Group #26131280 — subgroups

Subgroups — #26131280

Order 12032.

390 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 227 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×#15688899	C₂	1	2	yes	maximal
2	#20241466	C₂	1	4	yes	maximal
3	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
5	#15688899	C₄	1	8	yes	complemented
6	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₄³×C₄₇	C₄	1	1	yes	characteristic
11	C₂×C₄₇⋊C₁₆	—	2	8	no	—
12	C₂×C₄₇⋊C₁₆	C₂×C₄	1	8	yes	—
13	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	C₈	1	4	yes	—
17	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₄₇	—	2	4	no	—
20	C₄×C₈×C₄₇	—	2	4	no	—
21	C₄×C₈×C₄₇	C₈	1	4	yes	—
22	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	—	2	2	no	—
28	C₂×C₈×C₄₇	C₄²	1	2	yes	—
29	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
30	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
31	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
32	C₂×C₈×C₄₇	#41	1	4	yes	—
33	C₄²×C₄₇	—	2	8	no	—
34	C₄²×C₄₇	—	2	2	no	—
35	C₄²×C₄₇	—	2	2	no	—
36	C₄²×C₄₇	C₄²	1	2	yes	—
37	C₄²×C₄₇	C₂×C₈	1	2	yes	—
38	C₄₇⋊C₁₆	—	2	8	no	—
39	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₄×C₄₇	—	2	4	no	—
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	—	2	4	no	—
43	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
44	C₂×C₄×C₄₇	M₃₂	1	8	yes	—
45	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#117	1	2	yes	—
47	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
49	C₈×C₄₇	—	2	4	no	—
50	C₈×C₄₇	#97	1	4	yes	—
51	C₈×C₄₇	#117	1	4	yes	—
52	#12591	—	47	1	no	maximal, Sylow
53	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
54	C₂²×C₄₇	#431	1	2	yes	—
55	C₂²×C₄₇	C₂×M₃₂	1	4	yes	—
56	C₄×C₄₇	—	2	4	no	—
57	C₄×C₄₇	—	2	8	no	—
58	C₄×C₄₇	#376	1	1	yes	characteristic
59	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
60	C₄×C₄₇	#431	1	2	yes	—
61	C₂×#346	—	47	2	no	—
62	C₄²×C₈	D₄₇	1	1	yes	characteristic
63	#1374	—	47	4	no	—
64	C₂×C₄₇	#1744	1	1	yes	characteristic
65	C₂×C₄₇	C₄×M₃₂	1	2	yes	—
66	C₂×C₄₇	#2163	1	4	yes	—
67	C₂²×C₁₆	—	47	4	no	—
68	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
69	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
71	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
72	C₄³	Dic₄₇	1	1	yes	characteristic
73	#346	—	47	8	no	—
74	C₄₇	#12591	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
76	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
78	C₂×C₄²	C₄₇⋊C₈	1	4	yes	—
79	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
80	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
81	C₂×C₁₆	—	47	8	no	—
82	C₂×C₁₆	—	94	8	no	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	C₄₇⋊C₈	1	4	yes	—
86	C₄×C₈	C₄×D₄₇	1	4	yes	—
87	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
88	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
89	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
93	C₂×C₈	#10625329	1	4	yes	—
94	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
95	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
96	C₂×C₈	#10625350	1	4	yes	—
97	C₄²	—	2	8	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
101	C₄²	C₄×Dic₄₇	1	2	yes	—
102	C₁₆	—	94	8	no	—
103	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
108	C₂×C₄	#13093964	1	8	yes	—
109	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
110	C₂×C₄	#13093983	1	2	yes	—
111	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
112	C₂×C₄	C₂×#10625329	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	#13093935	1	4	yes	—
115	C₈	#13093981	1	4	yes	—
116	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
117	C₂²	#15689176	1	2	yes	—
118	C₂²	C₂×#13093964	1	4	yes	—
119	C₄	—	2	8	no	—
120	C₄	—	2	4	no	—
121	C₄	#15688969	1	1	yes	characteristic
122	C₄	C₂×#13093935	1	1	yes	characteristic
123	C₄	#15689176	1	2	yes	—
124	C₂	#20243134	1	1	yes	characteristic
125	C₂	C₄×#13093964	1	2	yes	—
126	C₂	#20245191	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688899

C₂

yes

maximal

#20241466

C₂

yes

maximal

C₄²×C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

#15688899

C₄

yes

complemented

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₄

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄³×C₄₇

C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₈

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₄²

yes

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

M₃₂

yes

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

#97

yes

—

C₈×C₄₇

#117

yes

—

#12591

—

maximal, Sylow

C₂²×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₄₇

#431

yes

—

C₂²×C₄₇

C₂×M₃₂

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#376

yes

characteristic

C₄×C₄₇

C₂×#97

yes

characteristic

C₄×C₄₇

#431

yes

—

C₂×#346

—

C₄²×C₈

D₄₇

yes

characteristic

#1374

—

C₂×C₄₇

#1744

yes

characteristic

C₂×C₄₇

C₄×M₃₂

yes

—

C₂×C₄₇

#2163

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₂×C₄×C₈

Dic₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄³

Dic₄₇

yes

characteristic

#346

—

C₄₇

#12591

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

C₄₇⋊C₈

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄₇⋊C₈

yes

—

C₄×C₈

C₄×D₄₇

yes

—

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

#10625329

yes

—

C₂×C₈

C₄×Dic₄₇

yes

—

C₂×C₈

C₂×C₄×D₄₇

yes

—

C₂×C₈

#10625350

yes

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂×C₄₇⋊C₈

yes

—

101

C₄²

C₄×Dic₄₇

yes

—

102

C₁₆

—

103

C₂³

C₂²×C₄₇⋊C₈

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×C₄₇⋊C₈

yes

—

108

C₂×C₄

#13093964

yes

—

109

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

110

C₂×C₄

#13093983

yes

—

111

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

112

C₂×C₄

C₂×#10625329

yes

—

113

C₈

—

114

C₈

#13093935

yes

—

115

C₈

#13093981

yes

—

116

C₂²

C₂×C₄×C₄₇⋊C₈

yes

characteristic

117

C₂²

#15689176

yes

—

118

C₂²

C₂×#13093964

yes

—

119

C₄

—

120

C₄

—

121

C₄

#15688969

yes

characteristic

122

C₄

C₂×#13093935

yes

characteristic

123

C₄

#15689176

yes

—

124

C₂

#20243134

yes

characteristic

125

C₂

C₄×#13093964

yes

—

126

C₂

#20245191

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#12591
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂×C₄×C₈

Fitting

C₄²×C₈×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

C₂²×C₈

cosocle

C₂²×C₈×C₄₇

C₂³

hypercenter

C₄²×C₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
5. #15688899	C₄	semidirect
74. C₄₇	#12591	semidirect

H ≅ G/N

type

5. #15688899

C₄

semidirect

74. C₄₇

#12591

semidirect

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

subgroup	index
C₂×#15688899	2
#20241466	2
C₄²×C₈×C₄₇	2
#12591	47

subgroup

index

C₂×#15688899

#20241466

C₄²×C₈×C₄₇

#12591

Subgroups — #26131280

Conjugacy classes of subgroups

Complements

Maximal subgroups