Group #28362335 — subgroups

Subgroups — #28362335

Order 13952.

182 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₁₀₉⋊#564	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295392	C₂	1	1	yes	maximal, characteristic
3	#20295391	C₂	1	2	yes	maximal
4	#20294767	C₂	1	2	yes	maximal, complemented
5	#330×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812696	C₂²	1	1	yes	characteristic
7	#16812691	C₂²	1	2	yes	—
8	#16812690	—	2	2	no	—
9	#16812689	—	2	2	no	—
10	#16812689	—	2	2	no	—
11	#16812621	—	2	2	no	—
12	#16812621	—	2	2	no	—
13	#16812619	—	2	2	no	—
14	#16812610	C₂²	1	1	yes	characteristic
15	#125×C₁₀₉	C₂²	1	1	yes	characteristic
16	#97×C₁₀₉	C₂²	1	2	yes	—
17	C₂×D₄₃₆	—	2	1	no	—
18	C₂×D₄₃₆	—	2	2	no	—
19	C₂×#11724784	—	2	1	no	—
20	C₄⋊Dic₁₀₉	—	2	2	no	—
21	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
22	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
23	#13182008	—	2	2	no	—
24	#13182008	—	2	2	no	—
25	#13182008	—	2	2	no	—
26	#13182008	—	2	2	no	—
27	C₄×Dic₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
31	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₉	—	4	1	no	—
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	—	2	2	no	—
37	C₂×Dic₁₀₉	—	2	2	no	—
38	D₄₃₆	—	4	2	no	—
39	D₄₃₆	—	4	4	no	—
40	C₈×C₁₀₉	—	4	2	no	—
41	#11724784	—	4	1	no	—
42	C₂²×C₁₀₉	#120	1	1	yes	characteristic
43	C₂×D₁₀₉	—	8	2	no	—
44	C₂×D₁₀₉	—	8	1	no	—
45	C₄×C₁₀₉	—	2	1	no	—
46	C₄×C₁₀₉	—	2	2	no	—
47	C₄×C₁₀₉	—	4	2	no	—
48	Dic₁₀₉	—	4	2	no	—
49	Dic₁₀₉	—	4	2	no	—
50	C₂×C₁₀₉	#454	1	2	yes	—
51	C₂×C₁₀₉	#454	1	1	yes	characteristic
52	D₁₀₉	—	16	1	no	—
53	#1658	—	109	1	no	maximal, Sylow
54	C₁₀₉	#1658	1	1	yes	characteristic, complemented, Sylow
55	#564	—	109	1	no	—
56	#490	—	109	1	no	—
57	#490	—	109	2	no	—
58	#330	—	109	2	no	—
59	#330	D₁₀₉	1	1	yes	characteristic, complemented
60	#127	—	109	1	no	—
61	#125	—	109	2	no	—
62	#125	—	218	2	no	—
63	#125	—	218	2	no	—
64	#125	—	218	2	no	—
65	#125	C₂×D₁₀₉	1	1	yes	characteristic
66	#102	—	218	2	no	—
67	#102	—	218	2	no	—
68	#102	—	218	2	no	—
69	#97	—	109	1	no	—
70	#97	C₂×D₁₀₉	1	2	yes	—
71	C₂×D₄	—	218	2	no	—
72	C₂×D₄	—	218	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	218	1	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	109	2	no	—
77	C₄⋊C₄	—	109	2	no	—
78	C₄⋊C₄	—	218	2	no	—
79	C₄⋊C₄	—	218	2	no	—
80	C₄⋊C₄	—	218	2	no	—
81	C₄⋊C₄	—	218	2	no	—
82	C₄⋊C₄	—	218	2	no	—
83	C₄⋊C₄	D₄₃₆	1	2	yes	—
84	C₄²	—	218	2	no	—
85	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
86	C₂³	—	436	1	no	—
87	D₄	—	436	4	no	—
88	D₄	—	436	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	218	2	no	—
91	C₂×C₄	—	218	2	no	—
92	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
93	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
94	C₈	—	4	2	no	—
95	C₈	—	436	1	no	—
96	C₂²	—	872	2	no	—
97	C₂²	—	872	1	no	—
98	C₂²	#16812669	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	—	436	2	no	—
103	C₄	—	436	2	no	—
104	C₂	—	1744	1	no	—
105	C₂	#20295200	1	2	yes	—
106	C₂	#20295201	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉⋊#564

C₂

yes

maximal, characteristic, complemented

#20295392

C₂

yes

maximal, characteristic

#20295391

C₂

yes

maximal

#20294767

C₂

yes

maximal, complemented

#330×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812696

C₂²

yes

characteristic

#16812691

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

C₂²

yes

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₄⋊Dic₁₀₉

D₄

yes

—

—

—

—

—

C₄×Dic₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#120

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#454

yes

—

C₂×C₁₀₉

#454

yes

characteristic

D₁₀₉

—

#1658

—

109

maximal, Sylow

C₁₀₉

#1658

yes

characteristic, complemented, Sylow

#564

—

109

—

#490

—

109

—

#490

—

109

—

#330

—

109

—

#330

D₁₀₉

yes

characteristic, complemented

#127

—

109

—

#125

—

109

—

#125

—

218

—

#125

—

218

—

#125

—

218

—

#125

C₂×D₁₀₉

yes

characteristic

#102

—

218

—

#102

—

218

—

#102

—

218

—

#97

—

109

—

#97

C₂×D₁₀₉

yes

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

D₄₃₆

yes

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

436

—

D₄

—

436

—

D₄

—

436

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₄×D₁₀₉

yes

—

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₈

—

C₈

—

436

—

C₂²

—

872

—

C₂²

—

872

—

C₂²

#16812669

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

436

—

103

C₄

—

436

—

104

C₂

—

1744

—

105

C₂

#20295200

yes

—

106

C₂

#20295201

yes

characteristic

	subgroup	G/H
center	C₂²	#16812669
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#330×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#120
prime radical	#16812696	C₂²
hypercenter	#330	D₁₀₉
hypocenter	C₁₀₉	#1658
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#16812669

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#330×C₁₀₉

C₂

C₂²×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
1. C₁₀₉⋊#564	C₂	semidirect
4. #20294767	C₂	semidirect
5. #330×C₁₀₉	C₂	semidirect
54. C₁₀₉	#1658	semidirect
59. #330	D₁₀₉	semidirect

H ≅ G/N

type

1. C₁₀₉⋊#564

C₂

semidirect

4. #20294767

C₂

semidirect

5. #330×C₁₀₉

C₂

semidirect

54. C₁₀₉

#1658

semidirect

59. #330

D₁₀₉

semidirect

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

subgroup	index
C₁₀₉⋊#564	2
#20295392	2
#20295391	2
#20294767	2
#330×C₁₀₉	2
#1658	109

subgroup

index

C₁₀₉⋊#564

#330×C₁₀₉

109

Subgroups — #28362335

Conjugacy classes of subgroups

Complements

Maximal subgroups