Group #25001647 — subgroups

Subgroups — #25001647

Order 11008.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#19135401	C₂	1	3	yes	maximal
2	#19134755	C₂	1	3	yes	maximal, complemented
3	C₄₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063322	C₂²	1	1	yes	characteristic
5	C₂×#13063259	—	2	6	no	—
6	C₂×#13063247	—	2	6	no	—
7	#14582769	C₂²	1	3	yes	—
8	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
9	C₂²×D₁₇₂	—	2	3	no	—
10	#13063322	—	2	3	no	—
11	#13063322	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₄₃	—	2	3	no	—
13	C₂×C₄⋊Dic₄₃	—	2	6	no	—
14	#13063259	—	4	12	no	—
15	#13063247	—	4	12	no	—
16	C₂²×C₈×C₄₃	D₄	1	6	yes	—
17	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₄₃	—	2	6	no	—
19	C₂³×D₄₃	—	4	1	no	—
20	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
21	C₂²×Dic₄₃	—	4	3	no	—
22	C₂×D₁₇₂	—	2	6	no	—
23	C₂×D₁₇₂	—	4	3	no	—
24	C₂×D₁₇₂	—	4	6	no	—
25	C₂×D₁₇₂	—	4	6	no	—
26	C₂×C₈×C₄₃	—	2	6	no	—
27	C₂×C₈×C₄₃	—	2	6	no	—
28	C₂×C₈×C₄₃	D₈	1	12	yes	—
29	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₄₃	#41	1	3	yes	—
31	C₄⋊Dic₄₃	—	4	6	no	—
32	C₄⋊Dic₄₃	—	4	12	no	—
33	C₂³×C₄₃	#127	1	1	yes	characteristic
34	C₂²×D₄₃	—	4	2	no	—
35	C₂²×D₄₃	—	8	3	no	—
36	C₂²×D₄₃	—	8	3	no	—
37	C₂×C₄×C₄₃	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₄₃	#123	1	3	yes	—
39	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
40	C₂×C₄×C₄₃	#137	1	6	yes	—
41	C₂×Dic₄₃	—	8	3	no	—
42	C₂×Dic₄₃	—	8	6	no	—
43	D₁₇₂	—	4	12	no	—
44	D₁₇₂	—	8	6	no	—
45	C₈×C₄₃	—	2	12	no	—
46	#10673	—	43	1	no	maximal, Sylow
47	C₂²×C₄₃	#399	1	1	yes	characteristic
48	C₂²×C₄₃	#493	1	3	yes	—
49	C₂²×C₄₃	#497	1	3	yes	—
50	C₂×D₄₃	—	8	6	no	—
51	C₂×D₄₃	—	16	1	no	—
52	C₂×D₄₃	—	16	3	no	—
53	C₄×C₄₃	#474	1	6	yes	—
54	C₄×C₄₃	#480	1	6	yes	—
55	Dic₄₃	—	16	3	no	—
56	#1957	—	43	3	no	—
57	#1845	—	43	3	no	—
58	#1273	D₄₃	1	1	yes	characteristic, complemented
59	C₂×C₄₃	#1712	1	1	yes	characteristic
60	C₂×C₄₃	#1722	1	3	yes	—
61	C₂×C₄₃	#2093	1	3	yes	—
62	D₄₃	—	16	2	no	—
63	C₂×#127	—	43	1	no	—
64	C₂×#107	—	86	6	no	—
65	C₂×#102	—	86	6	no	—
66	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
67	#384	—	43	3	no	—
68	C₄₃	#10673	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	86	3	no	—
70	C₂²×C₈	D₁₇₂	1	6	yes	—
71	#127	—	43	2	no	—
72	#127	—	86	3	no	—
73	C₂×C₄⋊C₄	—	86	3	no	—
74	C₂×C₄⋊C₄	—	86	6	no	—
75	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
76	#107	—	172	12	no	—
77	#102	—	172	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	172	1	no	—
80	C₂×D₄	—	86	6	no	—
81	C₂×D₄	—	172	3	no	—
82	C₂×D₄	—	172	6	no	—
83	C₂×D₄	—	172	6	no	—
84	C₂²×C₄	—	172	3	no	—
85	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	D₃₄₄	1	12	yes	—
89	C₄⋊C₄	—	172	6	no	—
90	C₄⋊C₄	—	172	12	no	—
91	C₄²	#10621900	1	3	yes	—
92	C₄²	Q₈×D₄₃	1	1	yes	characteristic
93	C₂³	—	344	3	no	—
94	C₂³	—	344	3	no	—
95	C₂³	—	172	2	no	—
96	C₂³	#13063322	1	1	yes	characteristic
97	D₄	—	344	6	no	—
98	D₄	—	172	12	no	—
99	C₂×C₄	—	344	3	no	—
100	C₂×C₄	—	344	6	no	—
101	C₂×C₄	#13063305	1	3	yes	—
102	C₂×C₄	#13063309	1	3	yes	—
103	C₂×C₄	C₂×D₃₄₄	1	6	yes	—
104	C₂×C₄	#13063358	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	344	6	no	—
107	C₂²	—	688	1	no	—
108	C₂²	—	688	3	no	—
109	C₂²	#14582836	1	1	yes	characteristic
110	C₂²	#14583276	1	3	yes	—
111	C₂²	#14583293	1	3	yes	—
112	C₄	—	688	3	no	—
113	C₄	#14583180	1	6	yes	—
114	C₄	#14583213	1	6	yes	—
115	C₂	—	688	2	no	—
116	C₂	#19134210	1	1	yes	characteristic
117	C₂	#19134266	1	3	yes	—
118	C₂	#19136122	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135401

C₂

yes

maximal

#19134755

C₂

yes

maximal, complemented

C₄₃×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₂²×D₁₇₂

—

#13063322

—

#13063322

Q₈

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

#13063247

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂³×C₄₃

#127

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#497

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#474

yes

—

C₄×C₄₃

#480

yes

—

Dic₄₃

—

#1957

—

#1845

—

#1273

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#1712

yes

characteristic

C₂×C₄₃

#1722

yes

—

C₂×C₄₃

#2093

yes

—

D₄₃

—

C₂×#127

—

C₂×#107

—

C₂×#102

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

#384

—

C₄₃

#10673

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×C₈

D₁₇₂

yes

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

#107

—

172

—

#102

—

172

—

C₄×C₈

—

C₂⁴

—

172

—

C₂×D₄

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂²×C₄

—

172

—

C₂²×C₄

C₂×D₁₇₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₃₄₄

yes

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₄²

#10621900

yes

—

C₄²

Q₈×D₄₃

yes

characteristic

C₂³

—

344

—

C₂³

—

344

—

C₂³

—

172

—

C₂³

#13063322

yes

characteristic

D₄

—

344

—

D₄

—

172

—

C₂×C₄

—

344

—

100

C₂×C₄

—

344

—

101

C₂×C₄

#13063305

yes

—

102

C₂×C₄

#13063309

yes

—

103

C₂×C₄

C₂×D₃₄₄

yes

—

104

C₂×C₄

#13063358

yes

—

105

C₈

—

106

C₂²

—

344

—

107

C₂²

—

688

—

108

C₂²

—

688

—

109

C₂²

#14582836

yes

characteristic

110

C₂²

#14583276

yes

—

111

C₂²

#14583293

yes

—

112

C₄

—

688

—

113

C₄

#14583180

yes

—

114

C₄

#14583213

yes

—

115

C₂

—

688

—

116

C₂

#19134210

yes

characteristic

117

C₂

#19134266

yes

—

118

C₂

#19136122

yes

—

	subgroup	G/H
center	C₂³	#13063322
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1273	C₂
socle	C₂³×C₄₃	#127
prime radical	C₂×#13063322	C₂²
hypercenter	#1273	D₄₃
hypocenter	C₄₃	#10673
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#13063322

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1273

C₂

C₂³×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #19134755	C₂	semidirect
3. C₄₃×#1273	C₂	semidirect
58. #1273	D₄₃	semidirect
68. C₄₃	#10673	semidirect

H ≅ G/N

type

2. #19134755

C₂

semidirect

3. C₄₃×#1273

C₂

semidirect

58. #1273

D₄₃

semidirect

68. C₄₃

#10673

semidirect

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

subgroup	index
#19135401	2
#19134755	2
C₄₃×#1273	2
#10673	43

subgroup

index

#19135401

#19134755

C₄₃×#1273

#10673

Subgroups — #25001647

Conjugacy classes of subgroups

Complements

Maximal subgroups