Group #25010957 — subgroups

Subgroups — #25010957

Order 11008.

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

normal subgroups only

This group has 204 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	#19136870	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₈₆	C₂	1	1	yes	maximal, characteristic
3	#19133990	C₂	1	2	yes	maximal
4	#19132651	C₂	1	2	yes	maximal
5	C₄₃×#1371	C₂	1	1	yes	maximal, characteristic
6	#14583304	C₂²	1	2	yes	—
7	#14583110	—	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	C₁₆×Dic₄₃	—	2	1	no	—
14	#14582646	—	2	2	no	—
15	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#335	C₂²	1	2	yes	—
17	C₂×Dic₁₇₂	—	4	2	no	—
18	#13063325	—	2	2	no	—
19	C₄×Dic₈₆	D₄	1	1	yes	characteristic
20	C₂×C₄₃⋊C₁₆	—	2	1	no	—
21	#13063259	—	2	2	no	—
22	#13063259	—	2	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	Dic₁₇₂	—	8	4	no	—
44	C₁₆×C₄₃	—	2	1	no	—
45	C₁₆×C₄₃	Q₁₆	1	2	yes	complemented
46	C₄₃⋊C₁₆	—	4	1	no	—
47	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
50	C₂×Dic₄₃	—	2	1	no	—
51	C₂×Dic₄₃	—	2	1	no	—
52	C₂×Dic₄₃	—	8	2	no	—
53	Dic₈₆	—	2	1	no	—
54	Dic₈₆	—	8	2	no	—
55	Dic₈₆	—	8	2	no	—
56	Dic₈₆	D₁₆	1	2	yes	—
57	C₈×C₄₃	—	2	1	no	—
58	C₈×C₄₃	—	8	2	no	—
59	C₈×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₄₃	#137	1	1	yes	characteristic
61	C₄₃⋊C₈	—	4	1	no	—
62	#12457	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	#466	1	1	yes	characteristic
64	C₄×C₄₃	—	2	1	no	—
65	C₄×C₄₃	#462	1	1	yes	characteristic
66	C₄×C₄₃	#473	1	1	yes	characteristic
67	C₄×C₄₃	C₂×D₁₆	1	1	yes	characteristic
68	C₄×C₄₃	#508	1	1	yes	characteristic
69	Dic₄₃	—	2	2	no	—
70	Dic₄₃	—	4	1	no	—
71	Dic₄₃	—	16	2	no	—
72	#2252	—	43	1	no	—
73	Q₈×C₁₆	—	43	1	no	—
74	#1674	—	43	2	no	—
75	#1371	D₄₃	1	1	yes	characteristic
76	#1364	—	43	2	no	—
77	C₂×C₄₃	#1674	1	1	yes	characteristic
78	C₂×C₄₃	#2215	1	1	yes	characteristic
79	C₂×C₄₃	#2222	1	1	yes	characteristic
80	#500	—	43	2	no	—
81	#462	—	86	2	no	—
82	C₈×Q₈	—	43	1	no	—
83	#366	—	86	1	no	—
84	#366	—	86	1	no	—
85	C₄⋊C₁₆	—	43	1	no	—
86	C₄⋊C₁₆	—	86	1	no	—
87	C₄×C₁₆	—	86	1	no	—
88	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
89	#335	C₂×D₄₃	1	2	yes	—
90	#332	—	86	2	no	—
91	C₄₃	#12457	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	172	2	no	—
93	C₄⋊Q₈	—	86	2	no	—
94	C₄×Q₈	—	43	1	no	—
95	C₂×C₁₆	—	86	1	no	—
96	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
97	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
98	#107	—	86	2	no	—
99	#107	—	86	2	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	43	1	no	—
102	C₄⋊C₈	—	86	1	no	—
103	#103	—	172	2	no	—
104	C₄×C₈	—	86	1	no	—
105	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
106	C₂×Q₈	—	43	1	no	—
107	C₂×Q₈	—	172	2	no	—
108	Q₁₆	—	344	4	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	86	1	no	—
111	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
112	C₂×C₈	#10621902	1	1	yes	characteristic
113	C₄⋊C₄	—	43	1	no	—
114	C₄⋊C₄	—	86	1	no	—
115	C₄⋊C₄	—	172	2	no	—
116	C₄⋊C₄	—	172	2	no	—
117	C₄²	—	86	1	no	—
118	C₄²	D₄×D₄₃	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	172	1	no	—
121	C₁₆	Dic₁₇₂	1	2	yes	complemented
122	Q₈	—	43	2	no	—
123	Q₈	—	86	1	no	—
124	Q₈	—	344	2	no	—
125	Q₈	—	344	2	no	—
126	C₂×C₄	—	86	1	no	—
127	C₂×C₄	—	86	1	no	—
128	C₂×C₄	—	344	2	no	—
129	C₂×C₄	#13063300	1	1	yes	characteristic
130	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
131	C₂×C₄	#13063348	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	8	2	no	—
134	C₈	—	172	1	no	—
135	C₈	C₂×Dic₁₇₂	1	1	yes	characteristic
136	C₈	#13063358	1	1	yes	characteristic
137	C₂²	#14583136	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	86	2	no	—
140	C₄	—	172	1	no	—
141	C₄	—	688	2	no	—
142	C₄	#14583110	1	1	yes	characteristic
143	C₄	#14583175	1	1	yes	characteristic
144	C₄	D₁₆×D₄₃	1	1	yes	characteristic
145	C₄	#14583332	1	1	yes	characteristic
146	C₂	#19133990	1	1	yes	characteristic
147	C₂	#19136682	1	1	yes	characteristic
148	C₂	#19136721	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136870

C₂

yes

maximal, characteristic

C₁₆×Dic₈₆

C₂

yes

maximal, characteristic

#19133990

C₂

yes

maximal

#19132651

C₂

yes

maximal

C₄₃×#1371

C₂

yes

maximal, characteristic

#14583304

C₂²

yes

—

#14583110

—

C₈×Dic₈₆

C₂²

yes

characteristic

#14582725

—

#14582725

—

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

characteristic

Dic₄₃⋊C₁₆

—

C₁₆×Dic₄₃

—

#14582646

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄₃×#335

C₂²

yes

—

C₂×Dic₁₇₂

—

#13063325

—

C₄×Dic₈₆

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

#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

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₁₆×C₄₃

Q₁₆

yes

complemented

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

—

Dic₈₆

—

Dic₈₆

D₁₆

yes

—

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

C₂×D₁₆

yes

characteristic

C₄×C₄₃

#508

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#2252

—

Q₈×C₁₆

—

#1674

—

#1371

D₄₃

yes

characteristic

#1364

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

#335

C₂×D₄₃

yes

—

#332

—

C₄₃

#12457

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

172

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

D₁₇₂

yes

characteristic

C₂×C₁₆

D₁₇₂

yes

characteristic

#107

—

#107

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

#103

—

172

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₄₃

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

172

—

108

Q₁₆

—

344

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×D₁₇₂

yes

characteristic

112

C₂×C₈

#10621902

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

172

—

116

C₄⋊C₄

—

172

—

117

C₄²

—

118

C₄²

D₄×D₄₃

yes

characteristic

119

C₁₆

—

120

C₁₆

—

172

—

121

C₁₆

Dic₁₇₂

yes

complemented

122

Q₈

—

123

Q₈

—

124

Q₈

—

344

—

125

Q₈

—

344

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

344

—

129

C₂×C₄

#13063300

yes

characteristic

130

C₂×C₄

D₈×D₄₃

yes

characteristic

131

C₂×C₄

#13063348

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

172

—

135

C₈

C₂×Dic₁₇₂

yes

characteristic

136

C₈

#13063358

yes

characteristic

137

C₂²

#14583136

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

172

—

141

C₄

—

688

—

142

C₄

#14583110

yes

characteristic

143

C₄

#14583175

yes

characteristic

144

C₄

D₁₆×D₄₃

yes

characteristic

145

C₄

#14583332

yes

characteristic

146

C₂

#19133990

yes

characteristic

147

C₂

#19136682

yes

characteristic

148

C₂

#19136721

yes

characteristic

	subgroup	G/H
center	C₂²	#14583136
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1371	C₂
socle, prime radical	C₂²×C₄₃	#466
hypercenter	#1371	D₄₃
hypocenter	C₄₃	#12457
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#14583136

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1371

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
45. C₁₆×C₄₃	Q₁₆	semidirect
91. C₄₃	#12457	semidirect
121. C₁₆	Dic₁₇₂	semidirect

H ≅ G/N

type

45. C₁₆×C₄₃

Q₁₆

semidirect

91. C₄₃

#12457

semidirect

121. C₁₆

Dic₁₇₂

semidirect

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

subgroup	index
#19136870	2
C₁₆×Dic₈₆	2
#19133990	2
#19132651	2
C₄₃×#1371	2
#12457	43

subgroup

index

C₁₆×Dic₈₆

C₄₃×#1371

Subgroups — #25010957

Conjugacy classes of subgroups

Complements

Maximal subgroups