Group #16845341 — subgroups

Subgroups — #16845341

Order 3968.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14521445	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521425	C₂	1	1	yes	maximal, characteristic
3	#14521423	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521422	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521395	C₂	1	1	yes	maximal, characteristic, complemented
6	#14520771	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₂×#92481	—	2	2	no	—
11	C₂×#92479	—	2	2	no	—
12	C₂×#92470	—	2	2	no	—
13	#11758451	C₂²	1	1	yes	characteristic
14	#11758446	C₂²	1	1	yes	characteristic
15	C₃₁×#102	—	2	2	no	—
16	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	#11758420	—	2	2	no	—
18	#11758396	—	2	2	no	—
19	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₁	—	2	1	no	—
22	C₂×D₄×C₃₁	—	2	1	no	—
23	C₂×D₄×C₃₁	—	2	1	no	—
24	C₂×D₄×C₃₁	—	2	1	no	—
25	C₂×D₁₂₄	—	2	1	no	—
26	C₂×D₁₂₄	—	2	1	no	—
27	C₂×D₁₂₄	—	2	1	no	—
28	C₂×Dic₆₂	—	2	1	no	—
29	C₂×C₈×C₃₁	D₄	1	2	yes	—
30	C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	#92481	—	4	4	no	—
34	#92479	—	4	4	no	—
35	C₄⋊Dic₃₁	—	2	1	no	—
36	C₄⋊Dic₃₁	—	2	1	no	—
37	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
38	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
39	#92470	—	4	4	no	—
40	C₂³×C₃₁	—	4	1	no	—
41	C₂²×D₃₁	—	4	1	no	—
42	Q₈×C₃₁	—	4	2	no	—
43	D₄×C₃₁	—	4	2	no	—
44	D₄×C₃₁	—	4	2	no	—
45	D₄×C₃₁	—	4	2	no	—
46	C₂×C₄×C₃₁	—	4	1	no	—
47	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₃₁	—	4	1	no	—
51	D₁₂₄	—	4	2	no	—
52	D₁₂₄	—	4	2	no	—
53	D₁₂₄	—	4	2	no	—
54	Dic₆₂	—	4	2	no	—
55	C₈×C₃₁	—	2	2	no	—
56	C₃₁⋊C₈	—	2	2	no	—
57	C₃₁⋊C₈	SD₁₆	1	4	yes	complemented
58	#1715	—	31	1	no	maximal, Sylow
59	C₂²×C₃₁	—	8	1	no	—
60	C₂²×C₃₁	—	8	1	no	—
61	C₂²×C₃₁	—	8	1	no	—
62	C₂²×C₃₁	#127	1	1	yes	characteristic
63	C₂×D₃₁	—	8	1	no	—
64	C₂×D₃₁	—	8	1	no	—
65	C₂×D₃₁	—	8	1	no	—
66	C₄×C₃₁	—	8	1	no	—
67	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
68	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
69	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
70	Dic₃₁	—	8	1	no	—
71	#498	—	31	1	no	—
72	#493	—	31	1	no	—
73	#492	—	31	1	no	—
74	#492	—	31	1	no	—
75	#486	—	31	1	no	—
76	#486	D₃₁	1	1	yes	characteristic, complemented
77	#322	—	31	1	no	—
78	C₂×C₃₁	—	16	1	no	—
79	C₂×C₃₁	#492	1	1	yes	characteristic
80	C₂×C₃₁	#496	1	1	yes	characteristic
81	C₂×C₃₁	#496	1	1	yes	characteristic
82	D₃₁	—	16	1	no	—
83	C₂×SD₁₆	—	62	2	no	—
84	C₂×SD₁₆	—	62	2	no	—
85	C₂×D₈	—	62	2	no	—
86	C₄⋊Q₈	—	31	1	no	—
87	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
88	#127	—	31	1	no	—
89	#127	C₂×D₃₁	1	1	yes	characteristic
90	#106	—	62	2	no	—
91	#102	—	2	2	no	—
92	#102	—	62	2	no	—
93	C₄×C₈	—	31	1	no	—
94	C₄×C₈	—	31	1	no	—
95	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
96	C₃₁	#1715	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	62	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	62	1	no	—
103	C₂×D₄	—	62	1	no	—
104	C₂×D₄	—	62	1	no	—
105	SD₁₆	—	124	4	no	—
106	SD₁₆	—	124	4	no	—
107	D₈	—	124	4	no	—
108	C₂×C₈	—	31	2	no	—
109	C₂×C₈	—	31	2	no	—
110	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	62	1	no	—
114	C₄⋊C₄	—	62	1	no	—
115	C₄²	C₂²×D₃₁	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	124	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	124	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	124	2	no	—
124	D₄	—	124	2	no	—
125	D₄	—	124	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	124	1	no	—
128	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	31	4	no	—
133	C₈	—	62	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	248	1	no	—
138	C₂²	—	248	1	no	—
139	C₂²	—	248	1	no	—
140	C₂²	#11758517	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	248	1	no	—
143	C₄	#11758488	1	2	yes	—
144	C₄	SD₁₆×D₃₁	1	2	yes	—
145	C₄	#11758526	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	496	1	no	—
148	C₂	#14521420	1	1	yes	characteristic
149	C₂	#14521437	1	1	yes	characteristic
150	C₂	#14521439	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521445

C₂

yes

maximal, characteristic, complemented

#14521425

C₂

yes

maximal, characteristic

#14521423

C₂

yes

maximal, characteristic, complemented

#14521422

C₂

yes

maximal, characteristic, complemented

#14521395

C₂

yes

maximal, characteristic, complemented

#14520771

C₂

yes

maximal, characteristic, complemented

C₃₁×#486

C₂

yes

maximal, characteristic, complemented

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₃₁×#127

C₂²

yes

—

—

—

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₃₁×#102

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758420

—

#11758396

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92481

—

#92479

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

#92470

—

C₂³×C₃₁

—

C₂²×D₃₁

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

SD₁₆

yes

complemented

#1715

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#127

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×SD₁₆

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

Dic₃₁

—

—

—

—

—

—

D₃₁

yes

characteristic, complemented

#322

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

D₃₁

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

#127

—

#127

C₂×D₃₁

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1715

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

SD₁₆

—

124

—

106

SD₁₆

—

124

—

107

D₈

—

124

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂²⋊D₃₁

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

C₂²×D₃₁

yes

characteristic

116

C₂³

—

117

C₂³

—

124

—

118

Q₈

—

119

Q₈

—

124

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

—

124

D₄

—

124

—

125

D₄

—

124

—

126

C₂×C₄

—

127

C₂×C₄

—

124

—

128

C₂×C₄

D₄×D₃₁

yes

characteristic

129

C₂×C₄

D₄×D₃₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

248

—

138

C₂²

—

248

—

139

C₂²

—

248

—

140

C₂²

#11758517

yes

characteristic

141

C₄

—

142

C₄

—

248

—

143

C₄

#11758488

yes

—

144

C₄

SD₁₆×D₃₁

yes

—

145

C₄

#11758526

yes

—

146

C₂

—

147

C₂

—

496

—

148

C₂

#14521420

yes

characteristic

149

C₂

#14521437

yes

characteristic

150

C₂

#14521439

yes

characteristic

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#486	C₂
socle	C₂²×C₃₁	#127
prime radical	#14521425	C₂
hypercenter	#486	D₃₁
hypocenter	C₃₁	#1715
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#11758517

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#486

C₂

C₂²×C₃₁

C₂

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #14521445	C₂	semidirect
3. #14521423	C₂	semidirect
4. #14521422	C₂	semidirect
5. #14521395	C₂	semidirect
6. #14520771	C₂	semidirect
7. C₃₁×#486	C₂	semidirect
57. C₃₁⋊C₈	SD₁₆	semidirect
76. #486	D₃₁	semidirect
96. C₃₁	#1715	semidirect

H ≅ G/N

type

1. #14521445

C₂

semidirect

3. #14521423

C₂

semidirect

4. #14521422

C₂

semidirect

5. #14521395

C₂

semidirect

6. #14520771

C₂

semidirect

7. C₃₁×#486

C₂

semidirect

57. C₃₁⋊C₈

SD₁₆

semidirect

76. #486

D₃₁

semidirect

96. C₃₁

#1715

semidirect

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

subgroup	index
#14521445	2
#14521425	2
#14521423	2
#14521422	2
#14521395	2
#14520771	2
C₃₁×#486	2
#1715	31

subgroup

index

C₃₁×#486

Subgroups — #16845341

Conjugacy classes of subgroups

Complements

Maximal subgroups