Group #16845169 — subgroups

Subgroups — #16845169

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	#14521400	C₂	1	1	yes	maximal, characteristic
2	#14521344	C₂	1	1	yes	maximal, characteristic
3	#14521334	C₂	1	1	yes	maximal, characteristic
4	#14520802	C₂	1	1	yes	maximal, characteristic
5	#14520797	C₂	1	1	yes	maximal, characteristic
6	#14520796	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	#11758446	C₂²	1	1	yes	characteristic
10	#11758446	C₂²	1	1	yes	characteristic
11	C₃₁×C₄⋊C₈	—	2	1	no	—
12	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₃₁	—	2	1	no	—
14	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	#11758423	—	2	1	no	—
16	#11758423	—	2	1	no	—
17	#11758404	—	2	1	no	—
18	#11758404	—	2	1	no	—
19	#11758396	—	2	1	no	—
20	#11758395	—	2	1	no	—
21	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂×Dic₆₂	—	2	1	no	—
25	C₂×Dic₆₂	—	2	1	no	—
26	C₂×C₈×C₃₁	—	2	1	no	—
27	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
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	C₄⋊Dic₃₁	—	2	1	no	—
36	C₄⋊Dic₃₁	—	2	1	no	—
37	C₂×C₃₁⋊C₈	—	2	1	no	—
38	C₂×C₃₁⋊C₈	—	2	1	no	—
39	Q₈×C₃₁	—	2	1	no	—
40	Q₈×C₃₁	Q₁₆	1	2	yes	—
41	C₂×C₄×C₃₁	—	2	1	no	—
42	C₂×C₄×C₃₁	—	2	1	no	—
43	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
46	C₂×Dic₃₁	—	4	1	no	—
47	C₂×Dic₃₁	—	4	1	no	—
48	Dic₆₂	—	4	1	no	—
49	Dic₆₂	—	4	1	no	—
50	Dic₆₂	—	8	1	no	—
51	C₈×C₃₁	—	2	1	no	—
52	C₈×C₃₁	—	2	1	no	—
53	C₈×C₃₁	—	4	1	no	—
54	C₃₁⋊C₈	—	4	1	no	—
55	C₃₁⋊C₈	—	4	1	no	—
56	#1684	—	31	1	no	maximal, Sylow
57	C₂²×C₃₁	#121	1	1	yes	characteristic
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	—	2	1	no	—
60	C₄×C₃₁	—	4	1	no	—
61	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₃₁	#135	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	Dic₃₁	—	8	1	no	—
67	#487	—	31	1	no	—
68	#477	—	31	1	no	—
69	#475	—	31	1	no	—
70	C₈×Q₈	D₃₁	1	1	yes	characteristic
71	#335	—	31	1	no	—
72	#333	—	31	1	no	—
73	#332	—	31	1	no	—
74	C₂×C₃₁	#464	1	1	yes	characteristic
75	C₂×C₃₁	#467	1	1	yes	characteristic
76	C₂×C₃₁	#471	1	1	yes	characteristic
77	C₄⋊Q₈	—	31	1	no	—
78	C₄⋊Q₈	—	31	1	no	—
79	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
80	#107	—	62	1	no	—
81	#106	—	62	1	no	—
82	C₄⋊C₈	—	2	1	no	—
83	C₄⋊C₈	—	31	1	no	—
84	C₄⋊C₈	—	31	1	no	—
85	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
86	#103	—	62	1	no	—
87	#103	—	62	1	no	—
88	#103	—	62	1	no	—
89	#103	—	62	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
92	C₃₁	#1684	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₈	—	62	1	no	—
94	C₂×Q₈	—	62	1	no	—
95	C₂×Q₈	D₁₂₄	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	62	1	no	—
98	C₂×C₈	—	62	1	no	—
99	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
100	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	62	1	no	—
103	C₄⋊C₄	—	62	1	no	—
104	C₄⋊C₄	—	62	1	no	—
105	C₄⋊C₄	—	62	1	no	—
106	C₄⋊C₄	D₁₂₄	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₃₁	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	124	1	no	—
111	Q₈	—	124	1	no	—
112	Q₈	—	248	1	no	—
113	Q₈	Dic₁₂₄	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	124	1	no	—
117	C₂×C₄	—	124	1	no	—
118	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
119	C₂×C₄	#92495	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	124	1	no	—
125	C₈	—	124	1	no	—
126	C₂²	#11758507	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	248	1	no	—
131	C₄	—	248	1	no	—
132	C₄	#11758481	1	1	yes	characteristic
133	C₄	C₂×Dic₁₂₄	1	1	yes	characteristic
134	C₄	#11758519	1	1	yes	characteristic
135	C₄	#11758527	1	1	yes	characteristic
136	C₂	#14521271	1	1	yes	characteristic
137	C₂	#14521290	1	1	yes	characteristic
138	C₂	#14521312	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521400

C₂

yes

maximal, characteristic

#14521344

C₂

yes

maximal, characteristic

#14521334

C₂

yes

maximal, characteristic

#14520802

C₂

yes

maximal, characteristic

#14520797

C₂

yes

maximal, characteristic

#14520796

C₂

yes

maximal, characteristic

C₈×Q₈×C₃₁

C₂

yes

maximal, characteristic

C₄×Q₈×C₃₁

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

C₂²

yes

—

—

—

—

—

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

Q₈×C₃₁

—

Q₈×C₃₁

Q₁₆

yes

—

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

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×Q₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

#487

—

#477

—

#475

—

C₈×Q₈

D₃₁

yes

—

—

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₃₁

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1684

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₂₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

100

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

D₁₂₄

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₃₁

yes

characteristic

109

Q₈

—

110

Q₈

—

124

—

111

Q₈

—

124

—

112

Q₈

—

248

—

113

Q₈

Dic₁₂₄

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

124

—

117

C₂×C₄

—

124

—

118

C₂×C₄

C₂×D₁₂₄

yes

characteristic

119

C₂×C₄

#92495

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

124

—

125

C₈

—

124

—

126

C₂²

#11758507

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

248

—

131

C₄

—

248

—

132

C₄

#11758481

yes

characteristic

133

C₄

C₂×Dic₁₂₄

yes

characteristic

134

C₄

#11758519

yes

characteristic

135

C₄

#11758527

yes

characteristic

136

C₂

#14521271

yes

characteristic

137

C₂

#14521290

yes

characteristic

138

C₂

#14521312

yes

characteristic

	subgroup	G/H
center	C₂²	#11758507
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₈×Q₈×C₃₁	C₂
socle, prime radical	C₂²×C₃₁	#121
hypercenter	C₈×Q₈	D₃₁
hypocenter	C₃₁	#1684
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#11758507

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₈×Q₈×C₃₁

C₂

socle, prime radical

C₂²×C₃₁

#121

hypercenter

C₈×Q₈

D₃₁

hypocenter

C₃₁

#1684

LCS

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
92. C₃₁	#1684	semidirect

H ≅ G/N

type

92. C₃₁

#1684

semidirect

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

subgroup	index
#14521400	2
#14521344	2
#14521334	2
#14520802	2
#14520797	2
#14520796	2
C₈×Q₈×C₃₁	2
#1684	31

subgroup

index

C₈×Q₈×C₃₁

Subgroups — #16845169

Conjugacy classes of subgroups

Complements

Maximal subgroups