Group #16846492 — subgroups

Subgroups — #16846492

Order 3968.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14521091	C₂	1	2	yes	maximal
2	C₂×#11758405	C₂	1	1	yes	maximal, characteristic
3	#14521004	C₂	1	1	yes	maximal, characteristic
4	#14520817	C₂	1	2	yes	maximal
5	C₂×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	#11758503	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
10	C₂×#92474	C₂²	1	1	yes	characteristic
11	C₃₁×#108	C₄	1	4	yes	complemented
12	#11758406	—	2	2	no	—
13	#11758405	—	2	1	no	—
14	#11758405	—	2	1	no	—
15	#11758405	—	2	2	no	—
16	#11758403	—	2	1	no	—
17	C₈×Dic₃₁	—	2	1	no	—
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₁	—	2	2	no	—
20	M₁₆×C₃₁	—	2	2	no	—
21	M₁₆×C₃₁	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₃₁	—	2	2	no	—
23	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₃₁	Q₈	1	2	yes	—
27	C₂²⋊Dic₃₁	—	2	1	no	—
28	C₄⋊Dic₃₁	—	2	2	no	—
29	C₄⋊Dic₃₁	—	2	2	no	—
30	C₄⋊Dic₃₁	D₄	1	2	yes	—
31	C₄×Dic₃₁	—	2	1	no	—
32	#92474	—	2	1	no	—
33	#92474	Q₈	1	2	yes	—
34	C₂×C₃₁⋊C₈	—	2	1	no	—
35	C₂³×C₃₁	#41	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
42	C₂×Dic₃₁	—	2	2	no	—
43	C₂×Dic₃₁	—	4	2	no	—
44	C₂×Dic₃₁	—	8	2	no	—
45	C₈×C₃₁	—	2	4	no	—
46	C₈×C₃₁	C₄⋊C₄	1	2	yes	—
47	C₈×C₃₁	C₄⋊C₄	1	2	yes	—
48	C₃₁⋊C₈	—	2	2	no	—
49	#1951	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	—	2	2	no	—
51	C₂²×C₃₁	C₄×Q₈	1	1	yes	characteristic
52	C₂²×C₃₁	#121	1	1	yes	characteristic
53	C₂²×C₃₁	#125	1	1	yes	characteristic
54	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
56	C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₃₁	—	4	2	no	—
59	Dic₃₁	—	8	2	no	—
60	C₂×#108	D₃₁	1	1	yes	characteristic
61	#428	—	31	2	no	—
62	C₂×#106	—	31	1	no	—
63	#405	—	31	1	no	—
64	#343	—	31	2	no	—
65	C₂×C₃₁	—	2	1	no	—
66	C₂×C₃₁	—	2	1	no	—
67	C₂×C₃₁	#389	1	1	yes	characteristic
68	C₂×C₃₁	#471	1	2	yes	—
69	C₂×M₁₆	—	31	1	no	—
70	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
71	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
72	#117	—	31	1	no	—
73	C₂×C₄⋊C₄	—	31	2	no	—
74	#108	Dic₃₁	1	4	yes	complemented
75	#107	—	62	2	no	—
76	#106	—	62	2	no	—
77	#106	—	62	1	no	—
78	#106	—	62	1	no	—
79	#97	—	62	1	no	—
80	C₄×C₈	—	62	1	no	—
81	C₃₁	#1951	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	62	2	no	—
83	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	31	2	no	—
86	M₁₆	—	62	1	no	—
87	M₁₆	C₂×Dic₃₁	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	62	1	no	—
90	C₂×C₈	Dic₆₂	1	2	yes	—
91	C₂×C₈	D₁₂₄	1	1	yes	characteristic
92	C₂×C₈	D₁₂₄	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
94	C₄⋊C₄	—	31	2	no	—
95	C₄⋊C₄	—	62	2	no	—
96	C₄⋊C₄	—	62	2	no	—
97	C₂²⋊C₄	—	62	1	no	—
98	C₄²	—	62	1	no	—
99	C₂³	#92499	1	1	yes	characteristic
100	C₂×C₄	—	62	2	no	—
101	C₂×C₄	—	124	2	no	—
102	C₂×C₄	—	248	2	no	—
103	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
104	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
105	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
106	C₂×C₄	#92497	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	62	2	no	—
111	C₈	C₄⋊Dic₃₁	1	2	yes	—
112	C₈	C₄⋊Dic₃₁	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	#11758467	1	1	yes	characteristic
115	C₂²	#11758472	1	1	yes	characteristic
116	C₂²	Q₈×Dic₃₁	1	1	yes	characteristic
117	C₄	—	124	2	no	—
118	C₄	—	248	2	no	—
119	C₄	#11758465	1	1	yes	characteristic
120	C₄	#11758474	1	1	yes	characteristic
121	C₄	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
122	C₄	D₄×Dic₃₁	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#14520948	1	1	yes	characteristic
126	C₂	#14521313	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521091

C₂

yes

maximal

C₂×#11758405

C₂

yes

maximal, characteristic

#14521004

C₂

yes

maximal, characteristic

#14520817

C₂

yes

maximal

C₂×C₃₁×#108

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×#92474

C₂²

yes

characteristic

C₃₁×#108

C₄

yes

—

—

—

—

—

C₈×Dic₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

Q₈

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

D₄

yes

—

C₄×Dic₃₁

—

#92474

—

#92474

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

#41

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₄⋊C₄

yes

—

C₈×C₃₁

C₄⋊C₄

yes

—

C₃₁⋊C₈

—

#1951

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×Q₈

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

characteristic

C₄×C₃₁

C₂²⋊Q₈

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂×#108

D₃₁

yes

—

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#389

yes

characteristic

C₂×C₃₁

#471

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#108

Dic₃₁

yes

—

—

—

—

—

C₄×C₈

—

C₃₁

#1951

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×Dic₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

D₁₂₄

yes

characteristic

C₂×C₈

D₁₂₄

yes

characteristic

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

#92499

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

124

—

102

C₂×C₄

—

248

—

103

C₂×C₄

C₂×Dic₆₂

yes

characteristic

104

C₂×C₄

C₂×D₁₂₄

yes

characteristic

105

C₂×C₄

D₄×D₃₁

yes

characteristic

106

C₂×C₄

#92497

yes

characteristic

107

C₂×C₄

Q₈×D₃₁

yes

characteristic

108

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

C₄⋊Dic₃₁

yes

—

112

C₈

C₄⋊Dic₃₁

yes

—

113

C₂²

—

114

C₂²

#11758467

yes

characteristic

115

C₂²

#11758472

yes

characteristic

116

C₂²

Q₈×Dic₃₁

yes

characteristic

117

C₄

—

124

—

118

C₄

—

248

—

119

C₄

#11758465

yes

characteristic

120

C₄

#11758474

yes

characteristic

121

C₄

C₂×C₄⋊Dic₃₁

yes

characteristic

122

C₄

D₄×Dic₃₁

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#14520948

yes

characteristic

126

C₂

#14521313

yes

—

	subgroup	G/H
center	C₂²	#11758472
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#108	C₂
socle	C₂²×C₃₁	#121
prime radical	C₂³×C₃₁	#41
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#108	D₃₁
hypocenter	C₃₁	#1951
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#11758472

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#108

C₂

C₂²×C₃₁

C₂³×C₃₁

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. C₃₁×#108	C₄	semidirect
74. #108	Dic₃₁	semidirect
81. C₃₁	#1951	semidirect

H ≅ G/N

type

11. C₃₁×#108

C₄

semidirect

74. #108

Dic₃₁

semidirect

81. C₃₁

#1951

semidirect

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

subgroup	index
#14521091	2
C₂×#11758405	2
#14521004	2
#14520817	2
C₂×C₃₁×#108	2
#1951	31

subgroup

index

C₂×C₃₁×#108

Subgroups — #16846492

Conjugacy classes of subgroups

Complements

Maximal subgroups