Group #25011320 — subgroups

Subgroups — #25011320

Order 11008.

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

normal subgroups only

This group has 330 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	#19136851	C₂	1	2	yes	maximal, complemented
2	#19136848	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136845	C₂	1	1	yes	maximal, characteristic
4	#19134201	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₄₄	—	2	2	no	—
7	C₂×#13063270	—	2	4	no	—
8	C₂×D₆₈₈	—	2	2	no	—
9	#14583279	C₂²	1	2	yes	—
10	#14583276	C₂²	1	2	yes	—
11	#14583246	—	2	4	no	—
12	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₈²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₇₂	—	2	2	no	—
16	C₂×Dic₁₇₂	—	2	2	no	—
17	C₂×D₃₄₄	—	2	2	no	—
18	C₂×D₃₄₄	—	2	2	no	—
19	#13063325	—	2	2	no	—
20	#13063322	—	2	2	no	—
21	Dic₃₄₄	—	4	4	no	—
22	#13063270	—	4	8	no	—
23	D₆₈₈	—	4	4	no	—
24	#13063247	—	4	8	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	4	no	—
31	C₂×Dic₈₆	—	4	2	no	—
32	C₂×C₈×C₄₃	D₈	1	4	yes	—
33	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₄₃	—	4	4	no	—
37	Dic₁₇₂	—	4	4	no	—
38	Dic₁₇₂	—	4	4	no	—
39	D₃₄₄	—	4	4	no	—
40	D₃₄₄	—	4	4	no	—
41	C₁₆×C₄₃	D₈	1	4	yes	—
42	C₁₆×C₄₃	D₈	1	4	yes	—
43	C₂²×D₄₃	—	8	2	no	—
44	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
46	C₂×Dic₄₃	—	8	2	no	—
47	D₁₇₂	—	8	2	no	—
48	D₁₇₂	—	8	2	no	—
49	D₁₇₂	—	8	8	no	—
50	Dic₈₆	—	8	2	no	—
51	Dic₈₆	—	8	2	no	—
52	C₈×C₄₃	—	2	4	no	—
53	C₈×C₄₃	C₂×D₈	1	2	yes	—
54	C₈×C₄₃	C₂×D₈	1	2	yes	—
55	#12518	—	43	1	no	maximal, Sylow
56	C₂²×C₄₃	#493	1	1	yes	characteristic
57	C₂×D₄₃	—	16	2	no	—
58	C₂×D₄₃	—	16	4	no	—
59	C₄×C₄₃	#493	1	1	yes	characteristic
60	C₄×C₄₃	#493	1	1	yes	characteristic
61	C₄×C₄₃	#508	1	4	yes	—
62	Dic₄₃	—	16	2	no	—
63	#2248	—	43	2	no	—
64	#2247	—	43	1	no	—
65	#2246	—	43	1	no	—
66	#1709	—	43	2	no	—
67	C₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
68	C₂×C₄₃	#1712	1	1	yes	characteristic
69	C₂×C₄₃	#2249	1	2	yes	—
70	D₄₃	—	32	2	no	—
71	C₂×Q₃₂	—	86	2	no	—
72	C₂×SD₃₂	—	86	4	no	—
73	C₂×D₁₆	—	86	2	no	—
74	#494	—	43	2	no	—
75	#493	—	43	2	no	—
76	#486	—	86	4	no	—
77	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
78	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
79	C₈²	C₂×D₄₃	1	1	yes	characteristic
80	C₄₃	#12518	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	86	2	no	—
82	C₂×Q₁₆	—	86	2	no	—
83	C₂×D₈	—	86	2	no	—
84	C₂×D₈	—	86	2	no	—
85	C₄⋊Q₈	—	86	2	no	—
86	#127	—	86	2	no	—
87	Q₃₂	—	172	4	no	—
88	SD₃₂	—	172	8	no	—
89	D₁₆	—	172	4	no	—
90	C₂×C₁₆	D₁₇₂	1	2	yes	—
91	C₂×C₁₆	D₁₇₂	1	2	yes	—
92	#102	—	172	8	no	—
93	C₄×C₈	D₁₇₂	1	2	yes	—
94	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
95	C₂×Q₈	—	172	2	no	—
96	C₂×D₄	—	172	2	no	—
97	C₂×D₄	—	172	4	no	—
98	Q₁₆	—	172	4	no	—
99	Q₁₆	—	172	4	no	—
100	D₈	—	172	4	no	—
101	D₈	—	172	4	no	—
102	C₂×C₈	D₃₄₄	1	4	yes	—
103	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
104	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
105	C₄⋊C₄	—	172	4	no	—
106	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
107	C₁₆	D₃₄₄	1	4	yes	—
108	C₁₆	D₃₄₄	1	4	yes	—
109	C₂³	—	344	2	no	—
110	Q₈	—	344	2	no	—
111	Q₈	—	344	2	no	—
112	D₄	—	344	8	no	—
113	D₄	—	344	2	no	—
114	D₄	—	344	2	no	—
115	C₂×C₄	—	344	2	no	—
116	C₂×C₄	#13063322	1	1	yes	characteristic
117	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
118	C₈	—	2	4	no	—
119	C₈	C₂×D₃₄₄	1	2	yes	—
120	C₈	C₂×D₃₄₄	1	2	yes	—
121	C₂²	—	688	2	no	—
122	C₂²	—	688	4	no	—
123	C₂²	#14583276	1	1	yes	characteristic
124	C₄	—	688	2	no	—
125	C₄	#14583276	1	1	yes	characteristic
126	C₄	#14583276	1	1	yes	characteristic
127	C₄	#14583331	1	4	yes	—
128	C₂	—	1376	2	no	—
129	C₂	#19134210	1	1	yes	characteristic
130	C₂	#19136856	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136851

C₂

yes

maximal, complemented

#19136848

C₂

yes

maximal, characteristic, complemented

#19136845

C₂

yes

maximal, characteristic

#19134201

C₂

yes

maximal, complemented

C₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₄₄

—

C₂×#13063270

—

C₂×D₆₈₈

—

#14583279

C₂²

yes

—

#14583276

C₂²

yes

—

#14583246

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

#13063325

—

#13063322

—

Dic₃₄₄

—

#13063270

—

D₆₈₈

—

#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×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

—

Dic₁₇₂

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

—

C₁₆×C₄₃

D₈

yes

—

C₁₆×C₄₃

D₈

yes

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₈

yes

—

C₈×C₄₃

C₂×D₈

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

—

—

—

—

C₈×C₁₆

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#1712

yes

characteristic

C₂×C₄₃

#2249

yes

—

D₄₃

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

—

#493

—

#486

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12518

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

172

—

SD₃₂

—

172

—

D₁₆

—

172

—

C₂×C₁₆

D₁₇₂

yes

—

C₂×C₁₆

D₁₇₂

yes

—

#102

—

172

—

C₄×C₈

D₁₇₂

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

Q₁₆

—

172

—

Q₁₆

—

172

—

100

D₈

—

172

—

101

D₈

—

172

—

102

C₂×C₈

D₃₄₄

yes

—

103

C₂×C₈

C₂×D₁₇₂

yes

characteristic

104

C₂×C₈

C₂×D₁₇₂

yes

characteristic

105

C₄⋊C₄

—

172

—

106

C₄²

C₂×D₁₇₂

yes

characteristic

107

C₁₆

D₃₄₄

yes

—

108

C₁₆

D₃₄₄

yes

—

109

C₂³

—

344

—

110

Q₈

—

344

—

111

Q₈

—

344

—

112

D₄

—

344

—

113

D₄

—

344

—

114

D₄

—

344

—

115

C₂×C₄

—

344

—

116

C₂×C₄

#13063322

yes

characteristic

117

C₂×C₄

C₂×D₃₄₄

yes

—

118

C₈

—

119

C₈

C₂×D₃₄₄

yes

—

120

C₈

C₂×D₃₄₄

yes

—

121

C₂²

—

688

—

122

C₂²

—

688

—

123

C₂²

#14583276

yes

characteristic

124

C₄

—

688

—

125

C₄

#14583276

yes

characteristic

126

C₄

#14583276

yes

characteristic

127

C₄

#14583331

yes

—

128

C₂

—

1376

—

129

C₂

#19134210

yes

characteristic

130

C₂

#19136856

yes

—

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₈×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	#493
prime radical	#19136845	C₂
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#12518
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

C₂²×C₄₃

C₂

C₈×C₁₆

D₄₃

hypocenter

C₄₃

#12518

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #19136851	C₂	semidirect
2. #19136848	C₂	semidirect
4. #19134201	C₂	semidirect
5. C₈×C₁₆×C₄₃	C₂	semidirect
67. C₈×C₁₆	D₄₃	semidirect
80. C₄₃	#12518	semidirect

H ≅ G/N

type

1. #19136851

C₂

semidirect

2. #19136848

C₂

semidirect

4. #19134201

C₂

semidirect

5. C₈×C₁₆×C₄₃

C₂

semidirect

67. C₈×C₁₆

D₄₃

semidirect

80. C₄₃

#12518

semidirect

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

subgroup	index
#19136851	2
#19136848	2
#19136845	2
#19134201	2
C₈×C₁₆×C₄₃	2
#12518	43

subgroup

index

C₈×C₁₆×C₄₃

Subgroups — #25011320

Conjugacy classes of subgroups

Complements

Maximal subgroups