Group #28371301 — subgroups

Subgroups — #28371301

Order 13952.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20294965	C₂	1	3	yes	maximal
2	#20294959	C₂	1	6	yes	maximal
3	#20294946	C₂	1	3	yes	maximal
4	#20294894	C₂	1	2	yes	maximal
5	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	—
8	C₂×#13182008	C₂²	1	6	yes	—
9	#16812606	C₂²	1	6	yes	—
10	#16812606	C₂²	1	3	yes	—
11	#16812606	C₂²	1	3	yes	—
12	#16812606	C₂²	1	6	yes	—
13	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₈	—	4	3	no	—
17	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
18	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
19	C₄⋊Dic₁₀₉	—	4	3	no	—
20	#13182008	—	4	6	no	—
21	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
23	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
24	C₄⋊C₄×C₁₀₉	—	4	3	no	—
25	C₄⋊C₄×C₁₀₉	—	4	3	no	—
26	C₄²×C₁₀₉	Q₈	1	4	yes	complemented
27	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	3	no	—
29	C₂×C₄×C₁₀₉	—	2	3	no	—
30	C₂×C₄×C₁₀₉	—	2	6	no	—
31	C₂×C₄×C₁₀₉	—	2	3	no	—
32	C₂×C₄×C₁₀₉	—	2	3	no	—
33	C₂×C₄×C₁₀₉	C₂×Q₈	1	6	yes	—
34	C₂×Dic₁₀₉	—	2	6	no	—
35	C₂×Dic₁₀₉	—	2	6	no	—
36	C₂×Dic₁₀₉	—	2	12	no	—
37	Dic₂₁₈	—	4	4	no	—
38	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₁₀₉	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₁₀₉	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₁₀₉	—	2	6	no	—
42	C₄×C₁₀₉	—	4	1	no	—
43	C₄×C₁₀₉	—	4	3	no	—
44	Dic₁₀₉	—	4	2	no	—
45	Dic₁₀₉	—	4	6	no	—
46	C₂×C₁₀₉	#544	1	3	yes	—
47	C₂×C₁₀₉	#560	1	1	yes	characteristic
48	C₂×C₁₀₉	#563	1	3	yes	—
49	#2843	—	109	1	no	maximal, Sylow
50	C₁₀₉	#2843	1	1	yes	characteristic, complemented, Sylow
51	#400	—	109	3	no	—
52	#398	—	109	6	no	—
53	#395	—	109	3	no	—
54	#384	D₁₀₉	1	1	yes	characteristic
55	#383	—	109	2	no	—
56	C₂²×Q₈	—	109	1	no	—
57	C₂×C₄⋊C₄	—	109	6	no	—
58	C₂×C₄⋊C₄	—	109	3	no	—
59	C₂×C₄⋊C₄	C₂×D₁₀₉	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₁₀₉	1	3	yes	—
61	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
62	#95	—	109	6	no	—
63	#95	—	109	6	no	—
64	#95	—	109	3	no	—
65	#95	—	109	3	no	—
66	C₂×Q₈	—	436	3	no	—
67	C₂²×C₄	—	109	6	no	—
68	C₂²×C₄	—	109	2	no	—
69	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
71	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	436	6	no	—
75	C₄⋊C₄	—	436	3	no	—
76	C₄²	Dic₂₁₈	1	4	yes	complemented
77	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
78	Q₈	—	436	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	218	6	no	—
85	C₂×C₄	—	218	6	no	—
86	C₂×C₄	—	218	12	no	—
87	C₂×C₄	C₂×Dic₂₁₈	1	6	yes	—
88	C₂²	C₂²×Dic₂₁₈	1	1	yes	characteristic
89	C₂²	#16812744	1	3	yes	—
90	C₂²	#16812747	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	436	2	no	—
95	C₄	—	436	6	no	—
96	C₂	#20295657	1	3	yes	—
97	C₂	#20295763	1	1	yes	characteristic
98	C₂	#20295775	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20294965

C₂

yes

maximal

#20294959

C₂

yes

maximal

#20294946

C₂

yes

maximal

#20294894

C₂

yes

maximal

#384×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

#16812606

C₂²

yes

—

#16812606

C₂²

yes

—

#16812606

C₂²

yes

—

#16812606

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

—

#13182008

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

Q₈

yes

complemented

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₉

H(5,𝔽₂)

yes

—

C₂²×C₁₀₉

2₋¹⁺⁴

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#544

yes

—

C₂×C₁₀₉

#560

yes

characteristic

C₂×C₁₀₉

#563

yes

—

#2843

—

109

maximal, Sylow

C₁₀₉

#2843

yes

characteristic, complemented, Sylow

#400

—

109

—

#398

—

109

—

#395

—

109

—

#384

D₁₀₉

yes

characteristic

#383

—

109

—

C₂²×Q₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#95

—

109

—

#95

—

109

—

#95

—

109

—

#95

—

109

—

C₂×Q₈

—

436

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄²

Dic₂₁₈

yes

complemented

C₂³

C₂³×D₁₀₉

yes

characteristic

Q₈

—

436

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×Dic₂₁₈

yes

—

C₂²

C₂²×Dic₂₁₈

yes

characteristic

C₂²

#16812744

yes

—

C₂²

#16812747

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

436

—

C₄

—

436

—

C₂

#20295657

yes

—

C₂

#20295763

yes

characteristic

C₂

#20295775

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, prime radical, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	#384×C₁₀₉	C₂
hypercenter	#384	D₁₀₉
hypocenter	C₁₀₉	#2843
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, prime radical, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

#384×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₁₀₉	Q₈	semidirect
50. C₁₀₉	#2843	semidirect
76. C₄²	Dic₂₁₈	semidirect

H ≅ G/N

type

26. C₄²×C₁₀₉

Q₈

semidirect

50. C₁₀₉

#2843

semidirect

76. C₄²

Dic₂₁₈

semidirect

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

subgroup	index
#20294965	2
#20294959	2
#20294946	2
#20294894	2
#384×C₁₀₉	2
#2843	109

subgroup

index

#384×C₁₀₉

109

Subgroups — #28371301

Conjugacy classes of subgroups

Complements

Maximal subgroups