Group #25011298 — subgroups

Subgroups — #25011298

Order 11008.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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	#19136824	C₂	1	2	yes	maximal, complemented
2	#19136820	C₂	1	2	yes	maximal
3	#19134201	C₂	1	1	yes	maximal, characteristic
4	#19132512	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583279	C₂²	1	1	yes	characteristic
7	#14583276	C₂²	1	1	yes	characteristic
8	#14583246	—	2	1	no	—
9	#14583246	—	2	1	no	—
10	#14582702	—	2	4	no	—
11	#14582699	—	2	4	no	—
12	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
13	C₄₃×#500	—	2	1	no	—
14	C₄₃×#500	C₂²	1	2	yes	—
15	C₄₃×#494	—	2	1	no	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	2	2	no	—
18	C₂×D₃₄₄	—	2	2	no	—
19	#13063325	—	2	1	no	—
20	#13063322	—	2	1	no	—
21	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	—
22	#13063247	—	4	2	no	—
23	#13063247	—	4	2	no	—
24	C₂×Q₁₆×C₄₃	—	4	2	no	—
25	C₄⋊Q₈×C₄₃	—	2	2	no	—
26	#107×C₄₃	—	2	2	no	—
27	#107×C₄₃	—	2	2	no	—
28	#107×C₄₃	—	4	2	no	—
29	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₄₃	—	4	2	no	—
33	C₂×D₁₇₂	—	4	1	no	—
34	C₂×D₁₇₂	—	4	1	no	—
35	C₂×D₁₇₂	—	4	1	no	—
36	C₂×Dic₈₆	—	4	1	no	—
37	Q₁₆×C₄₃	—	8	4	no	—
38	C₂×C₈×C₄₃	D₈	1	2	yes	—
39	C₂×C₈×C₄₃	D₈	1	2	yes	—
40	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₄₃	—	4	2	no	—
42	C₄⋊C₄×C₄₃	—	4	2	no	—
43	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₄⋊Dic₄₃	—	4	1	no	—
45	C₄⋊Dic₄₃	—	4	1	no	—
46	Dic₁₇₂	—	4	4	no	—
47	D₃₄₄	—	4	4	no	—
48	C₄₃⋊C₁₆	—	2	4	no	—
49	C₂²×D₄₃	—	8	1	no	—
50	Q₈×C₄₃	—	8	4	no	—
51	C₂×C₄×C₄₃	—	8	2	no	—
52	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₄₃	—	8	1	no	—
56	D₁₇₂	—	8	2	no	—
57	D₁₇₂	—	8	2	no	—
58	D₁₇₂	—	8	2	no	—
59	Dic₈₆	—	8	2	no	—
60	C₈×C₄₃	—	2	2	no	—
61	C₈×C₄₃	—	2	2	no	—
62	C₈×C₄₃	#135	1	4	yes	—
63	#12512	—	43	1	no	maximal, Sylow
64	C₂²×C₄₃	#493	1	1	yes	characteristic
65	C₂×D₄₃	—	16	1	no	—
66	C₂×D₄₃	—	16	1	no	—
67	C₂×D₄₃	—	16	1	no	—
68	C₄×C₄₃	—	16	2	no	—
69	C₄×C₄₃	#495	1	2	yes	—
70	C₄×C₄₃	#509	1	2	yes	—
71	C₄×C₄₃	#510	1	2	yes	—
72	Dic₄₃	—	16	1	no	—
73	#2241	—	43	2	no	—
74	#2240	—	43	2	no	—
75	#1713	D₄₃	1	1	yes	characteristic, complemented
76	#1709	—	43	1	no	—
77	#1311	—	43	1	no	—
78	C₂×C₄₃	#1709	1	1	yes	characteristic
79	C₂×C₄₃	#2250	1	1	yes	characteristic
80	C₂×C₄₃	#2251	1	1	yes	characteristic
81	D₄₃	—	32	1	no	—
82	#500	—	2	1	no	—
83	#500	C₂×D₄₃	1	2	yes	—
84	#494	—	2	1	no	—
85	#494	—	43	1	no	—
86	#493	—	43	1	no	—
87	#486	—	86	1	no	—
88	#486	—	86	1	no	—
89	#358	—	86	4	no	—
90	#357	—	86	4	no	—
91	C₄×C₁₆	—	43	2	no	—
92	C₈²	C₂×D₄₃	1	1	yes	characteristic
93	C₄₃	#12512	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	4	2	no	—
95	C₂×Q₁₆	—	86	2	no	—
96	C₂×D₈	—	86	2	no	—
97	C₄⋊Q₈	—	2	2	no	—
98	C₄⋊Q₈	—	86	1	no	—
99	#127	—	86	1	no	—
100	C₂×C₁₆	—	43	4	no	—
101	#107	—	2	2	no	—
102	#107	—	2	2	no	—
103	#107	—	4	2	no	—
104	#102	—	172	2	no	—
105	#102	—	172	2	no	—
106	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
109	C₂×Q₈	—	4	2	no	—
110	C₂×Q₈	—	172	1	no	—
111	C₂×D₄	—	172	1	no	—
112	C₂×D₄	—	172	1	no	—
113	C₂×D₄	—	172	1	no	—
114	Q₁₆	—	8	4	no	—
115	Q₁₆	—	172	4	no	—
116	D₈	—	172	4	no	—
117	C₂×C₈	#10621882	1	2	yes	—
118	C₂×C₈	#10621882	1	2	yes	—
119	C₂×C₈	D₄×D₄₃	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	172	1	no	—
123	C₄⋊C₄	—	172	1	no	—
124	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
125	C₁₆	—	86	4	no	—
126	C₂³	—	344	1	no	—
127	Q₈	—	8	4	no	—
128	Q₈	—	344	2	no	—
129	D₄	—	344	2	no	—
130	D₄	—	344	2	no	—
131	D₄	—	344	2	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	—	344	1	no	—
134	C₂×C₄	#13063321	1	1	yes	characteristic
135	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
136	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#13063348	1	4	yes	—
140	C₂²	—	688	1	no	—
141	C₂²	—	688	1	no	—
142	C₂²	—	688	1	no	—
143	C₂²	#14583275	1	1	yes	characteristic
144	C₄	—	16	2	no	—
145	C₄	—	688	1	no	—
146	C₄	#14583284	1	2	yes	—
147	C₄	#14583335	1	2	yes	—
148	C₄	#14583340	1	2	yes	—
149	C₂	—	1376	1	no	—
150	C₂	#19134202	1	1	yes	characteristic
151	C₂	#19136859	1	1	yes	characteristic
152	C₂	#19136864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136824

C₂

yes

maximal, complemented

#19136820

C₂

yes

maximal

#19134201

C₂

yes

maximal, characteristic

#19132512

C₂

yes

maximal, characteristic, complemented

C₄₃×#1713

C₂

yes

maximal, characteristic, complemented

#14583279

C₂²

yes

characteristic

#14583276

C₂²

yes

—

—

—

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#500

—

C₄₃×#500

C₂²

yes

—

C₄₃×#494

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×D₃₄₄

—

#13063325

—

#13063322

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

#13063247

—

#13063247

—

C₂×Q₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

—

#107×C₄₃

—

#107×C₄₃

—

#107×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

D₃₄₄

—

C₄₃⋊C₁₆

—

C₂²×D₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#135

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#495

yes

—

C₄×C₄₃

#509

yes

—

C₄×C₄₃

#510

yes

—

Dic₄₃

—

#2241

—

#2240

—

#1713

D₄₃

yes

characteristic, complemented

#1709

—

#1311

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#500

—

#500

C₂×D₄₃

yes

—

—

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12512

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

100

C₂×C₁₆

—

101

#107

—

102

#107

—

103

#107

—

104

#102

—

172

—

105

#102

—

172

—

106

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

107

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

108

C₄×C₈

C₂²×D₄₃

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

172

—

111

C₂×D₄

—

172

—

112

C₂×D₄

—

172

—

113

C₂×D₄

—

172

—

114

Q₁₆

—

115

Q₁₆

—

172

—

116

D₈

—

172

—

117

C₂×C₈

#10621882

yes

—

118

C₂×C₈

#10621882

yes

—

119

C₂×C₈

D₄×D₄₃

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

172

—

123

C₄⋊C₄

—

172

—

124

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

125

C₁₆

—

126

C₂³

—

344

—

127

Q₈

—

128

Q₈

—

344

—

129

D₄

—

344

—

130

D₄

—

344

—

131

D₄

—

344

—

132

C₂×C₄

—

133

C₂×C₄

—

344

—

134

C₂×C₄

#13063321

yes

characteristic

135

C₂×C₄

C₂×#10621882

yes

characteristic

136

C₂×C₄

C₂×#10621882

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#13063348

yes

—

140

C₂²

—

688

—

141

C₂²

—

688

—

142

C₂²

—

688

—

143

C₂²

#14583275

yes

characteristic

144

C₄

—

145

C₄

—

688

—

146

C₄

#14583284

yes

—

147

C₄

#14583335

yes

—

148

C₄

#14583340

yes

—

149

C₂

—

1376

—

150

C₂

#19134202

yes

characteristic

151

C₂

#19136859

yes

characteristic

152

C₂

#19136864

yes

characteristic

	subgroup	G/H
center	C₂²	#14583275
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1713	C₂
socle	C₂²×C₄₃	#493
prime radical	#14583276	C₂²
hypercenter	#1713	D₄₃
hypocenter	C₄₃	#12512
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#14583275

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1713

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. #19136824	C₂	semidirect
4. #19132512	C₂	semidirect
5. C₄₃×#1713	C₂	semidirect
75. #1713	D₄₃	semidirect
93. C₄₃	#12512	semidirect

H ≅ G/N

type

1. #19136824

C₂

semidirect

4. #19132512

C₂

semidirect

5. C₄₃×#1713

C₂

semidirect

75. #1713

D₄₃

semidirect

93. C₄₃

#12512

semidirect

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

subgroup	index
#19136824	2
#19136820	2
#19134201	2
#19132512	2
C₄₃×#1713	2
#12512	43

subgroup

index

C₄₃×#1713

Subgroups — #25011298

Conjugacy classes of subgroups

Complements

Maximal subgroups