Group #25010958 — subgroups

Subgroups — #25010958

Order 11008.

192 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	#19133996	C₂	1	2	yes	maximal
3	#19132667	C₂	1	1	yes	maximal, characteristic
4	#19132653	C₂	1	2	yes	maximal
5	C₄₃×#2252	C₂	1	1	yes	maximal, characteristic
6	#14583191	—	2	2	no	—
7	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
8	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₄₃⋊C₁₆	—	2	1	no	—
10	C₁₆×Dic₄₃	—	2	1	no	—
11	#14582652	C₂²	1	2	yes	—
12	#14582648	—	2	2	no	—
13	C₄₃×#500	C₂²	1	2	yes	—
14	C₄₃×#366	—	2	1	no	—
15	C₄₃×#366	—	2	1	no	—
16	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₈₆	D₄	1	1	yes	characteristic
18	C₂×C₄₃⋊C₁₆	—	2	1	no	—
19	#13063258	—	4	2	no	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₄₃⋊C₈	—	2	1	no	—
23	#13063248	—	4	2	no	—
24	C₈×Dic₄₃	—	2	1	no	—
25	C₄⋊Q₈×C₄₃	—	2	2	no	—
26	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
28	#107×C₄₃	—	2	2	no	—
29	#107×C₄₃	—	2	2	no	—
30	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₄₃	—	4	2	no	—
32	C₂×Dic₈₆	D₈	1	1	yes	characteristic
33	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₄₃	—	4	2	no	—
36	C₄⋊C₄×C₄₃	—	4	2	no	—
37	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
39	Dic₄₃⋊C₄	—	2	1	no	—
40	C₄×Dic₄₃	—	2	1	no	—
41	C₂×C₄₃⋊C₈	—	2	1	no	—
42	C₂×C₄₃⋊C₈	—	4	2	no	—
43	C₁₆×C₄₃	—	2	1	no	—
44	C₁₆×C₄₃	Q₁₆	1	2	yes	—
45	C₄₃⋊C₁₆	—	4	1	no	—
46	Q₈×C₄₃	—	16	2	no	—
47	C₂×C₄×C₄₃	—	8	2	no	—
48	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
51	C₂×Dic₄₃	—	2	1	no	—
52	C₂×Dic₄₃	—	2	1	no	—
53	Dic₈₆	—	2	1	no	—
54	Dic₈₆	Q₃₂	1	2	yes	—
55	C₈×C₄₃	—	2	1	no	—
56	C₈×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₄₃	#137	1	1	yes	characteristic
58	C₄₃⋊C₈	—	4	1	no	—
59	C₄₃⋊C₈	—	8	2	no	—
60	#12458	—	43	1	no	maximal, Sylow
61	C₂²×C₄₃	#466	1	1	yes	characteristic
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	—	16	2	no	—
64	C₄×C₄₃	#462	1	1	yes	characteristic
65	C₄×C₄₃	#473	1	1	yes	characteristic
66	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
67	C₄×C₄₃	#508	1	1	yes	characteristic
68	Dic₄₃	—	2	2	no	—
69	Dic₄₃	—	4	1	no	—
70	#2252	D₄₃	1	1	yes	characteristic
71	Q₈×C₁₆	—	43	1	no	—
72	#1676	—	43	2	no	—
73	#1371	—	43	1	no	—
74	#1365	—	43	2	no	—
75	C₂×C₄₃	#1674	1	1	yes	characteristic
76	C₂×C₄₃	#2216	1	1	yes	characteristic
77	C₂×C₄₃	#2221	1	1	yes	characteristic
78	#500	C₂×D₄₃	1	2	yes	—
79	#477	—	86	2	no	—
80	C₈×Q₈	—	43	1	no	—
81	#366	—	2	1	no	—
82	#366	—	2	1	no	—
83	C₄⋊C₁₆	—	43	1	no	—
84	C₄⋊C₁₆	—	86	1	no	—
85	C₄×C₁₆	—	86	1	no	—
86	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
87	#335	—	43	2	no	—
88	#333	—	86	2	no	—
89	C₄₃	#12458	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	43	1	no	—
92	C₂×C₁₆	—	86	1	no	—
93	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	172	2	no	—
98	C₄⋊C₈	—	43	1	no	—
99	C₄⋊C₈	—	86	1	no	—
100	C₄⋊C₈	—	86	2	no	—
101	#103	—	172	2	no	—
102	C₄×C₈	—	86	1	no	—
103	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	43	1	no	—
106	C₂×C₈	—	86	1	no	—
107	C₂×C₈	—	172	2	no	—
108	C₂×C₈	#10621900	1	1	yes	characteristic
109	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	43	1	no	—
113	C₄⋊C₄	—	86	1	no	—
114	C₄²	—	86	1	no	—
115	C₄²	D₄×D₄₃	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	172	1	no	—
118	C₁₆	#10621885	1	2	yes	—
119	Q₈	—	16	2	no	—
120	Q₈	—	43	2	no	—
121	Q₈	—	86	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	86	1	no	—
124	C₂×C₄	—	86	1	no	—
125	C₂×C₄	#13063296	1	1	yes	characteristic
126	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
127	C₂×C₄	#13063344	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	344	2	no	—
130	C₈	—	172	1	no	—
131	C₈	C₂×#10621885	1	1	yes	characteristic
132	C₈	#13063356	1	1	yes	characteristic
133	C₂²	#14583132	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	16	2	no	—
136	C₄	—	86	2	no	—
137	C₄	—	172	1	no	—
138	C₄	#14583106	1	1	yes	characteristic
139	C₄	#14583171	1	1	yes	characteristic
140	C₄	Q₃₂×D₄₃	1	1	yes	characteristic
141	C₄	#14583332	1	1	yes	characteristic
142	C₂	#19133986	1	1	yes	characteristic
143	C₂	#19136683	1	1	yes	characteristic
144	C₂	#19136712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₈₆

C₂

yes

maximal, characteristic

#19133996

C₂

yes

maximal

#19132667

C₂

yes

maximal, characteristic

#19132653

C₂

yes

maximal

C₄₃×#2252

C₂

yes

maximal, characteristic

#14583191

—

C₈×Dic₈₆

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

characteristic

Dic₄₃⋊C₁₆

—

C₁₆×Dic₄₃

—

#14582652

C₂²

yes

—

#14582648

—

C₄₃×#500

C₂²

yes

—

C₄₃×#366

—

C₄₃×#366

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×Dic₈₆

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

#13063258

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

characteristic

Dic₄₃⋊C₈

—

#13063248

—

C₈×Dic₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

characteristic

#107×C₄₃

—

#107×C₄₃

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

D₈

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₁₆×C₄₃

Q₁₆

yes

—

C₄₃⋊C₁₆

—

Q₈×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

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

Q₃₂

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₂×Q₁₆

yes

characteristic

C₈×C₄₃

#137

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×Q₃₂

yes

characteristic

C₄×C₄₃

#508

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2252

D₄₃

yes

characteristic

Q₈×C₁₆

—

#1676

—

#1371

—

#1365

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×D₄₃

yes

—

#477

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

#335

—

#333

—

C₄₃

#12458

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₄₃

yes

—

—

—

172

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#103

—

172

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₄₃

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×C₈

—

107

C₂×C₈

—

172

—

108

C₂×C₈

#10621900

yes

characteristic

109

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

D₄×D₄₃

yes

characteristic

116

C₁₆

—

117

C₁₆

—

172

—

118

C₁₆

#10621885

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#13063296

yes

characteristic

126

C₂×C₄

D₈×D₄₃

yes

characteristic

127

C₂×C₄

#13063344

yes

characteristic

128

C₈

—

129

C₈

—

344

—

130

C₈

—

172

—

131

C₈

C₂×#10621885

yes

characteristic

132

C₈

#13063356

yes

characteristic

133

C₂²

#14583132

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

172

—

138

C₄

#14583106

yes

characteristic

139

C₄

#14583171

yes

characteristic

140

C₄

Q₃₂×D₄₃

yes

characteristic

141

C₄

#14583332

yes

characteristic

142

C₂

#19133986

yes

characteristic

143

C₂

#19136683

yes

characteristic

144

C₂

#19136712

yes

characteristic

	subgroup	G/H
center	C₂²	#14583132
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#2252	C₂
socle, prime radical	C₂²×C₄₃	#466
hypercenter	#2252	D₄₃
hypocenter	C₄₃	#12458
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#14583132

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#2252

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
89. C₄₃	#12458	semidirect

H ≅ G/N

type

89. C₄₃

#12458

semidirect

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

subgroup	index
C₁₆×Dic₈₆	2
#19133996	2
#19132667	2
#19132653	2
C₄₃×#2252	2
#12458	43

subgroup

index

C₁₆×Dic₈₆

C₄₃×#2252

Subgroups — #25010958

Conjugacy classes of subgroups

Complements

Maximal subgroups