Group #16845113 — subgroups

Subgroups — #16845113

Order 3968.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14521452	C₂	1	1	yes	maximal, characteristic
2	#14521344	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14520801	C₂	1	1	yes	maximal, characteristic
5	#14520798	C₂	1	1	yes	maximal, characteristic
6	#14520797	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#477	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	#11758446	C₂²	1	1	yes	characteristic
11	C₃₁×#107	—	2	1	no	—
12	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃₁×#103	—	2	1	no	—
14	#11758423	—	2	1	no	—
15	#11758395	—	2	1	no	—
16	#11758395	—	2	1	no	—
17	#11758395	—	2	1	no	—
18	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
19	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄×C₃₁⋊C₈	—	2	1	no	—
22	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₁	—	2	1	no	—
24	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
25	C₂×Dic₆₂	—	2	1	no	—
26	C₂×C₈×C₃₁	—	2	1	no	—
27	C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₄²×C₃₁	—	2	1	no	—
32	C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₃₁	—	2	1	no	—
34	C₄⋊Dic₃₁	—	2	1	no	—
35	C₂×C₃₁⋊C₈	—	2	1	no	—
36	C₂×C₃₁⋊C₈	—	2	1	no	—
37	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
39	Q₈×C₃₁	—	2	1	no	—
40	Q₈×C₃₁	—	8	1	no	—
41	Q₈×C₃₁	Q₁₆	1	2	yes	—
42	C₂×C₄×C₃₁	—	2	1	no	—
43	C₂×C₄×C₃₁	—	2	1	no	—
44	C₂×C₄×C₃₁	—	4	1	no	—
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
48	C₂×Dic₃₁	—	4	1	no	—
49	Dic₆₂	—	8	1	no	—
50	C₈×C₃₁	—	4	1	no	—
51	C₃₁⋊C₈	—	2	1	no	—
52	C₃₁⋊C₈	—	4	1	no	—
53	C₃₁⋊C₈	—	4	1	no	—
54	C₃₁⋊C₈	Q₁₆	1	2	yes	—
55	#1676	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	#121	1	1	yes	characteristic
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	2	1	no	—
59	C₄×C₃₁	—	4	1	no	—
60	C₄×C₃₁	—	8	1	no	—
61	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₃₁	#135	1	1	yes	characteristic
64	C₄×C₃₁	#137	1	1	yes	characteristic
65	Dic₃₁	—	8	1	no	—
66	#500	—	31	1	no	—
67	#477	—	31	1	no	—
68	#477	D₃₁	1	1	yes	characteristic
69	C₈×Q₈	—	31	1	no	—
70	#335	—	31	1	no	—
71	#333	—	31	1	no	—
72	#333	—	31	1	no	—
73	C₂×C₃₁	#462	1	1	yes	characteristic
74	C₂×C₃₁	#467	1	1	yes	characteristic
75	C₂×C₃₁	#472	1	1	yes	characteristic
76	C₄⋊Q₈	—	31	1	no	—
77	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
78	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
79	#107	—	2	1	no	—
80	#107	—	62	1	no	—
81	#107	—	62	1	no	—
82	#107	—	62	1	no	—
83	C₄⋊C₈	—	31	1	no	—
84	C₄⋊C₈	—	31	1	no	—
85	C₄⋊C₈	—	62	1	no	—
86	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
87	#103	—	2	1	no	—
88	#103	—	62	1	no	—
89	C₄×C₈	—	31	1	no	—
90	C₄×C₈	—	62	1	no	—
91	C₃₁	#1676	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	62	1	no	—
94	C₂×Q₈	C₂²⋊D₃₁	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	31	1	no	—
97	C₂×C₈	—	31	1	no	—
98	C₂×C₈	—	62	1	no	—
99	C₂×C₈	—	62	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	62	1	no	—
104	C₄⋊C₄	—	62	1	no	—
105	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	C₂²×D₃₁	1	1	yes	characteristic
108	Q₈	—	2	1	no	—
109	Q₈	—	8	1	no	—
110	Q₈	—	248	1	no	—
111	Q₈	#92482	1	2	yes	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	124	1	no	—
116	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
117	C₂×C₄	#92497	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	31	2	no	—
121	C₈	—	62	1	no	—
122	C₈	—	124	1	no	—
123	C₈	—	124	1	no	—
124	C₂²	#11758516	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	4	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	248	1	no	—
130	C₄	Q₁₆×D₃₁	1	1	yes	characteristic
131	C₄	#11758495	1	1	yes	characteristic
132	C₄	C₂×#92482	1	1	yes	characteristic
133	C₄	#11758527	1	1	yes	characteristic
134	C₂	#14521257	1	1	yes	characteristic
135	C₂	#14521288	1	1	yes	characteristic
136	C₂	#14521317	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521452

C₂

yes

maximal, characteristic

#14521344

C₂

yes

maximal, characteristic

Q₈×C₃₁⋊C₈

C₂

yes

maximal, characteristic

#14520801

C₂

yes

maximal, characteristic

#14520798

C₂

yes

maximal, characteristic

#14520797

C₂

yes

maximal, characteristic

C₃₁×#477

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₄×Q₈×C₃₁

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₃₁×#107

—

C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₃₁×#103

—

—

—

—

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×Dic₆₂

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

D₄

yes

characteristic

C₂×C₃₁⋊C₈

D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

Q₁₆

yes

—

—

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×Q₁₆

yes

characteristic

C₄×C₃₁

C₂×Q₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

#500

—

#477

—

#477

D₃₁

yes

characteristic

C₈×Q₈

—

#335

—

#333

—

#333

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

C₄×Q₈

C₂×D₃₁

yes

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₃₁

#1676

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

C₂²⋊D₃₁

yes

characteristic

106

C₄²

—

107

C₄²

C₂²×D₃₁

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

248

—

111

Q₈

#92482

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

124

—

116

C₂×C₄

D₄×D₃₁

yes

characteristic

117

C₂×C₄

#92497

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

124

—

123

C₈

—

124

—

124

C₂²

#11758516

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

248

—

130

C₄

Q₁₆×D₃₁

yes

characteristic

131

C₄

#11758495

yes

characteristic

132

C₄

C₂×#92482

yes

characteristic

133

C₄

#11758527

yes

characteristic

134

C₂

#14521257

yes

characteristic

135

C₂

#14521288

yes

characteristic

136

C₂

#14521317

yes

characteristic

	subgroup	G/H
center	C₂²	#11758516
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#477	C₂
socle, prime radical	C₂²×C₃₁	#121
hypercenter	#477	D₃₁
hypocenter	C₃₁	#1676
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#11758516

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#477

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
91. C₃₁	#1676	semidirect

H ≅ G/N

type

91. C₃₁

#1676

semidirect

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

subgroup	index
#14521452	2
#14521344	2
Q₈×C₃₁⋊C₈	2
#14520801	2
#14520798	2
#14520797	2
C₃₁×#477	2
#1676	31

subgroup

index

Q₈×C₃₁⋊C₈

C₃₁×#477

Subgroups — #16845113

Conjugacy classes of subgroups

Complements

Maximal subgroups