Group #26105668 — subgroups

Subgroups — #26105668

Order 12032.

610 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 207 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×#15689174	C₂	1	3	yes	maximal
2	C₄₇⋊#1273	C₂	1	3	yes	maximal, complemented
3	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
5	#15689174	—	2	12	no	—
6	C₂×Dic₄₇⋊C₈	C₄	1	6	yes	—
7	C₂×(C₂×D₄₇)⋊C₈	C₄	1	6	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
9	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
10	C₂²×C₄×D₄₇	C₂×C₄	1	3	yes	—
11	C₂²×C₄₇⋊C₈	C₂×C₄	1	6	yes	—
12	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
13	C₄²×D₄₇	D₄	1	6	yes	—
14	C₄²×D₄₇	Q₈	1	2	yes	—
15	Dic₄₇⋊C₈	—	2	12	no	—
16	(C₂×D₄₇)⋊C₈	—	2	12	no	—
17	C₄×C₄₇⋊C₈	—	2	6	no	—
18	C₂²×C₈×C₄₇	C₂×C₄	1	6	yes	—
19	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₄₇	—	2	6	no	—
21	C₂³×D₄₇	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
23	C₂²×Dic₄₇	C₄²	1	3	yes	—
24	C₂×C₄×D₄₇	—	2	6	no	—
25	C₂×C₄×D₄₇	—	2	6	no	—
26	C₂×C₄×D₄₇	C₂²⋊C₄	1	6	yes	—
27	C₂×C₄×D₄₇	C₄⋊C₄	1	6	yes	—
28	C₂×C₈×C₄₇	—	2	6	no	—
29	C₂×C₈×C₄₇	—	2	6	no	—
30	C₂×C₈×C₄₇	—	2	6	no	—
31	C₄²×C₄₇	C₂×D₄	1	3	yes	—
32	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
33	C₄×Dic₄₇	C₂²⋊C₄	1	6	yes	—
34	C₄×Dic₄₇	C₄⋊C₄	1	6	yes	—
35	C₂×C₄₇⋊C₈	—	2	6	no	—
36	C₂×C₄₇⋊C₈	—	2	6	no	—
37	C₂×C₄₇⋊C₈	—	2	6	no	—
38	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
39	C₂²×D₄₇	—	2	3	no	—
40	C₂²×D₄₇	—	2	3	no	—
41	C₂²×D₄₇	#95	1	2	yes	—
42	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
43	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
44	C₂×C₄×C₄₇	#131	1	6	yes	—
45	C₂×C₄×C₄₇	#131	1	6	yes	—
46	C₂×Dic₄₇	—	2	3	no	—
47	C₂×Dic₄₇	—	2	3	no	—
48	C₂×Dic₄₇	#95	1	6	yes	—
49	C₄×D₄₇	—	2	12	no	—
50	C₄×D₄₇	—	2	12	no	—
51	C₈×C₄₇	—	4	6	no	—
52	C₄₇⋊C₈	—	4	6	no	—
53	#7851	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
55	C₂²×C₄₇	#405	1	3	yes	—
56	C₂²×C₄₇	#405	1	3	yes	—
57	C₂×D₄₇	—	2	6	no	—
58	C₂×D₄₇	—	4	1	no	—
59	C₂×D₄₇	—	4	3	no	—
60	C₄×C₄₇	#408	1	6	yes	—
61	C₄×C₄₇	#424	1	6	yes	—
62	Dic₄₇	—	2	6	no	—
63	C₂×#431	—	47	3	no	—
64	#1273	—	47	3	no	—
65	#1273	D₄₇	1	1	yes	characteristic, complemented
66	C₂×C₄₇	#1454	1	1	yes	characteristic
67	C₂×C₄₇	#1454	1	3	yes	—
68	C₂×C₄₇	#1729	1	3	yes	—
69	D₄₇	—	4	2	no	—
70	C₂²×C₄²	—	47	1	no	—
71	#431	—	94	12	no	—
72	C₂×C₄⋊C₈	—	47	6	no	—
73	C₂×C₂²⋊C₈	—	47	6	no	—
74	C₂×C₄×C₈	—	47	3	no	—
75	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
76	C₄₇	#7851	1	1	yes	characteristic, complemented, Sylow
77	C₂³×C₄	—	47	3	no	—
78	C₂²×C₈	—	47	6	no	—
79	C₂²×C₈	C₄×D₄₇	1	6	yes	—
80	C₂×C₄²	—	47	3	no	—
81	C₂×C₄²	—	47	6	no	—
82	C₂×C₄²	—	47	2	no	—
83	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
84	C₄⋊C₈	—	94	12	no	—
85	C₂²⋊C₈	—	94	12	no	—
86	C₄×C₈	—	2	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	—	94	6	no	—
93	C₂²×C₄	—	94	6	no	—
94	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
95	C₂×C₈	—	2	6	no	—
96	C₂×C₈	—	2	6	no	—
97	C₂×C₈	—	2	6	no	—
98	C₂×C₈	—	94	6	no	—
99	C₂×C₈	—	94	6	no	—
100	C₂×C₈	—	94	6	no	—
101	C₄²	—	47	6	no	—
102	C₄²	—	47	6	no	—
103	C₄²	D₄×D₄₇	1	3	yes	—
104	C₄²	Q₈×D₄₇	1	1	yes	characteristic
105	C₂³	—	47	2	no	—
106	C₂³	—	94	3	no	—
107	C₂³	—	94	3	no	—
108	C₂³	C₄²×D₄₇	1	1	yes	characteristic
109	C₂×C₄	—	47	6	no	—
110	C₂×C₄	—	94	3	no	—
111	C₂×C₄	—	94	3	no	—
112	C₂×C₄	—	94	12	no	—
113	C₂×C₄	—	94	12	no	—
114	C₂×C₄	C₂²⋊C₄×D₄₇	1	3	yes	—
115	C₂×C₄	C₄⋊C₄×D₄₇	1	3	yes	—
116	C₂×C₄	#13094030	1	6	yes	—
117	C₂×C₄	#13094032	1	6	yes	—
118	C₈	—	4	6	no	—
119	C₈	—	188	6	no	—
120	C₂²	—	94	6	no	—
121	C₂²	—	188	1	no	—
122	C₂²	—	188	3	no	—
123	C₂²	#95×D₄₇	1	1	yes	characteristic
124	C₂²	#15689077	1	3	yes	—
125	C₂²	#15689080	1	3	yes	—
126	C₄	—	94	6	no	—
127	C₄	#15689088	1	6	yes	—
128	C₄	#15689148	1	6	yes	—
129	C₂	—	188	2	no	—
130	C₂	#20241687	1	1	yes	characteristic
131	C₂	#20241689	1	3	yes	—
132	C₂	#20243083	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689174

C₂

yes

maximal

C₄₇⋊#1273

C₂

yes

maximal, complemented

C₄₇×#1273

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₄₇

C₂²

yes

characteristic

#15689174

—

C₂×Dic₄₇⋊C₈

C₄

yes

—

C₂×(C₂×D₄₇)⋊C₈

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄×D₄₇

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₄²×D₄₇

D₄

yes

—

C₄²×D₄₇

Q₈

yes

—

Dic₄₇⋊C₈

—

(C₂×D₄₇)⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×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

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄×Dic₄₇

C₂²⋊C₄

yes

—

C₄×Dic₄₇

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄²

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

#95

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#95

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7851

—

maximal, Sylow

C₂²×C₄₇

C₂×#95

yes

characteristic

C₂²×C₄₇

#405

yes

—

C₂²×C₄₇

#405

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#408

yes

—

C₄×C₄₇

#424

yes

—

Dic₄₇

—

C₂×#431

—

#1273

—

#1273

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#1454

yes

characteristic

C₂×C₄₇

#1454

yes

—

C₂×C₄₇

#1729

yes

—

D₄₇

—

C₂²×C₄²

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄₇

#7851

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₄×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄²

—

102

C₄²

—

103

C₄²

D₄×D₄₇

yes

—

104

C₄²

Q₈×D₄₇

yes

characteristic

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₄²×D₄₇

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²⋊C₄×D₄₇

yes

—

115

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

116

C₂×C₄

#13094030

yes

—

117

C₂×C₄

#13094032

yes

—

118

C₈

—

119

C₈

—

188

—

120

C₂²

—

121

C₂²

—

188

—

122

C₂²

—

188

—

123

C₂²

#95×D₄₇

yes

characteristic

124

C₂²

#15689077

yes

—

125

C₂²

#15689080

yes

—

126

C₄

—

127

C₄

#15689088

yes

—

128

C₄

#15689148

yes

—

129

C₂

—

188

—

130

C₂

#20241687

yes

characteristic

131

C₂

#20241689

yes

—

132

C₂

#20243083

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂×C₄²
Fitting	C₄₇×#1273	C₂
prime radical	C₂³×D₄₇	C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#7851
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₇

commutator, socle

C₂³×C₄₇

C₂×C₄²

Fitting

C₄₇×#1273

C₂

prime radical

C₂³×D₄₇

C₄²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. C₄₇⋊#1273	C₂	semidirect
3. C₄₇×#1273	C₂	semidirect
65. #1273	D₄₇	semidirect
76. C₄₇	#7851	semidirect

H ≅ G/N

type

2. C₄₇⋊#1273

C₂

semidirect

3. C₄₇×#1273

C₂

semidirect

65. #1273

D₄₇

semidirect

76. C₄₇

#7851

semidirect

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

subgroup	index
C₂×#15689174	2
C₄₇⋊#1273	2
C₄₇×#1273	2
#7851	47

subgroup

index

C₂×#15689174

C₄₇⋊#1273

C₄₇×#1273

#7851

Subgroups — #26105668

Conjugacy classes of subgroups

Complements

Maximal subgroups