Group #16854067 — subgroups

Subgroups — #16854067

Order 3968.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×#11758446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11758513	C₂	1	4	yes	maximal, complemented
3	#14520952	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₃₁	C₂²	1	4	yes	—
7	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
8	C₂²×Dic₆₂	C₂²	1	2	yes	—
9	C₃₁×#127	—	2	4	no	—
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₃₁	C₂²	1	16	yes	—
12	#11758513	—	2	16	no	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
14	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
16	#11758446	—	2	4	no	—
17	C₂⁴×C₃₁	C₂³	1	4	yes	—
18	C₂×D₄×C₃₁	—	2	8	no	—
19	C₂×D₄×C₃₁	—	2	8	no	—
20	C₂×D₄×C₃₁	—	2	4	no	—
21	C₂×D₄×C₃₁	—	2	16	no	—
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
24	C₂²×Dic₃₁	C₂³	1	4	yes	—
25	C₂²×Dic₃₁	C₂³	1	4	yes	—
26	C₂×Dic₆₂	—	2	4	no	—
27	C₂×Dic₆₂	—	2	8	no	—
28	C₄²×C₃₁	D₄	1	4	yes	—
29	C₂²⋊Dic₃₁	—	2	32	no	—
30	C₄⋊Dic₃₁	—	2	8	no	—
31	C₄×Dic₃₁	—	2	8	no	—
32	C₄×Dic₃₁	D₄	1	8	yes	complemented
33	C₂³×C₃₁	—	2	8	no	—
34	C₂³×C₃₁	—	2	4	no	—
35	C₂³×C₃₁	—	2	16	no	—
36	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
37	D₄×C₃₁	—	4	16	no	—
38	D₄×C₃₁	—	4	8	no	—
39	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₃₁	#41	1	8	yes	—
43	C₂×Dic₃₁	—	2	8	no	—
44	C₂×Dic₃₁	—	2	4	no	—
45	C₂×Dic₃₁	—	2	8	no	—
46	C₂×Dic₃₁	C₂×D₄	1	8	yes	—
47	Dic₆₂	—	4	8	no	—
48	#2858	—	31	1	no	maximal, Sylow
49	C₂²×C₃₁	—	4	8	no	—
50	C₂²×C₃₁	—	4	4	no	—
51	C₂²×C₃₁	—	4	16	no	—
52	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
53	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₃₁	C₂×#41	1	4	yes	—
55	C₄×C₃₁	#124	1	8	yes	complemented
56	C₄×C₃₁	#127	1	4	yes	—
57	Dic₃₁	—	2	8	no	—
58	Dic₃₁	—	4	4	no	—
59	C₂×C₄⋊Q₈	—	31	1	no	—
60	C₂×#127	D₃₁	1	1	yes	characteristic
61	C₂×#124	—	31	4	no	—
62	#390	—	31	8	no	—
63	C₄³	—	31	1	no	—
64	C₂×C₃₁	—	8	4	no	—
65	C₂×C₃₁	C₂×#124	1	2	yes	—
66	C₂×C₃₁	C₂×#127	1	1	yes	characteristic
67	C₂×C₃₁	#532	1	4	yes	—
68	C₂²×Q₈	—	31	2	no	—
69	C₂²×D₄	C₂×D₃₁	1	2	yes	—
70	C₂²×D₄	C₂×D₃₁	1	4	yes	—
71	C₄⋊Q₈	—	62	4	no	—
72	#127	—	2	4	no	—
73	#124	—	62	16	no	—
74	C₂×C₄⋊C₄	—	31	4	no	—
75	C₂×C₂²⋊C₄	—	31	16	no	—
76	C₂×C₄²	—	31	2	no	—
77	C₂×C₄²	—	31	4	no	—
78	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
79	C₃₁	#2858	1	1	yes	characteristic, complemented, Sylow
80	C₂⁴	C₂²×D₃₁	1	4	yes	—
81	C₂×Q₈	—	62	8	no	—
82	C₂×Q₈	—	62	4	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	16	no	—
85	C₂×D₄	—	2	8	no	—
86	C₂×D₄	—	2	4	no	—
87	C₂²×C₄	—	31	4	no	—
88	C₂²×C₄	—	31	4	no	—
89	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
91	C₄⋊C₄	—	62	8	no	—
92	C₂²⋊C₄	—	62	32	no	—
93	C₄²	—	31	8	no	—
94	C₄²	—	62	8	no	—
95	C₄²	C₂²⋊D₃₁	1	4	yes	—
96	C₂³	—	2	8	no	—
97	C₂³	—	2	16	no	—
98	C₂³	—	2	4	no	—
99	C₂³	C₂³×D₃₁	1	1	yes	characteristic
100	Q₈	—	124	8	no	—
101	D₄	—	4	16	no	—
102	D₄	—	4	8	no	—
103	C₂×C₄	—	31	8	no	—
104	C₂×C₄	—	62	8	no	—
105	C₂×C₄	—	62	8	no	—
106	C₂×C₄	—	62	4	no	—
107	C₂×C₄	D₄×D₃₁	1	4	yes	—
108	C₂×C₄	#92497	1	8	yes	—
109	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
111	C₂²	—	4	8	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	4	16	no	—
114	C₂²	C₂×D₄×D₃₁	1	2	yes	—
115	C₂²	C₂×#92497	1	4	yes	—
116	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
117	C₄	—	62	8	no	—
118	C₄	—	124	4	no	—
119	C₄	#11758513	1	8	yes	complemented
120	C₄	#11758517	1	4	yes	—
121	C₂	—	8	4	no	—
122	C₂	C₂×#11758513	1	2	yes	—
123	C₂	C₂×#11758517	1	1	yes	characteristic
124	C₂	#14521597	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic, complemented

C₂×#11758513

C₂

yes

maximal, complemented

#14520952

C₂

yes

maximal, complemented

C₄²×Dic₃₁

C₂

yes

maximal, characteristic, complemented

C₂×C₃₁×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₃₁

C₂²

yes

—

C₂²×D₄×C₃₁

C₂²

yes

—

C₂²×Dic₆₂

C₂²

yes

—

C₃₁×#127

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

#11758513

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

#11758446

—

C₂⁴×C₃₁

C₂³

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₄²×C₃₁

D₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

complemented

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂⁴

yes

characteristic

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

Dic₆₂

—

#2858

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂²×D₄

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂²×C₃₁

C₂×#41

yes

—

C₄×C₃₁

#124

yes

complemented

C₄×C₃₁

#127

yes

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×#127

D₃₁

yes

characteristic

C₂×#124

—

#390

—

C₄³

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#124

yes

—

C₂×C₃₁

C₂×#127

yes

characteristic

C₂×C₃₁

#532

yes

—

C₂²×Q₈

—

C₂²×D₄

C₂×D₃₁

yes

—

C₂²×D₄

C₂×D₃₁

yes

—

C₄⋊Q₈

—

#127

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₃₁

#2858

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₃₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₃₁

yes

characteristic

100

Q₈

—

124

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×D₃₁

yes

—

108

C₂×C₄

#92497

yes

—

109

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

110

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×D₄×D₃₁

yes

—

115

C₂²

C₂×#92497

yes

—

116

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

117

C₄

—

118

C₄

—

124

—

119

C₄

#11758513

yes

complemented

120

C₄

#11758517

yes

—

121

C₂

—

122

C₂

C₂×#11758513

yes

—

123

C₂

C₂×#11758517

yes

characteristic

124

C₂

#14521597

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator, socle, cosocle	C₂³×C₃₁	C₂⁴
Fitting, prime radical	C₂×C₃₁×#127	C₂
hypercenter	C₂×#127	D₃₁
hypocenter	C₃₁	#2858
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator, socle, cosocle

C₂³×C₃₁

C₂⁴

Fitting, prime radical

C₂×C₃₁×#127

C₂

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#11758446	C₂	semidirect
2. C₂×#11758513	C₂	semidirect
3. #14520952	C₂	semidirect
4. C₄²×Dic₃₁	C₂	semidirect
32. C₄×Dic₃₁	D₄	semidirect
55. C₄×C₃₁	#124	semidirect
79. C₃₁	#2858	semidirect
119. C₄	#11758513	semidirect

H ≅ G/N

type

1. C₂×#11758446

C₂

semidirect

2. C₂×#11758513

C₂

semidirect

3. #14520952

C₂

semidirect

4. C₄²×Dic₃₁

C₂

semidirect

32. C₄×Dic₃₁

D₄

semidirect

55. C₄×C₃₁

#124

semidirect

79. C₃₁

#2858

semidirect

119. C₄

#11758513

semidirect

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

subgroup	index
C₂×#11758446	2
C₂×#11758513	2
#14520952	2
C₄²×Dic₃₁	2
C₂×C₃₁×#127	2
#2858	31

subgroup

index

C₂×#11758446

C₂×#11758513

#14520952

C₄²×Dic₃₁

C₂×C₃₁×#127

#2858

Subgroups — #16854067

Conjugacy classes of subgroups

Complements

Maximal subgroups