Group #25010979 — subgroups

Subgroups — #25010979

Order 11008.

188 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 67 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	#19136870	C₂	1	1	yes	maximal, characteristic
2	#19136503	C₂	1	1	yes	maximal, characteristic
3	#19134000	C₂	1	2	yes	maximal
4	#19132654	C₂	1	2	yes	maximal
5	C₄₃×#1372	C₂	1	1	yes	maximal, characteristic
6	#14583304	C₂²	1	2	yes	—
7	#14583195	—	2	2	no	—
8	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
9	#14582725	—	2	1	no	—
10	#14582725	—	2	1	no	—
11	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₄₃⋊C₁₆	—	2	1	no	—
13	#14582674	—	2	1	no	—
14	#14582647	—	2	2	no	—
15	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#335	C₂²	1	2	yes	—
17	#13063325	—	2	2	no	—
18	C₄×Dic₈₆	D₄	1	1	yes	characteristic
19	C₂×C₄₃⋊C₁₆	—	2	1	no	—
20	#13063259	—	2	2	no	—
21	#13063259	—	2	2	no	—
22	#13063259	—	4	2	no	—
23	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
24	Dic₄₃⋊C₈	—	2	1	no	—
25	#13063250	—	4	2	no	—
26	C₈×Dic₄₃	—	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	2	no	—
30	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₈₆	—	4	2	no	—
32	C₂×Dic₈₆	D₈	1	1	yes	characteristic
33	C₂×C₈×C₄₃	—	4	2	no	—
34	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
36	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₄₃	—	4	2	no	—
38	C₄⋊Dic₄₃	—	4	2	no	—
39	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
40	Dic₄₃⋊C₄	—	2	1	no	—
41	C₄×Dic₄₃	—	2	1	no	—
42	C₂×C₄₃⋊C₈	—	2	1	no	—
43	C₁₆×C₄₃	—	2	1	no	—
44	C₁₆×C₄₃	Q₁₆	1	2	yes	—
45	C₄₃⋊C₁₆	—	4	1	no	—
46	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
49	C₂×Dic₄₃	—	2	1	no	—
50	C₂×Dic₄₃	—	2	1	no	—
51	C₂×Dic₄₃	—	8	2	no	—
52	Dic₈₆	—	2	2	no	—
53	Dic₈₆	—	16	2	no	—
54	C₈×C₄₃	—	2	1	no	—
55	C₈×C₄₃	—	8	2	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	#12461	—	43	1	no	maximal, Sylow
60	C₂²×C₄₃	#466	1	1	yes	characteristic
61	C₄×C₄₃	—	2	1	no	—
62	C₄×C₄₃	#462	1	1	yes	characteristic
63	C₄×C₄₃	#473	1	1	yes	characteristic
64	C₄×C₄₃	#510	1	2	yes	—
65	Dic₄₃	—	4	1	no	—
66	Dic₄₃	—	4	1	no	—
67	Dic₄₃	—	16	2	no	—
68	#2252	—	43	1	no	—
69	#2183	—	43	1	no	—
70	#1676	—	43	2	no	—
71	#1372	D₄₃	1	1	yes	characteristic
72	#1365	—	43	2	no	—
73	C₂×C₄₃	#1674	1	1	yes	characteristic
74	C₂×C₄₃	#2219	1	1	yes	characteristic
75	C₂×C₄₃	#2221	1	1	yes	characteristic
76	#500	—	43	2	no	—
77	#477	—	86	2	no	—
78	C₈×Q₈	—	43	1	no	—
79	#366	—	86	1	no	—
80	#366	—	86	1	no	—
81	C₄⋊C₁₆	—	43	1	no	—
82	C₄⋊C₁₆	—	86	1	no	—
83	#346	—	86	1	no	—
84	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
85	#335	C₂×D₄₃	1	2	yes	—
86	#333	—	86	2	no	—
87	C₄₃	#12461	1	1	yes	characteristic, complemented, Sylow
88	C₄⋊Q₈	—	86	2	no	—
89	C₄×Q₈	—	43	1	no	—
90	C₂×C₁₆	—	86	1	no	—
91	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
92	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
93	#107	—	86	2	no	—
94	#107	—	86	2	no	—
95	#107	—	172	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	43	1	no	—
98	C₄⋊C₈	—	86	1	no	—
99	#103	—	172	2	no	—
100	C₄×C₈	—	86	1	no	—
101	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
102	C₂×Q₈	—	43	1	no	—
103	C₂×Q₈	—	172	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	86	1	no	—
106	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
107	C₂×C₈	#10621902	1	1	yes	characteristic
108	C₄⋊C₄	—	43	1	no	—
109	C₄⋊C₄	—	86	1	no	—
110	C₄⋊C₄	—	172	2	no	—
111	C₄⋊C₄	—	172	2	no	—
112	C₄²	—	86	1	no	—
113	C₄²	D₄×D₄₃	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	172	1	no	—
116	C₁₆	Dic₁₇₂	1	2	yes	—
117	Q₈	—	86	2	no	—
118	Q₈	—	688	2	no	—
119	C₂×C₄	—	86	1	no	—
120	C₂×C₄	—	86	1	no	—
121	C₂×C₄	—	344	2	no	—
122	C₂×C₄	#13063300	1	1	yes	characteristic
123	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
124	C₂×C₄	#13063348	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	172	1	no	—
128	C₈	C₂×Dic₁₇₂	1	1	yes	characteristic
129	C₈	#13063358	1	1	yes	characteristic
130	C₂²	#14583136	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	172	1	no	—
133	C₄	—	172	1	no	—
134	C₄	—	688	2	no	—
135	C₄	#14583110	1	1	yes	characteristic
136	C₄	#14583175	1	1	yes	characteristic
137	C₄	#14583338	1	2	yes	—
138	C₂	#19133990	1	1	yes	characteristic
139	C₂	#19136704	1	1	yes	characteristic
140	C₂	#19136716	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136870

C₂

yes

maximal, characteristic

#19136503

C₂

yes

maximal, characteristic

#19134000

C₂

yes

maximal

#19132654

C₂

yes

maximal

C₄₃×#1372

C₂

yes

maximal, characteristic

#14583304

C₂²

yes

—

#14583195

—

C₈×Dic₈₆

C₂²

yes

characteristic

#14582725

—

#14582725

—

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

characteristic

Dic₄₃⋊C₁₆

—

#14582674

—

#14582647

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄₃×#335

C₂²

yes

—

#13063325

—

C₄×Dic₈₆

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

#13063259

—

#13063259

—

#13063259

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

characteristic

Dic₄₃⋊C₈

—

#13063250

—

C₈×Dic₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

D₈

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₈

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₁₆×C₄₃

Q₁₆

yes

—

C₄₃⋊C₁₆

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂×Q₁₆

yes

characteristic

C₈×C₄₃

#137

yes

characteristic

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

—

—

D₄₃

yes

characteristic

#1365

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

#335

C₂×D₄₃

yes

—

#333

—

C₄₃

#12461

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

D₁₇₂

yes

characteristic

C₂×C₁₆

D₁₇₂

yes

—

—

—

172

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

172

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₄₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

172

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×D₁₇₂

yes

characteristic

107

C₂×C₈

#10621902

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

172

—

111

C₄⋊C₄

—

172

—

112

C₄²

—

113

C₄²

D₄×D₄₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

172

—

116

C₁₆

Dic₁₇₂

yes

—

117

Q₈

—

118

Q₈

—

688

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

344

—

122

C₂×C₄

#13063300

yes

characteristic

123

C₂×C₄

D₈×D₄₃

yes

characteristic

124

C₂×C₄

#13063348

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

172

—

128

C₈

C₂×Dic₁₇₂

yes

characteristic

129

C₈

#13063358

yes

characteristic

130

C₂²

#14583136

yes

characteristic

131

C₄

—

132

C₄

—

172

—

133

C₄

—

172

—

134

C₄

—

688

—

135

C₄

#14583110

yes

characteristic

136

C₄

#14583175

yes

characteristic

137

C₄

#14583338

yes

—

138

C₂

#19133990

yes

characteristic

139

C₂

#19136704

yes

characteristic

140

C₂

#19136716

yes

characteristic

	subgroup	G/H
center	C₂²	#14583136
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1372	C₂
socle, prime radical	C₂²×C₄₃	#466
hypercenter	#1372	D₄₃
hypocenter	C₄₃	#12461
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#14583136

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1372

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
87. C₄₃	#12461	semidirect

H ≅ G/N

type

87. C₄₃

#12461

semidirect

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

subgroup	index
#19136870	2
#19136503	2
#19134000	2
#19132654	2
C₄₃×#1372	2
#12461	43

subgroup

index

C₄₃×#1372

Subgroups — #25010979

Conjugacy classes of subgroups

Complements

Maximal subgroups