Group #26216529 — subgroups

Subgroups — #26216529

Order 12032.

830 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 243 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	#376×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20243533	C₂	1	1	yes	maximal, characteristic, complemented
3	#20243532	C₂	1	6	yes	maximal, complemented
4	#20243080	C₂	1	6	yes	maximal
5	C₄₇×#1840	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10625324	C₂²	1	4	yes	—
7	C₂×C₄²×D₄₇	C₂²	1	3	yes	—
8	#15688969	C₂²	1	3	yes	—
9	#15688968	C₄	1	4	yes	—
10	#95×D₄₇	C₄	1	4	yes	—
11	#15688878	C₂²	1	6	yes	—
12	#15688877	C₂²	1	12	yes	—
13	C₄₇×#376	C₂²	1	1	yes	characteristic
14	C₄₇×#336	C₂²	1	6	yes	—
15	C₂²×C₄×D₄₇	C₂×C₄	1	6	yes	—
16	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
17	C₂×#10625324	—	2	24	no	—
18	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
19	C₂×C₄×Dic₄₇	C₂×C₄	1	6	yes	—
20	C₂×C₄×Dic₄₇	C₂³	1	3	yes	—
21	C₄²×D₄₇	—	2	12	no	—
22	C₄₇⋊#95	C₂×C₄	1	12	yes	—
23	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
24	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
25	#95×C₄₇	C₂×C₄	1	4	yes	—
26	C₂³×D₄₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
28	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
29	C₂²×Dic₄₇	C₂²×C₄	1	6	yes	—
30	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×D₄₇	—	2	12	no	—
32	C₂×C₄×D₄₇	—	4	12	no	—
33	C₂×C₄×D₄₇	#41	1	12	yes	—
34	C₂×C₈×C₄₇	—	2	12	no	—
35	C₄²×C₄₇	—	2	6	no	—
36	C₄×Dic₄₇	—	2	6	no	—
37	C₄×Dic₄₇	—	2	12	no	—
38	C₂×C₄₇⋊C₈	—	2	12	no	—
39	#10625324	—	4	16	no	—
40	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
41	C₂²×D₄₇	—	2	4	no	—
42	C₂²×D₄₇	#117	1	6	yes	—
43	C₂×C₄×C₄₇	—	2	6	no	—
44	C₂×C₄×C₄₇	—	2	12	no	—
45	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
46	C₂×Dic₄₇	—	2	6	no	—
47	C₂×Dic₄₇	—	2	12	no	—
48	C₂×Dic₄₇	#117	1	6	yes	—
49	C₄×D₄₇	—	4	24	no	—
50	C₄×D₄₇	#126	1	16	yes	—
51	C₈×C₄₇	—	4	4	no	—
52	C₄₇⋊C₈	—	4	4	no	—
53	#20773	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	C₂×#117	1	3	yes	—
55	C₂²×C₄₇	#568	1	4	yes	—
56	C₂×D₄₇	—	2	12	no	—
57	C₂×D₄₇	#383	1	4	yes	—
58	C₄×C₄₇	—	4	6	no	—
59	C₄×C₄₇	C₂×#126	1	4	yes	—
60	Dic₄₇	—	4	6	no	—
61	Dic₄₇	#383	1	4	yes	—
62	C₂×#376	—	47	1	no	—
63	#1840	—	47	1	no	—
64	#1840	—	47	6	no	—
65	#1840	D₄₇	1	1	yes	characteristic, complemented
66	#1728	—	47	6	no	—
67	C₂×C₄₇	C₂×#383	1	1	yes	characteristic
68	C₂×C₄₇	#2921	1	6	yes	—
69	D₄₇	—	2	4	no	—
70	C₂²×M₁₆	—	47	4	no	—
71	C₂²×C₄²	—	47	3	no	—
72	#376	—	47	3	no	—
73	#376	—	47	4	no	—
74	#376	C₂×D₄₇	1	1	yes	characteristic
75	C₂×#95	—	47	4	no	—
76	#336	—	47	12	no	—
77	#336	—	47	6	no	—
78	#336	C₂×D₄₇	1	6	yes	—
79	C₄₇	#20773	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₄	—	47	1	no	—
81	C₂³×C₄	—	47	6	no	—
82	C₂×M₁₆	—	94	24	no	—
83	C₂²×C₈	—	47	4	no	—
84	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
85	C₂×C₄²	—	47	6	no	—
86	C₂×C₄²	—	47	3	no	—
87	C₂×C₄²	—	94	12	no	—
88	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
89	#95	—	47	12	no	—
90	#95	C₄×D₄₇	1	4	yes	—
91	C₂⁴	—	47	1	no	—
92	C₂²×C₄	—	47	1	no	—
93	C₂²×C₄	—	47	12	no	—
94	C₂²×C₄	—	47	6	no	—
95	C₂²×C₄	—	94	12	no	—
96	C₂²×C₄	—	188	12	no	—
97	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
98	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
99	M₁₆	—	188	16	no	—
100	C₂×C₈	—	2	12	no	—
101	C₂×C₈	—	94	12	no	—
102	C₄²	—	2	6	no	—
103	C₄²	—	94	12	no	—
104	C₄²	—	94	6	no	—
105	C₂³	—	47	6	no	—
106	C₂³	—	94	4	no	—
107	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
108	C₂×C₄	—	2	6	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	47	6	no	—
111	C₂×C₄	—	47	16	no	—
112	C₂×C₄	—	94	6	no	—
113	C₂×C₄	—	94	12	no	—
114	C₂×C₄	—	188	24	no	—
115	C₂×C₄	#41×D₄₇	1	6	yes	—
116	C₈	—	4	4	no	—
117	C₈	—	188	4	no	—
118	C₂²	—	47	4	no	—
119	C₂²	—	94	12	no	—
120	C₂²	#117×D₄₇	1	3	yes	—
121	C₂²	#15689888	1	4	yes	—
122	C₄	—	4	6	no	—
123	C₄	—	47	4	no	—
124	C₄	—	188	6	no	—
125	C₄	#126×D₄₇	1	4	yes	—
126	C₂	—	94	4	no	—
127	C₂	#383×D₄₇	1	1	yes	characteristic
128	C₂	#20252022	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#376×D₄₇

C₂

yes

maximal, characteristic

#20243533

C₂

yes

maximal, characteristic, complemented

#20243532

C₂

yes

maximal, complemented

#20243080

C₂

yes

maximal

C₄₇×#1840

C₂

yes

maximal, characteristic, complemented

C₂²×#10625324

C₂²

yes

—

C₂×C₄²×D₄₇

C₂²

yes

—

#15688969

C₂²

yes

—

#15688968

C₄

yes

—

#95×D₄₇

C₄

yes

—

#15688878

C₂²

yes

—

#15688877

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₄₇×#336

C₂²

yes

—

C₂²×C₄×D₄₇

C₂×C₄

yes

—

C₂²×C₄×D₄₇

C₂³

yes

characteristic

C₂×#10625324

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂³

yes

—

C₄²×D₄₇

—

C₄₇⋊#95

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂³

yes

—

#95×C₄₇

C₂×C₄

yes

—

C₂³×D₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

#117

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#117

yes

—

C₄×D₄₇

—

C₄×D₄₇

#126

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#20773

—

maximal, Sylow

C₂²×C₄₇

C₂×#117

yes

—

C₂²×C₄₇

#568

yes

—

C₂×D₄₇

—

C₂×D₄₇

#383

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#126

yes

—

Dic₄₇

—

Dic₄₇

#383

yes

—

—

—

—

D₄₇

yes

characteristic, complemented

#1728

—

C₂×C₄₇

C₂×#383

yes

characteristic

C₂×C₄₇

#2921

yes

—

D₄₇

—

C₂²×M₁₆

—

C₂²×C₄²

—

#376

—

#376

—

#376

C₂×D₄₇

yes

characteristic

C₂×#95

—

#336

—

#336

—

#336

C₂×D₄₇

yes

—

C₄₇

#20773

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

—

#95

—

#95

C₄×D₄₇

yes

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×C₄×D₄₇

yes

—

C₂²×C₄

C₂³×D₄₇

yes

characteristic

M₁₆

—

188

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₂²×C₄×D₄₇

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

188

—

115

C₂×C₄

#41×D₄₇

yes

—

116

C₈

—

117

C₈

—

188

—

118

C₂²

—

119

C₂²

—

120

C₂²

#117×D₄₇

yes

—

121

C₂²

#15689888

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

188

—

125

C₄

#126×D₄₇

yes

—

126

C₂

—

127

C₂

#383×D₄₇

yes

characteristic

128

C₂

#20252022

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1840	C₂
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1840	D₄₇
hypocenter	C₄₇	#20773
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1840

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂³×C₄

Fitting

C₄₇×#1840

C₂

prime radical

C₂³×D₄₇

C₂²×C₄

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ #1840

N	H ≅ G/N	type
2. #20243533	C₂	semidirect
3. #20243532	C₂	semidirect
5. C₄₇×#1840	C₂	semidirect
65. #1840	D₄₇	semidirect
79. C₄₇	#20773	semidirect

H ≅ G/N

type

2. #20243533

C₂

semidirect

3. #20243532

C₂

semidirect

5. C₄₇×#1840

C₂

semidirect

65. #1840

D₄₇

semidirect

79. C₄₇

#20773

semidirect

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

subgroup	index
#376×D₄₇	2
#20243533	2
#20243532	2
#20243080	2
C₄₇×#1840	2
#20773	47

subgroup

index

#376×D₄₇

C₄₇×#1840

Subgroups — #26216529

Conjugacy classes of subgroups

Complements

Maximal subgroups