Group #14521802 — subgroups

Subgroups — #14521802

Order 1984.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₃₁×#123	C₂	1	1	yes	maximal, characteristic
2	#11758513	C₂	1	1	yes	maximal, characteristic
3	#11758512	C₂	1	1	yes	maximal, characteristic
4	#11758501	C₂	1	2	yes	maximal, complemented
5	#11758466	C₂	1	2	yes	maximal, complemented
6	#11758465	C₂	1	2	yes	maximal, complemented
7	#11758456	C₂	1	2	yes	maximal, complemented
8	#11758456	C₂	1	2	yes	maximal, complemented
9	#11758455	C₂	1	2	yes	maximal, complemented
10	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×Dic₃₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₂	C₂²	1	2	yes	—
14	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
19	C₂²⋊Dic₃₁	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
21	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
22	C₄⋊Dic₃₁	C₂²	1	2	yes	—
23	C₄⋊Dic₃₁	C₂²	1	2	yes	—
24	Dic₃₁⋊C₄	C₂²	1	2	yes	—
25	Dic₃₁⋊C₄	C₂²	1	2	yes	—
26	Dic₃₁⋊C₄	C₂²	1	2	yes	—
27	Dic₃₁⋊C₄	C₂²	1	2	yes	—
28	Dic₃₁⋊C₄	C₂²	1	2	yes	—
29	C₄×Dic₃₁	C₂²	1	2	yes	—
30	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
31	C₂³×C₃₁	C₂³	1	1	yes	characteristic
32	C₂³×C₃₁	C₂³	1	1	yes	characteristic
33	D₄×C₃₁	—	4	1	no	—
34	C₂×C₄×C₃₁	—	4	1	no	—
35	C₂×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂³	1	2	yes	—
38	C₂×Dic₃₁	—	4	1	no	—
39	C₂×Dic₃₁	C₂³	1	2	yes	—
40	C₂×Dic₃₁	C₂³	1	2	yes	—
41	C₂×Dic₃₁	C₂³	1	2	yes	—
42	C₂×Dic₃₁	C₂³	1	2	yes	—
43	Dic₆₂	—	4	1	no	—
44	Dic₆₂	—	4	2	no	—
45	C₂²×C₃₁	—	2	1	no	—
46	C₂²×C₃₁	—	2	1	no	—
47	C₂²×C₃₁	—	2	2	no	—
48	C₂²×C₃₁	—	2	2	no	—
49	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₄×C₃₁	—	2	2	no	—
51	C₄×C₃₁	—	2	1	no	—
52	C₄×C₃₁	—	2	2	no	—
53	Dic₃₁	—	2	2	no	—
54	Dic₃₁	—	2	2	no	—
55	Dic₃₁	—	2	2	no	—
56	Dic₃₁	—	2	2	no	—
57	#563	—	31	1	no	maximal, Sylow
58	C₂×C₃₁	—	4	1	no	—
59	C₂×C₃₁	—	4	1	no	—
60	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₃₁	2₋¹⁺⁴	1	2	yes	—
62	C₄⋊Q₈	—	31	2	no	—
63	#126	—	31	2	no	—
64	#126	—	31	2	no	—
65	#125	—	31	2	no	—
66	#124	—	31	1	no	—
67	#123	—	31	1	no	—
68	#123	D₃₁	1	1	yes	characteristic
69	C₂²⋊Q₈	—	31	2	no	—
70	C₂²⋊Q₈	—	31	2	no	—
71	C₃₁	#563	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₈	—	31	2	no	—
73	C₂×Q₈	—	31	1	no	—
74	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
75	C₂²×C₄	—	31	1	no	—
76	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
77	C₄⋊C₄	—	31	2	no	—
78	C₄⋊C₄	—	31	2	no	—
79	C₄⋊C₄	—	31	2	no	—
80	C₄⋊C₄	—	31	2	no	—
81	C₄⋊C₄	—	31	2	no	—
82	C₄⋊C₄	—	31	2	no	—
83	C₄⋊C₄	—	31	2	no	—
84	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
85	C₂²⋊C₄	—	31	2	no	—
86	C₂²⋊C₄	—	31	2	no	—
87	C₂²⋊C₄	—	31	2	no	—
88	C₂²⋊C₄	—	31	1	no	—
89	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
90	C₂²⋊C₄	C₂×D₃₁	1	1	yes	characteristic
91	C₄²	—	31	1	no	—
92	C₄²	—	31	2	no	—
93	C₂³	C₂²×D₃₁	1	1	yes	characteristic
94	C₂³	C₂²×D₃₁	1	1	yes	characteristic
95	Q₈	—	124	2	no	—
96	Q₈	—	124	1	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	31	2	no	—
100	C₂×C₄	—	31	2	no	—
101	C₂×C₄	—	31	2	no	—
102	C₂×C₄	—	31	2	no	—
103	C₂×C₄	—	124	1	no	—
104	C₂×C₄	C₂²×D₃₁	1	2	yes	—
105	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
106	C₂×C₄	C₂²×D₃₁	1	2	yes	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	2	no	—
111	C₂²	C₂³×D₃₁	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	2	no	—
115	C₄	—	62	2	no	—
116	C₄	—	62	2	no	—
117	C₄	—	62	2	no	—
118	C₄	—	62	2	no	—
119	C₂	—	4	1	no	—
120	C₂	—	4	1	no	—
121	C₂	#11758561	1	1	yes	characteristic
122	C₂	#11758567	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#123

C₂

yes

maximal, characteristic

#11758513

C₂

yes

maximal, characteristic

#11758512

C₂

yes

maximal, characteristic

#11758501

C₂

yes

maximal, complemented

#11758466

C₂

yes

maximal, complemented

#11758465

C₂

yes

maximal, complemented

#11758456

C₂

yes

maximal, complemented

#11758456

C₂

yes

maximal, complemented

#11758455

C₂

yes

maximal, complemented

C₂×D₄×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂²×Dic₃₁

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊C₄×C₃₁

C₂²

yes

characteristic

C₂²⋊Dic₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

characteristic

C₂²⋊Dic₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₂³×C₃₁

C₂³

yes

characteristic

C₂³×C₃₁

C₂³

yes

characteristic

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

Dic₆₂

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

#563

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

H(5,𝔽₂)

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

—

C₄⋊Q₈

—

—

—

—

—

—

D₃₁

yes

characteristic

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₃₁

#563

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

C₂×D₃₁

yes

characteristic

C₄²

—

C₄²

—

C₂³

C₂²×D₃₁

yes

characteristic

C₂³

C₂²×D₃₁

yes

characteristic

Q₈

—

124

—

Q₈

—

124

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

124

—

104

C₂×C₄

C₂²×D₃₁

yes

—

105

C₂×C₄

C₂²×D₃₁

yes

characteristic

106

C₂×C₄

C₂²×D₃₁

yes

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×D₃₁

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

120

C₂

—

121

C₂

#11758561

yes

characteristic

122

C₂

#11758567

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#123	C₂
prime radical	C₂×D₄×C₃₁	C₂²
hypercenter	#123	D₃₁
hypocenter	C₃₁	#563
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×#123

C₂

prime radical

C₂×D₄×C₃₁

C₂²

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
4. #11758501	C₂	semidirect
5. #11758466	C₂	semidirect
6. #11758465	C₂	semidirect
7. #11758456	C₂	semidirect
8. #11758456	C₂	semidirect
9. #11758455	C₂	semidirect
71. C₃₁	#563	semidirect

H ≅ G/N

type

4. #11758501

C₂

semidirect

5. #11758466

C₂

semidirect

6. #11758465

C₂

semidirect

7. #11758456

C₂

semidirect

8. #11758456

C₂

semidirect

9. #11758455

C₂

semidirect

71. C₃₁

#563

semidirect

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

subgroup	index
C₃₁×#123	2
#11758513	2
#11758512	2
#11758501	2
#11758466	2
#11758465	2
#11758456	2
#11758455	2
#563	31

subgroup

index

C₃₁×#123

Subgroups — #14521802

Conjugacy classes of subgroups

Complements

Maximal subgroups