Group #26103152 — subgroups

Subgroups — #26103152

Order 12032.

234 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 135 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×#15688874	C₂	1	2	yes	maximal
2	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
4	#15688874	—	2	2	no	—
5	#15688874	C₄	1	4	yes	—
6	C₂²×C₁₆×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#346	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	4	2	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₄₇	—	2	2	no	—
16	C₂×C₁₆×C₄₇	—	2	2	no	—
17	C₂×C₁₆×C₄₇	C₈	1	4	yes	complemented
18	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₇	D₄	1	2	yes	—
20	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
22	C₂×C₈×C₄₇	—	2	2	no	—
23	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
24	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
27	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
28	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
29	C₄²×C₄₇	D₈	1	1	yes	characteristic
30	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
31	C₂×C₄₇⋊C₈	—	4	4	no	—
32	C₂×C₄₇⋊C₈	—	8	4	no	—
33	C₁₆×C₄₇	—	2	4	no	—
34	C₂³×C₄₇	#106	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	2	4	no	—
36	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
37	C₂×C₄×C₄₇	#102	1	2	yes	—
38	C₂×C₄×C₄₇	#103	1	2	yes	—
39	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
40	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
41	C₂×C₄×C₄₇	#108	1	2	yes	—
42	C₈×C₄₇	—	2	2	no	—
43	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
44	C₈×C₄₇	#97	1	1	yes	characteristic
45	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
46	C₄₇⋊C₈	—	8	4	no	—
47	#7156	—	47	1	no	maximal, Sylow
48	C₂²×C₄₇	#334	1	2	yes	—
49	C₂²×C₄₇	#340	1	1	yes	characteristic
50	C₂²×C₄₇	#365	1	4	yes	—
51	C₄×C₄₇	—	2	2	no	—
52	C₄×C₄₇	—	2	2	no	—
53	C₄×C₄₇	#335	1	2	yes	—
54	C₄×C₄₇	#336	1	1	yes	characteristic
55	C₄×C₄₇	#341	1	1	yes	characteristic
56	C₂×#346	D₄₇	1	1	yes	characteristic
57	C₂×#334	—	47	2	no	—
58	C₂×C₄₇	#1277	1	1	yes	characteristic
59	C₂×C₄₇	#1368	1	4	yes	—
60	C₂×C₄₇	#1383	1	2	yes	—
61	C₂²×C₁₆	Dic₄₇	1	2	yes	—
62	C₂×C₄⋊C₈	—	94	2	no	—
63	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
64	#346	—	2	2	no	—
65	#334	—	47	4	no	—
66	#334	—	94	2	no	—
67	C₄₇	#7156	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	—	188	2	no	—
69	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
71	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	C₄₇⋊C₈	1	4	yes	complemented
75	C₄⋊C₈	—	94	4	no	—
76	C₄⋊C₈	—	188	2	no	—
77	C₄×C₈	C₄×D₄₇	1	2	yes	—
78	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
79	C₂²×C₄	#10625325	1	2	yes	—
80	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	376	4	no	—
83	C₂×C₈	—	188	4	no	—
84	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
85	C₂×C₈	#10625329	1	2	yes	—
86	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
87	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
88	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
89	C₄²	D₃₇₆	1	1	yes	characteristic
90	C₄²	Dic₁₈₈	1	1	yes	characteristic
91	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
92	C₁₆	—	2	4	no	—
93	C₂³	#13093956	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
96	C₂×C₄	#13093946	1	2	yes	—
97	C₂×C₄	#13093949	1	2	yes	—
98	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
99	C₂×C₄	#13093958	1	1	yes	characteristic
100	C₂×C₄	#13093960	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	376	4	no	—
103	C₈	C₄×C₄₇⋊C₈	1	1	yes	characteristic
104	C₈	#13093935	1	1	yes	characteristic
105	C₈	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
106	C₂²	#15688874	1	2	yes	—
107	C₂²	#15688886	1	1	yes	characteristic
108	C₂²	#15688946	1	4	yes	—
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	#15688876	1	2	yes	—
112	C₄	#15688878	1	1	yes	characteristic
113	C₄	#15688888	1	1	yes	characteristic
114	C₂	#20241223	1	1	yes	characteristic
115	C₂	#20241454	1	4	yes	—
116	C₂	#20241487	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688874

C₂

yes

maximal

C₂×C₄₇×#346

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

#15688874

—

#15688874

C₄

yes

—

C₂²×C₁₆×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#346

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₈

yes

complemented

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

Q₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

#106

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

#107

yes

characteristic

C₂×C₄×C₄₇

#108

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₄×C₈

yes

characteristic

C₈×C₄₇

#97

yes

characteristic

C₈×C₄₇

C₂²⋊C₈

yes

—

C₄₇⋊C₈

—

#7156

—

maximal, Sylow

C₂²×C₄₇

#334

yes

—

C₂²×C₄₇

#340

yes

characteristic

C₂²×C₄₇

#365

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#335

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

C₂×#334

—

C₂×C₄₇

#1277

yes

characteristic

C₂×C₄₇

#1368

yes

—

C₂×C₄₇

#1383

yes

—

C₂²×C₁₆

Dic₄₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

C₄₇

#7156

yes

characteristic, complemented, Sylow

C₂²×C₈

—

188

—

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄₇⋊C₈

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

188

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

#10625325

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

376

—

C₂×C₈

—

188

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

#10625329

yes

—

C₂×C₈

C₄×Dic₄₇

yes

characteristic

C₂×C₈

Dic₄₇⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

C₄²

D₃₇₆

yes

characteristic

C₄²

Dic₁₈₈

yes

characteristic

C₄²

(C₂×D₄₇)⋊C₄

yes

—

C₁₆

—

C₂³

#13093956

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄₇⋊#95

yes

characteristic

C₂×C₄

#13093946

yes

—

C₂×C₄

#13093949

yes

—

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

C₂×C₄

#13093958

yes

characteristic

100

C₂×C₄

#13093960

yes

—

101

C₈

—

102

C₈

—

376

—

103

C₈

C₄×C₄₇⋊C₈

yes

characteristic

104

C₈

#13093935

yes

characteristic

105

C₈

C₂²⋊(C₄₇⋊C₈)

yes

—

106

C₂²

#15688874

yes

—

107

C₂²

#15688886

yes

characteristic

108

C₂²

#15688946

yes

—

109

C₄

—

110

C₄

—

111

C₄

#15688876

yes

—

112

C₄

#15688878

yes

characteristic

113

C₄

#15688888

yes

characteristic

114

C₂

#20241223

yes

characteristic

115

C₂

#20241454

yes

—

116

C₂

#20241487

yes

—

	subgroup	G/H
center	C₂³	#13093956
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle, prime radical	C₂³×C₄₇	#106
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	#7156
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

#13093956

commutator

C₈×C₄₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₂×C₄₇×#346

C₂

socle, prime radical

C₂³×C₄₇

#106

cosocle

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
17. C₂×C₁₆×C₄₇	C₈	semidirect
67. C₄₇	#7156	semidirect
74. C₂×C₁₆	C₄₇⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₁₆×C₄₇

C₈

semidirect

67. C₄₇

#7156

semidirect

74. C₂×C₁₆

C₄₇⋊C₈

semidirect

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

subgroup	index
C₂×#15688874	2
C₂×C₄₇×#346	2
#7156	47

subgroup

index

C₂×#15688874

C₂×C₄₇×#346

#7156

Subgroups — #26103152

Conjugacy classes of subgroups

Complements

Maximal subgroups