Group #26130541 — subgroups

Subgroups — #26130541

Order 12032.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20245641	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₉₄	C₂	1	1	yes	maximal, characteristic
3	#20242759	C₂	1	2	yes	maximal
4	#20241436	C₂	1	2	yes	maximal
5	C₄₇×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	—	2	1	no	—
7	C₂×Dic₃₇₆	—	2	1	no	—
8	#15689503	C₂²	1	2	yes	—
9	#15689409	—	2	2	no	—
10	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
11	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₄₇⋊C₁₆	—	2	1	no	—
13	C₁₆×Dic₄₇	—	2	1	no	—
14	#15688868	—	2	2	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#335	C₂²	1	2	yes	—
17	C₂×Dic₁₈₈	—	2	2	no	—
18	C₂×Dic₁₈₈	—	2	2	no	—
19	#13094024	—	2	2	no	—
20	C₄×Dic₉₄	D₄	1	1	yes	characteristic
21	Dic₃₇₆	—	4	2	no	—
22	Dic₃₇₆	—	4	2	no	—
23	C₂×C₄₇⋊C₁₆	—	2	1	no	—
24	#13093956	—	4	2	no	—
25	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
26	Dic₄₇⋊C₈	—	2	1	no	—
27	#13093949	—	4	2	no	—
28	C₈×Dic₄₇	—	2	1	no	—
29	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₉₄	—	4	2	no	—
34	C₂×Dic₉₄	D₈	1	1	yes	characteristic
35	C₂×C₈×C₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
38	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₄₇	—	4	2	no	—
40	C₄⋊Dic₄₇	—	4	2	no	—
41	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
42	Dic₄₇⋊C₄	—	2	1	no	—
43	C₄×Dic₄₇	—	2	1	no	—
44	C₂×C₄₇⋊C₈	—	2	1	no	—
45	Dic₁₈₈	—	4	2	no	—
46	Dic₁₈₈	—	4	2	no	—
47	Dic₁₈₈	—	8	2	no	—
48	C₁₆×C₄₇	—	2	1	no	—
49	C₁₆×C₄₇	SD₁₆	1	2	yes	—
50	C₄₇⋊C₁₆	—	4	1	no	—
51	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
54	C₂×Dic₄₇	—	2	1	no	—
55	C₂×Dic₄₇	—	2	1	no	—
56	C₂×Dic₄₇	—	8	2	no	—
57	Dic₉₄	—	2	1	no	—
58	Dic₉₄	—	8	2	no	—
59	Dic₉₄	—	8	2	no	—
60	Dic₉₄	Q₃₂	1	2	yes	complemented
61	C₈×C₄₇	—	2	1	no	—
62	C₈×C₄₇	—	8	2	no	—
63	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄₇⋊C₈	—	4	1	no	—
66	#12440	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	#466	1	1	yes	characteristic
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	#460	1	1	yes	characteristic
70	C₄×C₄₇	#472	1	1	yes	characteristic
71	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
72	C₄×C₄₇	#508	1	1	yes	characteristic
73	Dic₄₇	—	2	2	no	—
74	Dic₄₇	—	4	1	no	—
75	Dic₄₇	—	16	2	no	—
76	#2247	—	47	1	no	—
77	Q₈×C₁₆	—	47	1	no	—
78	#1670	—	47	2	no	—
79	#1371	D₄₇	1	1	yes	characteristic
80	#1362	—	47	2	no	—
81	C₂×C₄₇	#1668	1	1	yes	characteristic
82	C₂×C₄₇	#2216	1	1	yes	characteristic
83	C₂×C₄₇	#2222	1	1	yes	characteristic
84	C₂×Q₃₂	—	94	1	no	—
85	C₂×Q₃₂	—	94	1	no	—
86	#494	—	47	2	no	—
87	#475	—	94	2	no	—
88	C₈×Q₈	—	47	1	no	—
89	C₄⋊C₁₆	—	47	1	no	—
90	C₄⋊C₁₆	—	94	1	no	—
91	C₄×C₁₆	—	94	1	no	—
92	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
93	#335	C₂×D₄₇	1	2	yes	—
94	#332	—	94	2	no	—
95	C₄₇	#12440	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	94	2	no	—
97	C₂×Q₁₆	—	94	2	no	—
98	C₄⋊Q₈	—	94	2	no	—
99	C₄×Q₈	—	47	1	no	—
100	Q₃₂	—	188	2	no	—
101	Q₃₂	—	188	2	no	—
102	C₂×C₁₆	—	94	1	no	—
103	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
104	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
105	#106	—	188	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	47	1	no	—
108	C₄⋊C₈	—	94	1	no	—
109	#103	—	188	2	no	—
110	C₄×C₈	—	94	1	no	—
111	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
112	C₂×Q₈	—	47	1	no	—
113	C₂×Q₈	—	188	2	no	—
114	Q₁₆	—	376	2	no	—
115	Q₁₆	—	188	2	no	—
116	Q₁₆	—	188	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	94	1	no	—
119	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
120	C₂×C₈	#10625354	1	1	yes	characteristic
121	C₄⋊C₄	—	47	1	no	—
122	C₄⋊C₄	—	94	1	no	—
123	C₄⋊C₄	—	188	2	no	—
124	C₄⋊C₄	—	188	2	no	—
125	C₄²	—	94	1	no	—
126	C₄²	D₄×D₄₇	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	188	1	no	—
129	C₁₆	#10625325	1	2	yes	—
130	Q₈	—	47	2	no	—
131	Q₈	—	94	1	no	—
132	Q₈	—	376	2	no	—
133	Q₈	—	376	2	no	—
134	C₂×C₄	—	94	1	no	—
135	C₂×C₄	—	94	1	no	—
136	C₂×C₄	—	376	2	no	—
137	C₂×C₄	#13093999	1	1	yes	characteristic
138	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
139	C₂×C₄	#13094047	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	188	1	no	—
143	C₈	C₂×#10625325	1	1	yes	characteristic
144	C₈	#13094052	1	1	yes	characteristic
145	C₂²	#15689360	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	94	2	no	—
148	C₄	—	188	1	no	—
149	C₄	—	752	2	no	—
150	C₄	#15689324	1	1	yes	characteristic
151	C₄	#15689394	1	1	yes	characteristic
152	C₄	Q₃₂×D₄₇	1	1	yes	characteristic
153	C₄	#15689552	1	1	yes	characteristic
154	C₂	#20242749	1	1	yes	characteristic
155	C₂	#20245480	1	1	yes	characteristic
156	C₂	#20245514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245641

C₂

yes

maximal, characteristic

C₁₆×Dic₉₄

C₂

yes

maximal, characteristic

#20242759

C₂

yes

maximal

#20241436

C₂

yes

maximal

C₄₇×#1371

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

—

C₂×Dic₃₇₆

—

#15689503

C₂²

yes

—

#15689409

—

C₈×Dic₉₄

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₁₆)

C₂²

yes

characteristic

Dic₄₇⋊C₁₆

—

C₁₆×Dic₄₇

—

#15688868

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×#335

C₂²

yes

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

C₄×Dic₉₄

D₄

yes

characteristic

Dic₃₇₆

—

Dic₃₇₆

—

C₂×C₄₇⋊C₁₆

—

#13093956

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

characteristic

Dic₄₇⋊C₈

—

#13093949

—

C₈×Dic₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

D₈

yes

characteristic

C₂×C₈×C₄₇

—

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

D₈

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

SD₁₆

yes

—

C₄₇⋊C₁₆

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

Q₃₂

yes

complemented

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×SD₁₆

yes

characteristic

C₈×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×Q₃₂

yes

characteristic

C₄×C₄₇

#508

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2247

—

Q₈×C₁₆

—

#1670

—

#1371

D₄₇

yes

characteristic

#1362

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

#475

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

C₂×D₄₇

yes

—

#332

—

C₄₇

#12440

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

100

Q₃₂

—

188

—

101

Q₃₂

—

188

—

102

C₂×C₁₆

—

103

C₂×C₁₆

D₁₈₈

yes

characteristic

104

C₂×C₁₆

D₁₈₈

yes

characteristic

105

#106

—

188

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

#103

—

188

—

110

C₄×C₈

—

111

C₄×C₈

C₂²×D₄₇

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

188

—

114

Q₁₆

—

376

—

115

Q₁₆

—

188

—

116

Q₁₆

—

188

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂×D₁₈₈

yes

characteristic

120

C₂×C₈

#10625354

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

188

—

124

C₄⋊C₄

—

188

—

125

C₄²

—

126

C₄²

D₄×D₄₇

yes

characteristic

127

C₁₆

—

128

C₁₆

—

188

—

129

C₁₆

#10625325

yes

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

376

—

133

Q₈

—

376

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

376

—

137

C₂×C₄

#13093999

yes

characteristic

138

C₂×C₄

D₈×D₄₇

yes

characteristic

139

C₂×C₄

#13094047

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

188

—

143

C₈

C₂×#10625325

yes

characteristic

144

C₈

#13094052

yes

characteristic

145

C₂²

#15689360

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

188

—

149

C₄

—

752

—

150

C₄

#15689324

yes

characteristic

151

C₄

#15689394

yes

characteristic

152

C₄

Q₃₂×D₄₇

yes

characteristic

153

C₄

#15689552

yes

characteristic

154

C₂

#20242749

yes

characteristic

155

C₂

#20245480

yes

characteristic

156

C₂

#20245514

yes

characteristic

	subgroup	G/H
center	C₂²	#15689360
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1371	C₂
socle, prime radical	C₂²×C₄₇	#466
hypercenter	#1371	D₄₇
hypocenter	C₄₇	#12440
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#15689360

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1371

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
60. Dic₉₄	Q₃₂	semidirect
95. C₄₇	#12440	semidirect

H ≅ G/N

type

60. Dic₉₄

Q₃₂

semidirect

95. C₄₇

#12440

semidirect

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

subgroup	index
#20245641	2
C₁₆×Dic₉₄	2
#20242759	2
#20241436	2
C₄₇×#1371	2
#12440	47

subgroup

index

C₁₆×Dic₉₄

C₄₇×#1371

Subgroups — #26130541

Conjugacy classes of subgroups

Complements

Maximal subgroups