Group #26130585 — subgroups

Subgroups — #26130585

Order 12032.

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

normal subgroups only

This group has 218 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	#20245541	C₂	1	2	yes	maximal
2	#20243006	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
4	#20241445	C₂	1	2	yes	maximal
5	C₄₇×#1370	C₂	1	1	yes	maximal, characteristic
6	#15689528	—	2	1	no	—
7	#15689528	C₂²	1	2	yes	—
8	#15689503	—	2	1	no	—
9	C₈×Dic₉₄	—	2	1	no	—
10	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	#15688949	—	2	2	no	—
12	#15688928	—	2	2	no	—
13	#15688876	C₂²	1	1	yes	characteristic
14	#15688855	—	2	1	no	—
15	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	4	2	no	—
18	C₂×Dic₁₈₈	D₄	1	1	yes	characteristic
19	#13094024	—	2	2	no	—
20	C₄×Dic₉₄	—	2	1	no	—
21	#13093958	—	2	2	no	—
22	#13093958	—	2	2	no	—
23	#13093958	—	4	1	no	—
24	#13093958	—	4	1	no	—
25	#13093958	D₄	1	1	yes	characteristic
26	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
27	Dic₄₇⋊C₈	—	4	1	no	—
28	#13093949	—	2	1	no	—
29	C₈×Dic₄₇	—	4	1	no	—
30	C₂×C₁₆×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×Dic₉₄	—	2	1	no	—
35	C₂×Dic₉₄	—	4	2	no	—
36	C₂×C₈×C₄₇	—	2	1	no	—
37	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
41	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₇	—	2	1	no	—
43	C₄⋊Dic₄₇	—	4	2	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	Dic₄₇⋊C₄	—	4	1	no	—
46	C₄×Dic₄₇	—	4	1	no	—
47	C₂×C₄₇⋊C₈	—	4	1	no	—
48	Dic₁₈₈	—	2	1	no	—
49	Dic₁₈₈	—	8	4	no	—
50	Dic₁₈₈	Q₁₆	1	2	yes	complemented
51	C₁₆×C₄₇	—	4	2	no	—
52	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×Dic₄₇	—	4	1	no	—
56	C₂×Dic₄₇	—	4	1	no	—
57	C₂×Dic₄₇	—	8	2	no	—
58	Dic₉₄	—	2	2	no	—
59	Dic₉₄	—	4	1	no	—
60	Dic₉₄	—	8	4	no	—
61	C₈×C₄₇	—	2	2	no	—
62	C₈×C₄₇	—	2	1	no	—
63	C₈×C₄₇	—	2	1	no	—
64	C₈×C₄₇	Q₃₂	1	2	yes	—
65	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₄₇	#135	1	1	yes	characteristic
67	C₄₇⋊C₈	—	8	1	no	—
68	#12448	—	47	1	no	maximal, Sylow
69	C₂²×C₄₇	#459	1	1	yes	characteristic
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	#467	1	1	yes	characteristic
72	C₄×C₄₇	#473	1	1	yes	characteristic
73	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
74	C₄×C₄₇	#510	1	1	yes	characteristic
75	Dic₄₇	—	4	2	no	—
76	Dic₄₇	—	8	1	no	—
77	Dic₄₇	—	16	2	no	—
78	#2227	—	47	2	no	—
79	#1713	—	47	1	no	—
80	C₈×Q₁₆	—	47	1	no	—
81	#1370	D₄₇	1	1	yes	characteristic
82	#1365	—	47	2	no	—
83	C₂×C₄₇	#1669	1	1	yes	characteristic
84	C₂×C₄₇	#2209	1	1	yes	characteristic
85	C₂×C₄₇	#2213	1	1	yes	characteristic
86	#500	—	47	2	no	—
87	#500	—	94	1	no	—
88	#494	—	94	1	no	—
89	C₈×Q₈	—	94	1	no	—
90	C₄×Q₁₆	—	47	1	no	—
91	#366	—	94	2	no	—
92	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
93	#358	—	94	2	no	—
94	#335	—	47	1	no	—
95	Q₈⋊C₈	—	94	1	no	—
96	C₈²	C₂×D₄₇	1	1	yes	characteristic
97	C₄₇	#12448	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	47	1	no	—
99	C₂×Q₁₆	—	188	2	no	—
100	C₄⋊Q₈	—	94	2	no	—
101	C₄×Q₈	—	94	1	no	—
102	C₂×C₁₆	—	2	2	no	—
103	#107	—	47	1	no	—
104	#107	—	94	2	no	—
105	#107	—	94	2	no	—
106	#107	—	188	1	no	—
107	#107	—	188	1	no	—
108	C₄⋊C₈	—	94	1	no	—
109	C₄⋊C₈	—	188	1	no	—
110	#103	—	94	1	no	—
111	C₄×C₈	—	188	1	no	—
112	C₄×C₈	D₁₈₈	1	1	yes	characteristic
113	C₄×C₈	D₁₈₈	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
115	C₂×Q₈	—	94	1	no	—
116	C₂×Q₈	—	188	2	no	—
117	Q₁₆	—	47	2	no	—
118	Q₁₆	—	94	1	no	—
119	Q₁₆	—	376	4	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	188	1	no	—
122	C₂×C₈	D₃₇₆	1	1	yes	characteristic
123	C₂×C₈	D₃₇₆	1	1	yes	characteristic
124	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
125	C₂×C₈	#10625354	1	1	yes	characteristic
126	C₄⋊C₄	—	94	1	no	—
127	C₄⋊C₄	—	188	2	no	—
128	C₄⋊C₄	—	188	2	no	—
129	C₄⋊C₄	—	188	1	no	—
130	C₄²	—	188	1	no	—
131	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	Q₈	—	94	2	no	—
134	Q₈	—	376	4	no	—
135	Q₈	—	188	1	no	—
136	C₂×C₄	—	376	2	no	—
137	C₂×C₄	—	188	1	no	—
138	C₂×C₄	—	188	1	no	—
139	C₂×C₄	#13093999	1	1	yes	characteristic
140	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
141	C₂×C₄	#13094052	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	376	1	no	—
146	C₈	Dic₃₇₆	1	2	yes	—
147	C₈	Q₁₆×D₄₇	1	1	yes	characteristic
148	C₈	#13094043	1	1	yes	characteristic
149	C₂²	#15689319	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	376	1	no	—
152	C₄	—	188	2	no	—
153	C₄	—	752	2	no	—
154	C₄	#15689365	1	1	yes	characteristic
155	C₄	#15689399	1	1	yes	characteristic
156	C₄	C₂×Dic₃₇₆	1	1	yes	characteristic
157	C₄	#15689566	1	1	yes	characteristic
158	C₂	#20242754	1	1	yes	characteristic
159	C₂	#20245439	1	1	yes	characteristic
160	C₂	#20245465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245541

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic

C₈×Dic₁₈₈

C₂

yes

maximal, characteristic

#20241445

C₂

yes

maximal

C₄₇×#1370

C₂

yes

maximal, characteristic

#15689528

—

#15689528

C₂²

yes

—

#15689503

—

C₈×Dic₉₄

—

C₄×Dic₁₈₈

C₂²

yes

—

—

C₂²

yes

characteristic

#15688855

—

C₄₇×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

D₄

yes

characteristic

#13094024

—

C₄×Dic₉₄

—

—

—

—

—

D₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093949

—

C₈×Dic₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

Q₁₆

yes

complemented

C₁₆×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

Q₃₂

yes

—

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

C₈×C₄₇

#135

yes

characteristic

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×Q₃₂

yes

characteristic

C₄×C₄₇

#510

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2227

—

#1713

—

C₈×Q₁₆

—

#1370

D₄₇

yes

characteristic

#1365

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

C₂×D₄₇

yes

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12448

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

188

—

100

C₄⋊Q₈

—

101

C₄×Q₈

—

102

C₂×C₁₆

—

103

#107

—

104

#107

—

105

#107

—

106

#107

—

188

—

107

#107

—

188

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

188

—

110

#103

—

111

C₄×C₈

—

188

—

112

C₄×C₈

D₁₈₈

yes

characteristic

113

C₄×C₈

D₁₈₈

yes

characteristic

114

C₄×C₈

C₂²×D₄₇

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

188

—

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

376

—

120

C₂×C₈

—

121

C₂×C₈

—

188

—

122

C₂×C₈

D₃₇₆

yes

characteristic

123

C₂×C₈

D₃₇₆

yes

characteristic

124

C₂×C₈

D₄×D₄₇

yes

characteristic

125

C₂×C₈

#10625354

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

188

—

128

C₄⋊C₄

—

188

—

129

C₄⋊C₄

—

188

—

130

C₄²

—

188

—

131

C₄²

C₂×D₁₈₈

yes

characteristic

132

C₁₆

—

133

Q₈

—

134

Q₈

—

376

—

135

Q₈

—

188

—

136

C₂×C₄

—

376

—

137

C₂×C₄

—

188

—

138

C₂×C₄

—

188

—

139

C₂×C₄

#13093999

yes

characteristic

140

C₂×C₄

C₂×D₃₇₆

yes

characteristic

141

C₂×C₄

#13094052

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

376

—

146

C₈

Dic₃₇₆

yes

—

147

C₈

Q₁₆×D₄₇

yes

characteristic

148

C₈

#13094043

yes

characteristic

149

C₂²

#15689319

yes

characteristic

150

C₄

—

151

C₄

—

376

—

152

C₄

—

188

—

153

C₄

—

752

—

154

C₄

#15689365

yes

characteristic

155

C₄

#15689399

yes

characteristic

156

C₄

C₂×Dic₃₇₆

yes

characteristic

157

C₄

#15689566

yes

characteristic

158

C₂

#20242754

yes

characteristic

159

C₂

#20245439

yes

characteristic

160

C₂

#20245465

yes

characteristic

	subgroup	G/H
center	C₂²	#15689319
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1370	C₂
socle, prime radical	C₂²×C₄₇	#459
hypercenter	#1370	D₄₇
hypocenter	C₄₇	#12448
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#15689319

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1370

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
50. Dic₁₈₈	Q₁₆	semidirect
97. C₄₇	#12448	semidirect

H ≅ G/N

type

50. Dic₁₈₈

Q₁₆

semidirect

97. C₄₇

#12448

semidirect

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

subgroup	index
#20245541	2
#20243006	2
C₈×Dic₁₈₈	2
#20241445	2
C₄₇×#1370	2
#12448	47

subgroup

index

C₈×Dic₁₈₈

C₄₇×#1370

Subgroups — #26130585

Conjugacy classes of subgroups

Complements

Maximal subgroups