Group #26128168 — subgroups

Subgroups — #26128168

Order 12032.

204 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20245230	C₂	1	2	yes	maximal, complemented
2	#20241456	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241364	C₂	1	2	yes	maximal, complemented
4	#20241308	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2179	C₂	1	1	yes	maximal, characteristic
6	#15689540	—	2	2	no	—
7	#15688944	C₂²	1	2	yes	—
8	#15688907	—	2	2	no	—
9	#15688899	C₂²	1	1	yes	characteristic
10	#15688844	C₂²	1	1	yes	characteristic
11	C₄₇×#508	C₄	1	1	yes	characteristic
12	C₄₇×#443	C₂²	1	2	yes	—
13	C₄₇×#368	C₄	1	1	yes	characteristic
14	C₄₇×#359	C₄	1	2	yes	—
15	C₄₇×#346	C₂²	1	1	yes	characteristic
16	#13093964	—	2	2	no	—
17	#13093964	—	4	2	no	—
18	C₂×C₄₇⋊C₁₆	—	4	2	no	—
19	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
20	C₄×C₄₇⋊C₈	—	2	1	no	—
21	#135×C₄₇	C₂×C₄	1	2	yes	—
22	#131×C₄₇	—	2	2	no	—
23	Q₃₂×C₄₇	C₈	1	1	yes	characteristic, complemented
24	SD₃₂×C₄₇	C₈	1	2	yes	complemented
25	D₁₆×C₄₇	C₈	1	1	yes	characteristic, complemented
26	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
28	#108×C₄₇	C₂×C₄	1	2	yes	—
29	#104×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	#41×C₄₇	—	2	2	no	—
32	Q₁₆×C₄₇	C₂×C₈	1	2	yes	—
33	SD₁₆×C₄₇	—	2	2	no	—
34	D₈×C₄₇	C₂×C₈	1	2	yes	—
35	M₁₆×C₄₇	—	2	2	no	—
36	M₁₆×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₄₇	#41	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	—	4	1	no	—
42	C₁₆×C₄₇	—	2	1	no	—
43	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
44	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₄₇⋊C₁₆	—	2	2	no	—
46	C₄₇⋊C₁₆	—	4	2	no	—
47	Q₈×C₄₇	—	2	2	no	—
48	D₄×C₄₇	—	2	2	no	—
49	D₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	4	2	no	—
52	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
53	C₈×C₄₇	—	4	2	no	—
54	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₄₇	#131	1	1	yes	characteristic
56	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₄₇⋊C₈	—	8	1	no	—
58	#12068	—	47	1	no	maximal, Sylow
59	C₂²×C₄₇	—	4	2	no	—
60	C₂²×C₄₇	#444	1	1	yes	characteristic
61	C₄×C₄₇	—	4	1	no	—
62	C₄×C₄₇	—	4	2	no	—
63	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
64	C₄×C₄₇	#442	1	1	yes	characteristic
65	#2179	D₄₇	1	1	yes	characteristic
66	#2171	—	47	2	no	—
67	#1369	—	47	1	no	—
68	#1336	—	47	2	no	—
69	#1313	—	47	1	no	—
70	C₂×C₄₇	—	2	1	no	—
71	C₂×C₄₇	—	8	2	no	—
72	C₂×C₄₇	#1580	1	1	yes	characteristic
73	#508	Dic₄₇	1	1	yes	characteristic
74	#504	—	94	2	no	—
75	#443	C₂×D₄₇	1	2	yes	—
76	#368	Dic₄₇	1	1	yes	characteristic
77	#364	—	47	2	no	—
78	#359	Dic₄₇	1	2	yes	—
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	C₂×Dic₄₇	1	2	yes	—
85	#131	—	2	2	no	—
86	Q₃₂	C₄₇⋊C₈	1	1	yes	characteristic, complemented
87	SD₃₂	C₄₇⋊C₈	1	2	yes	complemented
88	D₁₆	C₄₇⋊C₈	1	1	yes	characteristic, complemented
89	M₃₂	—	94	2	no	—
90	M₃₂	—	188	2	no	—
91	C₂×C₁₆	—	47	2	no	—
92	C₂×C₁₆	—	188	2	no	—
93	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
94	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
95	#108	C₂×Dic₄₇	1	2	yes	—
96	#104	—	2	2	no	—
97	C₄×C₈	—	94	1	no	—
98	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	#41	—	2	2	no	—
100	Q₁₆	C₂×C₄₇⋊C₈	1	2	yes	—
101	SD₁₆	—	2	2	no	—
102	D₈	C₂×C₄₇⋊C₈	1	2	yes	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	188	1	no	—
107	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
108	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
109	C₄²	#10625352	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	94	2	no	—
112	C₁₆	—	188	2	no	—
113	C₁₆	C₂×C₄₇⋊C₈	1	1	yes	characteristic
114	C₁₆	C₂×C₄₇⋊C₈	1	1	yes	characteristic
115	Q₈	—	2	2	no	—
116	D₄	—	2	2	no	—
117	D₄	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	376	1	no	—
123	C₈	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
124	C₈	#13094031	1	1	yes	characteristic
125	C₈	#13094048	1	2	yes	—
126	C₂²	—	4	2	no	—
127	C₂²	#15689245	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	2	no	—
130	C₄	D₄×C₄₇⋊C₈	1	1	yes	characteristic
131	C₄	#15689235	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	8	2	no	—
134	C₂	#20242212	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245230

C₂

yes

maximal, complemented

#20241456

C₂

yes

maximal, characteristic, complemented

#20241364

C₂

yes

maximal, complemented

#20241308

C₂

yes

maximal, characteristic, complemented

C₄₇×#2179

C₂

yes

maximal, characteristic

#15689540

—

#15688944

C₂²

yes

—

#15688907

—

#15688899

C₂²

yes

characteristic

#15688844

C₂²

yes

characteristic

C₄₇×#508

C₄

yes

characteristic

C₄₇×#443

C₂²

yes

—

C₄₇×#368

C₄

yes

characteristic

C₄₇×#359

C₄

yes

—

C₄₇×#346

C₂²

yes

characteristic

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₄×C₄₇⋊C₈

—

#135×C₄₇

C₂×C₄

yes

—

#131×C₄₇

—

Q₃₂×C₄₇

C₈

yes

characteristic, complemented

SD₃₂×C₄₇

C₈

yes

complemented

D₁₆×C₄₇

C₈

yes

characteristic, complemented

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

#108×C₄₇

C₂×C₄

yes

—

#104×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

#41×C₄₇

—

Q₁₆×C₄₇

C₂×C₈

yes

—

SD₁₆×C₄₇

—

D₈×C₄₇

C₂×C₈

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

characteristic

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×C₈

yes

characteristic

C₁₆×C₄₇

C₂×C₈

yes

characteristic

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₈

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

#12068

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#444

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈×D₄

yes

C₄×C₄₇

yes

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1580

yes

characteristic

#508

Dic₄₇

yes

characteristic

#504

—

#443

C₂×D₄₇

yes

—

#368

Dic₄₇

yes

characteristic

#364

—

#359

Dic₄₇

yes

—

#350

—

#346

—

#346

C₂×D₄₇

yes

characteristic

#322

—

C₄₇

#12068

yes

characteristic, complemented, Sylow

#135

C₂×Dic₄₇

yes

—

#131

—

Q₃₂

C₄₇⋊C₈

yes

characteristic, complemented

SD₃₂

C₄₇⋊C₈

yes

complemented

D₁₆

C₄₇⋊C₈

yes

characteristic, complemented

M₃₂

—

M₃₂

—

188

—

C₂×C₁₆

—

C₂×C₁₆

—

188

—

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

#108

C₂×Dic₄₇

yes

—

#104

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

#41

—

100

Q₁₆

C₂×C₄₇⋊C₈

yes

—

101

SD₁₆

—

102

D₈

C₂×C₄₇⋊C₈

yes

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

188

—

107

C₂×C₈

D₄×D₄₇

yes

characteristic

108

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

109

C₄²

#10625352

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

188

—

113

C₁₆

C₂×C₄₇⋊C₈

yes

characteristic

114

C₁₆

C₂×C₄₇⋊C₈

yes

characteristic

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×Dic₄₇

yes

characteristic

121

C₈

—

122

C₈

—

376

—

123

C₈

C₂²×C₄₇⋊C₈

yes

characteristic

124

C₈

#13094031

yes

characteristic

125

C₈

#13094048

yes

—

126

C₂²

—

127

C₂²

#15689245

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

D₄×C₄₇⋊C₈

yes

characteristic

131

C₄

#15689235

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#20242212

yes

characteristic

	subgroup	G/H
center	C₄	#15689235
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2179	C₂
socle	C₂×C₄₇	#1580
prime radical	C₄₇×#508	C₄
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#2179	D₄₇
hypocenter	C₄₇	#12068
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

subgroup

G/H

center

C₄

#15689235

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2179

C₂

socle

C₂×C₄₇

#1580

prime radical

C₄₇×#508

C₄

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

N	H ≅ G/N	type
1. #20245230	C₂	semidirect
2. #20241456	C₂	semidirect
3. #20241364	C₂	semidirect
4. #20241308	C₂	semidirect
23. Q₃₂×C₄₇	C₈	semidirect
24. SD₃₂×C₄₇	C₈	semidirect
25. D₁₆×C₄₇	C₈	semidirect
83. C₄₇	#12068	semidirect
86. Q₃₂	C₄₇⋊C₈	semidirect
87. SD₃₂	C₄₇⋊C₈	semidirect
88. D₁₆	C₄₇⋊C₈	semidirect

H ≅ G/N

type

1. #20245230

C₂

semidirect

2. #20241456

C₂

semidirect

3. #20241364

C₂

semidirect

4. #20241308

C₂

semidirect

23. Q₃₂×C₄₇

C₈

semidirect

24. SD₃₂×C₄₇

C₈

semidirect

25. D₁₆×C₄₇

C₈

semidirect

83. C₄₇

#12068

semidirect

86. Q₃₂

C₄₇⋊C₈

semidirect

87. SD₃₂

C₄₇⋊C₈

semidirect

88. D₁₆

C₄₇⋊C₈

semidirect

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

subgroup	index
#20245230	2
#20241456	2
#20241364	2
#20241308	2
C₄₇×#2179	2
#12068	47

subgroup

index

C₄₇×#2179

Subgroups — #26128168

Conjugacy classes of subgroups

Complements

Maximal subgroups