Group #26130575 — subgroups

Subgroups — #26130575

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	#20245531	C₂	1	2	yes	maximal
2	#20243006	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
4	#20241442	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	#15688951	—	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	1	no	—
18	C₂×Dic₁₈₈	—	4	1	no	—
19	C₂×Dic₁₈₈	D₄	1	1	yes	characteristic
20	#13094024	—	2	2	no	—
21	C₄×Dic₉₄	—	2	1	no	—
22	#13093958	—	2	2	no	—
23	#13093958	—	2	2	no	—
24	#13093958	—	4	2	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	2	no	—
50	Dic₁₈₈	—	8	2	no	—
51	Dic₁₈₈	Q₁₆	1	2	yes	complemented
52	C₁₆×C₄₇	—	4	2	no	—
53	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
56	C₂×Dic₄₇	—	4	1	no	—
57	C₂×Dic₄₇	—	4	1	no	—
58	C₂×Dic₄₇	—	8	2	no	—
59	Dic₉₄	—	2	2	no	—
60	Dic₉₄	—	4	1	no	—
61	Dic₉₄	—	8	2	no	—
62	Dic₉₄	—	8	2	no	—
63	C₈×C₄₇	—	2	2	no	—
64	C₈×C₄₇	—	2	1	no	—
65	C₈×C₄₇	—	2	1	no	—
66	C₈×C₄₇	SD₃₂	1	2	yes	—
67	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₄₇	#135	1	1	yes	characteristic
69	C₄₇⋊C₈	—	8	1	no	—
70	#12446	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	#459	1	1	yes	characteristic
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	#467	1	1	yes	characteristic
74	C₄×C₄₇	#473	1	1	yes	characteristic
75	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
76	C₄×C₄₇	#509	1	1	yes	characteristic
77	Dic₄₇	—	4	2	no	—
78	Dic₄₇	—	8	1	no	—
79	Dic₄₇	—	16	2	no	—
80	#2225	—	47	2	no	—
81	#1713	—	47	1	no	—
82	C₈×Q₁₆	—	47	1	no	—
83	#1370	D₄₇	1	1	yes	characteristic
84	#1364	—	47	2	no	—
85	C₂×C₄₇	#1669	1	1	yes	characteristic
86	C₂×C₄₇	#2207	1	1	yes	characteristic
87	C₂×C₄₇	#2214	1	1	yes	characteristic
88	#500	—	47	2	no	—
89	#500	—	94	1	no	—
90	#494	—	94	1	no	—
91	C₈×Q₈	—	94	1	no	—
92	C₄×Q₁₆	—	47	1	no	—
93	#367	—	94	2	no	—
94	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
95	#358	—	94	2	no	—
96	#335	—	47	1	no	—
97	Q₈⋊C₈	—	94	1	no	—
98	C₈²	C₂×D₄₇	1	1	yes	characteristic
99	C₄₇	#12446	1	1	yes	characteristic, complemented, Sylow
100	C₂×Q₁₆	—	47	1	no	—
101	C₂×Q₁₆	—	188	1	no	—
102	C₂×Q₁₆	—	188	1	no	—
103	C₄⋊Q₈	—	94	2	no	—
104	C₄×Q₈	—	94	1	no	—
105	C₂×C₁₆	—	2	2	no	—
106	#107	—	47	1	no	—
107	#107	—	94	2	no	—
108	#107	—	94	2	no	—
109	#107	—	188	2	no	—
110	C₄⋊C₈	—	94	1	no	—
111	C₄⋊C₈	—	188	1	no	—
112	#103	—	94	1	no	—
113	C₄×C₈	—	188	1	no	—
114	C₄×C₈	D₁₈₈	1	1	yes	characteristic
115	C₄×C₈	D₁₈₈	1	1	yes	characteristic
116	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
117	C₂×Q₈	—	94	1	no	—
118	C₂×Q₈	—	188	2	no	—
119	Q₁₆	—	47	2	no	—
120	Q₁₆	—	94	1	no	—
121	Q₁₆	—	376	2	no	—
122	Q₁₆	—	376	2	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	188	1	no	—
125	C₂×C₈	D₃₇₆	1	1	yes	characteristic
126	C₂×C₈	D₃₇₆	1	1	yes	characteristic
127	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
128	C₂×C₈	#10625354	1	1	yes	characteristic
129	C₄⋊C₄	—	94	1	no	—
130	C₄⋊C₄	—	188	2	no	—
131	C₄⋊C₄	—	188	2	no	—
132	C₄⋊C₄	—	188	1	no	—
133	C₄²	—	188	1	no	—
134	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
135	C₁₆	—	4	2	no	—
136	Q₈	—	94	2	no	—
137	Q₈	—	376	2	no	—
138	Q₈	—	376	2	no	—
139	Q₈	—	188	1	no	—
140	C₂×C₄	—	376	2	no	—
141	C₂×C₄	—	188	1	no	—
142	C₂×C₄	—	188	1	no	—
143	C₂×C₄	#13093999	1	1	yes	characteristic
144	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
145	C₂×C₄	#13094052	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	376	1	no	—
150	C₈	#13093969	1	2	yes	—
151	C₈	Q₁₆×D₄₇	1	1	yes	characteristic
152	C₈	#13094043	1	1	yes	characteristic
153	C₂²	#15689319	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	376	1	no	—
156	C₄	—	188	2	no	—
157	C₄	—	752	2	no	—
158	C₄	#15689365	1	1	yes	characteristic
159	C₄	#15689399	1	1	yes	characteristic
160	C₄	C₂×#13093969	1	1	yes	characteristic
161	C₄	#15689561	1	1	yes	characteristic
162	C₂	#20242754	1	1	yes	characteristic
163	C₂	#20245429	1	1	yes	characteristic
164	C₂	#20245470	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245531

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic

C₈×Dic₁₈₈

C₂

yes

maximal, characteristic

#20241442

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

—

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

—

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

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

SD₃₂

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×SD₃₂

yes

characteristic

C₄×C₄₇

#509

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2225

—

#1713

—

C₈×Q₁₆

—

#1370

D₄₇

yes

characteristic

#1364

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

#367

—

C₄⋊C₁₆

C₂×D₄₇

yes

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12446

yes

characteristic, complemented, Sylow

100

C₂×Q₁₆

—

101

C₂×Q₁₆

—

188

—

102

C₂×Q₁₆

—

188

—

103

C₄⋊Q₈

—

104

C₄×Q₈

—

105

C₂×C₁₆

—

106

#107

—

107

#107

—

108

#107

—

109

#107

—

188

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

188

—

112

#103

—

113

C₄×C₈

—

188

—

114

C₄×C₈

D₁₈₈

yes

characteristic

115

C₄×C₈

D₁₈₈

yes

characteristic

116

C₄×C₈

C₂²×D₄₇

yes

characteristic

117

C₂×Q₈

—

118

C₂×Q₈

—

188

—

119

Q₁₆

—

120

Q₁₆

—

121

Q₁₆

—

376

—

122

Q₁₆

—

376

—

123

C₂×C₈

—

124

C₂×C₈

—

188

—

125

C₂×C₈

D₃₇₆

yes

characteristic

126

C₂×C₈

D₃₇₆

yes

characteristic

127

C₂×C₈

D₄×D₄₇

yes

characteristic

128

C₂×C₈

#10625354

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

188

—

131

C₄⋊C₄

—

188

—

132

C₄⋊C₄

—

188

—

133

C₄²

—

188

—

134

C₄²

C₂×D₁₈₈

yes

characteristic

135

C₁₆

—

136

Q₈

—

137

Q₈

—

376

—

138

Q₈

—

376

—

139

Q₈

—

188

—

140

C₂×C₄

—

376

—

141

C₂×C₄

—

188

—

142

C₂×C₄

—

188

—

143

C₂×C₄

#13093999

yes

characteristic

144

C₂×C₄

C₂×D₃₇₆

yes

characteristic

145

C₂×C₄

#13094052

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

376

—

150

C₈

#13093969

yes

—

151

C₈

Q₁₆×D₄₇

yes

characteristic

152

C₈

#13094043

yes

characteristic

153

C₂²

#15689319

yes

characteristic

154

C₄

—

155

C₄

—

376

—

156

C₄

—

188

—

157

C₄

—

752

—

158

C₄

#15689365

yes

characteristic

159

C₄

#15689399

yes

characteristic

160

C₄

C₂×#13093969

yes

characteristic

161

C₄

#15689561

yes

characteristic

162

C₂

#20242754

yes

characteristic

163

C₂

#20245429

yes

characteristic

164

C₂

#20245470

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	#12446
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
51. Dic₁₈₈	Q₁₆	semidirect
99. C₄₇	#12446	semidirect

H ≅ G/N

type

51. Dic₁₈₈

Q₁₆

semidirect

99. C₄₇

#12446

semidirect

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

subgroup	index
#20245531	2
#20243006	2
C₈×Dic₁₈₈	2
#20241442	2
C₄₇×#1370	2
#12446	47

subgroup

index

C₈×Dic₁₈₈

C₄₇×#1370

Subgroups — #26130575

Conjugacy classes of subgroups

Complements

Maximal subgroups