Group #25011354 — subgroups

Subgroups — #25011354

Order 11008.

296 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 91 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	#19136851	C₂	1	2	yes	maximal
2	#19136823	C₂	1	2	yes	maximal, complemented
3	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134187	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063270	—	2	4	no	—
7	#14583304	—	2	1	no	—
8	#14583304	—	2	1	no	—
9	#14583304	C₂²	1	1	yes	characteristic
10	#14583279	C₂²	1	2	yes	—
11	#14583276	C₂²	1	1	yes	characteristic
12	#14583272	—	2	1	no	—
13	#14583246	—	2	1	no	—
14	#14582704	—	2	4	no	—
15	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×#10621873	—	4	2	no	—
20	C₂×D₃₄₄	—	2	2	no	—
21	#13063325	—	2	2	no	—
22	#13063325	—	2	1	no	—
23	#13063322	—	2	1	no	—
24	#13063270	—	4	8	no	—
25	#13063259	—	2	2	no	—
26	#13063259	—	4	2	no	—
27	#13063259	—	4	2	no	—
28	#13063247	—	4	2	no	—
29	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
30	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×D₁₇₂	—	4	1	no	—
34	C₂×D₁₇₂	—	4	1	no	—
35	C₂×D₁₇₂	—	4	1	no	—
36	C₂×Dic₈₆	—	4	2	no	—
37	C₂×Dic₈₆	—	4	1	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	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₃	—	4	1	no	—
43	C₄⋊Dic₄₃	—	4	1	no	—
44	C₄⋊Dic₄₃	—	4	2	no	—
45	C₄⋊Dic₄₃	—	4	2	no	—
46	Dic₁₇₂	—	4	4	no	—
47	Dic₁₇₂	—	4	4	no	—
48	D₃₄₄	—	4	4	no	—
49	#10621873	—	8	4	no	—
50	C₁₆×C₄₃	—	2	4	no	—
51	C₂²×D₄₃	—	8	1	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	2	no	—
56	C₂×Dic₄₃	—	8	1	no	—
57	D₁₇₂	—	8	2	no	—
58	D₁₇₂	—	8	2	no	—
59	D₁₇₂	—	8	2	no	—
60	Dic₈₆	—	8	2	no	—
61	Dic₈₆	—	8	2	no	—
62	Dic₈₆	—	8	2	no	—
63	C₈×C₄₃	—	2	2	no	—
64	C₈×C₄₃	SD₃₂	1	4	yes	complemented
65	C₈×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
66	#12526	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	#493	1	1	yes	characteristic
68	C₂×D₄₃	—	16	1	no	—
69	C₂×D₄₃	—	16	1	no	—
70	C₂×D₄₃	—	16	1	no	—
71	C₄×C₄₃	#496	1	2	yes	—
72	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
73	C₄×C₄₃	#510	1	2	yes	—
74	Dic₄₃	—	16	2	no	—
75	Dic₄₃	—	16	1	no	—
76	#2248	—	43	2	no	—
77	#2241	—	43	2	no	—
78	#1713	—	43	1	no	—
79	#1706	—	43	1	no	—
80	#1312	D₄₃	1	1	yes	characteristic, complemented
81	C₂×C₄₃	#1721	1	1	yes	characteristic
82	C₂×C₄₃	#2248	1	1	yes	characteristic
83	C₂×C₄₃	#2251	1	1	yes	characteristic
84	D₄₃	—	32	1	no	—
85	C₂×SD₃₂	—	86	4	no	—
86	#500	—	43	1	no	—
87	#500	—	86	1	no	—
88	#500	—	86	1	no	—
89	#494	—	43	2	no	—
90	#493	—	43	1	no	—
91	#492	—	86	1	no	—
92	#486	—	86	1	no	—
93	#358	—	86	4	no	—
94	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
95	C₈²	C₂×D₄₃	1	1	yes	characteristic
96	C₄₃	#12526	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₁₆	—	86	2	no	—
98	C₂×Q₁₆	—	86	2	no	—
99	C₂×SD₁₆	—	172	2	no	—
100	C₂×D₈	—	86	2	no	—
101	C₄⋊Q₈	—	86	1	no	—
102	C₄⋊Q₈	—	86	2	no	—
103	#127	—	86	1	no	—
104	SD₃₂	—	172	8	no	—
105	C₂×C₁₆	D₁₇₂	1	4	yes	—
106	#107	—	86	2	no	—
107	#107	—	172	2	no	—
108	#107	—	172	2	no	—
109	#102	—	172	2	no	—
110	C₄×C₈	D₁₇₂	1	1	yes	characteristic
111	C₄×C₈	D₁₇₂	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
113	C₂×Q₈	—	172	2	no	—
114	C₂×Q₈	—	172	1	no	—
115	C₂×D₄	—	172	1	no	—
116	C₂×D₄	—	172	1	no	—
117	C₂×D₄	—	172	1	no	—
118	Q₁₆	—	172	4	no	—
119	Q₁₆	—	172	4	no	—
120	SD₁₆	—	344	4	no	—
121	D₈	—	172	4	no	—
122	C₂×C₈	D₃₄₄	1	2	yes	—
123	C₂×C₈	D₃₄₄	1	2	yes	—
124	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
125	C₄⋊C₄	—	172	1	no	—
126	C₄⋊C₄	—	172	1	no	—
127	C₄⋊C₄	—	172	2	no	—
128	C₄⋊C₄	—	172	2	no	—
129	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
130	C₁₆	—	2	4	no	—
131	C₂³	—	344	1	no	—
132	Q₈	—	344	2	no	—
133	Q₈	—	344	2	no	—
134	Q₈	—	344	2	no	—
135	D₄	—	344	2	no	—
136	D₄	—	344	2	no	—
137	D₄	—	344	2	no	—
138	C₂×C₄	—	344	2	no	—
139	C₂×C₄	—	344	1	no	—
140	C₂×C₄	#13063322	1	1	yes	characteristic
141	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
142	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	#13063270	1	4	yes	complemented
145	C₈	#13063353	1	4	yes	—
146	C₂²	—	688	1	no	—
147	C₂²	—	688	1	no	—
148	C₂²	—	688	1	no	—
149	C₂²	#14583276	1	1	yes	characteristic
150	C₄	—	688	2	no	—
151	C₄	—	688	1	no	—
152	C₄	#14583288	1	2	yes	—
153	C₄	C₂×#13063270	1	2	yes	—
154	C₄	#14583342	1	2	yes	—
155	C₂	—	1376	1	no	—
156	C₂	#19134261	1	1	yes	characteristic
157	C₂	#19136851	1	1	yes	characteristic
158	C₂	#19136866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136851

C₂

yes

maximal

#19136823

C₂

yes

maximal, complemented

#19134213

C₂

yes

maximal, characteristic, complemented

#19134187

C₂

yes

maximal, characteristic

C₄₃×#1312

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#14583279

C₂²

yes

—

#14583276

C₂²

yes

—

—

—

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×#10621873

—

C₂×D₃₄₄

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

D₃₄₄

—

#10621873

—

C₁₆×C₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

SD₃₂

yes

complemented

C₈×C₄₃

AGL(1,ℤ/8)

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#496

yes

—

C₄×C₄₃

C₂×SD₃₂

yes

—

C₄×C₄₃

#510

yes

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×SD₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12526

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

172

—

100

C₂×D₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

#127

—

104

SD₃₂

—

172

—

105

C₂×C₁₆

D₁₇₂

yes

—

106

#107

—

107

#107

—

172

—

108

#107

—

172

—

109

#102

—

172

—

110

C₄×C₈

D₁₇₂

yes

characteristic

111

C₄×C₈

D₁₇₂

yes

characteristic

112

C₄×C₈

C₂²×D₄₃

yes

characteristic

113

C₂×Q₈

—

172

—

114

C₂×Q₈

—

172

—

115

C₂×D₄

—

172

—

116

C₂×D₄

—

172

—

117

C₂×D₄

—

172

—

118

Q₁₆

—

172

—

119

Q₁₆

—

172

—

120

SD₁₆

—

344

—

121

D₈

—

172

—

122

C₂×C₈

D₃₄₄

yes

—

123

C₂×C₈

D₃₄₄

yes

—

124

C₂×C₈

C₂×D₁₇₂

yes

—

125

C₄⋊C₄

—

172

—

126

C₄⋊C₄

—

172

—

127

C₄⋊C₄

—

172

—

128

C₄⋊C₄

—

172

—

129

C₄²

C₂×D₁₇₂

yes

characteristic

130

C₁₆

—

131

C₂³

—

344

—

132

Q₈

—

344

—

133

Q₈

—

344

—

134

Q₈

—

344

—

135

D₄

—

344

—

136

D₄

—

344

—

137

D₄

—

344

—

138

C₂×C₄

—

344

—

139

C₂×C₄

—

344

—

140

C₂×C₄

#13063322

yes

characteristic

141

C₂×C₄

C₂×D₃₄₄

yes

characteristic

142

C₂×C₄

C₂×D₃₄₄

yes

characteristic

143

C₈

—

144

C₈

#13063270

yes

complemented

145

C₈

#13063353

yes

—

146

C₂²

—

688

—

147

C₂²

—

688

—

148

C₂²

—

688

—

149

C₂²

#14583276

yes

characteristic

150

C₄

—

688

—

151

C₄

—

688

—

152

C₄

#14583288

yes

—

153

C₄

C₂×#13063270

yes

—

154

C₄

#14583342

yes

—

155

C₂

—

1376

—

156

C₂

#19134261

yes

characteristic

157

C₂

#19136851

yes

characteristic

158

C₂

#19136866

yes

characteristic

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1312	C₂
socle	C₂²×C₄₃	#493
prime radical	#14583276	C₂²
hypercenter	#1312	D₄₃
hypocenter	C₄₃	#12526
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#14583276

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1312

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
2. #19136823	C₂	semidirect
3. #19134213	C₂	semidirect
5. C₄₃×#1312	C₂	semidirect
64. C₈×C₄₃	SD₃₂	semidirect
80. #1312	D₄₃	semidirect
96. C₄₃	#12526	semidirect
144. C₈	#13063270	semidirect

H ≅ G/N

type

2. #19136823

C₂

semidirect

3. #19134213

C₂

semidirect

5. C₄₃×#1312

C₂

semidirect

64. C₈×C₄₃

SD₃₂

semidirect

80. #1312

D₄₃

semidirect

96. C₄₃

#12526

semidirect

144. C₈

#13063270

semidirect

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

subgroup	index
#19136851	2
#19136823	2
#19134213	2
#19134187	2
C₄₃×#1312	2
#12526	43

subgroup

index

C₄₃×#1312

Subgroups — #25011354

Conjugacy classes of subgroups

Complements

Maximal subgroups