Group #16847253 — subgroups

Subgroups — #16847253

Order 3968.

254 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 95 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×#11758405	C₂	1	2	yes	maximal
2	C₂×#11758404	C₂	1	2	yes	maximal
3	#14520955	C₂	1	1	yes	maximal, characteristic
4	#14520919	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	—	2	2	no	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
12	C₃₁⋊#95	—	2	2	no	—
13	#11758405	—	2	4	no	—
14	#11758404	—	2	4	no	—
15	C₂²×C₄×C₃₁	D₄	1	2	yes	—
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₁	—	2	1	no	—
18	C₂²×Dic₃₁	—	2	1	no	—
19	C₂²×Dic₃₁	—	2	2	no	—
20	C₂×Dic₆₂	—	2	2	no	—
21	C₂×Dic₆₂	—	2	2	no	—
22	C₂×Dic₆₂	—	2	1	no	—
23	C₂×Dic₆₂	Q₈	1	2	yes	—
24	C₂×C₈×C₃₁	—	2	2	no	—
25	C₂×C₈×C₃₁	—	2	2	no	—
26	C₂×C₈×C₃₁	D₄	1	4	yes	—
27	C₄²×C₃₁	—	2	2	no	—
28	C₄⋊Dic₃₁	—	2	2	no	—
29	C₄⋊Dic₃₁	—	2	2	no	—
30	C₄⋊Dic₃₁	—	2	2	no	—
31	C₄⋊Dic₃₁	—	4	4	no	—
32	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	—	2	2	no	—
35	C₂×C₄×C₃₁	SD₁₆	1	4	yes	—
36	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₃₁	#41	1	2	yes	—
39	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
40	C₂×Dic₃₁	—	2	2	no	—
41	C₂×Dic₃₁	—	4	1	no	—
42	C₂×Dic₃₁	—	4	1	no	—
43	C₂×Dic₃₁	—	4	2	no	—
44	C₂×Dic₃₁	—	4	1	no	—
45	C₂×Dic₃₁	—	4	4	no	—
46	C₂×Dic₃₁	—	4	2	no	—
47	Dic₆₂	—	2	4	no	—
48	Dic₆₂	—	4	2	no	—
49	C₈×C₃₁	—	2	4	no	—
50	#2096	—	31	1	no	maximal, Sylow
51	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₃₁	#123	1	1	yes	characteristic
53	C₂²×C₃₁	#127	1	1	yes	characteristic
54	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
55	C₂²×C₃₁	#137	1	2	yes	—
56	C₄×C₃₁	—	2	4	no	—
57	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₃₁	#123	1	2	yes	—
59	Dic₃₁	—	4	2	no	—
60	Dic₃₁	—	8	2	no	—
61	Dic₃₁	—	8	1	no	—
62	C₂×#106	—	31	2	no	—
63	C₂×#103	—	31	2	no	—
64	#391	—	31	1	no	—
65	#384	—	31	1	no	—
66	#336	D₃₁	1	1	yes	characteristic
67	C₂×C₃₁	#399	1	1	yes	characteristic
68	C₂×C₃₁	#475	1	2	yes	—
69	C₂×C₃₁	#483	1	2	yes	—
70	C₂×C₃₁	#492	1	1	yes	characteristic
71	C₂×C₃₁	#497	1	1	yes	characteristic
72	C₂²×Q₈	—	31	1	no	—
73	C₂²×C₈	C₂×D₃₁	1	2	yes	—
74	C₂×C₄⋊C₄	—	31	2	no	—
75	C₂×C₄⋊C₄	—	31	1	no	—
76	C₂×C₄⋊C₄	—	62	2	no	—
77	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
78	#106	—	62	4	no	—
79	#103	—	62	4	no	—
80	#95	—	62	2	no	—
81	C₃₁	#2096	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	31	2	no	—
83	C₂×Q₈	—	62	1	no	—
84	C₂×Q₈	—	62	2	no	—
85	C₂×Q₈	—	62	2	no	—
86	C₂²×C₄	—	62	1	no	—
87	C₂²×C₄	—	62	1	no	—
88	C₂²×C₄	—	62	2	no	—
89	C₂²×C₄	D₁₂₄	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₁₂₄	1	4	yes	—
94	C₄⋊C₄	—	62	2	no	—
95	C₄⋊C₄	—	62	2	no	—
96	C₄⋊C₄	—	62	2	no	—
97	C₄⋊C₄	—	124	4	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
100	Q₈	—	62	4	no	—
101	Q₈	—	124	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	62	2	no	—
105	C₂×C₄	—	124	1	no	—
106	C₂×C₄	—	124	1	no	—
107	C₂×C₄	—	124	2	no	—
108	C₂×C₄	—	124	1	no	—
109	C₂×C₄	—	124	4	no	—
110	C₂×C₄	—	124	2	no	—
111	C₂×C₄	#92470	1	4	yes	—
112	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
113	C₂×C₄	#92497	1	1	yes	characteristic
114	C₂×C₄	#92497	1	2	yes	—
115	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₂²	#11758451	1	1	yes	characteristic
118	C₂²	#11758463	1	1	yes	characteristic
119	C₂²	#11758474	1	1	yes	characteristic
120	C₂²	C₂×#92470	1	2	yes	—
121	C₂²	#11758486	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	124	2	no	—
124	C₄	—	248	2	no	—
125	C₄	—	248	1	no	—
126	C₄	#11758463	1	2	yes	—
127	C₄	#11758474	1	2	yes	—
128	C₂	#14520986	1	1	yes	characteristic
129	C₂	#14521335	1	2	yes	—
130	C₂	#14521380	1	2	yes	—
131	C₂	#14521422	1	1	yes	characteristic
132	C₂	#14521443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758405

C₂

yes

maximal

C₂×#11758404

C₂

yes

maximal

#14520955

C₂

yes

maximal, characteristic

#14520919

C₂

yes

maximal, characteristic

C₃₁×#336

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₃₁⋊#95

—

#11758405

—

#11758404

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

Q₈

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

#2096

—

maximal, Sylow

C₂²×C₃₁

C₂²⋊Q₈

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

C₂×SD₁₆

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

#123

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#483

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

—

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₃₁

#2096

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄²

—

C₂³

C₂×D₁₂₄

yes

characteristic

100

Q₈

—

101

Q₈

—

124

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

124

—

106

C₂×C₄

—

124

—

107

C₂×C₄

—

124

—

108

C₂×C₄

—

124

—

109

C₂×C₄

—

124

—

110

C₂×C₄

—

124

—

111

C₂×C₄

#92470

yes

—

112

C₂×C₄

C₂×D₁₂₄

yes

—

113

C₂×C₄

#92497

yes

characteristic

114

C₂×C₄

#92497

yes

—

115

C₂×C₄

Q₈×D₃₁

yes

characteristic

116

C₈

—

117

C₂²

#11758451

yes

characteristic

118

C₂²

#11758463

yes

characteristic

119

C₂²

#11758474

yes

characteristic

120

C₂²

C₂×#92470

yes

—

121

C₂²

#11758486

yes

—

122

C₄

—

123

C₄

—

124

—

124

C₄

—

248

—

125

C₄

—

248

—

126

C₄

#11758463

yes

—

127

C₄

#11758474

yes

—

128

C₂

#14520986

yes

characteristic

129

C₂

#14521335

yes

—

130

C₂

#14521380

yes

—

131

C₂

#14521422

yes

characteristic

132

C₂

#14521443

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#336	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#336	D₃₁
hypocenter	C₃₁	#2096
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#336

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
81. C₃₁	#2096	semidirect

H ≅ G/N

type

81. C₃₁

#2096

semidirect

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

subgroup	index
C₂×#11758405	2
C₂×#11758404	2
#14520955	2
#14520919	2
C₃₁×#336	2
#2096	31

subgroup

index

C₃₁×#336

Subgroups — #16847253

Conjugacy classes of subgroups

Complements

Maximal subgroups