Group #25008481 — subgroups

Subgroups — #25008481

Order 11008.

246 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19136472	C₂	1	1	yes	maximal, characteristic
2	#19133418	C₂	1	2	yes	maximal, complemented
3	#19132561	C₂	1	2	yes	maximal, complemented
4	#19132515	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₈₈	C₄	1	1	yes	characteristic
7	C₄×D₃₄₄	C₂²	1	2	yes	—
8	#14582964	—	2	2	no	—
9	#14582727	C₄	1	1	yes	characteristic
10	#14582701	C₄	1	2	yes	—
11	#14582675	C₂²	1	1	yes	characteristic
12	#14582626	—	2	2	no	—
13	#14582620	C₂²	1	1	yes	characteristic
14	C₄₃×#346	C₂²	1	1	yes	characteristic
15	C₄₃×#334	C₂²	1	2	yes	—
16	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₄₃	—	4	2	no	—
18	C₄×D₁₇₂	—	2	2	no	—
19	D₆₈₈	C₈	1	4	yes	complemented
20	C₂×C₄₃⋊C₁₆	—	2	1	no	—
21	#13063259	C₂×C₄	1	2	yes	—
22	#13063247	—	2	2	no	—
23	(C₂×D₄₃)⋊C₈	—	4	2	no	—
24	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₄₃	—	2	2	no	—
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×D₁₇₂	—	2	2	no	—
30	C₂×C₄×D₄₃	—	4	2	no	—
31	C₂×C₈×C₄₃	—	4	2	no	—
32	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
34	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
35	(C₂×D₄₃)⋊C₄	—	4	2	no	—
36	C₄⋊Dic₄₃	—	2	2	no	—
37	C₂×C₄₃⋊C₈	—	2	2	no	—
38	D₃₄₄	—	2	2	no	—
39	D₃₄₄	C₂×C₈	1	4	yes	complemented
40	C₈×D₄₃	—	8	4	no	—
41	C₁₆×C₄₃	—	2	1	no	—
42	C₁₆×C₄₃	C₂×C₈	1	2	yes	complemented
43	C₄₃⋊C₁₆	—	4	1	no	—
44	C₂²×D₄₃	—	4	2	no	—
45	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₄₃	—	4	2	no	—
48	D₁₇₂	—	2	4	no	—
49	D₁₇₂	—	4	2	no	—
50	C₄×D₄₃	—	8	2	no	—
51	C₄×D₄₃	—	8	2	no	—
52	C₈×C₄₃	—	2	1	no	—
53	C₈×C₄₃	—	8	2	no	—
54	C₈×C₄₃	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₄₃	#131	1	1	yes	characteristic
56	C₄₃⋊C₈	—	4	2	no	—
57	#12067	—	43	1	no	maximal, Sylow
58	C₂²×C₄₃	#442	1	1	yes	characteristic
59	C₂×D₄₃	—	4	4	no	—
60	C₂×D₄₃	—	8	2	no	—
61	C₂×D₄₃	—	8	2	no	—
62	C₄×C₄₃	—	2	2	no	—
63	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
64	C₄×C₄₃	#444	1	1	yes	characteristic
65	Dic₄₃	—	8	2	no	—
66	#2177	—	43	1	no	—
67	#1580	—	43	2	no	—
68	#1368	D₄₃	1	1	yes	characteristic, complemented
69	#1333	—	43	2	no	—
70	#1313	—	43	1	no	—
71	C₂×C₄₃	#1580	1	1	yes	characteristic
72	C₂×C₄₃	AGL(1,ℤ/16)	1	2	yes	—
73	D₄₃	—	8	4	no	—
74	C₂×D₁₆	—	43	1	no	—
75	C₄×D₈	—	43	2	no	—
76	C₈×D₄	—	86	2	no	—
77	#367	—	43	1	no	—
78	#357	—	43	2	no	—
79	#346	—	43	1	no	—
80	#346	C₂×D₄₃	1	1	yes	characteristic
81	#334	C₂×D₄₃	1	2	yes	—
82	D₄⋊C₈	—	86	2	no	—
83	#322	—	43	1	no	—
84	C₄₃	#12067	1	1	yes	characteristic, complemented, Sylow
85	C₂×D₈	—	43	2	no	—
86	C₂²×C₈	—	172	2	no	—
87	C₄×D₄	—	86	2	no	—
88	D₁₆	—	43	4	no	—
89	C₂×C₁₆	—	86	1	no	—
90	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
92	#107	—	43	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	#102	—	86	2	no	—
95	C₂²⋊C₈	—	172	2	no	—
96	C₄×C₈	—	43	2	no	—
97	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
98	C₂×D₄	—	86	2	no	—
99	C₂²×C₄	—	172	2	no	—
100	D₈	—	43	4	no	—
101	D₈	—	86	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	86	2	no	—
104	C₂×C₈	—	344	4	no	—
105	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
106	C₂×C₈	#10621902	1	1	yes	characteristic
107	C₄⋊C₄	—	86	2	no	—
108	C₂²⋊C₄	—	172	2	no	—
109	C₄²	D₄×D₄₃	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	172	1	no	—
112	C₁₆	C₈×D₄₃	1	2	yes	complemented
113	C₂³	—	172	2	no	—
114	D₄	—	86	4	no	—
115	D₄	—	172	2	no	—
116	C₂×C₄	—	344	2	no	—
117	C₂×C₄	—	344	2	no	—
118	C₂×C₄	—	172	2	no	—
119	C₂×C₄	#13063287	1	1	yes	characteristic
120	C₂×C₄	#13063352	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	8	2	no	—
123	C₈	—	172	2	no	—
124	C₈	C₂×C₈×D₄₃	1	1	yes	characteristic
125	C₈	#13063333	1	1	yes	characteristic
126	C₂²	—	344	2	no	—
127	C₂²	—	344	2	no	—
128	C₂²	—	172	4	no	—
129	C₂²	#14583010	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	344	2	no	—
132	C₄	#14582964	1	1	yes	characteristic
133	C₄	#14583020	1	1	yes	characteristic
134	C₂	—	344	4	no	—
135	C₂	#19133418	1	1	yes	characteristic
136	C₂	#19136492	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136472

C₂

yes

maximal, characteristic

#19133418

C₂

yes

maximal, complemented

#19132561

C₂

yes

maximal, complemented

#19132515

C₂

yes

maximal, characteristic, complemented

C₄₃×#1368

C₂

yes

maximal, characteristic, complemented

C₂×D₆₈₈

C₄

yes

characteristic

C₄×D₃₄₄

C₂²

yes

—

#14582964

—

#14582727

C₄

yes

characteristic

#14582701

C₄

yes

—

#14582675

C₂²

yes

characteristic

#14582626

—

#14582620

C₂²

yes

characteristic

C₄₃×#346

C₂²

yes

characteristic

C₄₃×#334

C₂²

yes

—

C₂×D₃₄₄

C₂×C₄

yes

—

C₂×C₈×D₄₃

—

C₄×D₁₇₂

—

D₆₈₈

C₈

yes

complemented

C₂×C₄₃⋊C₁₆

—

#13063259

C₂×C₄

yes

—

#13063247

—

(C₂×D₄₃)⋊C₈

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

D₃₄₄

—

D₃₄₄

C₂×C₈

yes

complemented

C₈×D₄₃

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×C₈

yes

complemented

C₄₃⋊C₁₆

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂²×C₈

yes

characteristic

C₈×C₄₃

#131

yes

characteristic

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₈×D₄

yes

characteristic

C₄×C₄₃

#444

yes

characteristic

Dic₄₃

—

#2177

—

#1580

—

#1368

D₄₃

yes

characteristic, complemented

#1333

—

#1313

—

C₂×C₄₃

#1580

yes

characteristic

C₂×C₄₃

AGL(1,ℤ/16)

yes

—

D₄₃

—

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

—

—

—

C₂×D₄₃

yes

characteristic

#334

C₂×D₄₃

yes

—

D₄⋊C₈

—

#322

—

C₄₃

#12067

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

172

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₃

yes

characteristic

C₂×C₁₆

C₄×D₄₃

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

172

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

172

—

100

D₈

—

101

D₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

344

—

105

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

106

C₂×C₈

#10621902

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

172

—

109

C₄²

D₄×D₄₃

yes

characteristic

110

C₁₆

—

111

C₁₆

—

172

—

112

C₁₆

C₈×D₄₃

yes

complemented

113

C₂³

—

172

—

114

D₄

—

115

D₄

—

172

—

116

C₂×C₄

—

344

—

117

C₂×C₄

—

344

—

118

C₂×C₄

—

172

—

119

C₂×C₄

#13063287

yes

characteristic

120

C₂×C₄

#13063352

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

172

—

124

C₈

C₂×C₈×D₄₃

yes

characteristic

125

C₈

#13063333

yes

characteristic

126

C₂²

—

344

—

127

C₂²

—

344

—

128

C₂²

—

172

—

129

C₂²

#14583010

yes

characteristic

130

C₄

—

131

C₄

—

344

—

132

C₄

#14582964

yes

characteristic

133

C₄

#14583020

yes

characteristic

134

C₂

—

344

—

135

C₂

#19133418

yes

characteristic

136

C₂

#19136492

yes

—

	subgroup	G/H
center	C₂²	#14583010
commutator	C₈×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1368	C₂
socle	C₂²×C₄₃	#442
prime radical	C₂×D₆₈₈	C₄
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1368	D₄₃
hypocenter	C₄₃	#12067
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#14583010

commutator

C₈×C₄₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1368

C₂

socle

C₂²×C₄₃

#442

prime radical

C₂×D₆₈₈

C₄

cosocle

C₄×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
2. #19133418	C₂	semidirect
3. #19132561	C₂	semidirect
4. #19132515	C₂	semidirect
5. C₄₃×#1368	C₂	semidirect
19. D₆₈₈	C₈	semidirect
39. D₃₄₄	C₂×C₈	semidirect
42. C₁₆×C₄₃	C₂×C₈	semidirect
68. #1368	D₄₃	semidirect
84. C₄₃	#12067	semidirect
112. C₁₆	C₈×D₄₃	semidirect

H ≅ G/N

type

2. #19133418

C₂

semidirect

3. #19132561

C₂

semidirect

4. #19132515

C₂

semidirect

5. C₄₃×#1368

C₂

semidirect

19. D₆₈₈

C₈

semidirect

39. D₃₄₄

C₂×C₈

semidirect

42. C₁₆×C₄₃

C₂×C₈

semidirect

68. #1368

D₄₃

semidirect

84. C₄₃

#12067

semidirect

112. C₁₆

C₈×D₄₃

semidirect

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

subgroup	index
#19136472	2
#19133418	2
#19132561	2
#19132515	2
C₄₃×#1368	2
#12067	43

subgroup

index

C₄₃×#1368

Subgroups — #25008481

Conjugacy classes of subgroups

Complements

Maximal subgroups