Group #26106698 — subgroups

Subgroups — #26106698

Order 12032.

520 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 151 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	#20252020	C₂	1	3	yes	maximal
2	C₄₇⋊#1309	C₂	1	3	yes	maximal, complemented
3	C₄₇×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
5	#15689179	—	2	6	no	—
6	#15689175	—	2	6	no	—
7	#15689147	C₂²	1	3	yes	—
8	#15689086	C₄	1	6	yes	—
9	M₁₆×Dic₄₇	C₄	1	6	yes	—
10	C₄₇×#423	C₂²	1	3	yes	—
11	C₂²×C₄×D₄₇	C₂×C₄	1	3	yes	—
12	C₂×#10625329	C₂×C₄	1	6	yes	—
13	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
14	C₄²×D₄₇	D₄	1	6	yes	—
15	C₄²×D₄₇	Q₈	1	2	yes	—
16	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
17	(C₂×D₄₇)⋊C₈	—	2	12	no	—
18	#13093934	—	2	6	no	—
19	C₈×Dic₄₇	—	2	6	no	—
20	C₂×M₁₆×C₄₇	C₂×C₄	1	6	yes	—
21	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	—	2	6	no	—
23	C₂³×D₄₇	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
25	C₂²×Dic₄₇	C₄²	1	3	yes	—
26	C₂×C₄×D₄₇	—	4	3	no	—
27	C₂×C₄×D₄₇	—	4	3	no	—
28	C₂×C₄×D₄₇	C₂²⋊C₄	1	6	yes	—
29	C₂×C₄×D₄₇	C₄⋊C₄	1	6	yes	—
30	M₁₆×C₄₇	—	4	6	no	—
31	C₂×C₈×C₄₇	—	2	6	no	—
32	C₄²×C₄₇	C₂×D₄	1	3	yes	—
33	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×Dic₄₇	C₂²⋊C₄	1	3	yes	—
35	C₄×Dic₄₇	C₂²⋊C₄	1	3	yes	—
36	C₄×Dic₄₇	C₄⋊C₄	1	6	yes	—
37	#10625329	—	4	6	no	—
38	C₂×C₄₇⋊C₈	—	2	6	no	—
39	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
40	C₂²×D₄₇	—	4	3	no	—
41	C₂²×D₄₇	#95	1	2	yes	—
42	C₂×C₄×C₄₇	—	2	3	no	—
43	C₂×C₄×C₄₇	—	2	3	no	—
44	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
45	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
46	C₂×Dic₄₇	—	4	3	no	—
47	C₂×Dic₄₇	#95	1	6	yes	—
48	C₄×D₄₇	—	2	6	no	—
49	C₄×D₄₇	—	2	6	no	—
50	C₄×D₄₇	—	4	6	no	—
51	C₈×C₄₇	—	4	6	no	—
52	C₄₇⋊C₈	—	4	6	no	—
53	#8130	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	—	2	3	no	—
55	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
56	C₂×D₄₇	—	2	6	no	—
57	C₂×D₄₇	—	4	4	no	—
58	C₄×C₄₇	—	2	3	no	—
59	C₄×C₄₇	#413	1	3	yes	—
60	C₄×C₄₇	#413	1	3	yes	—
61	Dic₄₇	—	2	6	no	—
62	#2920	—	47	3	no	—
63	#1309	—	47	3	no	—
64	#1309	D₄₇	1	1	yes	characteristic, complemented
65	C₂×C₄₇	—	4	1	no	—
66	C₂×C₄₇	#1742	1	3	yes	—
67	D₄₇	—	4	2	no	—
68	C₂²×C₄²	—	47	1	no	—
69	#432	—	94	6	no	—
70	#431	—	94	6	no	—
71	#423	—	47	3	no	—
72	#423	C₂×D₄₇	1	3	yes	—
73	#407	—	47	6	no	—
74	C₄×M₁₆	—	47	6	no	—
75	C₄₇	#8130	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₄	—	47	3	no	—
77	C₂×M₁₆	—	47	6	no	—
78	C₂×M₁₆	C₄×D₄₇	1	6	yes	—
79	C₂×C₄²	—	47	3	no	—
80	C₂×C₄²	—	47	6	no	—
81	C₂×C₄²	—	47	2	no	—
82	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
83	C₄⋊C₈	—	2	6	no	—
84	C₄⋊C₈	—	94	6	no	—
85	C₂²⋊C₈	—	94	12	no	—
86	#97	—	94	6	no	—
87	C₄×C₈	—	94	6	no	—
88	C₂⁴	—	47	1	no	—
89	C₂²×C₄	—	47	3	no	—
90	C₂²×C₄	—	47	6	no	—
91	C₂²×C₄	—	47	6	no	—
92	C₂²×C₄	—	188	3	no	—
93	C₂²×C₄	—	188	3	no	—
94	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
95	M₁₆	—	4	6	no	—
96	M₁₆	—	188	6	no	—
97	C₂×C₈	—	2	6	no	—
98	C₂×C₈	—	94	6	no	—
99	C₄²	—	47	3	no	—
100	C₄²	—	47	3	no	—
101	C₄²	—	47	6	no	—
102	C₄²	D₄×D₄₇	1	3	yes	—
103	C₄²	Q₈×D₄₇	1	1	yes	characteristic
104	C₂³	—	47	2	no	—
105	C₂³	—	188	3	no	—
106	C₂³	C₄²×D₄₇	1	1	yes	characteristic
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	47	6	no	—
110	C₂×C₄	—	94	6	no	—
111	C₂×C₄	—	94	6	no	—
112	C₂×C₄	—	188	6	no	—
113	C₂×C₄	—	188	3	no	—
114	C₂×C₄	C₂²⋊C₄×D₄₇	1	3	yes	—
115	C₂×C₄	C₄⋊C₄×D₄₇	1	3	yes	—
116	C₈	—	4	6	no	—
117	C₈	—	188	6	no	—
118	C₂²	—	2	3	no	—
119	C₂²	—	94	6	no	—
120	C₂²	—	188	4	no	—
121	C₂²	#95×D₄₇	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	94	6	no	—
124	C₄	#15689104	1	3	yes	—
125	C₄	#15689106	1	3	yes	—
126	C₂	—	4	1	no	—
127	C₂	—	188	2	no	—
128	C₂	#20243125	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252020

C₂

yes

maximal

C₄₇⋊#1309

C₂

yes

maximal, complemented

C₄₇×#1309

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₄₇

C₂²

yes

—

—

C₂²

yes

—

#15689086

C₄

yes

—

M₁₆×Dic₄₇

C₄

yes

—

C₄₇×#423

C₂²

yes

—

C₂²×C₄×D₄₇

C₂×C₄

yes

—

C₂×#10625329

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₄²×D₄₇

D₄

yes

—

C₄²×D₄₇

Q₈

yes

—

C₄⋊(C₄₇⋊C₈)

—

(C₂×D₄₇)⋊C₈

—

#13093934

—

C₈×Dic₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂³×D₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

C₄²

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂²⋊C₄

yes

—

C₂×C₄×D₄₇

C₄⋊C₄

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄×Dic₄₇

C₂²⋊C₄

yes

—

C₄×Dic₄₇

C₂²⋊C₄

yes

—

C₄×Dic₄₇

C₄⋊C₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄²

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

#95

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#95

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#8130

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#95

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#413

yes

—

C₄×C₄₇

#413

yes

—

Dic₄₇

—

#2920

—

#1309

—

#1309

D₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

#1742

yes

—

D₄₇

—

C₂²×C₄²

—

—

—

—

C₂×D₄₇

yes

—

#407

—

C₄×M₁₆

—

C₄₇

#8130

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×C₄×D₄₇

yes

—

M₁₆

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

D₄×D₄₇

yes

—

103

C₄²

Q₈×D₄₇

yes

characteristic

104

C₂³

—

105

C₂³

—

188

—

106

C₂³

C₄²×D₄₇

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

188

—

113

C₂×C₄

—

188

—

114

C₂×C₄

C₂²⋊C₄×D₄₇

yes

—

115

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

116

C₈

—

117

C₈

—

188

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

188

—

121

C₂²

#95×D₄₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

#15689104

yes

—

125

C₄

#15689106

yes

—

126

C₂

—

127

C₂

—

188

—

128

C₂

#20243125

yes

—

	subgroup	G/H
center	C₂²	#95×D₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1309	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂³×D₄₇	C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1309	D₄₇
hypocenter	C₄₇	#8130
LCS	G ▷ C₂³×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#95×D₄₇

commutator

C₂³×C₄₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1309

C₂

socle

C₂²×C₄₇

C₂×#95

prime radical

C₂³×D₄₇

C₄²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
2. C₄₇⋊#1309	C₂	semidirect
3. C₄₇×#1309	C₂	semidirect
64. #1309	D₄₇	semidirect
75. C₄₇	#8130	semidirect

H ≅ G/N

type

2. C₄₇⋊#1309

C₂

semidirect

3. C₄₇×#1309

C₂

semidirect

64. #1309

D₄₇

semidirect

75. C₄₇

#8130

semidirect

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

subgroup	index
#20252020	2
C₄₇⋊#1309	2
C₄₇×#1309	2
#8130	47

subgroup

index

#20252020

C₄₇⋊#1309

C₄₇×#1309

#8130

Subgroups — #26106698

Conjugacy classes of subgroups

Complements

Maximal subgroups