Group #25010859 — subgroups

Subgroups — #25010859

Order 11008.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#19136848	C₂	1	1	yes	maximal, characteristic
2	#19133965	C₂	1	2	yes	maximal
3	#19132667	C₂	1	1	yes	maximal, characteristic
4	#19132645	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₄	—	2	1	no	—
7	C₂×Dic₃₄₄	—	2	1	no	—
8	#14583279	C₂²	1	2	yes	—
9	#14583184	—	2	2	no	—
10	#14582652	C₂²	1	2	yes	—
11	#14582646	—	2	2	no	—
12	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×C₄⋊C₁₆	—	2	1	no	—
14	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₄₃	—	2	1	no	—
16	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	#13063325	—	2	2	no	—
20	Dic₃₄₄	—	4	2	no	—
21	Dic₃₄₄	—	4	2	no	—
22	#13063256	—	4	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	#13063249	—	4	2	no	—
25	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₃	—	2	1	no	—
27	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₄₃	—	2	1	no	—
30	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	—	2	1	no	—
32	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
34	C₂×Dic₈₆	—	4	2	no	—
35	C₂×C₈×C₄₃	—	2	1	no	—
36	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₄₃	—	2	1	no	—
39	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
40	C₄²×C₄₃	—	2	1	no	—
41	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₃	—	4	2	no	—
43	C₄⋊Dic₄₃	—	4	2	no	—
44	C₂×C₄₃⋊C₈	—	4	2	no	—
45	Dic₁₇₂	—	4	2	no	—
46	Dic₁₇₂	—	4	2	no	—
47	Dic₁₇₂	—	8	2	no	—
48	C₁₆×C₄₃	—	2	1	no	—
49	C₁₆×C₄₃	—	4	1	no	—
50	C₁₆×C₄₃	SD₁₆	1	2	yes	—
51	Q₈×C₄₃	—	2	1	no	—
52	Q₈×C₄₃	Q₃₂	1	2	yes	complemented
53	C₂×C₄×C₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	2	1	no	—
55	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
58	C₂×Dic₄₃	—	8	2	no	—
59	Dic₈₆	—	8	2	no	—
60	Dic₈₆	—	8	2	no	—
61	C₈×C₄₃	—	2	1	no	—
62	C₈×C₄₃	—	4	1	no	—
63	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄₃⋊C₈	—	8	2	no	—
66	#12440	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	#466	1	1	yes	characteristic
68	C₄×C₄₃	—	2	2	no	—
69	C₄×C₄₃	—	2	1	no	—
70	C₄×C₄₃	—	4	1	no	—
71	C₄×C₄₃	#460	1	1	yes	characteristic
72	C₄×C₄₃	#472	1	1	yes	characteristic
73	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
74	C₄×C₄₃	#508	1	1	yes	characteristic
75	Dic₄₃	—	16	2	no	—
76	#2247	—	43	1	no	—
77	Q₈×C₁₆	D₄₃	1	1	yes	characteristic
78	#1670	—	43	2	no	—
79	#1371	—	43	1	no	—
80	#1362	—	43	2	no	—
81	C₂×C₄₃	#1668	1	1	yes	characteristic
82	C₂×C₄₃	#2216	1	1	yes	characteristic
83	C₂×C₄₃	#2222	1	1	yes	characteristic
84	C₂×Q₃₂	—	86	1	no	—
85	C₂×Q₃₂	—	86	1	no	—
86	#494	—	43	2	no	—
87	#475	—	86	2	no	—
88	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
91	C₄×C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
93	#335	—	43	2	no	—
94	#332	—	86	2	no	—
95	C₄₃	#12440	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	86	2	no	—
97	C₂×Q₁₆	—	86	2	no	—
98	C₄⋊Q₈	—	86	2	no	—
99	C₄×Q₈	D₁₇₂	1	1	yes	characteristic
100	Q₃₂	—	172	2	no	—
101	Q₃₂	—	172	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
105	#106	—	172	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	86	2	no	—
108	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
109	#103	—	172	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
112	C₂×Q₈	—	172	2	no	—
113	C₂×Q₈	D₃₄₄	1	1	yes	characteristic
114	Q₁₆	—	344	2	no	—
115	Q₁₆	—	172	2	no	—
116	Q₁₆	—	172	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	172	2	no	—
119	C₂×C₈	#10621898	1	1	yes	characteristic
120	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	172	2	no	—
123	C₄⋊C₄	—	172	2	no	—
124	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	#10621884	1	2	yes	—
130	Q₈	—	2	1	no	—
131	Q₈	—	344	2	no	—
132	Q₈	—	344	2	no	—
133	Q₈	Dic₃₄₄	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	344	2	no	—
137	C₂×C₄	#13063299	1	1	yes	characteristic
138	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
139	C₂×C₄	#13063347	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	344	2	no	—
143	C₈	C₂×#10621884	1	1	yes	characteristic
144	C₈	#13063350	1	1	yes	characteristic
145	C₂²	#14583135	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	688	2	no	—
150	C₄	#14583099	1	1	yes	characteristic
151	C₄	#14583169	1	1	yes	characteristic
152	C₄	C₂×Dic₃₄₄	1	1	yes	characteristic
153	C₄	#14583331	1	1	yes	characteristic
154	C₂	#19133955	1	1	yes	characteristic
155	C₂	#19136686	1	1	yes	characteristic
156	C₂	#19136720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136848

C₂

yes

maximal, characteristic

#19133965

C₂

yes

maximal

#19132667

C₂

yes

maximal, characteristic

#19132645

C₂

yes

maximal

Q₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

—

C₂×Dic₃₄₄

—

#14583279

C₂²

yes

—

#14583184

—

#14582652

C₂²

yes

—

#14582646

—

C₈×Q₈×C₄₃

C₂²

yes

characteristic

C₄₃×C₄⋊C₁₆

—

C₄₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

Dic₃₄₄

—

Dic₃₄₄

—

#13063256

—

C₄⋊(C₄₃⋊C₈)

—

#13063249

—

C₄×Q₈×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

D₈

yes

characteristic

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₈

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₁₆×C₄₃

—

C₁₆×C₄₃

SD₁₆

yes

—

Q₈×C₄₃

—

Q₈×C₄₃

Q₃₂

yes

complemented

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂×SD₁₆

yes

characteristic

C₈×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×Q₃₂

yes

characteristic

C₄×C₄₃

#508

yes

characteristic

Dic₄₃

—

#2247

—

Q₈×C₁₆

D₄₃

yes

—

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

#475

—

C₈×Q₈

C₂×D₄₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₄₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

#335

—

#332

—

C₄₃

#12440

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₁₇₂

yes

characteristic

100

Q₃₂

—

172

—

101

Q₃₂

—

172

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

105

#106

—

172

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

D₁₇₂

yes

characteristic

109

#103

—

172

—

110

C₄×C₈

—

111

C₄×C₈

C₂²×D₄₃

yes

characteristic

112

C₂×Q₈

—

172

—

113

C₂×Q₈

D₃₄₄

yes

characteristic

114

Q₁₆

—

344

—

115

Q₁₆

—

172

—

116

Q₁₆

—

172

—

117

C₂×C₈

—

118

C₂×C₈

—

172

—

119

C₂×C₈

#10621898

yes

characteristic

120

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

172

—

123

C₄⋊C₄

—

172

—

124

C₄⋊C₄

D₃₄₄

yes

characteristic

125

C₄²

—

126

C₄²

C₂×D₁₇₂

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

#10621884

yes

—

130

Q₈

—

131

Q₈

—

344

—

132

Q₈

—

344

—

133

Q₈

Dic₃₄₄

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

344

—

137

C₂×C₄

#13063299

yes

characteristic

138

C₂×C₄

C₂×D₃₄₄

yes

characteristic

139

C₂×C₄

#13063347

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

344

—

143

C₈

C₂×#10621884

yes

characteristic

144

C₈

#13063350

yes

characteristic

145

C₂²

#14583135

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

688

—

150

C₄

#14583099

yes

characteristic

151

C₄

#14583169

yes

characteristic

152

C₄

C₂×Dic₃₄₄

yes

characteristic

153

C₄

#14583331

yes

characteristic

154

C₂

#19133955

yes

characteristic

155

C₂

#19136686

yes

characteristic

156

C₂

#19136720

yes

characteristic

	subgroup	G/H
center	C₂²	#14583135
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	Q₈×C₁₆×C₄₃	C₂
socle, prime radical	C₂²×C₄₃	#466
hypercenter	Q₈×C₁₆	D₄₃
hypocenter	C₄₃	#12440
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#14583135

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

Q₈×C₁₆×C₄₃

C₂

socle, prime radical

C₂²×C₄₃

#466

hypercenter

Q₈×C₁₆

D₄₃

hypocenter

C₄₃

#12440

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
52. Q₈×C₄₃	Q₃₂	semidirect
95. C₄₃	#12440	semidirect
133. Q₈	Dic₃₄₄	semidirect

H ≅ G/N

type

52. Q₈×C₄₃

Q₃₂

semidirect

95. C₄₃

#12440

semidirect

133. Q₈

Dic₃₄₄

semidirect

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

subgroup	index
#19136848	2
#19133965	2
#19132667	2
#19132645	2
Q₈×C₁₆×C₄₃	2
#12440	43

subgroup

index

Q₈×C₁₆×C₄₃

Subgroups — #25010859

Conjugacy classes of subgroups

Complements

Maximal subgroups