Group #28359503 — subgroups

Subgroups — C₁₀₉×#2249 #28359503

Order 13952.

184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 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×Q₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₃₂×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂×D₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#495×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
7	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
8	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
9	#124×C₁₀₉	—	2	2	no	—
10	Q₃₂×C₁₀₉	—	2	2	no	—
11	SD₃₂×C₁₀₉	—	2	4	no	—
12	D₁₆×C₁₀₉	—	2	2	no	—
13	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₀₉	—	2	2	no	—
17	C₂×D₄×C₁₀₉	—	2	2	no	—
18	Q₁₆×C₁₀₉	—	2	2	no	—
19	Q₁₆×C₁₀₉	—	2	2	no	—
20	SD₁₆×C₁₀₉	—	4	4	no	—
21	D₈×C₁₀₉	—	2	2	no	—
22	D₈×C₁₀₉	—	2	2	no	—
23	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
26	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
27	C₁₆×C₁₀₉	D₄	1	2	yes	—
28	C₁₆×C₁₀₉	D₄	1	2	yes	—
29	C₂³×C₁₀₉	—	4	2	no	—
30	Q₈×C₁₀₉	—	4	2	no	—
31	Q₈×C₁₀₉	—	4	2	no	—
32	D₄×C₁₀₉	—	4	2	no	—
33	D₄×C₁₀₉	—	4	2	no	—
34	C₂×C₄×C₁₀₉	—	4	2	no	—
35	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₀₉	D₈	1	2	yes	—
38	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₁₀₉	—	8	2	no	—
41	C₂²×C₁₀₉	—	8	4	no	—
42	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
43	C₄×C₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	—	8	2	no	—
45	C₄×C₁₀₉	#127	1	1	yes	characteristic
46	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₁₀₉	—	16	2	no	—
48	C₂×C₁₀₉	#493	1	1	yes	characteristic
49	C₂×C₁₀₉	#508	1	2	yes	—
50	#2249	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₁₀₉	#2249	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×Q₃₂	C₂×C₁₀₉	1	1	yes	characteristic, complemented
53	C₂×SD₃₂	C₂×C₁₀₉	1	2	yes	complemented
54	C₂×D₁₆	C₂×C₁₀₉	1	1	yes	characteristic
55	#495	C₂×C₁₀₉	1	2	yes	complemented
56	C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
57	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
58	C₂×SD₁₆	—	2	2	no	—
59	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
60	#124	—	2	2	no	—
61	Q₃₂	—	2	2	no	—
62	SD₃₂	—	2	4	no	—
63	D₁₆	—	2	2	no	—
64	C₂×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
65	C₂×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
66	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	2	2	no	—
69	Q₁₆	—	2	2	no	—
70	Q₁₆	—	2	2	no	—
71	SD₁₆	—	4	4	no	—
72	D₈	—	2	2	no	—
73	D₈	—	2	2	no	—
74	C₂×C₈	D₄×C₁₀₉	1	1	yes	characteristic
75	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
76	C₂²⋊C₄	—	4	4	no	—
77	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
78	C₁₆	D₄×C₁₀₉	1	2	yes	—
79	C₁₆	D₄×C₁₀₉	1	2	yes	—
80	C₂³	—	4	2	no	—
81	Q₈	—	4	2	no	—
82	Q₈	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	D₈×C₁₀₉	1	2	yes	—
87	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
88	C₈	D₈×C₁₀₉	1	2	yes	—
89	C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
90	C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
91	C₂²	—	8	4	no	—
92	C₂²	—	8	2	no	—
93	C₂²	C₂×D₈×C₁₀₉	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	8	2	no	—
96	C₄	#127×C₁₀₉	1	1	yes	characteristic
97	C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
98	C₂	—	16	2	no	—
99	C₂	#493×C₁₀₉	1	1	yes	characteristic
100	C₂	#508×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₃₂×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×SD₃₂×C₁₀₉

C₂

yes

maximal, complemented

C₂×D₁₆×C₁₀₉

C₂

yes

maximal, characteristic

#495×C₁₀₉

C₂

yes

maximal, complemented

C₄×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

C₂×SD₁₆×C₁₀₉

—

C₂×D₈×C₁₀₉

C₂²

yes

—

#124×C₁₀₉

—

Q₃₂×C₁₀₉

—

SD₃₂×C₁₀₉

—

D₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

SD₁₆×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₁₆×C₁₀₉

D₄

yes

—

C₁₆×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₈

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

D₈

yes

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#127

yes

characteristic

C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#493

yes

characteristic

C₂×C₁₀₉

#508

yes

—

#2249

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#2249

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂×C₁₀₉

yes

characteristic, complemented

C₂×SD₃₂

C₂×C₁₀₉

yes

complemented

C₂×D₁₆

C₂×C₁₀₉

yes

characteristic

#495

C₂×C₁₀₉

yes

complemented

C₄×C₁₆

C₂×C₁₀₉

yes

characteristic, complemented

C₂×Q₁₆

C₂²×C₁₀₉

yes

—

C₂×SD₁₆

—

C₂×D₈

C₂²×C₁₀₉

yes

—

#124

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₂×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₄×C₈

C₂²×C₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

D₄×C₁₀₉

yes

characteristic

C₂×C₈

C₂³×C₁₀₉

yes

characteristic

C₂²⋊C₄

—

C₄²

D₄×C₁₀₉

yes

characteristic

C₁₆

D₄×C₁₀₉

yes

—

C₁₆

D₄×C₁₀₉

yes

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₈×C₁₀₉

yes

—

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

C₈

D₈×C₁₀₉

yes

—

C₈

C₂×D₄×C₁₀₉

yes

characteristic

C₈

C₂×D₄×C₁₀₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×D₈×C₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

#127×C₁₀₉

yes

characteristic

C₄

C₂×D₈×C₁₀₉

yes

characteristic

C₂

—

C₂

#493×C₁₀₉

yes

characteristic

100

C₂

#508×C₁₀₉

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₀₉
prime radical	C₂×D₁₆×C₁₀₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₄×C₈×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₁₀₉

prime radical

C₂×D₁₆×C₁₀₉

C₂

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₄×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₂×Q₃₂×C₁₀₉	C₂	semidirect
2. C₂×SD₃₂×C₁₀₉	C₂	semidirect
4. #495×C₁₀₉	C₂	semidirect
5. C₄×C₁₆×C₁₀₉	C₂	semidirect
50. #2249	C₁₀₉	direct (N × H)
51. C₁₀₉	#2249	direct (N × H)
52. C₂×Q₃₂	C₂×C₁₀₉	semidirect
53. C₂×SD₃₂	C₂×C₁₀₉	semidirect
55. #495	C₂×C₁₀₉	semidirect
56. C₄×C₁₆	C₂×C₁₀₉	semidirect

H ≅ G/N

type

1. C₂×Q₃₂×C₁₀₉

C₂

semidirect

2. C₂×SD₃₂×C₁₀₉

C₂

semidirect

4. #495×C₁₀₉

C₂

semidirect

5. C₄×C₁₆×C₁₀₉

C₂

semidirect

50. #2249

C₁₀₉

direct (N × H)

51. C₁₀₉

#2249

direct (N × H)

52. C₂×Q₃₂

C₂×C₁₀₉

semidirect

53. C₂×SD₃₂

C₂×C₁₀₉

semidirect

55. #495

C₂×C₁₀₉

semidirect

56. C₄×C₁₆

C₂×C₁₀₉

semidirect

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

subgroup	index
C₂×Q₃₂×C₁₀₉	2
C₂×SD₃₂×C₁₀₉	2
C₂×D₁₆×C₁₀₉	2
#495×C₁₀₉	2
C₄×C₁₆×C₁₀₉	2
#2249	109

subgroup

index

C₂×Q₃₂×C₁₀₉

C₂×SD₃₂×C₁₀₉

C₂×D₁₆×C₁₀₉

#495×C₁₀₉

C₄×C₁₆×C₁₀₉

#2249

109

Subgroups — C₁₀₉×#2249 #28359503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#2249 #28359503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2249 #28359503