119	C₄²	—	31	2	no	—
120	C₄²	C₄×D₃₁	1	2	yes	—
121	C₄²	C₂²×D₃₁	1	2	yes	—
122	C₂³	—	31	2	no	—
123	C₂³	—	31	4	no	—
124	C₂³	—	124	2	no	—
125	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	31	4	no	—
128	C₂×C₄	—	31	2	no	—
129	C₂×C₄	—	31	8	no	—
130	C₂×C₄	—	31	4	no	—
131	C₂×C₄	—	31	4	no	—
132	C₂×C₄	—	31	4	no	—
133	C₂×C₄	—	124	2	no	—
134	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
136	C₂×C₄	D₄×D₃₁	1	2	yes	—
137	C₂×C₄	Q₈×D₃₁	1	2	yes	—
138	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	62	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	31	4	no	—
143	C₂²	—	62	4	no	—
144	C₂²	—	124	4	no	—
145	C₂²	C₄⋊C₄×D₃₁	1	2	yes	—
146	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	31	4	no	—
149	C₄	—	62	4	no	—
150	C₄	C₄⋊C₄×D₃₁	1	2	yes	—
151	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
152	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	62	4	no	—
155	C₂	C₂×C₄⋊C₄×D₃₁	1	1	yes	characteristic
156	C₂	#14521815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₃₁

C₂

yes

maximal

C₂×#11758449

C₂

yes

maximal, characteristic

#14521074

C₂

yes

maximal, complemented

#14521073

C₂

yes

maximal, characteristic, complemented

#14521072

C₂

yes

maximal, complemented

C₃₁×#423

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₁

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×(C₂×D₃₁)⋊C₄

C₄

yes

—

C₂×Dic₃₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

M₁₆×D₃₁

—

C₂×#92469

C₂²

yes

—

C₂×C₈×D₃₁

C₂²

yes

—

#11758449

C₄

yes

—

#11758449

C₂²

yes

—

C₃₁×C₄⋊C₈

C₂²

yes

complemented

Dic₃₁⋊C₈

C₂²

yes

complemented

C₄⋊(C₃₁⋊C₈)

C₂²

yes

complemented

C₂³×D₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂²×Dic₃₁

C₂×C₄

yes

characteristic

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

D₄

yes

—

C₂×C₄×D₃₁

Q₈

yes

—

C₂×C₄×D₃₁

C₂³

yes

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂³

yes

—

(C₂×D₃₁)⋊C₄

C₂×C₄

yes

—

Dic₃₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

#92474

—

C₂×C₃₁⋊C₈

C₂³

yes

—

#92469

—

C₈×D₃₁

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

C₄⋊C₄

yes

—

C₂²×D₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₂×Dic₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

C₂²×C₄

yes

—

C₄×D₃₁

C₄⋊C₄

yes

—

C₄×D₃₁

C₂×D₄

yes

—

C₄×D₃₁

C₂×Q₈

yes

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#2905

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×C₄⋊C₄

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₄×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

C₂²×Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

C₂×C₄⋊C₄

yes

—

C₂²×M₁₆

—

—

—

—

—

D₃₁

yes

characteristic, complemented

C₂×C₃₁

—

C₂×C₃₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₃₁

#568

yes

—

D₃₁

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

complemented

100

C₃₁

#2905

yes

characteristic, complemented, Sylow

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₃₁

yes

—

109

C₂²×C₄

C₂²×D₃₁

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²×D₃₁

yes

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₃₁

yes

—

121

C₄²

C₂²×D₃₁

yes

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

124

—

125

C₂³

C₂×C₄×D₃₁

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

124

—

134

C₂×C₄

C₂×C₄×D₃₁

yes

—

135

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

136

C₂×C₄

D₄×D₃₁

yes

—

137

C₂×C₄

Q₈×D₃₁

yes

—

138

C₂×C₄

C₂³×D₃₁

yes

characteristic

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

124

—

145

C₂²

C₄⋊C₄×D₃₁

yes

—

146

C₂²

C₂²×C₄×D₃₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₄⋊C₄×D₃₁

yes

—

151

C₄

C₂×D₄×D₃₁

yes

characteristic

152

C₄

C₂×Q₈×D₃₁

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

C₂×C₄⋊C₄×D₃₁

yes

characteristic

156

C₂

#14521815

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#423	C₂
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#423	D₃₁
hypocenter	C₃₁	#2905
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₃₁×#423

C₂

prime radical

C₂³×D₃₁

C₂×C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
3. #14521074	C₂	semidirect
4. #14521073	C₂	semidirect
5. #14521072	C₂	semidirect
6. C₃₁×#423	C₂	semidirect
19. C₃₁×C₄⋊C₈	C₂²	semidirect
20. Dic₃₁⋊C₈	C₂²	semidirect
21. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
80. #423	D₃₁	semidirect
99. C₄⋊C₈	C₂×D₃₁	semidirect
100. C₃₁	#2905	semidirect

H ≅ G/N

type

3. #14521074

C₂

semidirect

4. #14521073

C₂

semidirect

5. #14521072

C₂

semidirect

6. C₃₁×#423

C₂

semidirect

19. C₃₁×C₄⋊C₈

C₂²

semidirect

20. Dic₃₁⋊C₈

C₂²

semidirect

21. C₄⋊(C₃₁⋊C₈)

C₂²

semidirect

80. #423

D₃₁

semidirect

99. C₄⋊C₈

C₂×D₃₁

semidirect

100. C₃₁

#2905

semidirect

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

subgroup	index
C₂×M₁₆×D₃₁	2
C₂×#11758449	2
#14521074	2
#14521073	2
#14521072	2
C₃₁×#423	2
#2905	31

subgroup

index

C₂×M₁₆×D₃₁

C₃₁×#423

Subgroups — #16854273

Conjugacy classes of subgroups

Complements

Maximal subgroups