Group #26106707 — subgroups

Subgroups — #26106707

Order 12032.

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

normal subgroups only

This group has 446 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	#20243134	C₂	1	2	yes	maximal
2	C₄₇⋊#1273	C₂	1	4	yes	maximal
3	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
5	#15688878	C₄	1	8	yes	—
6	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
9	C₄³×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
11	C₄×C₄₇⋊C₈	—	2	8	no	—
12	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₄₇	—	2	4	no	—
18	C₄×C₈×C₄₇	—	2	4	no	—
19	C₄×C₈×C₄₇	C₈	1	8	yes	complemented
20	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
21	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
24	C₂×C₈×C₄₇	—	2	4	no	—
25	C₂×C₈×C₄₇	—	2	4	no	—
26	C₂×C₈×C₄₇	C₂×C₈	1	8	yes	—
27	C₄²×C₄₇	—	2	8	no	—
28	C₄²×C₄₇	C₂×C₈	1	4	yes	—
29	C₄²×C₄₇	M₁₆	1	4	yes	—
30	C₄²×C₄₇	#41	1	4	yes	—
31	C₂×C₄₇⋊C₈	—	2	8	no	—
32	C₂×C₄₇⋊C₈	—	2	16	no	—
33	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
34	C₂×C₄×C₄₇	—	2	4	no	—
35	C₂×C₄×C₄₇	—	2	4	no	—
36	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
37	C₂×C₄×C₄₇	#97	1	4	yes	—
38	C₂×C₄×C₄₇	#117	1	4	yes	—
39	C₂×C₄×C₄₇	#117	1	2	yes	—
40	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
41	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
42	C₂×C₄×C₄₇	#131	1	8	yes	—
43	C₈×C₄₇	—	2	8	no	—
44	C₄₇⋊C₈	—	4	8	no	—
45	#8133	—	47	1	no	maximal, Sylow
46	C₂²×C₄₇	#376	1	1	yes	characteristic
47	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
49	C₂²×C₄₇	#405	1	4	yes	—
50	C₄×C₄₇	—	2	4	no	—
51	C₄×C₄₇	—	2	4	no	—
52	C₄×C₄₇	#431	1	4	yes	—
53	C₄×C₄₇	#433	1	8	yes	—
54	#1744	—	47	2	no	—
55	C₄²×C₈	D₄₇	1	1	yes	characteristic
56	#1273	—	47	4	no	—
57	C₂×C₄₇	#1450	1	2	yes	—
58	C₂×C₄₇	#1452	1	2	yes	—
59	C₂×C₄₇	#1744	1	1	yes	characteristic
60	C₂×C₄₇	#1746	1	2	yes	—
61	C₂×C₄×C₈	—	47	4	no	—
62	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
63	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
64	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
65	C₄³	C₂×D₄₇	1	1	yes	characteristic
66	#336	—	47	8	no	—
67	C₄₇	#8133	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	—	47	8	no	—
69	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
70	C₂×C₄²	C₄×D₄₇	1	4	yes	—
71	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
72	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
73	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
74	C₄×C₈	—	2	4	no	—
75	C₄×C₈	—	2	4	no	—
76	C₄×C₈	—	94	8	no	—
77	C₄×C₈	C₄₇⋊C₈	1	8	yes	complemented
78	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
79	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
80	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
81	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	94	8	no	—
85	C₂×C₈	—	94	16	no	—
86	C₂×C₈	C₂×C₄₇⋊C₈	1	8	yes	—
87	C₄²	—	2	8	no	—
88	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
89	C₄²	#10625329	1	4	yes	—
90	C₄²	#10625350	1	4	yes	—
91	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
95	C₂×C₄	#13093935	1	4	yes	—
96	C₂×C₄	#13093981	1	4	yes	—
97	C₂×C₄	#13093983	1	2	yes	—
98	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
99	C₂×C₄	C₂×#10625329	1	2	yes	—
100	C₂×C₄	#13094030	1	8	yes	—
101	C₈	—	2	8	no	—
102	C₈	—	188	8	no	—
103	C₂²	#15688969	1	1	yes	characteristic
104	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
105	C₂²	C₂×#13093935	1	1	yes	characteristic
106	C₂²	#15689078	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	2	4	no	—
109	C₄	#15689176	1	4	yes	—
110	C₄	#15689182	1	8	yes	—
111	C₂	#20241671	1	2	yes	—
112	C₂	#20241680	1	2	yes	—
113	C₂	#20243134	1	1	yes	characteristic
114	C₂	#20243139	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243134

C₂

yes

maximal

C₄₇⋊#1273

C₂

yes

maximal

C₄²×C₈×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688878

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₈

yes

complemented

C₂²×C₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₄₇

C₂×#97

yes

characteristic

C₂²×C₄₇

#405

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#431

yes

—

C₄×C₄₇

#433

yes

—

#1744

—

C₄²×C₈

D₄₇

yes

characteristic

#1273

—

C₂×C₄₇

#1450

yes

—

C₂×C₄₇

#1452

yes

—

C₂×C₄₇

#1744

yes

characteristic

C₂×C₄₇

#1746

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

C₄₇

#8133

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄₇⋊C₈

yes

complemented

C₂²×C₄

C₄×Dic₄₇

yes

—

C₂²×C₄

C₄×Dic₄₇

yes

—

C₂²×C₄

C₂×C₄×D₄₇

yes

—

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

C₄²

—

C₄²

C₂×C₄₇⋊C₈

yes

—

C₄²

#10625329

yes

—

C₄²

#10625350

yes

—

C₂³

C₂×C₄×Dic₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₄₇⋊C₈

yes

—

C₂×C₄

#13093935

yes

—

C₂×C₄

#13093981

yes

—

C₂×C₄

#13093983

yes

—

C₂×C₄

C₂²×C₄₇⋊C₈

yes

—

C₂×C₄

C₂×#10625329

yes

—

100

C₂×C₄

#13094030

yes

—

101

C₈

—

102

C₈

—

188

—

103

C₂²

#15688969

yes

characteristic

104

C₂²

C₂×C₄×C₄₇⋊C₈

yes

characteristic

105

C₂²

C₂×#13093935

yes

characteristic

106

C₂²

#15689078

yes

—

107

C₄

—

108

C₄

—

109

C₄

#15689176

yes

—

110

C₄

#15689182

yes

—

111

C₂

#20241671

yes

—

112

C₂

#20241680

yes

—

113

C₂

#20243134

yes

characteristic

114

C₂

#20243139

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#8133
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂×C₄×C₈

Fitting

C₄²×C₈×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄²

cosocle

C₂×C₄²×C₄₇

C₂³

hypercenter

C₄²×C₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄² ◁ C₄²×C₈

N	H ≅ G/N	type
19. C₄×C₈×C₄₇	C₈	semidirect
67. C₄₇	#8133	semidirect
77. C₄×C₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

19. C₄×C₈×C₄₇

C₈

semidirect

67. C₄₇

#8133

semidirect

77. C₄×C₈

C₄₇⋊C₈

semidirect

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

subgroup	index
#20243134	2
C₄₇⋊#1273	2
C₄²×C₈×C₄₇	2
#8133	47

subgroup

index

#20243134

C₄₇⋊#1273

C₄²×C₈×C₄₇

#8133

Subgroups — #26106707

Conjugacy classes of subgroups

Complements

Maximal subgroups