Group #25008447 — subgroups

Subgroups — #25008447

Order 11008.

200 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 85 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	#19136500	C₂	1	2	yes	maximal
2	C₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#19132573	C₂	1	2	yes	maximal
4	#19132513	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1367	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₈₆	C₂²	1	2	yes	—
7	C₄×Dic₁₇₂	C₄	1	1	yes	characteristic
8	Dic₄₃⋊C₁₆	—	2	2	no	—
9	#14582674	—	2	2	no	—
10	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14582652	C₄	1	1	yes	characteristic
13	#14582631	C₄	1	2	yes	—
14	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₈²×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₂	C₈	1	1	yes	characteristic
17	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
18	C₂×C₄₃⋊C₁₆	—	2	1	no	—
19	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
20	#13063259	C₈	1	1	yes	characteristic
21	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
22	Dic₄₃⋊C₈	—	2	2	no	—
23	#13063250	C₈	1	2	yes	—
24	C₈×Dic₄₃	—	2	2	no	—
25	C₂×C₁₆×C₄₃	—	2	2	no	—
26	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₈₆	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₄₃	—	2	1	no	—
31	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
35	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
36	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
37	Dic₄₃⋊C₄	—	2	2	no	—
38	C₄×Dic₄₃	—	2	2	no	—
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	Dic₁₇₂	—	2	2	no	—
41	C₁₆×C₄₃	—	4	2	no	—
42	C₄₃⋊C₁₆	—	2	2	no	—
43	C₄₃⋊C₁₆	—	4	1	no	—
44	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
47	C₂×Dic₄₃	—	2	2	no	—
48	C₂×Dic₄₃	—	2	2	no	—
49	Dic₈₆	—	2	2	no	—
50	Dic₈₆	—	2	2	no	—
51	C₈×C₄₃	—	2	1	no	—
52	C₈×C₄₃	—	2	1	no	—
53	C₈×C₄₃	—	4	1	no	—
54	C₈×C₄₃	M₃₂	1	2	yes	—
55	C₈×C₄₃	#137	1	2	yes	—
56	C₄₃⋊C₈	—	4	2	no	—
57	#12061	—	43	1	no	maximal, Sylow
58	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
59	C₄×C₄₃	—	2	1	no	—
60	C₄×C₄₃	#441	1	1	yes	characteristic
61	C₄×C₄₃	#444	1	1	yes	characteristic
62	C₄×C₄₃	C₂×M₃₂	1	1	yes	characteristic
63	C₄×C₄₃	#504	1	1	yes	characteristic
64	Dic₄₃	—	4	2	no	—
65	Dic₄₃	—	4	2	no	—
66	#2183	—	43	2	no	—
67	C₈×Q₁₆	—	43	1	no	—
68	#1367	D₄₃	1	1	yes	characteristic
69	Q₈⋊C₁₆	—	43	2	no	—
70	#1312	—	43	1	no	—
71	C₂×C₄₃	#1579	1	1	yes	characteristic
72	C₂×C₄₃	#2169	1	1	yes	characteristic
73	C₂×C₄₃	#2171	1	1	yes	characteristic
74	C₈×Q₈	—	43	2	no	—
75	C₄×Q₁₆	—	43	1	no	—
76	C₄⋊C₁₆	—	86	2	no	—
77	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
78	#346	—	86	2	no	—
79	C₄×C₁₆	—	43	1	no	—
80	C₄×C₁₆	—	43	1	no	—
81	#335	—	43	1	no	—
82	Q₈⋊C₈	—	43	2	no	—
83	C₈²	C₂×D₄₃	1	1	yes	characteristic
84	C₄₃	#12061	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	43	1	no	—
86	C₄×Q₈	—	43	2	no	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	43	2	no	—
89	C₂×C₁₆	—	86	1	no	—
90	#107	—	43	1	no	—
91	C₄⋊C₈	—	43	2	no	—
92	C₄⋊C₈	—	86	2	no	—
93	#103	—	43	2	no	—
94	C₄×C₈	—	86	2	no	—
95	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
96	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
98	C₂×Q₈	—	43	2	no	—
99	Q₁₆	—	86	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	86	2	no	—
102	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
103	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
104	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
105	C₂×C₈	#10621902	1	1	yes	characteristic
106	C₄⋊C₄	—	43	2	no	—
107	C₄⋊C₄	—	86	2	no	—
108	C₄²	—	86	2	no	—
109	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
110	C₁₆	—	4	2	no	—
111	C₁₆	—	86	2	no	—
112	C₁₆	—	172	1	no	—
113	Q₈	—	86	2	no	—
114	Q₈	—	86	2	no	—
115	C₂×C₄	—	86	2	no	—
116	C₂×C₄	—	86	2	no	—
117	C₂×C₄	#13063287	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
119	C₂×C₄	#13063333	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	172	2	no	—
124	C₈	#13063264	1	2	yes	—
125	C₈	#13063354	1	2	yes	—
126	C₂²	#14582964	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	172	2	no	—
129	C₄	—	172	2	no	—
130	C₄	#14583005	1	1	yes	characteristic
131	C₄	#14583020	1	1	yes	characteristic
132	C₄	C₂×#13063264	1	1	yes	characteristic
133	C₄	#14583317	1	1	yes	characteristic
134	C₂	#19133413	1	1	yes	characteristic
135	C₂	#19136426	1	1	yes	characteristic
136	C₂	#19136436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136500

C₂

yes

maximal

C₈×Dic₁₇₂

C₂

yes

maximal, characteristic

#19132573

C₂

yes

maximal

#19132513

C₂

yes

maximal, characteristic

C₄₃×#1367

C₂

yes

maximal, characteristic

C₈×Dic₈₆

C₂²

yes

—

C₄×Dic₁₇₂

C₄

yes

characteristic

Dic₄₃⋊C₁₆

—

#14582674

—

C₄×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₄×C₄₃⋊C₁₆

C₂²

yes

characteristic

#14582652

C₄

yes

characteristic

#14582631

C₄

yes

—

C₄₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

C₈

yes

characteristic

C₄×Dic₈₆

C₂×C₄

yes

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

#13063259

C₈

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂×C₄

yes

—

Dic₄₃⋊C₈

—

#13063250

C₈

yes

—

C₈×Dic₄₃

—

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

C₂×C₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₄₃

C₂×C₈

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

—

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₃

#131

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

M₃₂

yes

—

C₈×C₄₃

#137

yes

—

C₄₃⋊C₈

—

#12061

—

maximal, Sylow

C₂²×C₄₃

C₈×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×M₃₂

yes

characteristic

C₄×C₄₃

#504

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2183

—

C₈×Q₁₆

—

#1367

D₄₃

yes

characteristic

Q₈⋊C₁₆

—

#1312

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₄₃

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12061

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₄₃

yes

characteristic

C₄×C₈

C₄×D₄₃

yes

characteristic

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₈×D₄₃

yes

characteristic

103

C₂×C₈

C₈×D₄₃

yes

characteristic

104

C₂×C₈

D₄×D₄₃

yes

characteristic

105

C₂×C₈

#10621902

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

C₂×C₄×D₄₃

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

172

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#13063287

yes

characteristic

118

C₂×C₄

C₂×C₈×D₄₃

yes

characteristic

119

C₂×C₄

#13063333

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

172

—

124

C₈

#13063264

yes

—

125

C₈

#13063354

yes

—

126

C₂²

#14582964

yes

characteristic

127

C₄

—

128

C₄

—

172

—

129

C₄

—

172

—

130

C₄

#14583005

yes

characteristic

131

C₄

#14583020

yes

characteristic

132

C₄

C₂×#13063264

yes

characteristic

133

C₄

#14583317

yes

characteristic

134

C₂

#19133413

yes

characteristic

135

C₂

#19136426

yes

characteristic

136

C₂

#19136436

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1367	C₂
socle, prime radical	C₂²×C₄₃	C₈×D₄
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1367	D₄₃
hypocenter	C₄₃	#12061
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1367

C₂

socle, prime radical

C₂²×C₄₃

C₈×D₄

cosocle

C₄×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
84. C₄₃	#12061	semidirect

H ≅ G/N

type

84. C₄₃

#12061

semidirect

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

subgroup	index
#19136500	2
C₈×Dic₁₇₂	2
#19132573	2
#19132513	2
C₄₃×#1367	2
#12061	43

subgroup

index

C₈×Dic₁₇₂

C₄₃×#1367

Subgroups — #25008447

Conjugacy classes of subgroups

Complements

Maximal subgroups