Group #26108734 — subgroups

Subgroups — #26108734

Order 12032.

656 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20244358	C₂	1	3	yes	maximal
2	C₂×#15688865	C₂	1	3	yes	maximal
3	C₂×C₄₇×#564	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093935	C₂²	1	3	yes	—
5	#15688993	C₂²	1	1	yes	characteristic
6	#15688885	—	2	6	no	—
7	#15688865	—	2	6	no	—
8	C₄₇×#564	—	2	6	no	—
9	C₄₇×#564	C₄	1	4	yes	complemented
10	C₂×#125×C₄₇	—	2	6	no	—
11	C₂×#125×C₄₇	C₂²	1	3	yes	—
12	C₂²×C₄₇⋊C₈	—	2	3	no	—
13	C₂×C₄⋊Dic₄₇	—	2	3	no	—
14	#13093935	—	2	6	no	—
15	#125×C₄₇	—	2	24	no	—
16	#125×C₄₇	—	2	24	no	—
17	#125×C₄₇	—	2	6	no	—
18	#125×C₄₇	C₂×C₄	1	6	yes	—
19	#125×C₄₇	D₄	1	6	yes	—
20	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
21	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
22	C₂×C₄⋊C₄×C₄₇	D₄	1	6	yes	—
23	C₂×C₄²×C₄₇	—	2	2	no	—
24	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	6	no	—
26	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
27	C₂²×Dic₄₇	—	4	1	no	—
28	C₄⋊C₄×C₄₇	—	2	24	no	—
29	C₄⋊C₄×C₄₇	—	2	12	no	—
30	C₄⋊C₄×C₄₇	—	2	12	no	—
31	C₄⋊C₄×C₄₇	—	2	6	no	—
32	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	12	yes	—
33	C₄²×C₄₇	—	2	8	no	—
34	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₄₇	C₂×D₄	1	3	yes	—
36	C₄⋊Dic₄₇	—	4	6	no	—
37	C₂×C₄₇⋊C₈	—	2	6	no	—
38	C₂×C₄₇⋊C₈	—	4	3	no	—
39	C₂×C₄₇⋊C₈	—	4	3	no	—
40	C₂³×C₄₇	#120	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	3	no	—
42	C₂×C₄×C₄₇	—	2	3	no	—
43	C₂×C₄×C₄₇	—	2	12	no	—
44	C₂×C₄×C₄₇	—	4	6	no	—
45	C₂×C₄×C₄₇	—	4	6	no	—
46	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
47	C₂×C₄×C₄₇	#120	1	3	yes	—
48	C₂×Dic₄₇	—	8	3	no	—
49	C₄₇⋊C₈	—	4	6	no	—
50	#8509	—	47	1	no	maximal, Sylow
51	C₂²×C₄₇	#379	1	1	yes	characteristic
52	C₂²×C₄₇	#454	1	3	yes	—
53	C₂²×C₄₇	#456	1	3	yes	—
54	C₄×C₄₇	—	2	3	no	—
55	C₄×C₄₇	—	2	3	no	—
56	C₄×C₄₇	—	4	12	no	—
57	Dic₄₇	—	16	1	no	—
58	C₂×#564	D₄₇	1	1	yes	characteristic
59	#1992	—	47	3	no	—
60	C₂×#330	—	47	3	no	—
61	C₂×C₄₇	#1658	1	1	yes	characteristic
62	C₂×C₄₇	#1663	1	3	yes	—
63	C₂×C₄₇	#1793	1	3	yes	—
64	#564	—	2	6	no	—
65	#564	Dic₄₇	1	4	yes	complemented
66	C₂×#125	—	2	6	no	—
67	C₂×#125	C₂×D₄₇	1	3	yes	—
68	C₂×#97	—	47	3	no	—
69	#384	—	47	1	no	—
70	#340	—	94	6	no	—
71	#330	—	94	6	no	—
72	C₄₇	#8509	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	94	3	no	—
74	#125	—	2	24	no	—
75	#125	—	2	24	no	—
76	#125	—	2	6	no	—
77	#125	C₂×Dic₄₇	1	6	yes	—
78	#125	C₂²⋊D₄₇	1	6	yes	—
79	C₂×C₄⋊C₄	—	2	6	no	—
80	C₂×C₄⋊C₄	—	2	6	no	—
81	C₂×C₄⋊C₄	—	94	3	no	—
82	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	6	yes	—
83	C₂×C₄²	—	2	2	no	—
84	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
85	#97	—	94	6	no	—
86	C₂²×C₄	—	2	6	no	—
87	C₂²×C₄	—	188	1	no	—
88	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
89	C₂×C₈	—	94	6	no	—
90	C₂×C₈	—	188	3	no	—
91	C₂×C₈	—	188	3	no	—
92	C₄⋊C₄	—	2	24	no	—
93	C₄⋊C₄	—	2	12	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	—	188	6	no	—
97	C₄⋊C₄	C₂²⋊Dic₄₇	1	12	yes	—
98	C₄²	—	2	8	no	—
99	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
100	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
101	C₂³	#13093993	1	1	yes	characteristic
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	—	376	3	no	—
108	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
109	C₂×C₄	#13093993	1	3	yes	—
110	C₈	—	188	6	no	—
111	C₂²	#15688979	1	1	yes	characteristic
112	C₂²	#15689296	1	3	yes	—
113	C₂²	#15689306	1	3	yes	—
114	C₄	—	2	3	no	—
115	C₄	—	2	3	no	—
116	C₄	—	4	12	no	—
117	C₄	—	752	1	no	—
118	C₂	#20242690	1	1	yes	characteristic
119	C₂	#20242716	1	3	yes	—
120	C₂	#20243347	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244358

C₂

yes

maximal

C₂×#15688865

C₂

yes

maximal

C₂×C₄₇×#564

C₂

yes

maximal, characteristic

C₂×#13093935

C₂²

yes

—

#15688993

C₂²

yes

characteristic

#15688885

—

#15688865

—

C₄₇×#564

—

C₄₇×#564

C₄

yes

complemented

C₂×#125×C₄₇

—

C₂×#125×C₄₇

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂×C₄⋊Dic₄₇

—

#13093935

—

#125×C₄₇

—

#125×C₄₇

—

#125×C₄₇

—

#125×C₄₇

C₂×C₄

yes

—

#125×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

#120

yes

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#456

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

C₂×#564

D₄₇

yes

characteristic

#1992

—

C₂×#330

—

C₂×C₄₇

#1658

yes

characteristic

C₂×C₄₇

#1663

yes

—

C₂×C₄₇

#1793

yes

—

#564

—

#564

Dic₄₇

yes

complemented

C₂×#125

—

C₂×#125

C₂×D₄₇

yes

—

C₂×#97

—

#384

—

#340

—

#330

—

C₄₇

#8509

yes

characteristic, complemented, Sylow

C₂²×C₈

—

—

—

—

C₂×Dic₄₇

yes

—

#125

C₂²⋊D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#97

—

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄⋊C₄

C₂²⋊Dic₄₇

yes

—

C₄²

—

C₄²

C₂²×Dic₄₇

yes

characteristic

100

C₄²

C₂×C₂²⋊D₄₇

yes

—

101

C₂³

#13093993

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

376

—

108

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

109

C₂×C₄

#13093993

yes

—

110

C₈

—

188

—

111

C₂²

#15688979

yes

characteristic

112

C₂²

#15689296

yes

—

113

C₂²

#15689306

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

752

—

118

C₂

#20242690

yes

characteristic

119

C₂

#20242716

yes

—

120

C₂

#20243347

yes

—

	subgroup	G/H
center	C₂³	#13093993
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#564	C₂
socle, prime radical	C₂³×C₄₇	#120
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#564	D₄₇
hypocenter	C₄₇	#8509
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

subgroup

G/H

center

C₂³

#13093993

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#564

C₂

socle, prime radical

C₂³×C₄₇

#120

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

N	H ≅ G/N	type
9. C₄₇×#564	C₄	semidirect
65. #564	Dic₄₇	semidirect
72. C₄₇	#8509	semidirect

H ≅ G/N

type

9. C₄₇×#564

C₄

semidirect

65. #564

Dic₄₇

semidirect

72. C₄₇

#8509

semidirect

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

subgroup	index
#20244358	2
C₂×#15688865	2
C₂×C₄₇×#564	2
#8509	47

subgroup

index

#20244358

C₂×#15688865

C₂×C₄₇×#564

#8509

Subgroups — #26108734

Conjugacy classes of subgroups

Complements

Maximal subgroups