Group #25011267 — subgroups

Subgroups — #25011267

Order 11008.

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

normal subgroups only

This group has 302 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	#19136877	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136845	C₂	1	1	yes	maximal, characteristic
3	#19136819	C₂	1	2	yes	maximal, complemented
4	#19134187	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₈₈	—	2	2	no	—
7	#14583304	C₂²	1	2	yes	—
8	#14583276	C₂²	1	2	yes	—
9	#14583272	—	2	2	no	—
10	#14583246	—	2	2	no	—
11	#14582727	—	2	2	no	—
12	#14582701	—	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×#10621873	—	4	4	no	—
17	C₂×D₃₄₄	—	2	2	no	—
18	C₂×D₃₄₄	—	2	2	no	—
19	#13063325	—	2	2	no	—
20	#13063322	—	2	2	no	—
21	D₆₈₈	—	4	4	no	—
22	#13063259	—	2	2	no	—
23	#13063259	—	2	2	no	—
24	#13063247	—	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	2	yes	—
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×D₁₇₂	—	4	2	no	—
30	C₂×D₁₇₂	—	4	2	no	—
31	C₂×D₁₇₂	—	4	2	no	—
32	C₂×Dic₈₆	—	4	2	no	—
33	C₂×C₈×C₄₃	D₈	1	4	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	D₃₄₄	—	4	4	no	—
40	D₃₄₄	—	4	4	no	—
41	#10621873	—	8	8	no	—
42	C₁₆×C₄₃	—	2	2	no	—
43	C₁₆×C₄₃	SD₁₆	1	4	yes	complemented
44	C₂²×D₄₃	—	8	2	no	—
45	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
47	C₂×Dic₄₃	—	8	2	no	—
48	D₁₇₂	—	8	2	no	—
49	D₁₇₂	—	8	2	no	—
50	D₁₇₂	—	8	4	no	—
51	D₁₇₂	—	8	4	no	—
52	Dic₈₆	—	8	4	no	—
53	C₈×C₄₃	—	2	4	no	—
54	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
55	C₈×C₄₃	#135	1	2	yes	—
56	#12506	—	43	1	no	maximal, Sylow
57	C₂²×C₄₃	#493	1	1	yes	characteristic
58	C₂×D₄₃	—	16	2	no	—
59	C₂×D₄₃	—	16	2	no	—
60	C₂×D₄₃	—	16	2	no	—
61	C₄×C₄₃	#492	1	1	yes	characteristic
62	C₄×C₄₃	#495	1	1	yes	characteristic
63	C₄×C₄₃	#509	1	4	yes	—
64	Dic₄₃	—	16	2	no	—
65	#2254	—	43	1	no	—
66	#2246	—	43	1	no	—
67	#2240	—	43	2	no	—
68	#1706	—	43	2	no	—
69	#1311	D₄₃	1	1	yes	characteristic, complemented
70	C₂×C₄₃	#1706	1	1	yes	characteristic
71	C₂×C₄₃	#2250	1	2	yes	—
72	D₄₃	—	32	2	no	—
73	C₂×D₁₆	—	86	2	no	—
74	#500	—	43	2	no	—
75	#493	—	43	2	no	—
76	#492	—	86	2	no	—
77	#486	—	86	2	no	—
78	#367	—	86	2	no	—
79	#357	—	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	#12506	1	1	yes	characteristic, complemented, Sylow
84	C₂×SD₁₆	—	172	4	no	—
85	C₂×D₈	—	86	2	no	—
86	C₂×D₈	—	86	2	no	—
87	C₄⋊Q₈	—	86	2	no	—
88	#127	—	86	2	no	—
89	D₁₆	—	172	4	no	—
90	C₂×C₁₆	D₁₇₂	1	2	yes	—
91	C₂×C₁₆	D₁₇₂	1	2	yes	—
92	#107	—	86	2	no	—
93	#107	—	86	2	no	—
94	#102	—	172	4	no	—
95	C₄×C₈	D₁₇₂	1	2	yes	—
96	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
97	C₂×Q₈	—	172	2	no	—
98	C₂×D₄	—	172	2	no	—
99	C₂×D₄	—	172	2	no	—
100	C₂×D₄	—	172	2	no	—
101	SD₁₆	—	344	8	no	—
102	D₈	—	172	4	no	—
103	D₈	—	172	4	no	—
104	C₂×C₈	D₃₄₄	1	4	yes	—
105	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
106	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
107	C₄⋊C₄	—	172	2	no	—
108	C₄⋊C₄	—	172	2	no	—
109	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	#10621873	1	4	yes	complemented
112	C₂³	—	344	2	no	—
113	Q₈	—	344	4	no	—
114	D₄	—	344	4	no	—
115	D₄	—	344	4	no	—
116	D₄	—	344	2	no	—
117	D₄	—	344	2	no	—
118	C₂×C₄	—	344	2	no	—
119	C₂×C₄	#13063322	1	1	yes	characteristic
120	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	C₂×#10621873	1	2	yes	—
123	C₈	#13063347	1	2	yes	—
124	C₂²	—	688	2	no	—
125	C₂²	—	688	2	no	—
126	C₂²	—	688	2	no	—
127	C₂²	#14583276	1	1	yes	characteristic
128	C₄	—	688	2	no	—
129	C₄	#14583272	1	1	yes	characteristic
130	C₄	#14583283	1	1	yes	characteristic
131	C₄	#14583337	1	4	yes	—
132	C₂	—	1376	2	no	—
133	C₂	#19134187	1	1	yes	characteristic
134	C₂	#19136861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136877

C₂

yes

maximal, characteristic, complemented

#19136845

C₂

yes

maximal, characteristic

#19136819

C₂

yes

maximal, complemented

#19134187

C₂

yes

maximal, complemented

C₄₃×#1311

C₂

yes

maximal, characteristic, complemented

C₂×D₆₈₈

—

#14583304

C₂²

yes

—

#14583276

C₂²

yes

—

—

—

—

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

C₂²

yes

characteristic

C₂×#10621873

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

#13063325

—

#13063322

—

D₆₈₈

—

#13063259

—

#13063259

—

#13063247

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

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

—

D₃₄₄

—

D₃₄₄

—

#10621873

—

C₁₆×C₄₃

—

C₁₆×C₄₃

SD₁₆

yes

complemented

C₂²×D₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×SD₁₆

yes

—

C₈×C₄₃

#135

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#1706

yes

characteristic

C₂×C₄₃

#2250

yes

—

D₄₃

—

C₂×D₁₆

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12506

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

172

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

172

—

C₂×C₁₆

D₁₇₂

yes

—

C₂×C₁₆

D₁₇₂

yes

—

#107

—

#107

—

#102

—

172

—

C₄×C₈

D₁₇₂

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

100

C₂×D₄

—

172

—

101

SD₁₆

—

344

—

102

D₈

—

172

—

103

D₈

—

172

—

104

C₂×C₈

D₃₄₄

yes

—

105

C₂×C₈

C₂×D₁₇₂

yes

characteristic

106

C₂×C₈

C₂×D₁₇₂

yes

characteristic

107

C₄⋊C₄

—

172

—

108

C₄⋊C₄

—

172

—

109

C₄²

C₂×D₁₇₂

yes

characteristic

110

C₁₆

—

111

C₁₆

#10621873

yes

complemented

112

C₂³

—

344

—

113

Q₈

—

344

—

114

D₄

—

344

—

115

D₄

—

344

—

116

D₄

—

344

—

117

D₄

—

344

—

118

C₂×C₄

—

344

—

119

C₂×C₄

#13063322

yes

characteristic

120

C₂×C₄

C₂×D₃₄₄

yes

—

121

C₈

—

122

C₈

C₂×#10621873

yes

—

123

C₈

#13063347

yes

—

124

C₂²

—

688

—

125

C₂²

—

688

—

126

C₂²

—

688

—

127

C₂²

#14583276

yes

characteristic

128

C₄

—

688

—

129

C₄

#14583272

yes

characteristic

130

C₄

#14583283

yes

characteristic

131

C₄

#14583337

yes

—

132

C₂

—

1376

—

133

C₂

#19134187

yes

characteristic

134

C₂

#19136861

yes

—

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1311	C₂
socle	C₂²×C₄₃	#493
prime radical	#19136845	C₂
hypercenter	#1311	D₄₃
hypocenter	C₄₃	#12506
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#14583276

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1311

C₂

C₂²×C₄₃

C₂

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #19136877	C₂	semidirect
3. #19136819	C₂	semidirect
4. #19134187	C₂	semidirect
5. C₄₃×#1311	C₂	semidirect
43. C₁₆×C₄₃	SD₁₆	semidirect
69. #1311	D₄₃	semidirect
83. C₄₃	#12506	semidirect
111. C₁₆	#10621873	semidirect

H ≅ G/N

type

1. #19136877

C₂

semidirect

3. #19136819

C₂

semidirect

4. #19134187

C₂

semidirect

5. C₄₃×#1311

C₂

semidirect

43. C₁₆×C₄₃

SD₁₆

semidirect

69. #1311

D₄₃

semidirect

83. C₄₃

#12506

semidirect

111. C₁₆

#10621873

semidirect

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

subgroup	index
#19136877	2
#19136845	2
#19136819	2
#19134187	2
C₄₃×#1311	2
#12506	43

subgroup

index

C₄₃×#1311

Subgroups — #25011267

Conjugacy classes of subgroups

Complements

Maximal subgroups