Group #25011291 — subgroups

Subgroups — #25011291

Order 11008.

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

normal subgroups only

This group has 260 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	#19136823	C₂	1	2	yes	maximal, complemented
2	#19136819	C₂	1	2	yes	maximal
3	#19134205	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134201	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583304	C₂²	1	2	yes	—
7	#14583279	C₂²	1	1	yes	characteristic
8	#14583276	C₂²	1	1	yes	characteristic
9	#14583250	—	2	2	no	—
10	#14583246	—	2	2	no	—
11	#14582704	—	2	4	no	—
12	#14582701	—	2	4	no	—
13	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
14	C₈²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₇₂	—	2	2	no	—
16	C₂×D₃₄₄	—	2	2	no	—
17	#13063325	—	2	2	no	—
18	#13063325	—	2	1	no	—
19	#13063322	—	2	1	no	—
20	#13063259	—	2	2	no	—
21	#13063259	—	2	2	no	—
22	#13063250	—	4	4	no	—
23	#13063247	—	4	4	no	—
24	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
25	C₄×C₈×C₄₃	D₄	1	2	yes	—
26	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
27	C₂×D₁₇₂	—	4	1	no	—
28	C₂×D₁₇₂	—	4	2	no	—
29	C₂×Dic₈₆	—	4	1	no	—
30	C₂×Dic₈₆	—	4	2	no	—
31	C₂×C₈×C₄₃	D₈	1	4	yes	—
32	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
33	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₄₃	—	4	2	no	—
35	C₄⋊Dic₄₃	—	4	2	no	—
36	C₄⋊Dic₄₃	—	4	2	no	—
37	Dic₁₇₂	—	4	4	no	—
38	D₃₄₄	—	4	4	no	—
39	C₁₆×C₄₃	—	2	4	no	—
40	C₂²×D₄₃	—	8	1	no	—
41	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
42	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
43	C₂×Dic₄₃	—	8	1	no	—
44	C₂×Dic₄₃	—	8	2	no	—
45	D₁₇₂	—	8	2	no	—
46	D₁₇₂	—	8	4	no	—
47	Dic₈₆	—	8	2	no	—
48	Dic₈₆	—	8	4	no	—
49	C₈×C₄₃	—	2	4	no	—
50	C₈×C₄₃	#135	1	4	yes	—
51	#12511	—	43	1	no	maximal, Sylow
52	C₂²×C₄₃	#493	1	1	yes	characteristic
53	C₂×D₄₃	—	16	1	no	—
54	C₂×D₄₃	—	16	2	no	—
55	C₄×C₄₃	#495	1	2	yes	—
56	C₄×C₄₃	#508	1	4	yes	—
57	Dic₄₃	—	16	1	no	—
58	Dic₄₃	—	16	2	no	—
59	#2241	—	43	2	no	—
60	#2240	—	43	2	no	—
61	#1710	—	43	1	no	—
62	#1709	—	43	1	no	—
63	C₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
64	C₂×C₄₃	#1709	1	1	yes	characteristic
65	C₂×C₄₃	#2249	1	2	yes	—
66	D₄₃	—	32	1	no	—
67	#500	—	43	2	no	—
68	#494	—	43	1	no	—
69	#493	—	43	1	no	—
70	#487	—	86	2	no	—
71	#486	—	86	2	no	—
72	#358	—	86	4	no	—
73	#357	—	86	4	no	—
74	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
75	C₈²	C₂×D₄₃	1	1	yes	characteristic
76	C₄₃	#12511	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	86	2	no	—
78	C₂×D₈	—	86	2	no	—
79	C₄⋊Q₈	—	86	1	no	—
80	C₄⋊Q₈	—	86	2	no	—
81	#127	—	86	1	no	—
82	C₂×C₁₆	D₁₇₂	1	4	yes	—
83	#107	—	86	2	no	—
84	#107	—	86	2	no	—
85	#103	—	172	4	no	—
86	#102	—	172	4	no	—
87	C₄×C₈	D₁₇₂	1	2	yes	—
88	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
89	C₂×Q₈	—	172	1	no	—
90	C₂×Q₈	—	172	2	no	—
91	C₂×D₄	—	172	1	no	—
92	C₂×D₄	—	172	2	no	—
93	Q₁₆	—	172	4	no	—
94	D₈	—	172	4	no	—
95	C₂×C₈	D₃₄₄	1	4	yes	—
96	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
97	C₄⋊C₄	—	172	2	no	—
98	C₄⋊C₄	—	172	2	no	—
99	C₄⋊C₄	—	172	2	no	—
100	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₂³	—	344	1	no	—
103	Q₈	—	344	4	no	—
104	Q₈	—	344	2	no	—
105	D₄	—	344	4	no	—
106	D₄	—	344	2	no	—
107	C₂×C₄	—	344	2	no	—
108	C₂×C₄	—	344	1	no	—
109	C₂×C₄	#13063322	1	1	yes	characteristic
110	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#13063347	1	4	yes	—
113	C₂²	—	688	2	no	—
114	C₂²	—	688	1	no	—
115	C₂²	#14583276	1	1	yes	characteristic
116	C₄	—	688	2	no	—
117	C₄	—	688	1	no	—
118	C₄	#14583283	1	2	yes	—
119	C₄	#14583331	1	4	yes	—
120	C₂	—	1376	1	no	—
121	C₂	#19134201	1	1	yes	characteristic
122	C₂	#19136856	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136823

C₂

yes

maximal, complemented

#19136819

C₂

yes

maximal

#19134205

C₂

yes

maximal, characteristic, complemented

#19134201

C₂

yes

maximal, characteristic

C₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

#14583304

C₂²

yes

—

#14583279

C₂²

yes

characteristic

#14583276

C₂²

yes

—

—

—

—

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×D₃₄₄

—

—

—

—

—

—

—

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

D₃₄₄

—

C₁₆×C₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

#135

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#495

yes

—

C₄×C₄₃

#508

yes

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

C₈×C₁₆

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#1709

yes

characteristic

C₂×C₄₃

#2249

yes

—

D₄₃

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12511

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₁₇₂

yes

—

#107

—

#107

—

#103

—

172

—

#102

—

172

—

C₄×C₈

D₁₇₂

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

C₂×Q₈

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

Q₁₆

—

172

—

D₈

—

172

—

C₂×C₈

D₃₄₄

yes

—

C₂×C₈

C₂×D₁₇₂

yes

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

100

C₄²

C₂×D₁₇₂

yes

characteristic

101

C₁₆

—

102

C₂³

—

344

—

103

Q₈

—

344

—

104

Q₈

—

344

—

105

D₄

—

344

—

106

D₄

—

344

—

107

C₂×C₄

—

344

—

108

C₂×C₄

—

344

—

109

C₂×C₄

#13063322

yes

characteristic

110

C₂×C₄

C₂×D₃₄₄

yes

—

111

C₈

—

112

C₈

#13063347

yes

—

113

C₂²

—

688

—

114

C₂²

—

688

—

115

C₂²

#14583276

yes

characteristic

116

C₄

—

688

—

117

C₄

—

688

—

118

C₄

#14583283

yes

—

119

C₄

#14583331

yes

—

120

C₂

—

1376

—

121

C₂

#19134201

yes

characteristic

122

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	#14583276	C₂²
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#12511
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

C₂²

C₈×C₁₆

D₄₃

hypocenter

C₄₃

#12511

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #19136823	C₂	semidirect
3. #19134205	C₂	semidirect
5. C₈×C₁₆×C₄₃	C₂	semidirect
63. C₈×C₁₆	D₄₃	semidirect
76. C₄₃	#12511	semidirect

H ≅ G/N

type

1. #19136823

C₂

semidirect

3. #19134205

C₂

semidirect

5. C₈×C₁₆×C₄₃

C₂

semidirect

63. C₈×C₁₆

D₄₃

semidirect

76. C₄₃

#12511

semidirect

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

subgroup	index
#19136823	2
#19136819	2
#19134205	2
#19134201	2
C₈×C₁₆×C₄₃	2
#12511	43

subgroup

index

C₈×C₁₆×C₄₃

Subgroups — #25011291

Conjugacy classes of subgroups

Complements

Maximal subgroups