Group #26128033 — subgroups

Subgroups — #26128033

Order 12032.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20245270	C₂	1	1	yes	maximal, characteristic
2	#20245224	C₂	1	2	yes	maximal, complemented
3	#20241360	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689555	C₄	1	1	yes	characteristic
7	#15689541	—	2	2	no	—
8	#15689241	C₂²	1	2	yes	—
9	#15688953	C₄	1	1	yes	characteristic
10	#15688931	C₄	1	2	yes	—
11	#15688908	—	2	2	no	—
12	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₄₇×#364	C₂²	1	2	yes	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	#13094046	C₂×C₄	1	2	yes	—
17	#13094030	—	2	2	no	—
18	Dic₃₇₆	C₈	1	1	yes	characteristic, complemented
19	#13093969	C₈	1	2	yes	complemented
20	D₇₅₂	C₈	1	1	yes	characteristic, complemented
21	#13093963	—	4	2	no	—
22	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₄₇	—	4	2	no	—
25	#13093960	C₂×C₄	1	2	yes	—
26	#13093952	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	1	no	—
28	M₃₂×C₄₇	—	2	2	no	—
29	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	#10625350	—	2	2	no	—
33	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₄₇	#41	1	1	yes	characteristic
36	#10625329	—	2	2	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	Dic₁₈₈	C₂×C₈	1	2	yes	—
39	D₃₇₆	C₂×C₈	1	2	yes	—
40	#10625325	—	2	2	no	—
41	#10625324	—	4	2	no	—
42	C₈×D₄₇	—	4	2	no	—
43	C₁₆×C₄₇	—	2	1	no	—
44	C₁₆×C₄₇	—	4	2	no	—
45	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
47	C₄₇⋊C₁₆	—	2	1	no	—
48	C₄₇⋊C₁₆	D₈	1	1	yes	characteristic
49	C₄₇⋊C₁₆	D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
52	C₂²⋊D₄₇	—	4	2	no	—
53	D₁₈₈	—	2	2	no	—
54	C₄×D₄₇	—	4	2	no	—
55	Dic₉₄	—	2	2	no	—
56	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₄₇	#131	1	1	yes	characteristic
58	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
59	C₈×C₄₇	#135	1	1	yes	characteristic
60	C₄₇⋊C₈	—	2	4	no	—
61	C₄₇⋊C₈	—	4	2	no	—
62	#12046	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#443	1	1	yes	characteristic
64	C₂×D₄₇	—	4	2	no	—
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
67	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
68	Dic₄₇	—	4	2	no	—
69	#2178	—	47	1	no	—
70	#2170	—	47	2	no	—
71	#1373	D₄₇	1	1	yes	characteristic, complemented
72	#1335	—	47	2	no	—
73	C₈×C₁₆	—	47	1	no	—
74	C₂×C₄₇	—	2	1	no	—
75	C₂×C₄₇	C₈×D₈	1	1	yes	characteristic
76	D₄₇	—	8	2	no	—
77	#508	—	47	1	no	—
78	#504	—	94	2	no	—
79	#443	—	47	2	no	—
80	#368	—	47	1	no	—
81	#364	C₂×D₄₇	1	2	yes	—
82	#359	—	47	2	no	—
83	#350	—	94	2	no	—
84	C₄×C₁₆	—	47	1	no	—
85	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
86	C₈²	—	47	1	no	—
87	C₄₇	#12046	1	1	yes	characteristic, complemented, Sylow
88	#135	—	47	2	no	—
89	#131	—	94	2	no	—
90	Q₃₂	—	47	1	no	—
91	SD₃₂	—	47	2	no	—
92	D₁₆	—	47	1	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	188	2	no	—
95	C₂×C₁₆	—	47	1	no	—
96	C₂×C₁₆	—	47	1	no	—
97	C₂×C₁₆	—	188	2	no	—
98	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
100	#108	—	47	2	no	—
101	#104	—	94	2	no	—
102	C₄×C₈	—	94	1	no	—
103	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	#41	—	94	2	no	—
105	Q₁₆	—	47	2	no	—
106	SD₁₆	—	94	2	no	—
107	D₈	—	47	2	no	—
108	M₁₆	—	94	2	no	—
109	M₁₆	—	188	2	no	—
110	C₂×C₈	—	94	2	no	—
111	C₂×C₈	—	188	2	no	—
112	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
113	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
114	C₄²	#10625354	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	—	47	1	no	—
118	C₁₆	—	47	1	no	—
119	C₁₆	—	94	1	no	—
120	C₁₆	C₈×D₄₇	1	1	yes	characteristic
121	C₁₆	C₈×D₄₇	1	1	yes	characteristic
122	Q₈	—	94	2	no	—
123	D₄	—	94	2	no	—
124	D₄	—	188	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	188	2	no	—
127	C₂×C₄	#13093986	1	1	yes	characteristic
128	C₈	—	94	4	no	—
129	C₈	—	188	2	no	—
130	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
131	C₈	#13094032	1	1	yes	characteristic
132	C₈	D₈×D₄₇	1	1	yes	characteristic
133	C₈	#13094047	1	1	yes	characteristic
134	C₂²	—	188	2	no	—
135	C₂²	#15689239	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	188	2	no	—
138	C₄	#15689188	1	1	yes	characteristic
139	C₄	#15689205	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	376	2	no	—
142	C₂	#20242182	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245270

C₂

yes

maximal, characteristic

#20245224

C₂

yes

maximal, complemented

#20241360

C₂

yes

maximal, complemented

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#1373

C₂

yes

maximal, characteristic, complemented

#15689555

C₄

yes

characteristic

#15689541

—

#15689241

C₂²

yes

—

#15688953

C₄

yes

characteristic

#15688931

C₄

yes

—

#15688908

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₄₇×#364

C₂²

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

#13094046

C₂×C₄

yes

—

#13094030

—

Dic₃₇₆

C₈

yes

characteristic, complemented

#13093969

C₈

yes

complemented

D₇₅₂

C₈

yes

characteristic, complemented

#13093963

—

C₂×C₄₇⋊C₁₆

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

D₄

yes

characteristic

C₁₆×D₄₇

—

#13093960

C₂×C₄

yes

—

#13093952

—

C₄×C₄₇⋊C₈

—

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

#10625350

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

C₂×C₈

yes

—

D₃₇₆

C₂×C₈

yes

—

#10625325

—

#10625324

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×C₈

yes

characteristic

C₁₆×C₄₇

C₂×C₈

yes

characteristic

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

D₈

yes

characteristic

C₄₇⋊C₁₆

D₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂²⋊D₄₇

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

#131

yes

characteristic

C₈×C₄₇

C₂×D₈

yes

characteristic

C₈×C₄₇

#135

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

C₄×D₈

yes

characteristic

Dic₄₇

—

#2178

—

#2170

—

#1373

D₄₇

yes

characteristic, complemented

#1335

—

C₈×C₁₆

—

C₂×C₄₇

—

C₂×C₄₇

C₈×D₈

yes

characteristic

D₄₇

—

—

—

—

—

C₂×D₄₇

yes

—

#359

—

#350

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12046

yes

characteristic, complemented, Sylow

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

188

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

188

—

C₂×C₁₆

C₄×D₄₇

yes

characteristic

C₂×C₁₆

C₄×D₄₇

yes

characteristic

100

#108

—

101

#104

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₄₇

yes

characteristic

104

#41

—

105

Q₁₆

—

106

SD₁₆

—

107

D₈

—

108

M₁₆

—

109

M₁₆

—

188

—

110

C₂×C₈

—

111

C₂×C₈

—

188

—

112

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

113

C₂×C₈

D₄×D₄₇

yes

characteristic

114

C₄²

#10625354

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

C₈×D₄₇

yes

characteristic

121

C₁₆

C₈×D₄₇

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

188

—

125

C₂×C₄

—

126

C₂×C₄

—

188

—

127

C₂×C₄

#13093986

yes

characteristic

128

C₈

—

129

C₈

—

188

—

130

C₈

C₂×C₈×D₄₇

yes

characteristic

131

C₈

#13094032

yes

characteristic

132

C₈

D₈×D₄₇

yes

characteristic

133

C₈

#13094047

yes

characteristic

134

C₂²

—

188

—

135

C₂²

#15689239

yes

characteristic

136

C₄

—

137

C₄

—

188

—

138

C₄

#15689188

yes

characteristic

139

C₄

#15689205

yes

characteristic

140

C₂

—

141

C₂

—

376

—

142

C₂

#20242182

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₄₇
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1373	C₂
socle	C₂×C₄₇	C₈×D₈
prime radical	#15689555	C₄
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1373	D₄₇
hypocenter	C₄₇	#12046
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₄₇

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1373

C₂

socle

C₂×C₄₇

C₈×D₈

prime radical

#15689555

C₄

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #20245224	C₂	semidirect
3. #20241360	C₂	semidirect
4. C₁₆×C₄₇⋊C₈	C₂	semidirect
5. C₄₇×#1373	C₂	semidirect
18. Dic₃₇₆	C₈	semidirect
19. #13093969	C₈	semidirect
20. D₇₅₂	C₈	semidirect
71. #1373	D₄₇	semidirect
87. C₄₇	#12046	semidirect

H ≅ G/N

type

2. #20245224

C₂

semidirect

3. #20241360

C₂

semidirect

4. C₁₆×C₄₇⋊C₈

C₂

semidirect

5. C₄₇×#1373

C₂

semidirect

18. Dic₃₇₆

C₈

semidirect

19. #13093969

C₈

semidirect

20. D₇₅₂

C₈

semidirect

71. #1373

D₄₇

semidirect

87. C₄₇

#12046

semidirect

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

subgroup	index
#20245270	2
#20245224	2
#20241360	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#1373	2
#12046	47

subgroup

index

#20245270

#20245224

#20241360

C₁₆×C₄₇⋊C₈

C₄₇×#1373

#12046

Subgroups — #26128033

Conjugacy classes of subgroups

Complements

Maximal subgroups