Group #26128167 — subgroups

Subgroups — #26128167

Order 12032.

204 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 73 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	#20245275	C₂	1	1	yes	maximal, characteristic
2	#20245229	C₂	1	2	yes	maximal, complemented
3	#20241363	C₂	1	2	yes	maximal, complemented
4	#20241308	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1369	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	#15688899	C₂²	1	1	yes	characteristic
13	#15688844	C₂²	1	1	yes	characteristic
14	C₄₇×#364	C₂²	1	2	yes	—
15	C₄₇×#346	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	2	yes	—
23	C₁₆×D₄₇	—	4	2	no	—
24	#13093960	C₂×C₄	1	2	yes	—
25	#13093952	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	1	no	—
27	M₃₂×C₄₇	—	2	2	no	—
28	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	#10625350	—	2	2	no	—
32	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₄₇	#41	1	1	yes	characteristic
35	#10625329	—	2	2	no	—
36	C₂×C₄₇⋊C₈	—	4	1	no	—
37	Dic₁₈₈	C₂×C₈	1	2	yes	—
38	D₃₇₆	C₂×C₈	1	2	yes	—
39	#10625325	—	2	2	no	—
40	#10625324	—	4	2	no	—
41	C₈×D₄₇	—	4	2	no	—
42	C₁₆×C₄₇	—	2	1	no	—
43	C₁₆×C₄₇	—	4	2	no	—
44	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
46	C₄₇⋊C₁₆	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
49	C₂²⋊D₄₇	—	4	2	no	—
50	D₁₈₈	—	2	2	no	—
51	C₄×D₄₇	—	4	2	no	—
52	Dic₉₄	—	2	2	no	—
53	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₄₇	#131	1	1	yes	characteristic
55	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
56	C₄₇⋊C₈	—	4	2	no	—
57	C₄₇⋊C₈	—	8	1	no	—
58	#12068	—	47	1	no	maximal, Sylow
59	C₂²×C₄₇	#444	1	1	yes	characteristic
60	C₂×D₄₇	—	4	2	no	—
61	C₄×C₄₇	—	4	1	no	—
62	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
63	C₄×C₄₇	#442	1	1	yes	characteristic
64	Dic₄₇	—	4	2	no	—
65	#2179	—	47	1	no	—
66	#2171	—	47	2	no	—
67	#1369	D₄₇	1	1	yes	characteristic, complemented
68	#1336	—	47	2	no	—
69	#1313	—	47	1	no	—
70	C₂×C₄₇	—	2	1	no	—
71	C₂×C₄₇	#1580	1	1	yes	characteristic
72	D₄₇	—	8	2	no	—
73	#508	—	47	1	no	—
74	#504	—	94	2	no	—
75	#443	—	47	2	no	—
76	#368	—	47	1	no	—
77	#364	C₂×D₄₇	1	2	yes	—
78	#359	—	47	2	no	—
79	#350	—	94	2	no	—
80	#346	—	47	1	no	—
81	#346	C₂×D₄₇	1	1	yes	characteristic
82	#322	—	47	1	no	—
83	C₄₇	#12068	1	1	yes	characteristic, complemented, Sylow
84	#135	—	47	2	no	—
85	#131	—	94	2	no	—
86	Q₃₂	—	47	1	no	—
87	SD₃₂	—	47	2	no	—
88	D₁₆	—	47	1	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	—	188	2	no	—
91	C₂×C₁₆	—	47	2	no	—
92	C₂×C₁₆	—	188	2	no	—
93	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
94	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
95	#108	—	47	2	no	—
96	#104	—	94	2	no	—
97	C₄×C₈	—	94	1	no	—
98	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	#41	—	94	2	no	—
100	Q₁₆	—	47	2	no	—
101	SD₁₆	—	94	2	no	—
102	D₈	—	47	2	no	—
103	M₁₆	—	94	2	no	—
104	M₁₆	—	188	2	no	—
105	C₂×C₈	—	188	1	no	—
106	C₂×C₈	—	188	2	no	—
107	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
108	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
109	C₄²	#10625354	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	2	no	—
112	C₁₆	—	94	2	no	—
113	C₁₆	C₈×D₄₇	1	1	yes	characteristic
114	C₁₆	C₈×D₄₇	1	1	yes	characteristic
115	Q₈	—	94	2	no	—
116	D₄	—	94	2	no	—
117	D₄	—	188	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	188	2	no	—
120	C₂×C₄	#13093986	1	1	yes	characteristic
121	C₈	—	376	1	no	—
122	C₈	—	188	2	no	—
123	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
124	C₈	#13094032	1	1	yes	characteristic
125	C₈	#13094051	1	2	yes	—
126	C₂²	—	188	2	no	—
127	C₂²	#15689244	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	188	2	no	—
130	C₄	#15689188	1	1	yes	characteristic
131	C₄	#15689234	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	376	2	no	—
134	C₂	#20242211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245275

C₂

yes

maximal, characteristic

#20245229

C₂

yes

maximal, complemented

#20241363

C₂

yes

maximal, complemented

#20241308

C₂

yes

maximal, characteristic, complemented

C₄₇×#1369

C₂

yes

maximal, characteristic, complemented

#15689555

C₄

yes

characteristic

#15689541

—

#15689241

C₂²

yes

—

#15688953

C₄

yes

characteristic

#15688931

C₄

yes

—

#15688908

—

#15688899

C₂²

yes

characteristic

#15688844

C₂²

yes

characteristic

C₄₇×#364

C₂²

yes

—

C₄₇×#346

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

—

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

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂²⋊D₄₇

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂²×C₈

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

#442

yes

characteristic

Dic₄₇

—

#2179

—

#2171

—

#1369

D₄₇

yes

characteristic, complemented

#1336

—

#1313

—

C₂×C₄₇

—

C₂×C₄₇

#1580

yes

characteristic

D₄₇

—

—

—

—

—

C₂×D₄₇

yes

—

—

—

—

C₂×D₄₇

yes

characteristic

#322

—

C₄₇

#12068

yes

characteristic, complemented, Sylow

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

188

—

C₂×C₁₆

—

C₂×C₁₆

—

188

—

C₂×C₁₆

C₄×D₄₇

yes

characteristic

C₂×C₁₆

C₄×D₄₇

yes

characteristic

#108

—

#104

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

#41

—

100

Q₁₆

—

101

SD₁₆

—

102

D₈

—

103

M₁₆

—

104

M₁₆

—

188

—

105

C₂×C₈

—

188

—

106

C₂×C₈

—

188

—

107

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

108

C₂×C₈

D₄×D₄₇

yes

characteristic

109

C₄²

#10625354

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₈×D₄₇

yes

characteristic

114

C₁₆

C₈×D₄₇

yes

characteristic

115

Q₈

—

116

D₄

—

117

D₄

—

188

—

118

C₂×C₄

—

119

C₂×C₄

—

188

—

120

C₂×C₄

#13093986

yes

characteristic

121

C₈

—

376

—

122

C₈

—

188

—

123

C₈

C₂×C₈×D₄₇

yes

characteristic

124

C₈

#13094032

yes

characteristic

125

C₈

#13094051

yes

—

126

C₂²

—

188

—

127

C₂²

#15689244

yes

characteristic

128

C₄

—

129

C₄

—

188

—

130

C₄

#15689188

yes

characteristic

131

C₄

#15689234

yes

characteristic

132

C₂

—

133

C₂

—

376

—

134

C₂

#20242211

yes

characteristic

	subgroup	G/H
center	C₄	#15689234
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1369	C₂
socle	C₂×C₄₇	#1580
prime radical	#15689555	C₄
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1369	D₄₇
hypocenter	C₄₇	#12068
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

subgroup

G/H

center

C₄

#15689234

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1369

C₂

C₂×C₄₇

C₄

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

N	H ≅ G/N	type
2. #20245229	C₂	semidirect
3. #20241363	C₂	semidirect
4. #20241308	C₂	semidirect
5. C₄₇×#1369	C₂	semidirect
18. Dic₃₇₆	C₈	semidirect
19. #13093969	C₈	semidirect
20. D₇₅₂	C₈	semidirect
67. #1369	D₄₇	semidirect
83. C₄₇	#12068	semidirect

H ≅ G/N

type

2. #20245229

C₂

semidirect

3. #20241363

C₂

semidirect

4. #20241308

C₂

semidirect

5. C₄₇×#1369

C₂

semidirect

18. Dic₃₇₆

C₈

semidirect

19. #13093969

C₈

semidirect

20. D₇₅₂

C₈

semidirect

67. #1369

D₄₇

semidirect

83. C₄₇

#12068

semidirect

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

subgroup	index
#20245275	2
#20245229	2
#20241363	2
#20241308	2
C₄₇×#1369	2
#12068	47

subgroup

index

C₄₇×#1369

Subgroups — #26128167

Conjugacy classes of subgroups

Complements

Maximal subgroups