Group #28371414 — subgroups

Subgroups — #28371414

Order 13952.

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

normal subgroups only

This group has 408 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	#20294932	C₂	1	3	yes	maximal
2	#20294924	C₂	1	3	yes	maximal
3	#20294910	C₂	1	6	yes	maximal
4	#20294894	C₂	1	2	yes	maximal
5	C₄³×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	6	yes	—
11	#16812606	C₂²	1	6	yes	—
12	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	2	6	no	—
16	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
17	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
18	C₄⋊Dic₁₀₉	—	2	6	no	—
19	#13182008	—	2	12	no	—
20	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
21	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
22	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₄²×C₁₀₉	—	2	6	no	—
24	C₄²×C₁₀₉	—	2	6	no	—
25	C₄²×C₁₀₉	Q₈	1	4	yes	complemented
26	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	2	3	no	—
28	C₂×C₄×C₁₀₉	—	2	3	no	—
29	C₂×C₄×C₁₀₉	—	2	3	no	—
30	C₂×C₄×C₁₀₉	—	2	3	no	—
31	C₂×C₄×C₁₀₉	C₂×Q₈	1	6	yes	—
32	C₂×C₄×C₁₀₉	#41	1	6	yes	—
33	C₂×C₄×C₁₀₉	#41	1	6	yes	—
34	C₂×Dic₁₀₉	—	2	12	no	—
35	C₂×Dic₁₀₉	—	2	6	no	—
36	C₂×Dic₁₀₉	—	2	6	no	—
37	Dic₂₁₈	—	4	4	no	—
38	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
40	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
41	C₄×C₁₀₉	—	2	6	no	—
42	C₄×C₁₀₉	—	2	6	no	—
43	C₄×C₁₀₉	#126	1	4	yes	—
44	Dic₁₀₉	—	4	2	no	—
45	Dic₁₀₉	—	4	6	no	—
46	C₂×C₁₀₉	C₂×#126	1	1	yes	characteristic
47	C₂×C₁₀₉	#529	1	3	yes	—
48	C₂×C₁₀₉	#533	1	3	yes	—
49	#2863	—	109	1	no	maximal, Sylow
50	C₁₀₉	#2863	1	1	yes	characteristic, complemented, Sylow
51	#391	—	109	3	no	—
52	#389	—	109	3	no	—
53	#387	—	109	6	no	—
54	#383	—	109	2	no	—
55	C₄³	D₁₀₉	1	1	yes	characteristic
56	C₂²×Q₈	—	109	1	no	—
57	C₂×C₄⋊C₄	—	109	6	no	—
58	C₂×C₄⋊C₄	—	109	3	no	—
59	C₂×C₄²	C₂×D₁₀₉	1	3	yes	—
60	C₂×C₄²	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	6	no	—
65	C₂×Q₈	—	218	6	no	—
66	C₂²×C₄	—	109	2	no	—
67	C₂²×C₄	—	109	6	no	—
68	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
69	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
70	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
71	C₄⋊C₄	—	218	12	no	—
72	C₄⋊C₄	—	218	6	no	—
73	C₄²	—	2	6	no	—
74	C₄²	—	2	6	no	—
75	C₄²	Dic₂₁₈	1	4	yes	complemented
76	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
77	Q₈	—	436	4	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	218	6	no	—
83	C₂×C₄	—	218	12	no	—
84	C₂×C₄	—	218	6	no	—
85	C₂×C₄	C₂×Dic₂₁₈	1	6	yes	—
86	C₂×C₄	#13182029	1	6	yes	—
87	C₂×C₄	#13182029	1	6	yes	—
88	C₂²	C₂²×Dic₂₁₈	1	1	yes	characteristic
89	C₂²	C₂×#13182029	1	3	yes	—
90	C₂²	C₂×#13182029	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	2	6	no	—
93	C₄	—	436	2	no	—
94	C₄	—	436	6	no	—
95	C₄	#16812693	1	4	yes	—
96	C₂	C₂×#16812693	1	1	yes	characteristic
97	C₂	#20295560	1	3	yes	—
98	C₂	#20295583	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20294932

C₂

yes

maximal

#20294924

C₂

yes

maximal

#20294910

C₂

yes

maximal

#20294894

C₂

yes

maximal

C₄³×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

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

C₂×C₄²×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

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

—

C₄²×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×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

C₂×#41

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#126

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₂×#126

yes

characteristic

C₂×C₁₀₉

#529

yes

—

C₂×C₁₀₉

#533

yes

—

#2863

—

109

maximal, Sylow

C₁₀₉

#2863

yes

characteristic, complemented, Sylow

#391

—

109

—

#389

—

109

—

#387

—

109

—

#383

—

109

—

C₄³

D₁₀₉

yes

characteristic

C₂²×Q₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#95

—

109

—

#95

—

109

—

#95

—

109

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

C₄²

—

C₄²

Dic₂₁₈

yes

complemented

C₂³

C₂³×D₁₀₉

yes

characteristic

Q₈

—

436

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×Dic₂₁₈

yes

—

C₂×C₄

#13182029

yes

—

C₂×C₄

#13182029

yes

—

C₂²

C₂²×Dic₂₁₈

yes

characteristic

C₂²

C₂×#13182029

yes

—

C₂²

C₂×#13182029

yes

—

C₄

—

C₄

—

C₄

—

436

—

C₄

—

436

—

C₄

#16812693

yes

—

C₂

C₂×#16812693

yes

characteristic

C₂

#20295560

yes

—

C₂

#20295583

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, prime radical, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	C₄³×C₁₀₉	C₂
hypercenter	C₄³	D₁₀₉
hypocenter	C₁₀₉	#2863
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, prime radical, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

C₄³×C₁₀₉

C₂

hypercenter

C₄³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
25. C₄²×C₁₀₉	Q₈	semidirect
50. C₁₀₉	#2863	semidirect
75. C₄²	Dic₂₁₈	semidirect

H ≅ G/N

type

25. C₄²×C₁₀₉

Q₈

semidirect

50. C₁₀₉

#2863

semidirect

75. C₄²

Dic₂₁₈

semidirect

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

subgroup	index
#20294932	2
#20294924	2
#20294910	2
#20294894	2
C₄³×C₁₀₉	2
#2863	109

subgroup

index

C₄³×C₁₀₉

109

Subgroups — #28371414

Conjugacy classes of subgroups

Complements

Maximal subgroups