Group #28362358 — subgroups

Subgroups — #28362358

Order 13952.

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

normal subgroups only

This group has 172 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
2	#20295432	C₂	1	2	yes	maximal
3	#20294768	C₂	1	1	yes	maximal, characteristic
4	#20294767	C₂	1	2	yes	maximal
5	#491×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812691	C₂²	1	2	yes	—
7	#16812690	—	2	2	no	—
8	#16812689	—	2	2	no	—
9	#16812689	—	2	2	no	—
10	#16812632	—	2	2	no	—
11	#16812630	—	2	2	no	—
12	#16812610	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#125×C₁₀₉	C₂²	1	1	yes	characteristic
15	#103×C₁₀₉	—	2	2	no	—
16	#97×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#11724784	—	2	2	no	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
20	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
21	#13182008	—	2	2	no	—
22	#13182008	—	2	2	no	—
23	#13182008	—	2	2	no	—
24	#13182008	—	2	2	no	—
25	C₄×Dic₁₀₉	—	2	2	no	—
26	C₂×Q₈×C₁₀₉	—	2	1	no	—
27	C₂×C₈×C₁₀₉	—	2	1	no	—
28	C₄⋊C₄×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	Q₈×C₁₀₉	—	4	2	no	—
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	4	1	no	—
35	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	—	2	2	no	—
38	C₂×Dic₁₀₉	—	2	2	no	—
39	C₈×C₁₀₉	—	4	1	no	—
40	#11724784	—	2	4	no	—
41	C₂²×C₁₀₉	#120	1	1	yes	characteristic
42	C₄×C₁₀₉	—	2	1	no	—
43	C₄×C₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	—	4	2	no	—
45	C₄×C₁₀₉	—	8	1	no	—
46	Dic₁₀₉	—	4	2	no	—
47	Dic₁₀₉	—	4	2	no	—
48	C₂×C₁₀₉	#454	1	1	yes	characteristic
49	C₂×C₁₀₉	#456	1	2	yes	—
50	#1663	—	109	1	no	maximal, Sylow
51	C₁₀₉	#1663	1	1	yes	characteristic, complemented, Sylow
52	#564	—	109	1	no	—
53	#501	—	109	2	no	—
54	#491	D₁₀₉	1	1	yes	characteristic
55	#330	—	109	1	no	—
56	#330	—	109	2	no	—
57	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
58	#125	—	109	2	no	—
59	#125	—	218	2	no	—
60	#125	—	218	2	no	—
61	#125	—	218	2	no	—
62	#125	C₂×D₁₀₉	1	1	yes	characteristic
63	#107	—	218	2	no	—
64	#106	—	218	2	no	—
65	#103	—	2	2	no	—
66	#97	—	109	2	no	—
67	#97	C₂×D₁₀₉	1	1	yes	characteristic
68	C₂×Q₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	218	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	109	2	no	—
74	C₄⋊C₄	—	109	2	no	—
75	C₄⋊C₄	—	218	2	no	—
76	C₄⋊C₄	—	218	2	no	—
77	C₄⋊C₄	—	218	2	no	—
78	C₄⋊C₄	—	218	2	no	—
79	C₄⋊C₄	—	218	2	no	—
80	C₄⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
81	C₄²	—	218	2	no	—
82	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
83	Q₈	—	4	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	218	2	no	—
87	C₂×C₄	—	218	2	no	—
88	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
90	C₈	—	4	1	no	—
91	C₈	—	218	4	no	—
92	C₂²	#16812667	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	8	1	no	—
97	C₄	—	436	2	no	—
98	C₄	—	436	2	no	—
99	C₂	#20295198	1	1	yes	characteristic
100	C₂	#20295207	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉⋊#564

C₂

yes

maximal, characteristic

#20295432

C₂

yes

maximal

#20294768

C₂

yes

maximal, characteristic

#20294767

C₂

yes

maximal

#491×C₁₀₉

C₂

yes

maximal, characteristic

#16812691

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#125×C₁₀₉

C₂²

yes

characteristic

#103×C₁₀₉

—

#97×C₁₀₉

C₂²

yes

characteristic

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₄⋊Dic₁₀₉

D₄

yes

—

—

—

—

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#454

yes

characteristic

C₂×C₁₀₉

#456

yes

—

#1663

—

109

maximal, Sylow

C₁₀₉

#1663

yes

characteristic, complemented, Sylow

#564

—

109

—

#501

—

109

—

#491

D₁₀₉

yes

characteristic

#330

—

109

—

#330

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

#125

—

109

—

#125

—

218

—

#125

—

218

—

#125

—

218

—

#125

C₂×D₁₀₉

yes

characteristic

#107

—

218

—

#106

—

218

—

#103

—

#97

—

109

—

#97

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

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

C₂²⋊D₁₀₉

yes

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₄×D₁₀₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₂²

#16812667

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

436

—

C₄

—

436

—

C₂

#20295198

yes

characteristic

100

C₂

#20295207

yes

—

	subgroup	G/H
center	C₂²	#16812667
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#491×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#120
hypercenter	#491	D₁₀₉
hypocenter	C₁₀₉	#1663
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

#16812667

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#491×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
51. C₁₀₉	#1663	semidirect

H ≅ G/N

type

51. C₁₀₉

#1663

semidirect

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

subgroup	index
C₁₀₉⋊#564	2
#20295432	2
#20294768	2
#20294767	2
#491×C₁₀₉	2
#1663	109

subgroup

index

C₁₀₉⋊#564

#491×C₁₀₉

109

Subgroups — #28362358

Conjugacy classes of subgroups

Complements

Maximal subgroups