Group #16848293 — subgroups

Subgroups — #16848293

Order 3968.

1104 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 309 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×#11758503	C₂	1	2	yes	maximal, complemented
2	#14520952	C₂	1	8	yes	maximal, complemented
3	#14520905	C₂	1	4	yes	maximal, complemented
4	C₂³×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁵×C₃₁	C₂²	1	2	yes	—
6	C₂²×D₄×C₃₁	—	2	8	no	—
7	C₂²×D₄×C₃₁	C₄	1	8	yes	complemented
8	C₂²×D₄×C₃₁	C₂²	1	4	yes	—
9	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₃₁	C₂²	1	16	yes	complemented
12	#11758503	—	2	8	no	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	complemented
14	C₂×C₄×Dic₃₁	C₂²	1	4	yes	complemented
15	C₂⁴×C₃₁	—	2	8	no	—
16	C₂⁴×C₃₁	—	4	8	no	—
17	C₂⁴×C₃₁	C₂×C₄	1	8	yes	—
18	C₂⁴×C₃₁	C₂³	1	4	yes	—
19	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₃₁	—	2	32	no	—
21	C₂×D₄×C₃₁	—	2	8	no	—
22	C₂×D₄×C₃₁	—	2	8	no	—
23	C₂×D₄×C₃₁	C₂×C₄	1	16	yes	—
24	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₃₁	D₄	1	8	yes	—
26	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
27	C₂²×Dic₃₁	C₂³	1	8	yes	—
28	C₂²⋊Dic₃₁	—	2	8	no	—
29	C₂²⋊Dic₃₁	—	4	16	no	—
30	C₄⋊Dic₃₁	—	2	8	no	—
31	C₄×Dic₃₁	—	2	8	no	—
32	C₂³×C₃₁	—	2	4	no	—
33	C₂³×C₃₁	—	2	16	no	—
34	C₂³×C₃₁	—	2	8	no	—
35	C₂³×C₃₁	—	2	16	no	—
36	C₂³×C₃₁	—	4	16	no	—
37	C₂³×C₃₁	—	4	32	no	—
38	C₂³×C₃₁	C₂²×C₄	1	8	yes	—
39	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
40	C₂³×C₃₁	C₂×D₄	1	4	yes	—
41	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
42	D₄×C₃₁	—	2	32	no	—
43	C₂×C₄×C₃₁	C₂²⋊C₄	1	16	yes	complemented
44	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
47	C₂×Dic₃₁	—	2	8	no	—
48	C₂×Dic₃₁	—	2	16	no	—
49	#2309	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	—	2	4	no	—
51	C₂²×C₃₁	—	2	8	no	—
52	C₂²×C₃₁	—	2	8	no	—
53	C₂²×C₃₁	—	2	16	no	—
54	C₂²×C₃₁	—	4	32	no	—
55	C₂²×C₃₁	—	4	16	no	—
56	C₂²×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
57	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
58	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
59	C₂²×C₃₁	H(5,𝔽₂)	1	4	yes	—
60	C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
61	Dic₃₁	—	4	8	no	—
62	C₂³×D₄	D₃₁	1	1	yes	characteristic
63	C₂×#117	—	31	2	no	—
64	#390	—	31	8	no	—
65	#379	—	31	4	no	—
66	C₂×C₃₁	—	2	4	no	—
67	C₂×C₃₁	—	4	8	no	—
68	C₂×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃₁	#518	1	2	yes	—
70	C₂×C₃₁	#535	1	4	yes	—
71	C₂⁵	C₂×D₃₁	1	2	yes	—
72	C₂²×D₄	—	2	8	no	—
73	C₂²×D₄	Dic₃₁	1	8	yes	complemented
74	C₂²×D₄	C₂×D₃₁	1	4	yes	—
75	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
76	#117	—	62	8	no	—
77	C₂×C₄⋊C₄	—	31	4	no	—
78	C₂×C₂²⋊C₄	—	31	4	no	—
79	C₂×C₂²⋊C₄	—	31	16	no	—
80	C₂×C₄²	—	31	4	no	—
81	C₃₁	#2309	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	—	2	8	no	—
83	C₂⁴	—	4	8	no	—
84	C₂⁴	C₂×Dic₃₁	1	8	yes	—
85	C₂⁴	C₂²×D₃₁	1	4	yes	—
86	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
87	C₂×D₄	—	2	32	no	—
88	C₂×D₄	—	2	8	no	—
89	C₂×D₄	—	2	8	no	—
90	C₂×D₄	C₂×Dic₃₁	1	16	yes	—
91	C₂²×C₄	—	31	8	no	—
92	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
93	C₂²×C₄	C₂²⋊D₃₁	1	8	yes	—
94	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
95	C₄⋊C₄	—	62	8	no	—
96	C₂²⋊C₄	—	62	8	no	—
97	C₂²⋊C₄	—	124	16	no	—
98	C₄²	—	62	8	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	2	16	no	—
101	C₂³	—	2	16	no	—
102	C₂³	—	2	8	no	—
103	C₂³	—	4	16	no	—
104	C₂³	—	4	32	no	—
105	C₂³	C₂²×Dic₃₁	1	8	yes	—
106	C₂³	C₂²×Dic₃₁	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₃₁	1	4	yes	—
108	C₂³	C₂³×D₃₁	1	1	yes	characteristic
109	D₄	—	2	32	no	—
110	C₂×C₄	—	62	16	no	—
111	C₂×C₄	—	62	8	no	—
112	C₂×C₄	C₂²⋊Dic₃₁	1	16	yes	complemented
113	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
115	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
116	C₂²	—	2	8	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	16	no	—
120	C₂²	—	4	32	no	—
121	C₂²	—	4	16	no	—
122	C₂²	C₂×C₂²⋊Dic₃₁	1	4	yes	—
123	C₂²	#11758561	1	4	yes	—
124	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
125	C₂²	C₂²×C₂²⋊D₃₁	1	2	yes	—
126	C₄	—	124	8	no	—
127	C₄	C₂×C₂²⋊Dic₃₁	1	8	yes	—
128	C₂	—	2	4	no	—
129	C₂	—	4	8	no	—
130	C₂	C₂²×C₂²⋊Dic₃₁	1	1	yes	characteristic
131	C₂	#14521524	1	2	yes	—
132	C₂	#14521616	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758503

C₂

yes

maximal, complemented

#14520952

C₂

yes

maximal, complemented

#14520905

C₂

yes

maximal, complemented

C₂³×D₄×C₃₁

C₂

yes

maximal, characteristic

C₂⁵×C₃₁

C₂²

yes

—

C₂²×D₄×C₃₁

—

C₂²×D₄×C₃₁

C₄

yes

complemented

C₂²×D₄×C₃₁

C₂²

yes

—

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

complemented

#11758503

—

C₂×C₄⋊Dic₃₁

C₂²

yes

complemented

C₂×C₄×Dic₃₁

C₂²

yes

complemented

C₂⁴×C₃₁

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

C₂×C₄

yes

—

C₂⁴×C₃₁

C₂³

yes

—

C₂⁴×C₃₁

C₂³

yes

characteristic

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

D₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

complemented

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

#2309

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

C₂²×D₄

yes

—

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

—

Dic₃₁

—

C₂³×D₄

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₃₁

#518

yes

—

C₂×C₃₁

#535

yes

—

C₂⁵

C₂×D₃₁

yes

—

C₂²×D₄

—

C₂²×D₄

Dic₃₁

yes

complemented

C₂²×D₄

C₂×D₃₁

yes

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₃₁

#2309

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×Dic₃₁

yes

—

C₂⁴

C₂²×D₃₁

yes

—

C₂⁴

C₂²×D₃₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

124

—

C₄²

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂²×Dic₃₁

yes

—

106

C₂³

C₂²×Dic₃₁

yes

—

107

C₂³

C₂×C₂²⋊D₃₁

yes

—

108

C₂³

C₂³×D₃₁

yes

characteristic

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²⋊Dic₃₁

yes

complemented

113

C₂×C₄

C₂²×Dic₃₁

yes

—

114

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

115

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₂²⋊Dic₃₁

yes

—

123

C₂²

#11758561

yes

—

124

C₂²

C₂³×Dic₃₁

yes

characteristic

125

C₂²

C₂²×C₂²⋊D₃₁

yes

—

126

C₄

—

124

—

127

C₄

C₂×C₂²⋊Dic₃₁

yes

—

128

C₂

—

129

C₂

—

130

C₂

C₂²×C₂²⋊Dic₃₁

yes

characteristic

131

C₂

#14521524

yes

—

132

C₂

#14521616

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting, prime radical	C₂³×D₄×C₃₁	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂³×D₄	D₃₁
hypocenter	C₃₁	#2309
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting, prime radical

C₂³×D₄×C₃₁

C₂

socle, cosocle

C₂³×C₃₁

C₂⁴

hypercenter

C₂³×D₄

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂³×D₄

N	H ≅ G/N	type
1. C₂×#11758503	C₂	semidirect
2. #14520952	C₂	semidirect
3. #14520905	C₂	semidirect
7. C₂²×D₄×C₃₁	C₄	semidirect
11. C₂×C₂²⋊Dic₃₁	C₂²	semidirect
13. C₂×C₄⋊Dic₃₁	C₂²	semidirect
14. C₂×C₄×Dic₃₁	C₂²	semidirect
43. C₂×C₄×C₃₁	C₂²⋊C₄	semidirect
73. C₂²×D₄	Dic₃₁	semidirect
81. C₃₁	#2309	semidirect
112. C₂×C₄	C₂²⋊Dic₃₁	semidirect

H ≅ G/N

type

1. C₂×#11758503

C₂

semidirect

2. #14520952

C₂

semidirect

3. #14520905

C₂

semidirect

7. C₂²×D₄×C₃₁

C₄

semidirect

11. C₂×C₂²⋊Dic₃₁

C₂²

semidirect

13. C₂×C₄⋊Dic₃₁

C₂²

semidirect

14. C₂×C₄×Dic₃₁

C₂²

semidirect

43. C₂×C₄×C₃₁

C₂²⋊C₄

semidirect

73. C₂²×D₄

Dic₃₁

semidirect

81. C₃₁

#2309

semidirect

112. C₂×C₄

C₂²⋊Dic₃₁

semidirect

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

subgroup	index
C₂×#11758503	2
#14520952	2
#14520905	2
C₂³×D₄×C₃₁	2
#2309	31

subgroup

index

C₂³×D₄×C₃₁

Subgroups — #16848293

Conjugacy classes of subgroups

Complements

Maximal subgroups