Group #25011308 — subgroups

Subgroups — #25011308

Order 11008.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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	#19136844	C₂	1	4	yes	maximal, complemented
2	#19134210	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063266	—	2	8	no	—
6	#14583276	—	2	2	no	—
7	#14583276	C₂²	1	2	yes	—
8	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	C₄₃×#493	—	2	2	no	—
10	C₄₃×#493	C₂²	1	2	yes	—
11	C₈²×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×D₃₄₄	—	2	4	no	—
13	C₂×D₃₄₄	—	4	4	no	—
14	#13063322	—	2	2	no	—
15	#13063266	—	4	16	no	—
16	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	—
17	C₂×D₈×C₄₃	—	2	4	no	—
18	C₂×D₈×C₄₃	—	4	4	no	—
19	#127×C₄₃	—	2	2	no	—
20	C₄×C₈×C₄₃	D₄	1	2	yes	—
21	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×D₄×C₄₃	—	4	2	no	—
23	C₂×D₄×C₄₃	—	4	4	no	—
24	C₂×D₁₇₂	—	4	2	no	—
25	C₂×D₁₇₂	—	4	4	no	—
26	D₈×C₄₃	—	4	8	no	—
27	D₈×C₄₃	—	8	8	no	—
28	C₂×C₈×C₄₃	D₈	1	4	yes	—
29	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
30	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
31	D₃₄₄	—	4	8	no	—
32	D₃₄₄	—	8	8	no	—
33	C₄₃⋊C₁₆	D₈	1	8	yes	complemented
34	C₂³×C₄₃	—	8	2	no	—
35	C₂²×D₄₃	—	8	2	no	—
36	D₄×C₄₃	—	8	4	no	—
37	D₄×C₄₃	—	8	8	no	—
38	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
39	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
40	D₁₇₂	—	8	4	no	—
41	D₁₇₂	—	8	8	no	—
42	C₈×C₄₃	D₁₆	1	8	yes	complemented
43	C₈×C₄₃	C₂×D₈	1	4	yes	—
44	#12515	—	43	1	no	maximal, Sylow
45	C₂²×C₄₃	—	16	2	no	—
46	C₂²×C₄₃	—	16	4	no	—
47	C₂²×C₄₃	#493	1	1	yes	characteristic
48	C₂×D₄₃	—	16	2	no	—
49	C₂×D₄₃	—	16	4	no	—
50	C₄×C₄₃	#493	1	2	yes	—
51	C₄×C₄₃	C₂×D₁₆	1	4	yes	—
52	#2246	—	43	4	no	—
53	#1712	—	43	1	no	—
54	#1712	D₄₃	1	1	yes	characteristic, complemented
55	C₈×C₁₆	—	43	1	no	—
56	C₂×C₄₃	—	32	2	no	—
57	C₂×C₄₃	#1712	1	1	yes	characteristic
58	C₂×C₄₃	#2246	1	2	yes	—
59	D₄₃	—	32	2	no	—
60	C₂×D₁₆	—	86	8	no	—
61	#493	—	2	2	no	—
62	#493	—	43	2	no	—
63	#493	—	86	2	no	—
64	#493	C₂×D₄₃	1	2	yes	—
65	C₄×C₁₆	—	43	2	no	—
66	C₈²	C₂×D₄₃	1	1	yes	characteristic
67	C₄₃	#12515	1	1	yes	characteristic, complemented, Sylow
68	C₂×D₈	—	2	4	no	—
69	C₂×D₈	—	4	4	no	—
70	C₂×D₈	—	86	4	no	—
71	C₂×D₈	—	172	4	no	—
72	#127	—	2	2	no	—
73	#127	—	86	2	no	—
74	D₁₆	—	172	16	no	—
75	C₂×C₁₆	—	43	4	no	—
76	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
77	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	4	no	—
80	C₂×D₄	—	172	2	no	—
81	C₂×D₄	—	172	4	no	—
82	D₈	—	4	8	no	—
83	D₈	—	8	8	no	—
84	D₈	—	344	8	no	—
85	D₈	—	172	8	no	—
86	C₂×C₈	#10621882	1	4	yes	—
87	C₂×C₈	D₄×D₄₃	1	2	yes	—
88	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
89	C₁₆	—	43	8	no	—
90	C₂³	—	8	2	no	—
91	C₂³	—	344	2	no	—
92	D₄	—	8	8	no	—
93	D₄	—	8	4	no	—
94	D₄	—	344	8	no	—
95	D₄	—	344	4	no	—
96	C₂×C₄	#13063321	1	1	yes	characteristic
97	C₂×C₄	C₂×#10621882	1	2	yes	—
98	C₈	#13063266	1	8	yes	complemented
99	C₈	D₈×D₄₃	1	4	yes	—
100	C₂²	—	16	2	no	—
101	C₂²	—	16	4	no	—
102	C₂²	—	688	2	no	—
103	C₂²	—	688	4	no	—
104	C₂²	#14583275	1	1	yes	characteristic
105	C₄	#14583274	1	2	yes	—
106	C₄	C₂×#13063266	1	4	yes	—
107	C₂	—	32	2	no	—
108	C₂	—	1376	2	no	—
109	C₂	#19134209	1	1	yes	characteristic
110	C₂	#19136844	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136844

C₂

yes

maximal, complemented

#19134210

C₂

yes

maximal, characteristic, complemented

C₈×C₄₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄₃×#1712

C₂

yes

maximal, characteristic, complemented

C₂×#13063266

—

#14583276

—

#14583276

C₂²

yes

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#493

—

C₄₃×#493

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×D₃₄₄

—

C₂×D₃₄₄

—

#13063322

—

#13063266

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

#127×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

D₈×C₄₃

—

D₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

D₃₄₄

—

D₃₄₄

—

C₄₃⋊C₁₆

D₈

yes

complemented

C₂³×C₄₃

—

C₂²×D₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

D₁₆

yes

complemented

C₈×C₄₃

C₂×D₈

yes

—

#12515

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#493

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#493

yes

—

C₄×C₄₃

C₂×D₁₆

yes

—

#2246

—

#1712

—

#1712

D₄₃

yes

characteristic, complemented

C₈×C₁₆

—

C₂×C₄₃

—

C₂×C₄₃

#1712

yes

characteristic

C₂×C₄₃

#2246

yes

—

D₄₃

—

C₂×D₁₆

—

—

—

—

C₂×D₄₃

yes

—

C₄×C₁₆

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12515

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

172

—

#127

—

#127

—

D₁₆

—

172

—

C₂×C₁₆

—

C₄×C₈

C₂²⋊D₄₃

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

D₈

—

D₈

—

D₈

—

344

—

D₈

—

172

—

C₂×C₈

#10621882

yes

—

C₂×C₈

D₄×D₄₃

yes

—

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

344

—

D₄

—

D₄

—

D₄

—

344

—

D₄

—

344

—

C₂×C₄

#13063321

yes

characteristic

C₂×C₄

C₂×#10621882

yes

—

C₈

#13063266

yes

complemented

C₈

D₈×D₄₃

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

688

—

103

C₂²

—

688

—

104

C₂²

#14583275

yes

characteristic

105

C₄

#14583274

yes

—

106

C₄

C₂×#13063266

yes

—

107

C₂

—

108

C₂

—

1376

—

109

C₂

#19134209

yes

characteristic

110

C₂

#19136844

yes

—

	subgroup	G/H
center	C₂²	#14583275
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1712	C₂
socle	C₂²×C₄₃	#493
hypercenter	#1712	D₄₃
hypocenter	C₄₃	#12515
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#14583275

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1712

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #19136844	C₂	semidirect
2. #19134210	C₂	semidirect
3. C₈×C₄₃⋊C₁₆	C₂	semidirect
4. C₄₃×#1712	C₂	semidirect
33. C₄₃⋊C₁₆	D₈	semidirect
42. C₈×C₄₃	D₁₆	semidirect
54. #1712	D₄₃	semidirect
67. C₄₃	#12515	semidirect
98. C₈	#13063266	semidirect

H ≅ G/N

type

1. #19136844

C₂

semidirect

2. #19134210

C₂

semidirect

3. C₈×C₄₃⋊C₁₆

C₂

semidirect

4. C₄₃×#1712

C₂

semidirect

33. C₄₃⋊C₁₆

D₈

semidirect

42. C₈×C₄₃

D₁₆

semidirect

54. #1712

D₄₃

semidirect

67. C₄₃

#12515

semidirect

98. C₈

#13063266

semidirect

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

subgroup	index
#19136844	2
#19134210	2
C₈×C₄₃⋊C₁₆	2
C₄₃×#1712	2
#12515	43

subgroup

index

#19136844

#19134210

C₈×C₄₃⋊C₁₆

C₄₃×#1712

#12515

Subgroups — #25011308

Conjugacy classes of subgroups

Complements

Maximal subgroups