Group #26106740 — subgroups

Subgroups — #26106740

Order 12032.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20241680	C₂	1	1	yes	maximal, characteristic
2	#20241671	C₂	1	1	yes	maximal, characteristic
3	#20241219	C₂	1	2	yes	maximal
4	#20241217	C₂	1	2	yes	maximal
5	C₄₇×#1746	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	#15689176	C₂²	1	2	yes	—
8	#15688845	C₄	1	4	yes	complemented
9	#15688844	C₄	1	2	yes	complemented
10	C₈×C₄₇⋊C₈	C₄	1	2	yes	complemented
11	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#376	C₂²	1	1	yes	characteristic
14	C₄₇×#336	C₄	1	4	yes	—
15	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	#97×C₄₇	—	2	2	no	—
26	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
29	#95×C₄₇	—	2	2	no	—
30	C₂²×C₄×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	#41	1	2	yes	—
33	C₂×C₈×C₄₇	—	2	4	no	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	4	no	—
36	C₂×C₈×C₄₇	C₄²	1	2	yes	—
37	C₂×C₈×C₄₇	C₄²	1	2	yes	—
38	C₄²×C₄₇	—	2	4	no	—
39	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₄₇	#41	1	2	yes	—
42	C₂×C₄₇⋊C₈	—	2	4	no	—
43	C₂×C₄₇⋊C₈	—	2	4	no	—
44	C₂³×C₄₇	#117	1	1	yes	characteristic
45	C₂×C₄×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#117	1	2	yes	—
50	C₂×C₄×C₄₇	#117	1	2	yes	—
51	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#131	1	4	yes	—
53	C₈×C₄₇	—	2	2	no	—
54	C₈×C₄₇	—	2	2	no	—
55	C₈×C₄₇	—	4	2	no	—
56	C₄₇⋊C₈	—	4	4	no	—
57	C₄₇⋊C₈	—	4	4	no	—
58	#8142	—	47	1	no	maximal, Sylow
59	C₂²×C₄₇	—	2	2	no	—
60	C₂²×C₄₇	#376	1	1	yes	characteristic
61	C₂²×C₄₇	#433	1	2	yes	—
62	C₄×C₄₇	—	2	2	no	—
63	C₄×C₄₇	—	2	2	no	—
64	C₄×C₄₇	—	8	2	no	—
65	C₄×C₄₇	#405	1	2	yes	—
66	C₄×C₄₇	#433	1	2	yes	—
67	#1746	D₄₇	1	1	yes	characteristic
68	#1452	—	47	1	no	—
69	#1450	—	47	1	no	—
70	#1275	—	47	2	no	—
71	#1274	—	47	2	no	—
72	C₂×C₄₇	—	2	2	no	—
73	C₂×C₄₇	#1746	1	1	yes	characteristic
74	C₂×C₄₇	#1747	1	2	yes	—
75	#431	—	47	2	no	—
76	#431	—	47	2	no	—
77	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
79	#376	C₂×D₄₇	1	1	yes	characteristic
80	#336	Dic₄₇	1	4	yes	—
81	#322	—	47	2	no	—
82	#322	—	47	4	no	—
83	C₈²	—	47	2	no	—
84	C₄₇	#8142	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
86	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
87	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
88	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
89	C₄⋊C₈	—	94	2	no	—
90	C₄⋊C₈	—	94	2	no	—
91	C₂²⋊C₈	—	94	2	no	—
92	C₂²⋊C₈	—	94	2	no	—
93	#97	—	2	2	no	—
94	C₄×C₈	—	47	4	no	—
95	C₄×C₈	—	47	4	no	—
96	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
97	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
98	C₄×C₈	C₄×D₄₇	1	2	yes	—
99	#95	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	#10625350	1	2	yes	—
102	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	94	4	no	—
107	C₂×C₈	—	94	4	no	—
108	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
109	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
110	C₄²	—	2	4	no	—
111	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
112	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
113	C₄²	#10625350	1	2	yes	—
114	C₂³	#13093983	1	1	yes	characteristic
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
119	C₂×C₄	#13093981	1	2	yes	—
120	C₂×C₄	#13093981	1	2	yes	—
121	C₂×C₄	#13093983	1	1	yes	characteristic
122	C₂×C₄	#13094031	1	4	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	2	no	—
126	C₈	—	188	4	no	—
127	C₈	—	188	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#15688969	1	1	yes	characteristic
130	C₂²	#15689184	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	8	2	no	—
134	C₄	#15689079	1	2	yes	—
135	C₄	#15689184	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#20243140	1	1	yes	characteristic
138	C₂	#20243143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20241680

C₂

yes

maximal, characteristic

#20241671

C₂

yes

maximal, characteristic

#20241219

C₂

yes

maximal

#20241217

C₂

yes

maximal

C₄₇×#1746

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

#15689176

C₂²

yes

—

#15688845

C₄

yes

complemented

#15688844

C₄

yes

complemented

C₈×C₄₇⋊C₈

C₄

yes

complemented

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#376

C₂²

yes

characteristic

C₄₇×#336

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#97×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

—

C₂×C₈×C₄₇

C₄²

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8142

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#376

yes

characteristic

C₂²×C₄₇

#433

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#405

yes

—

C₄×C₄₇

#433

yes

—

#1746

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#1746

yes

characteristic

C₂×C₄₇

#1747

yes

—

#431

—

#431

—

C₂×#97

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#376

C₂×D₄₇

yes

characteristic

#336

Dic₄₇

yes

—

#322

—

#322

—

C₈²

—

C₄₇

#8142

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

C₄×D₄₇

yes

—

#95

—

100

C₂²×C₄

—

101

C₂²×C₄

#10625350

yes

—

102

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₄×Dic₄₇

yes

—

109

C₂×C₈

C₄×Dic₄₇

yes

—

110

C₄²

—

111

C₄²

C₂×C₄×D₄₇

yes

characteristic

112

C₄²

C₂×C₄×D₄₇

yes

characteristic

113

C₄²

#10625350

yes

—

114

C₂³

#13093983

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

119

C₂×C₄

#13093981

yes

—

120

C₂×C₄

#13093981

yes

—

121

C₂×C₄

#13093983

yes

characteristic

122

C₂×C₄

#13094031

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

188

—

127

C₈

—

188

—

128

C₂²

—

129

C₂²

#15688969

yes

characteristic

130

C₂²

#15689184

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#15689079

yes

—

135

C₄

#15689184

yes

—

136

C₂

—

137

C₂

#20243140

yes

characteristic

138

C₂

#20243143

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1746	C₂
socle	C₂²×C₄₇	#376
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1746	D₄₇
hypocenter	C₄₇	#8142
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #1746

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₄₇

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1746

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #1746

N	H ≅ G/N	type
8. #15688845	C₄	semidirect
9. #15688844	C₄	semidirect
10. C₈×C₄₇⋊C₈	C₄	semidirect
84. C₄₇	#8142	semidirect

H ≅ G/N

type

8. #15688845

C₄

semidirect

9. #15688844

C₄

semidirect

10. C₈×C₄₇⋊C₈

C₄

semidirect

84. C₄₇

#8142

semidirect

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

subgroup	index
#20241680	2
#20241671	2
#20241219	2
#20241217	2
C₄₇×#1746	2
#8142	47

subgroup

index

C₄₇×#1746

Subgroups — #26106740

Conjugacy classes of subgroups

Complements

Maximal subgroups