Group #25011326 — subgroups

Subgroups — #25011326

Order 11008.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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	#19136852	C₂	1	2	yes	maximal, complemented
2	#19136844	C₂	1	2	yes	maximal, complemented
3	#19134210	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132512	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063268	—	2	4	no	—
7	C₂×#13063266	—	2	4	no	—
8	#14583276	—	2	2	no	—
9	#14583276	C₂²	1	2	yes	—
10	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
11	C₄₃×#494	C₂²	1	1	yes	characteristic
12	C₄₃×#493	C₂²	1	1	yes	characteristic
13	C₄₃×#486	—	2	2	no	—
14	C₈²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×D₃₄₄	—	2	2	no	—
16	C₂×D₃₄₄	—	2	2	no	—
17	C₂×D₃₄₄	—	4	4	no	—
18	#13063322	—	2	2	no	—
19	#13063268	—	4	8	no	—
20	#13063266	—	4	8	no	—
21	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	—
22	C₂×Q₁₆×C₄₃	—	2	2	no	—
23	C₂×D₈×C₄₃	—	2	2	no	—
24	C₄⋊Q₈×C₄₃	—	2	1	no	—
25	#127×C₄₃	—	2	1	no	—
26	#102×C₄₃	—	4	4	no	—
27	C₄×C₈×C₄₃	D₄	1	2	yes	—
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₃	—	4	1	no	—
30	C₂×D₄×C₄₃	—	4	1	no	—
31	C₂×D₄×C₄₃	—	4	2	no	—
32	C₂×D₁₇₂	—	4	2	no	—
33	C₂×D₁₇₂	—	4	4	no	—
34	Q₁₆×C₄₃	—	4	4	no	—
35	D₈×C₄₃	—	4	4	no	—
36	C₂×C₈×C₄₃	D₈	1	4	yes	—
37	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₄₃	—	4	2	no	—
39	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
40	D₃₄₄	—	4	4	no	—
41	D₃₄₄	—	4	4	no	—
42	D₃₄₄	—	8	8	no	—
43	C₄₃⋊C₁₆	D₈	1	8	yes	complemented
44	C₂³×C₄₃	—	8	1	no	—
45	C₂²×D₄₃	—	8	2	no	—
46	Q₈×C₄₃	—	8	2	no	—
47	D₄×C₄₃	—	8	2	no	—
48	D₄×C₄₃	—	8	4	no	—
49	C₂×C₄×C₄₃	—	8	1	no	—
50	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
52	D₁₇₂	—	8	2	no	—
53	D₁₇₂	—	8	2	no	—
54	D₁₇₂	—	8	8	no	—
55	C₈×C₄₃	—	2	4	no	—
56	C₈×C₄₃	C₂×D₈	1	4	yes	—
57	#12519	—	43	1	no	maximal, Sylow
58	C₂²×C₄₃	—	16	1	no	—
59	C₂²×C₄₃	—	16	2	no	—
60	C₂²×C₄₃	#493	1	1	yes	characteristic
61	C₂×D₄₃	—	16	2	no	—
62	C₂×D₄₃	—	16	4	no	—
63	C₄×C₄₃	—	16	1	no	—
64	C₄×C₄₃	#493	1	2	yes	—
65	C₄×C₄₃	#509	1	4	yes	—
66	#2248	—	43	2	no	—
67	#2246	—	43	2	no	—
68	#1712	—	43	1	no	—
69	#1709	D₄₃	1	1	yes	characteristic, complemented
70	#1311	—	43	1	no	—
71	C₂×C₄₃	—	32	1	no	—
72	C₂×C₄₃	#1712	1	1	yes	characteristic
73	C₂×C₄₃	#2250	1	2	yes	—
74	D₄₃	—	32	2	no	—
75	C₂×SD₃₂	—	86	4	no	—
76	C₂×D₁₆	—	86	4	no	—
77	#494	C₂×D₄₃	1	1	yes	characteristic
78	#493	—	43	2	no	—
79	#493	—	86	2	no	—
80	#493	C₂×D₄₃	1	1	yes	characteristic
81	#486	—	2	2	no	—
82	C₄×C₁₆	—	43	2	no	—
83	C₈²	C₂×D₄₃	1	1	yes	characteristic
84	C₄₃	#12519	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₂×D₈	—	86	2	no	—
88	C₂×D₈	—	86	2	no	—
89	C₂×D₈	—	172	4	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	#127	—	2	1	no	—
92	#127	—	86	2	no	—
93	SD₃₂	—	172	8	no	—
94	D₁₆	—	172	8	no	—
95	C₂×C₁₆	—	43	4	no	—
96	#102	—	4	4	no	—
97	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
98	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
99	C₂×Q₈	—	4	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	172	2	no	—
103	C₂×D₄	—	172	4	no	—
104	Q₁₆	—	4	4	no	—
105	D₈	—	4	4	no	—
106	D₈	—	344	8	no	—
107	D₈	—	172	4	no	—
108	D₈	—	172	4	no	—
109	C₂×C₈	#10621882	1	4	yes	—
110	C₂×C₈	D₄×D₄₃	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
113	C₁₆	—	43	8	no	—
114	C₂³	—	8	1	no	—
115	C₂³	—	344	2	no	—
116	Q₈	—	8	2	no	—
117	D₄	—	8	4	no	—
118	D₄	—	8	2	no	—
119	D₄	—	344	8	no	—
120	D₄	—	344	2	no	—
121	D₄	—	344	2	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	#13063321	1	1	yes	characteristic
124	C₂×C₄	C₂×#10621882	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	D₈×D₄₃	1	4	yes	—
127	C₂²	—	16	1	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	688	2	no	—
130	C₂²	—	688	4	no	—
131	C₂²	#14583275	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	#14583274	1	2	yes	—
134	C₄	#14583334	1	4	yes	—
135	C₂	—	32	1	no	—
136	C₂	—	1376	2	no	—
137	C₂	#19134209	1	1	yes	characteristic
138	C₂	#19136862	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136852

C₂

yes

maximal, complemented

#19136844

C₂

yes

maximal, complemented

#19134210

C₂

yes

maximal, characteristic, complemented

#19132512

C₂

yes

maximal, characteristic, complemented

C₄₃×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#494

C₂²

yes

characteristic

C₄₃×#493

C₂²

yes

characteristic

C₄₃×#486

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×D₃₄₄

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

#13063322

—

#13063268

—

#13063266

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

C₂×Q₁₆×C₄₃

—

C₂×D₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

#127×C₄₃

—

#102×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

Q₁₆×C₄₃

—

D₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

C₄₃⋊C₁₆

D₈

yes

complemented

C₂³×C₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₈

yes

—

#12519

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#493

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#493

yes

—

C₄×C₄₃

#509

yes

—

—

—

—

D₄₃

yes

characteristic, complemented

#1311

—

C₂×C₄₃

—

C₂×C₄₃

#1712

yes

characteristic

C₂×C₄₃

#2250

yes

—

D₄₃

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₄₃

yes

—

—

C₂×D₄₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12519

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

172

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

172

—

D₁₆

—

172

—

C₂×C₁₆

—

#102

—

C₄×C₈

C₂²⋊D₄₃

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

172

—

103

C₂×D₄

—

172

—

104

Q₁₆

—

105

D₈

—

106

D₈

—

344

—

107

D₈

—

172

—

108

D₈

—

172

—

109

C₂×C₈

#10621882

yes

—

110

C₂×C₈

D₄×D₄₃

yes

—

111

C₄⋊C₄

—

112

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

113

C₁₆

—

114

C₂³

—

115

C₂³

—

344

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

344

—

120

D₄

—

344

—

121

D₄

—

344

—

122

C₂×C₄

—

123

C₂×C₄

#13063321

yes

characteristic

124

C₂×C₄

C₂×#10621882

yes

—

125

C₈

—

126

C₈

D₈×D₄₃

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

688

—

130

C₂²

—

688

—

131

C₂²

#14583275

yes

characteristic

132

C₄

—

133

C₄

#14583274

yes

—

134

C₄

#14583334

yes

—

135

C₂

—

136

C₂

—

1376

—

137

C₂

#19134209

yes

characteristic

138

C₂

#19136862

yes

—

	subgroup	G/H
center	C₂²	#14583275
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1709	C₂
socle	C₂²×C₄₃	#493
hypercenter	#1709	D₄₃
hypocenter	C₄₃	#12519
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#14583275

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1709

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #19136852	C₂	semidirect
2. #19136844	C₂	semidirect
3. #19134210	C₂	semidirect
4. #19132512	C₂	semidirect
5. C₄₃×#1709	C₂	semidirect
43. C₄₃⋊C₁₆	D₈	semidirect
69. #1709	D₄₃	semidirect
84. C₄₃	#12519	semidirect

H ≅ G/N

type

1. #19136852

C₂

semidirect

2. #19136844

C₂

semidirect

3. #19134210

C₂

semidirect

4. #19132512

C₂

semidirect

5. C₄₃×#1709

C₂

semidirect

43. C₄₃⋊C₁₆

D₈

semidirect

69. #1709

D₄₃

semidirect

84. C₄₃

#12519

semidirect

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

subgroup	index
#19136852	2
#19136844	2
#19134210	2
#19132512	2
C₄₃×#1709	2
#12519	43

subgroup

index

C₄₃×#1709

Subgroups — #25011326

Conjugacy classes of subgroups

Complements

Maximal subgroups