Group #25011264 — subgroups

Subgroups — #25011264

Order 11008.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#19136818	C₂	1	2	yes	maximal, complemented
3	#19134184	C₂	1	2	yes	maximal, complemented
4	#19132511	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#2246	C₂	1	1	yes	maximal, characteristic
6	#14583304	C₂²	1	2	yes	—
7	#14583271	—	2	2	no	—
8	#14583245	—	2	2	no	—
9	#14582727	—	2	2	no	—
10	#14582700	—	2	4	no	—
11	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₄₃	—	2	2	no	—
14	C₄₃×#493	C₂²	1	2	yes	—
15	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×#10621883	—	4	4	no	—
17	#13063325	—	2	2	no	—
18	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
19	#13063259	—	2	2	no	—
20	#13063259	—	2	2	no	—
21	#13063246	—	4	4	no	—
22	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
23	C₂×D₈×C₄₃	—	2	2	no	—
24	C₂×D₈×C₄₃	—	2	2	no	—
25	#127×C₄₃	—	2	2	no	—
26	D₁₆×C₄₃	—	4	4	no	—
27	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₄₃	—	4	2	no	—
30	C₂×D₄×C₄₃	—	4	2	no	—
31	C₂×D₄×C₄₃	—	4	2	no	—
32	C₂×Dic₈₆	—	4	2	no	—
33	D₈×C₄₃	—	4	4	no	—
34	D₈×C₄₃	—	4	4	no	—
35	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
38	#10621883	—	8	8	no	—
39	C₄⋊Dic₄₃	—	4	2	no	—
40	C₄⋊Dic₄₃	—	4	2	no	—
41	C₂×C₄₃⋊C₈	D₈	1	4	yes	—
42	C₁₆×C₄₃	SD₁₆	1	4	yes	complemented
43	C₄₃⋊C₁₆	—	2	2	no	—
44	C₂³×C₄₃	—	8	2	no	—
45	D₄×C₄₃	—	8	2	no	—
46	D₄×C₄₃	—	8	2	no	—
47	D₄×C₄₃	—	8	4	no	—
48	D₄×C₄₃	—	8	4	no	—
49	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
51	C₂×Dic₄₃	—	8	2	no	—
52	Dic₈₆	—	8	4	no	—
53	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
54	C₈×C₄₃	#135	1	2	yes	—
55	C₄₃⋊C₈	—	2	4	no	—
56	#12506	—	43	1	no	maximal, Sylow
57	C₂²×C₄₃	—	16	2	no	—
58	C₂²×C₄₃	—	16	2	no	—
59	C₂²×C₄₃	—	16	2	no	—
60	C₂²×C₄₃	#493	1	1	yes	characteristic
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	D₄₃	1	1	yes	characteristic
67	#2240	—	43	2	no	—
68	#1706	—	43	2	no	—
69	#1311	—	43	1	no	—
70	C₂×C₄₃	—	32	2	no	—
71	C₂×C₄₃	#1706	1	1	yes	characteristic
72	C₂×C₄₃	#2250	1	2	yes	—
73	C₂×D₁₆	—	2	2	no	—
74	#500	—	43	2	no	—
75	#493	C₂×D₄₃	1	2	yes	—
76	#492	—	86	2	no	—
77	#486	—	86	2	no	—
78	#367	—	86	2	no	—
79	#357	—	86	4	no	—
80	C₄×C₁₆	—	43	1	no	—
81	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
82	C₈²	—	43	1	no	—
83	C₄₃	#12506	1	1	yes	characteristic, complemented, Sylow
84	C₂×SD₁₆	—	172	4	no	—
85	C₂×D₈	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₄⋊Q₈	—	86	2	no	—
88	#127	—	2	2	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	—	43	2	no	—
91	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
92	#107	—	86	2	no	—
93	#107	—	86	2	no	—
94	#102	—	172	4	no	—
95	C₄×C₈	—	43	2	no	—
96	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
97	C₂×Q₈	—	172	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	—	4	2	no	—
101	SD₁₆	—	344	8	no	—
102	D₈	—	4	4	no	—
103	D₈	—	4	4	no	—
104	C₂×C₈	—	43	4	no	—
105	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
107	C₄⋊C₄	—	172	2	no	—
108	C₄⋊C₄	—	172	2	no	—
109	C₄²	D₄×D₄₃	1	1	yes	characteristic
110	C₁₆	—	86	2	no	—
111	C₁₆	#10621883	1	4	yes	complemented
112	C₂³	—	8	2	no	—
113	Q₈	—	344	4	no	—
114	D₄	—	8	4	no	—
115	D₄	—	8	4	no	—
116	D₄	—	8	2	no	—
117	D₄	—	8	2	no	—
118	C₂×C₄	—	344	2	no	—
119	C₂×C₄	#13063321	1	1	yes	characteristic
120	C₂×C₄	D₈×D₄₃	1	2	yes	—
121	C₈	—	86	4	no	—
122	C₈	C₂×#10621883	1	2	yes	—
123	C₈	#13063344	1	2	yes	—
124	C₂²	—	16	2	no	—
125	C₂²	—	16	2	no	—
126	C₂²	—	16	2	no	—
127	C₂²	#14583274	1	1	yes	characteristic
128	C₄	—	688	2	no	—
129	C₄	#14583271	1	1	yes	characteristic
130	C₄	#14583282	1	1	yes	characteristic
131	C₄	#14583333	1	4	yes	—
132	C₂	—	32	2	no	—
133	C₂	#19134184	1	1	yes	characteristic
134	C₂	#19136858	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136877

C₂

yes

maximal, characteristic, complemented

#19136818

C₂

yes

maximal, complemented

#19134184

C₂

yes

maximal, complemented

#19132511

C₂

yes

maximal, characteristic, complemented

C₄₃×#2246

C₂

yes

maximal, characteristic

#14583304

C₂²

yes

—

—

—

—

—

C₄×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×D₁₆×C₄₃

—

C₄₃×#493

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#10621883

—

#13063325

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

#13063259

—

#13063259

—

#13063246

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

#127×C₄₃

—

D₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×Dic₈₆

—

D₈×C₄₃

—

D₈×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

#10621883

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

D₈

yes

—

C₁₆×C₄₃

SD₁₆

yes

complemented

C₄₃⋊C₁₆

—

C₂³×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

C₂×SD₁₆

yes

—

C₈×C₄₃

#135

yes

—

C₄₃⋊C₈

—

#12506

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

#2254

—

#2246

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

#1706

yes

characteristic

C₂×C₄₃

#2250

yes

—

C₂×D₁₆

—

#500

—

#493

C₂×D₄₃

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₈²

—

C₄₃

#12506

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

172

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₃

yes

—

#107

—

#107

—

#102

—

172

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

SD₁₆

—

344

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₄₃

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

107

C₄⋊C₄

—

172

—

108

C₄⋊C₄

—

172

—

109

C₄²

D₄×D₄₃

yes

characteristic

110

C₁₆

—

111

C₁₆

#10621883

yes

complemented

112

C₂³

—

113

Q₈

—

344

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

344

—

119

C₂×C₄

#13063321

yes

characteristic

120

C₂×C₄

D₈×D₄₃

yes

—

121

C₈

—

122

C₈

C₂×#10621883

yes

—

123

C₈

#13063344

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#14583274

yes

characteristic

128

C₄

—

688

—

129

C₄

#14583271

yes

characteristic

130

C₄

#14583282

yes

characteristic

131

C₄

#14583333

yes

—

132

C₂

—

133

C₂

#19134184

yes

characteristic

134

C₂

#19136858

yes

—

	subgroup	G/H
center	C₂²	#14583274
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting, prime radical	C₄₃×#2246	C₂
socle	C₂²×C₄₃	#493
hypercenter	#2246	D₄₃
hypocenter	C₄₃	#12506
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#14583274

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting, prime radical

C₄₃×#2246

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #19136877	C₂	semidirect
2. #19136818	C₂	semidirect
3. #19134184	C₂	semidirect
4. #19132511	C₂	semidirect
42. C₁₆×C₄₃	SD₁₆	semidirect
83. C₄₃	#12506	semidirect
111. C₁₆	#10621883	semidirect

H ≅ G/N

type

1. #19136877

C₂

semidirect

2. #19136818

C₂

semidirect

3. #19134184

C₂

semidirect

4. #19132511

C₂

semidirect

42. C₁₆×C₄₃

SD₁₆

semidirect

83. C₄₃

#12506

semidirect

111. C₁₆

#10621883

semidirect

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

subgroup	index
#19136877	2
#19136818	2
#19134184	2
#19132511	2
C₄₃×#2246	2
#12506	43

subgroup

index

C₄₃×#2246

Subgroups — #25011264

Conjugacy classes of subgroups

Complements

Maximal subgroups