Group #25085926 — subgroups

Subgroups — #25085926

Order 11008.

736 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 179 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	#19145340	C₂	1	3	yes	maximal, complemented
2	#19144970	C₂	1	3	yes	maximal
3	#19134255	C₂	1	6	yes	maximal, complemented
4	#19134207	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583456	C₂²	1	3	yes	—
7	C₂×#13063322	C₂²	1	1	yes	characteristic
8	#14583297	C₂²	1	3	yes	complemented
9	#14583279	C₂²	1	3	yes	complemented
10	#14583264	—	2	6	no	—
11	#14583250	C₂²	1	6	yes	complemented
12	#14583246	C₂²	1	6	yes	—
13	#14583246	C₂²	1	6	yes	—
14	#14583157	—	2	6	no	—
15	#14583007	—	2	6	no	—
16	#14582905	—	2	6	no	—
17	C₄×M₁₆×C₄₃	C₂²	1	3	yes	—
18	C₄₃×#322	C₂²	1	3	yes	complemented
19	C₈²×C₄₃	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₁₇₂	—	2	3	no	—
21	C₂×Dic₁₇₂	—	2	6	no	—
22	#13063325	C₂³	1	6	yes	—
23	#13063322	—	2	3	no	—
24	#13063322	C₂³	1	2	yes	—
25	#13063317	—	2	3	no	—
26	#13063304	—	2	6	no	—
27	C₄×Dic₈₆	—	2	6	no	—
28	#13063284	—	2	3	no	—
29	#13063257	—	2	6	no	—
30	#13063250	—	2	12	no	—
31	#13063247	—	2	12	no	—
32	#13063247	—	2	12	no	—
33	C₂×M₁₆×C₄₃	D₄	1	6	yes	—
34	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
35	#97×C₄₃	D₄	1	6	yes	—
36	C₄×C₈×C₄₃	C₂³	1	3	yes	—
37	C₄×C₈×C₄₃	C₂³	1	3	yes	—
38	C₂³×D₄₃	—	4	1	no	—
39	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
40	C₂×D₁₇₂	—	2	6	no	—
41	C₂×D₁₇₂	—	4	3	no	—
42	C₂×D₁₇₂	—	4	12	no	—
43	C₂×Dic₈₆	—	2	6	no	—
44	M₁₆×C₄₃	—	4	6	no	—
45	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
46	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
47	C₄²×C₄₃	C₂×D₄	1	3	yes	—
48	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
49	C₂²⋊Dic₄₃	—	4	3	no	—
50	C₄⋊Dic₄₃	—	2	6	no	—
51	C₄⋊Dic₄₃	—	2	6	no	—
52	Dic₄₃⋊C₄	—	4	6	no	—
53	C₄×Dic₄₃	—	4	3	no	—
54	Dic₁₇₂	—	4	12	no	—
55	C₂³×C₄₃	#127	1	1	yes	characteristic
56	C₂²×D₄₃	—	4	2	no	—
57	C₂²×D₄₃	—	8	6	no	—
58	C₂×C₄×C₄₃	—	2	6	no	—
59	C₂×C₄×C₄₃	#127	1	3	yes	—
60	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
61	C₂×Dic₄₃	—	4	6	no	—
62	D₁₇₂	—	4	12	no	—
63	D₁₇₂	—	8	6	no	—
64	Dic₈₆	—	4	12	no	—
65	C₈×C₄₃	—	2	6	no	—
66	C₈×C₄₃	—	2	6	no	—
67	#19753	—	43	1	no	maximal, Sylow
68	C₂²×C₄₃	—	2	3	no	—
69	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
70	C₂×D₄₃	—	8	6	no	—
71	C₂×D₄₃	—	16	4	no	—
72	C₄×C₄₃	—	2	3	no	—
73	C₄×C₄₃	#575	1	6	yes	—
74	Dic₄₃	—	8	6	no	—
75	#3185	—	43	3	no	—
76	#3138	—	43	3	no	—
77	#1720	—	43	6	no	—
78	#1711	—	43	2	no	—
79	#1455	D₄₃	1	1	yes	characteristic, complemented
80	C₂×C₄₃	—	4	1	no	—
81	C₂×C₄₃	#3153	1	3	yes	—
82	D₄₃	—	16	2	no	—
83	#533	—	43	3	no	—
84	C₂×#127	—	43	1	no	—
85	#498	—	43	3	no	—
86	#494	—	43	3	no	—
87	#490	—	86	6	no	—
88	#487	—	43	6	no	—
89	#486	—	43	6	no	—
90	#486	—	43	6	no	—
91	#470	—	86	6	no	—
92	#441	—	86	6	no	—
93	#418	—	86	6	no	—
94	C₄×M₁₆	C₂×D₄₃	1	3	yes	—
95	#322	C₂×D₄₃	1	3	yes	complemented
96	C₈²	C₂×D₄₃	1	1	yes	characteristic, complemented
97	C₄₃	#19753	1	1	yes	characteristic, complemented, Sylow
98	C₂²×D₄	—	86	3	no	—
99	C₂×Q₁₆	—	86	6	no	—
100	C₂×M₁₆	D₁₇₂	1	6	yes	—
101	C₄⋊Q₈	—	43	6	no	—
102	#127	—	43	2	no	—
103	#127	—	86	3	no	—
104	#125	—	86	3	no	—
105	C₂²⋊Q₈	—	86	6	no	—
106	C₄×Q₈	—	86	6	no	—
107	#117	—	86	3	no	—
108	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
109	#106	—	86	6	no	—
110	#103	—	86	12	no	—
111	#102	—	86	12	no	—
112	#102	—	86	12	no	—
113	#97	D₁₇₂	1	6	yes	—
114	C₄×C₈	C₂²×D₄₃	1	3	yes	—
115	C₄×C₈	C₂²×D₄₃	1	3	yes	—
116	C₂⁴	—	172	1	no	—
117	C₂×Q₈	—	86	6	no	—
118	C₂×D₄	—	86	6	no	—
119	C₂×D₄	—	172	12	no	—
120	C₂×D₄	—	172	3	no	—
121	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
122	Q₁₆	—	172	12	no	—
123	M₁₆	—	4	6	no	—
124	C₂×C₈	C₂×D₁₇₂	1	6	yes	—
125	C₂×C₈	C₂×D₁₇₂	1	6	yes	—
126	C₄⋊C₄	—	86	6	no	—
127	C₄⋊C₄	—	86	6	no	—
128	C₄⋊C₄	—	172	6	no	—
129	C₂²⋊C₄	—	172	3	no	—
130	C₄²	—	172	3	no	—
131	C₄²	C₂×D₁₇₂	1	3	yes	—
132	C₄²	C₂³×D₄₃	1	1	yes	characteristic
133	C₂³	—	344	6	no	—
134	C₂³	—	172	2	no	—
135	C₂³	#13063322	1	1	yes	characteristic
136	Q₈	—	172	12	no	—
137	D₄	—	344	6	no	—
138	D₄	—	172	12	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	172	6	no	—
141	C₂×C₄	#13063322	1	3	yes	—
142	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
143	C₈	—	2	6	no	—
144	C₈	—	2	6	no	—
145	C₂²	—	2	3	no	—
146	C₂²	—	344	6	no	—
147	C₂²	—	688	4	no	—
148	C₂²	C₂×#13063322	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	344	6	no	—
151	C₄	#14583699	1	6	yes	—
152	C₂	—	4	1	no	—
153	C₂	—	688	2	no	—
154	C₂	#19145061	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145340

C₂

yes

maximal, complemented

#19144970

C₂

yes

maximal

#19134255

C₂

yes

maximal, complemented

#19134207

C₂

yes

maximal, complemented

C₄₃×#1455

C₂

yes

maximal, characteristic, complemented

#14583456

C₂²

yes

—

C₂×#13063322

C₂²

yes

characteristic

#14583297

C₂²

yes

complemented

#14583279

C₂²

yes

complemented

#14583264

—

#14583250

C₂²

yes

complemented

#14583246

C₂²

yes

—

#14583246

C₂²

yes

—

#14583157

—

#14583007

—

#14582905

—

C₄×M₁₆×C₄₃

C₂²

yes

—

C₄₃×#322

C₂²

yes

complemented

C₈²×C₄₃

C₂²

yes

characteristic, complemented

C₂²×D₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

C₂³

yes

—

#13063322

—

#13063322

C₂³

yes

—

#13063317

—

#13063304

—

C₄×Dic₈₆

—

—

—

—

—

—

C₂×M₁₆×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#97×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

—

C₄×C₈×C₄₃

C₂³

yes

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂⁴

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

—

C₂³×C₄₃

#127

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

#19753

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#127

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#575

yes

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

—

C₂×C₄₃

#3153

yes

—

D₄₃

—

—

—

—

—

—

—

—

—

—

—

—

C₄×M₁₆

C₂×D₄₃

yes

—

#322

C₂×D₄₃

yes

complemented

C₈²

C₂×D₄₃

yes

characteristic, complemented

C₄₃

#19753

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×Q₁₆

—

100

C₂×M₁₆

D₁₇₂

yes

—

101

C₄⋊Q₈

—

102

#127

—

103

#127

—

104

#125

—

105

C₂²⋊Q₈

—

106

C₄×Q₈

—

107

#117

—

108

C₂×C₄²

C₂²×D₄₃

yes

characteristic

109

#106

—

110

#103

—

111

#102

—

112

#102

—

113

#97

D₁₇₂

yes

—

114

C₄×C₈

C₂²×D₄₃

yes

—

115

C₄×C₈

C₂²×D₄₃

yes

—

116

C₂⁴

—

172

—

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

172

—

120

C₂×D₄

—

172

—

121

C₂²×C₄

C₂×D₁₇₂

yes

—

122

Q₁₆

—

172

—

123

M₁₆

—

124

C₂×C₈

C₂×D₁₇₂

yes

—

125

C₂×C₈

C₂×D₁₇₂

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

172

—

129

C₂²⋊C₄

—

172

—

130

C₄²

—

172

—

131

C₄²

C₂×D₁₇₂

yes

—

132

C₄²

C₂³×D₄₃

yes

characteristic

133

C₂³

—

344

—

134

C₂³

—

172

—

135

C₂³

#13063322

yes

characteristic

136

Q₈

—

172

—

137

D₄

—

344

—

138

D₄

—

172

—

139

C₂×C₄

—

140

C₂×C₄

—

172

—

141

C₂×C₄

#13063322

yes

—

142

C₂×C₄

C₂²×D₁₇₂

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

344

—

147

C₂²

—

688

—

148

C₂²

C₂×#13063322

yes

characteristic

149

C₄

—

150

C₄

—

344

—

151

C₄

#14583699

yes

—

152

C₂

—

153

C₂

—

688

—

154

C₂

#19145061

yes

—

	subgroup	G/H
center	C₂²	C₂×#13063322
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1455	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	C₂×#13063322	C₂²
hypercenter	#1455	D₄₃
hypocenter	C₄₃	#19753
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

subgroup

G/H

center

C₂²

C₂×#13063322

commutator, cosocle

C₄²×C₄₃

C₂⁴

Frattini

C₄²

C₂³×D₄₃

Fitting

C₄₃×#1455

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

N	H ≅ G/N	type
1. #19145340	C₂	semidirect
3. #19134255	C₂	semidirect
4. #19134207	C₂	semidirect
5. C₄₃×#1455	C₂	semidirect
8. #14583297	C₂²	semidirect
9. #14583279	C₂²	semidirect
11. #14583250	C₂²	semidirect
18. C₄₃×#322	C₂²	semidirect
19. C₈²×C₄₃	C₂²	semidirect
79. #1455	D₄₃	semidirect
95. #322	C₂×D₄₃	semidirect
96. C₈²	C₂×D₄₃	semidirect
97. C₄₃	#19753	semidirect

H ≅ G/N

type

1. #19145340

C₂

semidirect

3. #19134255

C₂

semidirect

4. #19134207

C₂

semidirect

5. C₄₃×#1455

C₂

semidirect

8. #14583297

C₂²

semidirect

9. #14583279

C₂²

semidirect

11. #14583250

C₂²

semidirect

18. C₄₃×#322

C₂²

semidirect

19. C₈²×C₄₃

C₂²

semidirect

79. #1455

D₄₃

semidirect

95. #322

C₂×D₄₃

semidirect

96. C₈²

C₂×D₄₃

semidirect

97. C₄₃

#19753

semidirect

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

subgroup	index
#19145340	2
#19144970	2
#19134255	2
#19134207	2
C₄₃×#1455	2
#19753	43

subgroup

index

C₄₃×#1455

Subgroups — #25085926

Conjugacy classes of subgroups

Complements

Maximal subgroups