Group #26127954 — subgroups

Subgroups — #26127954

Order 12032.

200 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20245191	C₂	1	1	yes	maximal, characteristic
2	#20242151	C₂	1	1	yes	maximal, characteristic
3	#20241453	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
8	#15688899	C₂²	1	2	yes	—
9	#15688876	—	2	1	no	—
10	#15688874	C₂²	1	2	yes	—
11	C₄₇×#431	—	2	1	no	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#335	—	2	1	no	—
14	C₄₇×#334	C₂²	1	2	yes	—
15	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
16	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
18	#13093935	—	4	1	no	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	—	2	2	no	—
23	C₂²⋊C₈×C₄₇	—	4	1	no	—
24	C₄×C₈×C₄₇	—	4	1	no	—
25	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	4	1	no	—
34	C₂×C₈×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
41	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄₇⋊C₈	—	4	2	no	—
44	C₄₇⋊C₁₆	—	2	4	no	—
45	C₂³×C₄₇	#106	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	4	1	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₈×C₄₇	—	2	1	no	—
57	C₈×C₄₇	—	2	1	no	—
58	C₈×C₄₇	—	2	1	no	—
59	C₈×C₄₇	—	8	2	no	—
60	C₈×C₄₇	#131	1	2	yes	—
61	C₄₇⋊C₈	—	8	2	no	—
62	#12031	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	#424	1	1	yes	characteristic
73	C₄×C₄₇	#430	1	1	yes	characteristic
74	#2163	—	47	1	no	—
75	#1568	—	47	1	no	—
76	#1564	D₄₇	1	1	yes	characteristic
77	#1368	—	47	4	no	—
78	C₂×C₄₇	—	4	1	no	—
79	C₂×C₄₇	#1570	1	1	yes	characteristic
80	C₂×C₄₇	#2155	1	1	yes	characteristic
81	C₂×C₄₇	#2155	1	1	yes	characteristic
82	#432	—	94	1	no	—
83	#431	—	2	1	no	—
84	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
85	C₄⋊C₁₆	—	47	2	no	—
86	C₂²⋊C₁₆	—	47	2	no	—
87	#346	—	47	2	no	—
88	#335	—	2	1	no	—
89	#335	—	94	1	no	—
90	#334	—	47	2	no	—
91	#334	C₂×D₄₇	1	2	yes	—
92	C₄₇	#12031	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
96	C₂×C₁₆	—	47	4	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	94	2	no	—
99	C₂²⋊C₈	—	4	1	no	—
100	C₂²⋊C₈	—	188	1	no	—
101	#97	—	188	1	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	Dic₉₄	1	1	yes	characteristic
104	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
107	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
108	C₂²×C₄	#10625335	1	1	yes	characteristic
109	C₂²×C₄	#10625336	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	188	2	no	—
114	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
115	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
117	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
118	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
119	C₄²	#10625335	1	1	yes	characteristic
120	C₄²	#10625336	1	1	yes	characteristic
121	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
122	C₁₆	—	94	4	no	—
123	C₂³	#13093955	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#13093955	1	1	yes	characteristic
129	C₂×C₄	#13093955	1	1	yes	characteristic
130	C₂×C₄	#13093955	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
133	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	2	no	—
138	C₈	—	376	2	no	—
139	C₈	#13094030	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#13093955	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#15689149	1	1	yes	characteristic
150	C₄	#15689171	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	#20242158	1	1	yes	characteristic
153	C₂	#20245160	1	1	yes	characteristic
154	C₂	#20245162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245191

C₂

yes

maximal, characteristic

#20242151

C₂

yes

maximal, characteristic

#20241453

C₂

yes

maximal, complemented

C₄₇×#1564

C₂

yes

maximal, characteristic

#15689180

—

C₄⋊(C₄₇⋊C₁₆)

C₄

yes

—

C₂²⋊(C₄₇⋊C₁₆)

C₄

yes

—

#15688899

C₂²

yes

—

#15688876

—

#15688874

C₂²

yes

—

C₄₇×#431

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#335

—

C₄₇×#334

C₂²

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#106

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#131

yes

—

C₄₇⋊C₈

—

#12031

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#106

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

—

D₄₇

yes

characteristic

#1368

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

C₂×D₄₇

yes

—

C₄₇

#12031

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

188

—

101

#97

—

188

—

102

C₄×C₈

—

103

C₄×C₈

Dic₉₄

yes

characteristic

104

C₄×C₈

C₄×D₄₇

yes

characteristic

105

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

106

C₄×C₈

C₂²×D₄₇

yes

characteristic

107

C₂²×C₄

Dic₄₇⋊C₄

yes

characteristic

108

C₂²×C₄

#10625335

yes

characteristic

109

C₂²×C₄

#10625336

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

188

—

114

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

115

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

116

C₂×C₈

C₂×Dic₉₄

yes

characteristic

117

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

118

C₄²

Dic₄₇⋊C₄

yes

characteristic

119

C₄²

#10625335

yes

characteristic

120

C₄²

#10625336

yes

characteristic

121

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

122

C₁₆

—

123

C₂³

#13093955

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#13093955

yes

characteristic

129

C₂×C₄

#13093955

yes

characteristic

130

C₂×C₄

#13093955

yes

characteristic

131

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

132

C₂×C₄

C₂×#10625336

yes

characteristic

133

C₂×C₄

C₂×#10625335

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

376

—

139

C₈

#13094030

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#13093955

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#15689149

yes

characteristic

150

C₄

#15689171

yes

characteristic

151

C₂

—

152

C₂

#20242158

yes

characteristic

153

C₂

#20245160

yes

characteristic

154

C₂

#20245162

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13093955
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	C₂×#106
prime radical	C₂³×C₄₇	#106
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#12031
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

subgroup

G/H

center

C₂²

C₂×#13093955

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1564

C₂

C₂²×C₄₇

C₂³×C₄₇

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

N	H ≅ G/N	type
3. #20241453	C₂	semidirect
92. C₄₇	#12031	semidirect

H ≅ G/N

type

3. #20241453

C₂

semidirect

92. C₄₇

#12031

semidirect

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

subgroup	index
#20245191	2
#20242151	2
#20241453	2
C₄₇×#1564	2
#12031	47

subgroup

index

C₄₇×#1564

Subgroups — #26127954

Conjugacy classes of subgroups

Complements

Maximal subgroups