Group #16845748 — subgroups

Subgroups — #16845748

Order 3968.

292 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758410	C₂	1	2	yes	maximal, complemented
3	#14521012	C₂	1	2	yes	maximal
4	#14520814	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×#92474	C₂²	1	2	yes	—
10	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₃₁	—	2	4	no	—
12	C₂×C₈×D₃₁	C₄	1	4	yes	—
13	#11758410	—	2	4	no	—
14	(C₂×D₃₁)⋊C₈	—	2	4	no	—
15	C₂³×D₃₁	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₁	Q₈	1	1	yes	characteristic
18	C₂×C₄×D₃₁	—	2	2	no	—
19	C₂×C₄×D₃₁	—	2	2	no	—
20	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₃₁	D₄	1	2	yes	—
22	M₁₆×C₃₁	—	2	2	no	—
23	M₁₆×C₃₁	—	2	2	no	—
24	C₂×C₈×C₃₁	—	2	2	no	—
25	C₂×C₈×C₃₁	—	2	2	no	—
26	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
27	#92474	—	2	2	no	—
28	#92474	—	2	2	no	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
32	C₈×D₃₁	—	2	8	no	—
33	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
34	C₂²×D₃₁	—	2	2	no	—
35	C₂²×D₃₁	—	2	4	no	—
36	C₂²×D₃₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	#41	1	2	yes	—
41	C₂×Dic₃₁	—	2	2	no	—
42	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
43	C₄×D₃₁	—	2	2	no	—
44	C₄×D₃₁	—	2	2	no	—
45	C₄×D₃₁	—	2	4	no	—
46	C₈×C₃₁	—	2	2	no	—
47	C₈×C₃₁	—	4	2	no	—
48	C₃₁⋊C₈	—	4	2	no	—
49	C₃₁⋊C₈	C₂²⋊C₄	1	4	yes	—
50	#1810	—	31	1	no	maximal, Sylow
51	C₂²×C₃₁	—	2	2	no	—
52	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
54	C₂×D₃₁	—	2	8	no	—
55	C₂×D₃₁	—	2	2	no	—
56	C₂×D₃₁	—	2	2	no	—
57	C₂×D₃₁	#108	1	4	yes	complemented
58	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₃₁	#117	1	1	yes	characteristic
60	C₄×C₃₁	#121	1	2	yes	—
61	Dic₃₁	—	2	2	no	—
62	C₂³×C₈	—	31	1	no	—
63	C₂×#108	—	31	2	no	—
64	#407	—	31	2	no	—
65	#341	—	31	1	no	—
66	#341	D₃₁	1	1	yes	characteristic, complemented
67	C₂×C₃₁	—	2	2	no	—
68	C₂×C₃₁	#380	1	1	yes	characteristic
69	C₂×C₃₁	C₂×#108	1	2	yes	—
70	D₃₁	—	2	4	no	—
71	C₂³×C₄	—	31	1	no	—
72	C₂×M₁₆	—	31	2	no	—
73	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
74	C₂²×C₈	—	31	1	no	—
75	C₂²×C₈	—	31	4	no	—
76	C₂²×C₈	—	62	4	no	—
77	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
78	#108	—	62	4	no	—
79	C₂²⋊C₈	—	62	4	no	—
80	C₃₁	#1810	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	—	31	1	no	—
82	C₂²×C₄	—	31	1	no	—
83	C₂²×C₄	—	31	2	no	—
84	C₂²×C₄	—	31	2	no	—
85	C₂²×C₄	—	62	2	no	—
86	C₂²×C₄	—	62	2	no	—
87	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	62	2	no	—
91	M₁₆	—	62	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	31	2	no	—
95	C₂×C₈	—	31	4	no	—
96	C₂×C₈	—	62	8	no	—
97	C₂×C₈	—	62	2	no	—
98	C₂×C₈	C₄×D₃₁	1	2	yes	—
99	C₂³	—	31	2	no	—
100	C₂³	—	62	4	no	—
101	C₂³	—	62	2	no	—
102	C₂³	Q₈×D₃₁	1	1	yes	characteristic
103	C₂×C₄	—	31	2	no	—
104	C₂×C₄	—	62	2	no	—
105	C₂×C₄	—	62	2	no	—
106	C₂×C₄	—	62	2	no	—
107	C₂×C₄	—	62	4	no	—
108	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
109	C₂×C₄	#92495	1	2	yes	—
110	C₂×C₄	D₄×D₃₁	1	2	yes	—
111	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	31	4	no	—
115	C₈	—	124	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	31	4	no	—
118	C₂²	—	62	8	no	—
119	C₂²	—	62	2	no	—
120	C₂²	—	62	2	no	—
121	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
122	C₂²	#11758473	1	2	yes	—
123	C₄	—	62	2	no	—
124	C₄	#11758449	1	1	yes	characteristic
125	C₄	C₂²⋊C₄×D₃₁	1	1	yes	characteristic
126	C₄	#11758461	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	62	4	no	—
129	C₂	#14520908	1	1	yes	characteristic
130	C₂	#108×D₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₃₁

C₂

yes

maximal, characteristic

C₂×#11758410

C₂

yes

maximal, complemented

#14521012

C₂

yes

maximal

#14520814

C₂

yes

maximal, characteristic, complemented

C₃₁×#341

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₁

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₃₁

—

C₂×C₈×D₃₁

C₄

yes

—

#11758410

—

(C₂×D₃₁)⋊C₈

—

C₂³×D₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

Q₈

yes

characteristic

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

D₄

yes

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₈×D₃₁

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₂²⋊C₄

yes

—

#1810

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

#108

yes

complemented

C₄×C₃₁

C₂×C₂²⋊C₄

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

Dic₃₁

—

C₂³×C₈

—

—

—

—

D₃₁

yes

characteristic, complemented

C₂×C₃₁

—

C₂×C₃₁

#380

yes

characteristic

C₂×C₃₁

C₂×#108

yes

—

D₃₁

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

#108

—

C₂²⋊C₈

—

C₃₁

#1810

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₁

yes

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

Q₈×D₃₁

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

109

C₂×C₄

#92495

yes

—

110

C₂×C₄

D₄×D₃₁

yes

—

111

C₂×C₄

D₄×D₃₁

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

124

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

122

C₂²

#11758473

yes

—

123

C₄

—

124

C₄

#11758449

yes

characteristic

125

C₄

C₂²⋊C₄×D₃₁

yes

characteristic

126

C₄

#11758461

yes

—

127

C₂

—

128

C₂

—

129

C₂

#14520908

yes

characteristic

130

C₂

#108×D₃₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#341	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×D₃₁	Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#341	D₃₁
hypocenter	C₃₁	#1810
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#341

C₂

socle

C₂²×C₃₁

C₂×C₄⋊C₄

prime radical

C₂³×D₃₁

Q₈

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#11758410	C₂	semidirect
4. #14520814	C₂	semidirect
5. C₃₁×#341	C₂	semidirect
57. C₂×D₃₁	#108	semidirect
66. #341	D₃₁	semidirect
80. C₃₁	#1810	semidirect

H ≅ G/N

type

2. C₂×#11758410

C₂

semidirect

4. #14520814

C₂

semidirect

5. C₃₁×#341

C₂

semidirect

57. C₂×D₃₁

#108

semidirect

66. #341

D₃₁

semidirect

80. C₃₁

#1810

semidirect

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

subgroup	index
C₂²×C₈×D₃₁	2
C₂×#11758410	2
#14521012	2
#14520814	2
C₃₁×#341	2
#1810	31

subgroup

index

C₂²×C₈×D₃₁

C₃₁×#341

Subgroups — #16845748

Conjugacy classes of subgroups

Complements

Maximal subgroups