Group #26130625 — subgroups

Subgroups — #26130625

Order 12032.

206 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20245663	C₂	1	1	yes	maximal, characteristic, complemented
2	#20242787	C₂	1	2	yes	maximal
3	#20241462	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241445	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2169	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689413	—	2	2	no	—
8	#15688949	—	2	1	no	—
9	#15688949	—	2	1	no	—
10	#15688876	C₂²	1	2	yes	—
11	#15688871	—	2	2	no	—
12	C₂×M₃₂×C₄₇	—	2	1	no	—
13	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄₇×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	#13094024	—	2	2	no	—
18	#13093958	—	2	2	no	—
19	#13093958	—	2	2	no	—
20	#13093955	—	4	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	#13093945	—	4	2	no	—
23	C₂²×C₈×C₄₇	—	2	1	no	—
24	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
25	M₃₂×C₄₇	—	4	2	no	—
26	C₂×C₁₆×C₄₇	—	2	1	no	—
27	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²⋊C₈×C₄₇	—	2	1	no	—
31	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	—	2	1	no	—
34	C₂×Dic₉₄	—	4	2	no	—
35	C₂×C₈×C₄₇	—	2	1	no	—
36	C₂×C₈×C₄₇	—	4	1	no	—
37	C₂×C₈×C₄₇	—	4	1	no	—
38	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
41	C₂²⋊C₄×C₄₇	—	2	1	no	—
42	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₄₇	—	4	2	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	C₂×C₄₇⋊C₈	—	4	2	no	—
46	C₁₆×C₄₇	—	2	1	no	—
47	C₁₆×C₄₇	—	4	1	no	—
48	C₁₆×C₄₇	SD₁₆	1	2	yes	—
49	C₂³×C₄₇	—	2	1	no	—
50	D₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	4	1	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
57	C₂×Dic₄₇	—	8	2	no	—
58	Dic₉₄	—	16	2	no	—
59	C₈×C₄₇	—	2	1	no	—
60	C₈×C₄₇	—	4	1	no	—
61	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₄₇	#137	1	1	yes	characteristic
63	C₄₇⋊C₈	—	8	2	no	—
64	#12455	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	4	1	no	—
66	C₂²×C₄₇	—	4	1	no	—
67	C₂²×C₄₇	—	4	1	no	—
68	C₂²×C₄₇	#466	1	1	yes	characteristic
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	4	1	no	—
71	C₄×C₄₇	#461	1	1	yes	characteristic
72	C₄×C₄₇	#473	1	1	yes	characteristic
73	C₄×C₄₇	#510	1	2	yes	—
74	Dic₄₇	—	16	2	no	—
75	#2252	—	47	1	no	—
76	#2169	D₄₇	1	1	yes	characteristic
77	#1675	—	47	2	no	—
78	#1372	—	47	1	no	—
79	#1365	—	47	2	no	—
80	C₂×C₄₇	—	8	1	no	—
81	C₂×C₄₇	#1673	1	1	yes	characteristic
82	C₂×C₄₇	#2219	1	1	yes	characteristic
83	C₂×C₄₇	#2221	1	1	yes	characteristic
84	C₂×M₃₂	—	2	1	no	—
85	#500	—	47	2	no	—
86	#476	—	94	2	no	—
87	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
88	#366	—	94	1	no	—
89	#366	—	94	1	no	—
90	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
91	C₂²⋊C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
93	#335	—	47	2	no	—
94	#333	—	94	2	no	—
95	C₄₇	#12455	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	2	1	no	—
97	C₄⋊Q₈	—	94	2	no	—
98	C₄×D₄	D₁₈₈	1	1	yes	characteristic
99	M₃₂	—	4	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
103	#107	—	94	2	no	—
104	#107	—	94	2	no	—
105	#106	—	188	2	no	—
106	C₄⋊C₈	—	94	2	no	—
107	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
108	#102	—	188	2	no	—
109	C₂²⋊C₈	—	2	1	no	—
110	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
111	C₂×Q₈	—	188	2	no	—
112	C₂×D₄	D₃₇₆	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	188	2	no	—
118	C₂×C₈	#10625350	1	1	yes	characteristic
119	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
120	C₄⋊C₄	—	188	2	no	—
121	C₄⋊C₄	—	188	2	no	—
122	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	4	1	no	—
127	C₁₆	#10625335	1	2	yes	—
128	C₂³	—	2	1	no	—
129	Q₈	—	752	2	no	—
130	D₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	376	2	no	—
135	C₂×C₄	#13093998	1	1	yes	characteristic
136	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
137	C₂×C₄	#13094046	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	376	2	no	—
141	C₈	C₂×#10625335	1	1	yes	characteristic
142	C₈	#13094054	1	1	yes	characteristic
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#15689359	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	752	2	no	—
150	C₄	#15689328	1	1	yes	characteristic
151	C₄	#15689398	1	1	yes	characteristic
152	C₄	#15689566	1	2	yes	—
153	C₂	—	8	1	no	—
154	C₂	#20242777	1	1	yes	characteristic
155	C₂	#20245496	1	1	yes	characteristic
156	C₂	#20245508	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245663

C₂

yes

maximal, characteristic, complemented

#20242787

C₂

yes

maximal

#20241462

C₂

yes

maximal, characteristic, complemented

#20241445

C₂

yes

maximal, complemented

C₄₇×#2169

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

—

—

—

C₂²

yes

—

#15688871

—

C₂×M₃₂×C₄₇

—

C₈×D₄×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

C₂²

yes

characteristic

C₄₇×C₂²⋊C₁₆

—

C₄×C₁₆×C₄₇

C₂²

yes

—

—

—

—

C₄⋊(C₄₇⋊C₈)

—

#13093945

—

C₂²×C₈×C₄₇

—

C₄×D₄×C₄₇

D₄

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

D₄

yes

characteristic

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

D₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

D₈

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

SD₁₆

yes

—

C₂³×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×SD₁₆

yes

characteristic

C₈×C₄₇

#137

yes

characteristic

C₄₇⋊C₈

—

#12455

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#466

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2252

—

#2169

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×M₃₂

—

#500

—

#476

—

C₈×D₄

C₂×D₄₇

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

C₂×D₄₇

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

—

#333

—

C₄₇

#12455

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄×D₄

D₁₈₈

yes

characteristic

M₃₂

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

103

#107

—

104

#107

—

105

#106

—

188

—

106

C₄⋊C₈

—

107

C₄⋊C₈

D₁₈₈

yes

characteristic

108

#102

—

188

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂²×D₄₇

yes

characteristic

111

C₂×Q₈

—

188

—

112

C₂×D₄

D₃₇₆

yes

characteristic

113

C₂²×C₄

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

188

—

118

C₂×C₈

#10625350

yes

characteristic

119

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

120

C₄⋊C₄

—

188

—

121

C₄⋊C₄

—

188

—

122

C₄⋊C₄

D₃₇₆

yes

characteristic

123

C₂²⋊C₄

—

124

C₄²

C₂×D₁₈₈

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

#10625335

yes

—

128

C₂³

—

129

Q₈

—

752

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

376

—

135

C₂×C₄

#13093998

yes

characteristic

136

C₂×C₄

C₂×D₃₇₆

yes

characteristic

137

C₂×C₄

#13094046

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

376

—

141

C₈

C₂×#10625335

yes

characteristic

142

C₈

#13094054

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#15689359

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

752

—

150

C₄

#15689328

yes

characteristic

151

C₄

#15689398

yes

characteristic

152

C₄

#15689566

yes

—

153

C₂

—

154

C₂

#20242777

yes

characteristic

155

C₂

#20245496

yes

characteristic

156

C₂

#20245508

yes

characteristic

	subgroup	G/H
center	C₂²	#15689359
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2169	C₂
socle	C₂²×C₄₇	#466
prime radical	C₂×D₄×C₄₇	D₈
hypercenter	#2169	D₄₇
hypocenter	C₄₇	#12455
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

subgroup

G/H

center

C₂²

#15689359

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2169

C₂

socle

C₂²×C₄₇

#466

prime radical

C₂×D₄×C₄₇

D₈

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

N	H ≅ G/N	type
1. #20245663	C₂	semidirect
3. #20241462	C₂	semidirect
4. #20241445	C₂	semidirect
95. C₄₇	#12455	semidirect

H ≅ G/N

type

1. #20245663

C₂

semidirect

3. #20241462

C₂

semidirect

4. #20241445

C₂

semidirect

95. C₄₇

#12455

semidirect

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

subgroup	index
#20245663	2
#20242787	2
#20241462	2
#20241445	2
C₄₇×#2169	2
#12455	47

subgroup

index

C₄₇×#2169

Subgroups — #26130625

Conjugacy classes of subgroups

Complements

Maximal subgroups