Group #25008404 — subgroups

Subgroups — #25008404

Order 11008.

206 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 91 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	#19136503	C₂	1	2	yes	maximal
2	C₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#19132665	C₂	1	1	yes	maximal, characteristic
4	#19132575	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₈₆	C₂²	1	2	yes	—
7	C₄×Dic₁₇₂	C₄	1	1	yes	characteristic
8	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
9	Dic₄₃⋊C₁₆	—	2	2	no	—
10	#14582674	—	2	2	no	—
11	#14582652	C₄	1	1	yes	characteristic
12	#14582631	C₄	1	2	yes	—
13	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
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	2	no	—
19	#13063259	C₈	1	1	yes	characteristic
20	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
21	Dic₄₃⋊C₈	—	2	2	no	—
22	#13063250	C₈	1	2	yes	—
23	C₈×Dic₄₃	—	2	2	no	—
24	C₂×C₁₆×C₄₃	—	2	1	no	—
25	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₈₆	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₄₃	—	2	1	no	—
32	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
36	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
38	Dic₄₃⋊C₄	—	2	2	no	—
39	C₄×Dic₄₃	—	2	2	no	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	Dic₁₇₂	—	2	2	no	—
42	C₁₆×C₄₃	—	2	2	no	—
43	C₁₆×C₄₃	—	2	1	no	—
44	C₁₆×C₄₃	Q₁₆	1	2	yes	complemented
45	C₄₃⋊C₁₆	—	4	2	no	—
46	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
49	C₂×Dic₄₃	—	2	2	no	—
50	C₂×Dic₄₃	—	2	2	no	—
51	Dic₈₆	—	2	2	no	—
52	Dic₈₆	—	2	2	no	—
53	C₈×C₄₃	—	2	2	no	—
54	C₈×C₄₃	—	2	1	no	—
55	C₈×C₄₃	—	2	1	no	—
56	C₈×C₄₃	M₃₂	1	2	yes	—
57	C₈×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₄₃	#135	1	1	yes	characteristic
59	C₄₃⋊C₈	—	4	2	no	—
60	#12054	—	43	1	no	maximal, Sylow
61	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
64	C₄×C₄₃	#443	1	1	yes	characteristic
65	C₄×C₄₃	C₂×M₃₂	1	1	yes	characteristic
66	C₄×C₄₃	#504	1	1	yes	characteristic
67	Dic₄₃	—	4	2	no	—
68	Dic₄₃	—	4	2	no	—
69	#2183	—	43	2	no	—
70	C₈×Q₁₆	—	43	1	no	—
71	#1370	—	43	1	no	—
72	Q₈⋊C₁₆	—	43	2	no	—
73	C₈×C₁₆	D₄₃	1	1	yes	characteristic
74	C₂×C₄₃	C₈×Q₁₆	1	1	yes	characteristic
75	C₂×C₄₃	#2169	1	1	yes	characteristic
76	C₂×C₄₃	#2170	1	1	yes	characteristic
77	C₈×Q₈	—	43	2	no	—
78	C₄×Q₁₆	—	43	1	no	—
79	C₄⋊C₁₆	—	43	2	no	—
80	C₄⋊C₁₆	—	86	2	no	—
81	#346	—	86	2	no	—
82	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
83	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
84	#335	—	43	1	no	—
85	Q₈⋊C₈	—	43	2	no	—
86	C₈²	C₂×D₄₃	1	1	yes	characteristic
87	C₄₃	#12054	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	43	1	no	—
89	C₄×Q₈	—	43	2	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	86	2	no	—
92	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
93	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
94	#107	—	43	1	no	—
95	C₄⋊C₈	—	43	2	no	—
96	C₄⋊C₈	—	86	2	no	—
97	#103	—	43	2	no	—
98	C₄×C₈	—	86	2	no	—
99	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
100	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
102	C₂×Q₈	—	43	2	no	—
103	Q₁₆	—	86	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	86	2	no	—
106	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
107	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
108	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
109	C₂×C₈	#10621898	1	1	yes	characteristic
110	C₄⋊C₄	—	43	2	no	—
111	C₄⋊C₄	—	86	2	no	—
112	C₄²	—	86	2	no	—
113	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	—	172	2	no	—
117	C₁₆	Dic₁₇₂	1	2	yes	complemented
118	Q₈	—	86	2	no	—
119	Q₈	—	86	2	no	—
120	C₂×C₄	—	86	2	no	—
121	C₂×C₄	—	86	2	no	—
122	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
124	C₂×C₄	#13063331	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	172	2	no	—
129	C₈	#13063264	1	2	yes	—
130	C₈	C₂×Dic₁₇₂	1	1	yes	characteristic
131	C₈	#13063347	1	1	yes	characteristic
132	C₂²	C₈×D₁₇₂	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	172	2	no	—
135	C₄	—	172	2	no	—
136	C₄	C₄×Dic₁₇₂	1	1	yes	characteristic
137	C₄	#14583017	1	1	yes	characteristic
138	C₄	C₂×#13063264	1	1	yes	characteristic
139	C₄	#14583315	1	1	yes	characteristic
140	C₂	C₈×Dic₁₇₂	1	1	yes	characteristic
141	C₂	#19136428	1	1	yes	characteristic
142	C₂	#19136433	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136503

C₂

yes

maximal

C₈×Dic₁₇₂

C₂

yes

maximal, characteristic

#19132665

C₂

yes

maximal, characteristic

#19132575

C₂

yes

maximal

C₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic

C₈×Dic₈₆

C₂²

yes

—

C₄×Dic₁₇₂

C₄

yes

characteristic

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

—

Dic₄₃⋊C₁₆

—

#14582674

—

#14582652

C₄

yes

characteristic

#14582631

C₄

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

C₈

yes

characteristic

C₄×Dic₈₆

C₂×C₄

yes

—

C₂×C₄₃⋊C₁₆

—

#13063259

C₈

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂×C₄

yes

—

Dic₄₃⋊C₈

—

#13063250

C₈

yes

—

C₈×Dic₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

characteristic

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

—

C₁₆×C₄₃

—

C₁₆×C₄₃

Q₁₆

yes

complemented

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

C₂×Q₁₆

yes

characteristic

C₈×C₄₃

#135

yes

characteristic

C₄₃⋊C₈

—

#12054

—

maximal, Sylow

C₂²×C₄₃

C₈×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₄×Q₁₆

yes

characteristic

C₄×C₄₃

#443

yes

characteristic

C₄×C₄₃

C₂×M₃₂

yes

characteristic

C₄×C₄₃

#504

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2183

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

D₄₃

yes

characteristic

C₂×C₄₃

C₈×Q₁₆

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄×C₁₆

C₂×D₄₃

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12054

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₁₇₂

yes

characteristic

C₂×C₁₆

D₁₇₂

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₄₃

yes

characteristic

100

C₄×C₈

C₄×D₄₃

yes

characteristic

101

C₄×C₈

C₂²×D₄₃

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₈×D₄₃

yes

characteristic

107

C₂×C₈

C₈×D₄₃

yes

characteristic

108

C₂×C₈

C₂×D₁₇₂

yes

characteristic

109

C₂×C₈

#10621898

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂×C₄×D₄₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

172

—

117

C₁₆

Dic₁₇₂

yes

complemented

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₄×D₁₇₂

yes

characteristic

123

C₂×C₄

C₂×C₈×D₄₃

yes

characteristic

124

C₂×C₄

#13063331

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

172

—

129

C₈

#13063264

yes

—

130

C₈

C₂×Dic₁₇₂

yes

characteristic

131

C₈

#13063347

yes

characteristic

132

C₂²

C₈×D₁₇₂

yes

characteristic

133

C₄

—

134

C₄

—

172

—

135

C₄

—

172

—

136

C₄

C₄×Dic₁₇₂

yes

characteristic

137

C₄

#14583017

yes

characteristic

138

C₄

C₂×#13063264

yes

characteristic

139

C₄

#14583315

yes

characteristic

140

C₂

C₈×Dic₁₇₂

yes

characteristic

141

C₂

#19136428

yes

characteristic

142

C₂

#19136433

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₈×C₁₆×C₄₃	C₂
socle, prime radical	C₂²×C₄₃	C₈×D₄
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#12054
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆

subgroup

G/H

center

C₂×C₈

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₈×C₁₆×C₄₃

C₂

socle, prime radical

C₂²×C₄₃

C₈×D₄

cosocle

C₄×C₈×C₄₃

C₂³

hypercenter

C₈×C₁₆

D₄₃

hypocenter

C₄₃

#12054

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆

N	H ≅ G/N	type
44. C₁₆×C₄₃	Q₁₆	semidirect
87. C₄₃	#12054	semidirect
117. C₁₆	Dic₁₇₂	semidirect

H ≅ G/N

type

44. C₁₆×C₄₃

Q₁₆

semidirect

87. C₄₃

#12054

semidirect

117. C₁₆

Dic₁₇₂

semidirect

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

subgroup	index
#19136503	2
C₈×Dic₁₇₂	2
#19132665	2
#19132575	2
C₈×C₁₆×C₄₃	2
#12054	43

subgroup

index

C₈×Dic₁₇₂

C₈×C₁₆×C₄₃

Subgroups — #25008404

Conjugacy classes of subgroups

Complements

Maximal subgroups