Group #25011255 — subgroups

Subgroups — #25011255

Order 11008.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#19136848	C₂	1	1	yes	maximal, characteristic
3	#19136823	C₂	1	2	yes	maximal
4	#19134192	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₄	—	2	2	no	—
7	#14583304	C₂²	1	2	yes	—
8	#14583297	—	2	2	no	—
9	#14583279	C₂²	1	2	yes	—
10	#14583250	—	2	2	no	—
11	#14582725	—	2	2	no	—
12	#14582704	—	2	4	no	—
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	—	2	2	no	—
17	C₂×Dic₁₇₂	—	2	2	no	—
18	#13063325	—	2	2	no	—
19	#13063325	—	2	2	no	—
20	Dic₃₄₄	—	4	4	no	—
21	#13063259	—	2	2	no	—
22	#13063259	—	2	2	no	—
23	#13063257	—	4	4	no	—
24	#13063250	—	4	4	no	—
25	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
27	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₈₆	—	4	2	no	—
31	C₂×Dic₈₆	—	4	2	no	—
32	C₂×C₈×C₄₃	D₈	1	2	yes	—
33	C₂×C₈×C₄₃	D₈	1	2	yes	—
34	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₃	C₂×D₄	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	—	4	2	no	—
40	C₄⋊Dic₄₃	—	4	2	no	—
41	Dic₁₇₂	—	4	4	no	—
42	Dic₁₇₂	—	4	4	no	—
43	C₁₆×C₄₃	—	2	2	no	—
44	C₁₆×C₄₃	SD₁₆	1	4	yes	—
45	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₄₃	—	8	2	no	—
49	C₂×Dic₄₃	—	8	2	no	—
50	Dic₈₆	—	8	2	no	—
51	Dic₈₆	—	8	2	no	—
52	Dic₈₆	—	8	4	no	—
53	C₈×C₄₃	—	2	2	no	—
54	C₈×C₄₃	Q₃₂	1	4	yes	complemented
55	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
56	C₈×C₄₃	#135	1	2	yes	—
57	#12504	—	43	1	no	maximal, Sylow
58	C₂²×C₄₃	#493	1	1	yes	characteristic
59	C₄×C₄₃	#492	1	1	yes	characteristic
60	C₄×C₄₃	#495	1	1	yes	characteristic
61	C₄×C₄₃	C₂×Q₃₂	1	2	yes	—
62	C₄×C₄₃	#508	1	2	yes	—
63	Dic₄₃	—	16	2	no	—
64	Dic₄₃	—	16	2	no	—
65	#2252	—	43	1	no	—
66	#2247	—	43	1	no	—
67	#2241	—	43	2	no	—
68	#1707	—	43	2	no	—
69	C₈×C₁₆	D₄₃	1	1	yes	characteristic
70	C₂×C₄₃	#1706	1	1	yes	characteristic
71	C₂×C₄₃	#2247	1	1	yes	characteristic
72	C₂×C₄₃	#2249	1	1	yes	characteristic
73	C₂×Q₃₂	—	86	2	no	—
74	#500	—	43	2	no	—
75	#498	—	86	2	no	—
76	#494	—	43	2	no	—
77	#487	—	86	2	no	—
78	#366	—	86	2	no	—
79	#358	—	86	4	no	—
80	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
82	C₈²	C₂×D₄₃	1	1	yes	characteristic
83	C₄₃	#12504	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₁₆	—	86	2	no	—
85	C₂×Q₁₆	—	86	2	no	—
86	C₄⋊Q₈	—	86	2	no	—
87	C₄⋊Q₈	—	86	2	no	—
88	Q₃₂	—	172	4	no	—
89	C₂×C₁₆	D₁₇₂	1	2	yes	—
90	C₂×C₁₆	D₁₇₂	1	2	yes	—
91	#107	—	86	2	no	—
92	#107	—	86	2	no	—
93	#106	—	172	4	no	—
94	#103	—	172	4	no	—
95	C₄×C₈	D₁₇₂	1	1	yes	characteristic
96	C₄×C₈	D₁₇₂	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
98	C₂×Q₈	—	172	2	no	—
99	C₂×Q₈	—	172	2	no	—
100	Q₁₆	—	172	4	no	—
101	Q₁₆	—	172	4	no	—
102	C₂×C₈	D₃₄₄	1	2	yes	—
103	C₂×C₈	D₃₄₄	1	2	yes	—
104	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
105	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
106	C₄⋊C₄	—	172	2	no	—
107	C₄⋊C₄	—	172	2	no	—
108	C₄⋊C₄	—	172	2	no	—
109	C₄⋊C₄	—	172	2	no	—
110	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	#10621873	1	4	yes	—
113	Q₈	—	344	4	no	—
114	Q₈	—	344	2	no	—
115	Q₈	—	344	2	no	—
116	C₂×C₄	—	344	2	no	—
117	C₂×C₄	—	344	2	no	—
118	C₂×C₄	#13063322	1	1	yes	characteristic
119	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
120	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Dic₃₄₄	1	4	yes	complemented
123	C₈	C₂×#10621873	1	2	yes	—
124	C₈	#13063347	1	2	yes	—
125	C₂²	#14583276	1	1	yes	characteristic
126	C₄	—	688	2	no	—
127	C₄	—	688	2	no	—
128	C₄	#14583272	1	1	yes	characteristic
129	C₄	#14583283	1	1	yes	characteristic
130	C₄	C₂×Dic₃₄₄	1	2	yes	—
131	C₄	#14583331	1	2	yes	—
132	C₂	#19134187	1	1	yes	characteristic
133	C₂	#19136848	1	1	yes	characteristic
134	C₂	#19136856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136870

C₂

yes

maximal, characteristic

#19136848

C₂

yes

maximal, characteristic

#19136823

C₂

yes

maximal

#19134192

C₂

yes

maximal

C₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

—

#14583304

C₂²

yes

—

#14583297

—

#14583279

C₂²

yes

—

#14583250

—

#14582725

—

#14582704

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

#13063325

—

Dic₃₄₄

—

—

—

—

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₁₆×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

Q₃₂

yes

complemented

C₈×C₄₃

C₂×SD₁₆

yes

—

C₈×C₄₃

#135

yes

—

—

C₂²×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×Q₃₂

yes

—

C₄×C₄₃

#508

yes

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

C₈×C₁₆

D₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×Q₃₂

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12504

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

172

—

C₂×C₁₆

D₁₇₂

yes

—

C₂×C₁₆

D₁₇₂

yes

—

#107

—

#107

—

#106

—

172

—

#103

—

172

—

C₄×C₈

D₁₇₂

yes

characteristic

C₄×C₈

D₁₇₂

yes

characteristic

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

C₂×Q₈

—

172

—

100

Q₁₆

—

172

—

101

Q₁₆

—

172

—

102

C₂×C₈

D₃₄₄

yes

—

103

C₂×C₈

D₃₄₄

yes

—

104

C₂×C₈

C₂×D₁₇₂

yes

characteristic

105

C₂×C₈

C₂×D₁₇₂

yes

characteristic

106

C₄⋊C₄

—

172

—

107

C₄⋊C₄

—

172

—

108

C₄⋊C₄

—

172

—

109

C₄⋊C₄

—

172

—

110

C₄²

C₂×D₁₇₂

yes

characteristic

111

C₁₆

—

112

C₁₆

#10621873

yes

—

113

Q₈

—

344

—

114

Q₈

—

344

—

115

Q₈

—

344

—

116

C₂×C₄

—

344

—

117

C₂×C₄

—

344

—

118

C₂×C₄

#13063322

yes

characteristic

119

C₂×C₄

C₂×D₃₄₄

yes

characteristic

120

C₂×C₄

C₂×D₃₄₄

yes

characteristic

121

C₈

—

122

C₈

Dic₃₄₄

yes

complemented

123

C₈

C₂×#10621873

yes

—

124

C₈

#13063347

yes

—

125

C₂²

#14583276

yes

characteristic

126

C₄

—

688

—

127

C₄

—

688

—

128

C₄

#14583272

yes

characteristic

129

C₄

#14583283

yes

characteristic

130

C₄

C₂×Dic₃₄₄

yes

—

131

C₄

#14583331

yes

—

132

C₂

#19134187

yes

characteristic

133

C₂

#19136848

yes

characteristic

134

C₂

#19136856

yes

characteristic

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₈×C₁₆×C₄₃	C₂
socle, prime radical	C₂²×C₄₃	#493
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#12504
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#14583276

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₈×C₁₆×C₄₃

C₂

socle, prime radical

C₂²×C₄₃

#493

hypercenter

C₈×C₁₆

D₄₃

hypocenter

C₄₃

#12504

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
54. C₈×C₄₃	Q₃₂	semidirect
83. C₄₃	#12504	semidirect
122. C₈	Dic₃₄₄	semidirect

H ≅ G/N

type

54. C₈×C₄₃

Q₃₂

semidirect

83. C₄₃

#12504

semidirect

122. C₈

Dic₃₄₄

semidirect

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

subgroup	index
#19136870	2
#19136848	2
#19136823	2
#19134192	2
C₈×C₁₆×C₄₃	2
#12504	43

subgroup

index

C₈×C₁₆×C₄₃

Subgroups — #25011255

Conjugacy classes of subgroups

Complements

Maximal subgroups