Group #26132645 — subgroups

Subgroups — #26132645

Order 12032.

234 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 111 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	#20245154	C₂	1	1	yes	maximal, characteristic
2	#20245147	C₂	1	1	yes	maximal, characteristic
3	#20241477	C₂	1	2	yes	maximal
4	#20241474	C₂	1	2	yes	maximal
5	C₄₇×#1747	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	1	yes	characteristic
7	C₂×#13093964	C₂²	1	2	yes	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	#15688944	—	2	2	no	—
10	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
11	C₄₇×#405	C₂²	1	2	yes	—
12	C₄₇×#405	C₂²	1	1	yes	characteristic
13	C₄₇×#338	C₄	1	1	yes	characteristic
14	C₄₇×H(3,ℤ/4)	C₄	1	1	yes	characteristic
15	C₄₇×H(3,ℤ/4)	C₄	1	2	yes	—
16	#13093964	—	2	2	no	—
17	#13093964	—	2	2	no	—
18	#13093964	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	#117×C₄₇	C₂×C₄	1	2	yes	—
27	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
28	#97×C₄₇	—	2	2	no	—
29	#97×C₄₇	Q₈	1	2	yes	—
30	C₄×C₈×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	D₄	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₄₇	—	2	2	no	—
34	M₁₆×C₄₇	—	2	2	no	—
35	M₁₆×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₇	#41	1	2	yes	—
41	C₂×C₈×C₄₇	#41	1	2	yes	—
42	C₄⋊C₄×C₄₇	—	2	1	no	—
43	C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂²⋊C₄×C₄₇	—	2	1	no	—
45	C₂²⋊C₄×C₄₇	—	2	2	no	—
46	C₄²×C₄₇	—	2	2	no	—
47	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₄₇⋊C₁₆	—	4	1	no	—
49	C₄₇⋊C₁₆	—	4	1	no	—
50	C₄₇⋊C₁₆	—	4	2	no	—
51	C₂³×C₄₇	#131	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	4	2	no	—
54	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#117	1	2	yes	—
56	C₂×C₄×C₄₇	#131	1	2	yes	—
57	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
58	C₈×C₄₇	—	2	2	no	—
59	C₈×C₄₇	—	4	2	no	—
60	C₈×C₄₇	#124	1	1	yes	characteristic
61	C₈×C₄₇	#124	1	1	yes	characteristic
62	C₈×C₄₇	#125	1	2	yes	—
63	#12885	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	4	1	no	—
65	C₂²×C₄₇	#424	1	1	yes	characteristic
66	C₂²×C₄₇	#433	1	2	yes	—
67	C₄×C₄₇	—	4	2	no	—
68	C₄×C₄₇	—	8	2	no	—
69	C₄×C₄₇	#382	1	1	yes	characteristic
70	C₄×C₄₇	#424	1	1	yes	characteristic
71	C₄×C₄₇	#433	1	2	yes	—
72	#2153	—	47	1	no	—
73	#2151	—	47	1	no	—
74	#1747	D₄₇	1	1	yes	characteristic
75	#1379	—	47	2	no	—
76	#1378	—	47	2	no	—
77	C₂×C₄₇	—	2	1	no	—
78	C₂×C₄₇	—	2	2	no	—
79	C₂×C₄₇	#1832	1	1	yes	characteristic
80	C₂×M₃₂	—	47	1	no	—
81	C₂×M₃₂	—	47	2	no	—
82	C₂²×C₁₆	—	47	1	no	—
83	#405	C₂×D₄₇	1	2	yes	—
84	#405	C₂×D₄₇	1	1	yes	characteristic
85	#364	—	94	2	no	—
86	C₄⋊C₁₆	—	94	2	no	—
87	#338	Dic₄₇	1	1	yes	characteristic
88	H(3,ℤ/4)	Dic₄₇	1	1	yes	characteristic
89	H(3,ℤ/4)	Dic₄₇	1	2	yes	—
90	C₄₇	#12885	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
92	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
94	#117	C₂×Dic₄₇	1	2	yes	—
95	#117	C₂×Dic₄₇	1	1	yes	characteristic
96	M₃₂	—	94	2	no	—
97	M₃₂	—	94	2	no	—
98	M₃₂	—	94	2	no	—
99	C₂×C₁₆	—	94	1	no	—
100	C₂×C₁₆	—	94	1	no	—
101	C₂×C₁₆	—	94	2	no	—
102	C₂×C₁₆	—	94	2	no	—
103	#97	—	2	2	no	—
104	#97	Dic₉₄	1	2	yes	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	D₁₈₈	1	2	yes	—
107	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
113	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
114	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
115	C₂×C₈	#10625350	1	2	yes	—
116	C₂×C₈	#10625350	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₄⋊Dic₄₇	1	2	yes	—
123	C₁₆	—	188	1	no	—
124	C₁₆	—	188	1	no	—
125	C₁₆	—	188	2	no	—
126	C₂³	#13094031	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
130	C₂×C₄	#13093983	1	2	yes	—
131	C₂×C₄	#13094031	1	2	yes	—
132	C₂×C₄	#13094031	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	#13094012	1	1	yes	characteristic
136	C₈	#13094012	1	1	yes	characteristic
137	C₈	#13094016	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#15689150	1	1	yes	characteristic
140	C₂²	#15689184	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	#15688989	1	1	yes	characteristic
144	C₄	#15689150	1	1	yes	characteristic
145	C₄	#15689184	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#20243508	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245154

C₂

yes

maximal, characteristic

#20245147

C₂

yes

maximal, characteristic

#20241477

C₂

yes

maximal

#20241474

C₂

yes

maximal

C₄₇×#1747

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

characteristic

C₂×#13093964

C₂²

yes

—

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15688944

—

C₄⋊(C₄₇⋊C₁₆)

—

C₄₇×#405

C₂²

yes

—

C₄₇×#405

C₂²

yes

characteristic

C₄₇×#338

C₄

yes

characteristic

C₄₇×H(3,ℤ/4)

C₄

yes

characteristic

C₄₇×H(3,ℤ/4)

C₄

yes

—

#13093964

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

C₂×C₄

yes

—

#117×C₄₇

C₂×C₄

yes

characteristic

#97×C₄₇

—

#97×C₄₇

Q₈

yes

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#131

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

#12885

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#424

yes

characteristic

C₂²×C₄₇

#433

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#2153

—

#2151

—

#1747

D₄₇

yes

characteristic

#1379

—

#1378

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1832

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

C₂×D₄₇

yes

—

#405

C₂×D₄₇

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

Dic₄₇

yes

characteristic

H(3,ℤ/4)

Dic₄₇

yes

characteristic

H(3,ℤ/4)

Dic₄₇

yes

—

C₄₇

#12885

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×Dic₄₇

yes

—

C₂×M₁₆

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

#117

C₂×Dic₄₇

yes

—

#117

C₂×Dic₄₇

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

#97

—

104

#97

Dic₉₄

yes

—

105

C₄×C₈

—

106

C₄×C₈

D₁₈₈

yes

—

107

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₄⋊Dic₄₇

yes

—

113

C₂×C₈

C₂×Dic₉₄

yes

characteristic

114

C₂×C₈

C₂×D₁₈₈

yes

characteristic

115

C₂×C₈

#10625350

yes

—

116

C₂×C₈

#10625350

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₄⋊Dic₄₇

yes

—

123

C₁₆

—

188

—

124

C₁₆

—

188

—

125

C₁₆

—

188

—

126

C₂³

#13094031

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

130

C₂×C₄

#13093983

yes

—

131

C₂×C₄

#13094031

yes

—

132

C₂×C₄

#13094031

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

#13094012

yes

characteristic

136

C₈

#13094012

yes

characteristic

137

C₈

#13094016

yes

—

138

C₂²

—

139

C₂²

#15689150

yes

characteristic

140

C₂²

#15689184

yes

—

141

C₄

—

142

C₄

—

143

C₄

#15688989

yes

characteristic

144

C₄

#15689150

yes

characteristic

145

C₄

#15689184

yes

—

146

C₂

—

147

C₂

—

148

C₂

#20243508

yes

characteristic

	subgroup	G/H
center	C₈	#13094012
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1747	C₂
socle	C₂×C₄₇	#1832
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1747	D₄₇
hypocenter	C₄₇	#12885
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

subgroup

G/H

center

C₈

#13094012

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1747

C₂

C₂×C₄₇

C₂³×C₄₇

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

N	H ≅ G/N	type
90. C₄₇	#12885	semidirect

H ≅ G/N

type

90. C₄₇

#12885

semidirect

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

subgroup	index
#20245154	2
#20245147	2
#20241477	2
#20241474	2
C₄₇×#1747	2
#12885	47

subgroup

index

C₄₇×#1747

Subgroups — #26132645

Conjugacy classes of subgroups

Complements

Maximal subgroups