Group #28362937 — subgroups

Subgroups — #28362937

Order 13952.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#16812634	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₀₉	C₂	1	2	yes	maximal
3	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
4	#20294791	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812634	C₄	1	8	yes	complemented
10	#16812609	—	2	2	no	—
11	#16812609	C₄	1	4	yes	—
12	C₈×Dic₁₀₉	—	2	2	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	Q₈	1	2	yes	—
16	#13182013	—	2	2	no	—
17	#13182013	C₂×C₄	1	4	yes	—
18	C₂×#11724784	—	2	2	no	—
19	C₂×#11724784	C₂×C₄	1	2	yes	—
20	C₂×#11724784	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₀₉	D₄	1	2	yes	—
23	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₀₉	—	2	2	no	—
25	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₁₀₉	—	2	2	no	—
27	C₂×C₈×C₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	#41	1	2	yes	—
34	C₂×Dic₁₀₉	—	4	2	no	—
35	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
36	C₈×C₁₀₉	—	2	2	no	—
37	C₈×C₁₀₉	—	2	4	no	—
38	#11724784	—	2	4	no	—
39	#11724784	C₄²	1	4	yes	—
40	C₂²×C₁₀₉	—	2	2	no	—
41	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
43	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₀₉	#117	1	1	yes	characteristic
45	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
46	Dic₁₀₉	—	2	4	no	—
47	C₂×C₁₀₉	—	2	2	no	—
48	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₁₀₉	#430	1	2	yes	—
50	#1778	—	109	1	no	maximal, Sylow
51	C₁₀₉	#1778	1	1	yes	characteristic, complemented, Sylow
52	C₂×#108	—	109	2	no	—
53	C₄×M₁₆	—	109	2	no	—
54	C₂×#97	—	109	1	no	—
55	#341	—	109	1	no	—
56	#341	D₁₀₉	1	1	yes	characteristic
57	C₂×M₁₆	—	109	2	no	—
58	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
59	C₂²×C₈	—	109	1	no	—
60	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
61	C₂×C₄²	—	109	1	no	—
62	#108	—	109	8	no	—
63	#97	—	109	4	no	—
64	#97	—	218	2	no	—
65	C₄×C₈	—	218	2	no	—
66	C₂²×C₄	—	109	2	no	—
67	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	109	4	no	—
70	M₁₆	—	218	2	no	—
71	M₁₆	C₄×D₁₀₉	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	109	2	no	—
75	C₂×C₈	—	109	4	no	—
76	C₂×C₈	—	218	2	no	—
77	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
78	C₄²	—	109	2	no	—
79	C₄²	—	109	2	no	—
80	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
81	C₂×C₄	—	109	4	no	—
82	C₂×C₄	—	436	2	no	—
83	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
85	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
86	C₂×C₄	#13182028	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	109	4	no	—
90	C₈	—	218	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
93	C₂²	#16812664	1	2	yes	—
94	C₄	—	218	4	no	—
95	C₄	C₄²×D₁₀₉	1	1	yes	characteristic
96	C₄	#16812655	1	1	yes	characteristic
97	C₄	#16812659	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#20294877	1	1	yes	characteristic
100	C₂	#20295073	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812634

C₂

yes

maximal

M₁₆×Dic₁₀₉

C₂

yes

maximal

C₂×#16812609

C₂

yes

maximal, characteristic

#20294791

C₂

yes

maximal, characteristic

#341×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812634

C₄

yes

complemented

#16812609

—

#16812609

C₄

yes

—

C₈×Dic₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

—

#13182013

—

#13182013

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

C₄²

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#430

yes

—

#1778

—

109

maximal, Sylow

C₁₀₉

#1778

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

C₄×M₁₆

—

109

—

C₂×#97

—

109

—

#341

—

109

—

#341

D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

#108

—

109

—

#97

—

109

—

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

C₄×D₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

Q₈×D₁₀₉

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

C₂×C₄

D₄×D₁₀₉

yes

characteristic

C₂×C₄

#13182028

yes

—

C₈

—

C₈

—

C₈

—

109

—

C₈

—

218

—

C₂²

—

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

C₂²

#16812664

yes

—

C₄

—

218

—

C₄

C₄²×D₁₀₉

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

#20294877

yes

characteristic

100

C₂

#20295073

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#341×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#341	D₁₀₉
hypocenter	C₁₀₉	#1778
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#341×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #16812634	C₄	semidirect
51. C₁₀₉	#1778	semidirect

H ≅ G/N

type

9. #16812634

C₄

semidirect

51. C₁₀₉

#1778

semidirect

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

subgroup	index
C₂×#16812634	2
M₁₆×Dic₁₀₉	2
C₂×#16812609	2
#20294791	2
#341×C₁₀₉	2
#1778	109

subgroup

index

M₁₆×Dic₁₀₉

#341×C₁₀₉

109

Subgroups — #28362937

Conjugacy classes of subgroups

Complements

Maximal subgroups