Group #26127831 — subgroups

Subgroups — #26127831

Order 12032.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#346×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20242133	C₂	1	2	yes	maximal
3	#20241454	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241453	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₄₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₄₇	C₄	1	1	yes	characteristic
8	#15689175	—	2	2	no	—
9	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
10	#15688899	C₂²	1	1	yes	characteristic
11	#15688898	C₄	1	1	yes	characteristic
12	#15688898	C₄	1	1	yes	characteristic
13	#15688875	—	2	2	no	—
14	#15688874	C₂²	1	2	yes	—
15	C₄₇×#346	C₂²	1	1	yes	characteristic
16	C₄₇×#334	C₂²	1	2	yes	—
17	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₄₇	Q₈	1	1	yes	characteristic
19	C₄²×D₄₇	D₄	1	1	yes	characteristic
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₁₆×D₄₇	—	2	2	no	—
23	C₁₆×D₄₇	C₈	1	4	yes	complemented
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	(C₂×D₄₇)⋊C₈	—	4	2	no	—
26	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
27	C₈×Dic₄₇	—	4	2	no	—
28	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₄₇	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	—	2	2	no	—
33	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₄₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₄₇	SD₁₆	1	2	yes	—
42	C₂×C₄₇⋊C₈	—	4	2	no	—
43	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₈×D₄₇	—	2	2	no	—
46	C₈×D₄₇	C₂×C₈	1	2	yes	—
47	C₈×D₄₇	M₁₆	1	2	yes	—
48	C₁₆×C₄₇	—	2	1	no	—
49	C₁₆×C₄₇	C₂×C₈	1	2	yes	—
50	C₄₇⋊C₁₆	—	2	1	no	—
51	C₄₇⋊C₁₆	C₂×C₈	1	2	yes	—
52	C₂²×D₄₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
55	C₂×Dic₄₇	#106	1	2	yes	—
56	C₂×Dic₄₇	#106	1	1	yes	characteristic
57	C₄×D₄₇	—	4	2	no	—
58	C₄×D₄₇	C₄⋊C₈	1	2	yes	—
59	C₄×D₄₇	#108	1	2	yes	—
60	C₈×C₄₇	—	2	1	no	—
61	C₈×C₄₇	—	8	2	no	—
62	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
64	C₄₇⋊C₈	—	2	1	no	—
65	C₄₇⋊C₈	—	8	2	no	—
66	C₄₇⋊C₈	C₄⋊C₈	1	2	yes	—
67	#11999	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
69	C₂×D₄₇	—	2	2	no	—
70	C₂×D₄₇	#334	1	2	yes	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
74	Dic₄₇	—	2	2	no	—
75	Dic₄₇	#334	1	2	yes	—
76	C₂×#346	—	47	1	no	—
77	#1564	—	47	2	no	—
78	#1368	—	47	1	no	—
79	#1368	—	47	2	no	—
80	#1368	D₄₇	1	1	yes	characteristic, complemented
81	C₂×C₄₇	C₂×#334	1	1	yes	characteristic
82	C₂×C₄₇	#2155	1	2	yes	—
83	D₄₇	—	2	2	no	—
84	C₂²×C₁₆	—	47	1	no	—
85	C₂²×C₁₆	—	47	1	no	—
86	#431	—	94	2	no	—
87	C₂×C₄×C₈	—	47	1	no	—
88	#346	—	47	1	no	—
89	#346	—	47	1	no	—
90	#346	—	47	1	no	—
91	#346	C₂×D₄₇	1	1	yes	characteristic
92	#335	—	94	2	no	—
93	#334	—	47	2	no	—
94	#334	C₂×D₄₇	1	2	yes	—
95	C₄₇	#11999	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	47	1	no	—
97	C₂²×C₈	—	47	1	no	—
98	C₂×C₄²	—	47	1	no	—
99	C₂×C₁₆	—	47	1	no	—
100	C₂×C₁₆	—	47	4	no	—
101	C₂×C₁₆	—	47	1	no	—
102	C₂×C₁₆	—	94	2	no	—
103	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
104	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	94	2	no	—
107	C₂²⋊C₈	—	188	2	no	—
108	C₄×C₈	—	47	1	no	—
109	C₄×C₈	—	47	1	no	—
110	C₄×C₈	—	47	1	no	—
111	C₄×C₈	—	188	2	no	—
112	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
113	C₂²×C₄	—	47	1	no	—
114	C₂²×C₄	—	47	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	47	1	no	—
117	C₂×C₈	—	47	2	no	—
118	C₂×C₈	—	47	2	no	—
119	C₂×C₈	—	47	1	no	—
120	C₂×C₈	—	94	2	no	—
121	C₂×C₈	—	188	2	no	—
122	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
123	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
124	C₄²	—	47	1	no	—
125	C₄²	—	47	2	no	—
126	C₄²	D₄×D₄₇	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	47	2	no	—
129	C₁₆	—	94	1	no	—
130	C₁₆	C₈×D₄₇	1	2	yes	—
131	C₂³	—	47	1	no	—
132	C₂×C₄	—	47	1	no	—
133	C₂×C₄	—	47	2	no	—
134	C₂×C₄	—	47	2	no	—
135	C₂×C₄	—	47	2	no	—
136	C₂×C₄	—	188	2	no	—
137	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
138	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	—	47	2	no	—
142	C₈	—	94	1	no	—
143	C₈	—	376	2	no	—
144	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
145	C₈	M₁₆×D₄₇	1	1	yes	characteristic
146	C₂²	—	47	2	no	—
147	C₂²	—	94	2	no	—
148	C₂²	#106×D₄₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	47	2	no	—
151	C₄	—	94	2	no	—
152	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
153	C₄	#108×D₄₇	1	1	yes	characteristic
154	C₂	—	94	2	no	—
155	C₂	#334×D₄₇	1	1	yes	characteristic
156	C₂	#20245159	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#346×D₄₇

C₂

yes

maximal, characteristic

#20242133

C₂

yes

maximal

#20241454

C₂

yes

maximal, characteristic, complemented

#20241453

C₂

yes

maximal, complemented

C₄₇×#1368

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₄₇

C₄

yes

characteristic

C₂×C₁₆×D₄₇

C₄

yes

characteristic

#15689175

—

C₄×C₈×D₄₇

C₂²

yes

characteristic

#15688899

C₂²

yes

characteristic

#15688898

C₄

yes

characteristic

#15688898

C₄

yes

characteristic

#15688875

—

#15688874

C₂²

yes

—

C₄₇×#346

C₂²

yes

characteristic

C₄₇×#334

C₂²

yes

—

C₂×C₈×D₄₇

C₂×C₄

yes

characteristic

C₂×C₈×D₄₇

Q₈

yes

characteristic

C₄²×D₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₄₇

—

C₁₆×D₄₇

C₈

yes

complemented

C₄⋊(C₄₇⋊C₈)

—

(C₂×D₄₇)⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₈×Dic₄₇

—

C₈×Dic₄₇

C₂×C₄

yes

characteristic

C₈×Dic₄₇

Q₈

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄×D₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄×Dic₄₇

C₄⋊C₄

yes

characteristic

C₄×Dic₄₇

SD₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₄₇

—

C₈×D₄₇

C₂×C₈

yes

—

C₈×D₄₇

M₁₆

yes

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×C₈

yes

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

C₂×C₈

yes

—

C₂²×D₄₇

#106

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×Dic₄₇

#106

yes

—

C₂×Dic₄₇

#106

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

C₄⋊C₈

yes

—

C₄×D₄₇

#108

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

C₂×M₁₆

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊C₈

yes

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

#334

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₄₇

C₂×#108

yes

characteristic

Dic₄₇

—

Dic₄₇

#334

yes

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

C₂×#334

yes

characteristic

C₂×C₄₇

#2155

yes

—

D₄₇

—

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

—

—

—

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

C₄₇

#11999

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₄×D₄₇

yes

characteristic

104

C₂×C₁₆

C₄×D₄₇

yes

characteristic

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

188

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

188

—

112

C₄×C₈

C₂²×D₄₇

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

188

—

122

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

123

C₂×C₈

Q₈×D₄₇

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

D₄×D₄₇

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

130

C₁₆

C₈×D₄₇

yes

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

188

—

137

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

138

C₂×C₄

SD₁₆×D₄₇

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

376

—

144

C₈

C₂×C₈×D₄₇

yes

characteristic

145

C₈

M₁₆×D₄₇

yes

characteristic

146

C₂²

—

147

C₂²

—

148

C₂²

#106×D₄₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄⋊C₈×D₄₇

yes

characteristic

153

C₄

#108×D₄₇

yes

characteristic

154

C₂

—

155

C₂

#334×D₄₇

yes

characteristic

156

C₂

#20245159

yes

—

	subgroup	G/H
center	C₂²	#106×D₄₇
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1368	C₂
socle	C₂²×C₄₇	C₂×#106
prime radical	C₂²×D₄₇	#106
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1368	D₄₇
hypocenter	C₄₇	#11999
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#106×D₄₇

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1368

C₂

C₂²×C₄₇

C₂²×D₄₇

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
3. #20241454	C₂	semidirect
4. #20241453	C₂	semidirect
5. C₄₇×#1368	C₂	semidirect
23. C₁₆×D₄₇	C₈	semidirect
80. #1368	D₄₇	semidirect
95. C₄₇	#11999	semidirect

H ≅ G/N

type

3. #20241454

C₂

semidirect

4. #20241453

C₂

semidirect

5. C₄₇×#1368

C₂

semidirect

23. C₁₆×D₄₇

C₈

semidirect

80. #1368

D₄₇

semidirect

95. C₄₇

#11999

semidirect

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

subgroup	index
#346×D₄₇	2
#20242133	2
#20241454	2
#20241453	2
C₄₇×#1368	2
#11999	47

subgroup

index

#346×D₄₇

C₄₇×#1368

Subgroups — #26127831

Conjugacy classes of subgroups

Complements

Maximal subgroups