Group #28359418 — subgroups

Subgroups — C₁₀₉×#2164 #28359418

Order 13952.

138 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 96 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	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₁₆×C₁₀₉	C₂	1	2	yes	maximal
4	#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	#117×C₁₀₉	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
11	#97×C₁₀₉	C₄	1	1	yes	characteristic
12	#97×C₁₀₉	C₄	1	1	yes	characteristic
13	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	M₁₆×C₁₀₉	—	2	2	no	—
17	C₂×C₈×C₁₀₉	—	2	2	no	—
18	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₉	C₈	1	2	yes	—
23	C₂²⋊C₄×C₁₀₉	C₈	1	2	yes	—
24	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₁₆×C₁₀₉	—	2	2	no	—
27	C₁₆×C₁₀₉	—	2	2	no	—
28	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₀₉	—	2	1	no	—
36	C₈×C₁₀₉	—	2	1	no	—
37	C₈×C₁₀₉	M₁₆	1	2	yes	—
38	C₈×C₁₀₉	#41	1	2	yes	—
39	C₂²×C₁₀₉	—	2	1	no	—
40	C₂²×C₁₀₉	—	2	2	no	—
41	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₀₉	—	2	1	no	—
43	C₄×C₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	—	2	2	no	—
45	C₄×C₁₀₉	#117	1	1	yes	characteristic
46	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₁₀₉	—	4	1	no	—
48	C₂×C₁₀₉	#431	1	1	yes	characteristic
49	C₂×C₁₀₉	#504	1	2	yes	—
50	#2164	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₁₀₉	#2164	1	1	yes	characteristic, direct product factor, Sylow
52	#405	C₂×C₁₀₉	1	1	yes	characteristic
53	C₄⋊C₁₆	C₂×C₁₀₉	1	2	yes	complemented
54	C₂²⋊C₁₆	C₂×C₁₀₉	1	2	yes	—
55	#346	C₂×C₁₀₉	1	1	yes	characteristic, complemented
56	C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
57	C₂×M₁₆	C₄×C₁₀₉	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
59	#117	C₄×C₁₀₉	1	1	yes	characteristic
60	C₂×C₁₆	C₂²×C₁₀₉	1	2	yes	—
61	C₂×C₁₆	C₂²×C₁₀₉	1	2	yes	—
62	#97	C₄×C₁₀₉	1	1	yes	characteristic
63	#97	C₄×C₁₀₉	1	1	yes	characteristic
64	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
65	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
66	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
67	M₁₆	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
70	C₂×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
71	C₂×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
72	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
73	C₄⋊C₄	C₈×C₁₀₉	1	2	yes	—
74	C₂²⋊C₄	C₈×C₁₀₉	1	2	yes	—
75	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
76	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	2	2	no	—
79	C₂³	C₂×C₈×C₁₀₉	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
83	C₂×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
84	C₂×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
85	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	M₁₆×C₁₀₉	1	2	yes	—
89	C₈	#41×C₁₀₉	1	2	yes	—
90	C₂²	—	2	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	#117×C₁₀₉	1	1	yes	characteristic
97	C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#431×C₁₀₉	1	1	yes	characteristic
100	C₂	#504×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#405×C₁₀₉

C₂

yes

maximal, characteristic

C₄⋊C₁₆×C₁₀₉

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₁₀₉

C₂

yes

maximal

#346×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₄×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×M₁₆×C₁₀₉

C₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₄

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

—

C₂×C₁₆×C₁₀₉

C₂²

yes

—

#97×C₁₀₉

C₄

yes

characteristic

#97×C₁₀₉

C₄

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₈

yes

—

C₂²⋊C₄×C₁₀₉

C₈

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#431

yes

characteristic

C₂×C₁₀₉

#504

yes

—

#2164

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#2164

yes

characteristic, direct product factor, Sylow

#405

C₂×C₁₀₉

yes

characteristic

C₄⋊C₁₆

C₂×C₁₀₉

yes

complemented

C₂²⋊C₁₆

C₂×C₁₀₉

yes

—

#346

C₂×C₁₀₉

yes

characteristic, complemented

C₄×C₁₆

C₂×C₁₀₉

yes

characteristic, complemented

C₂×M₁₆

C₄×C₁₀₉

yes

characteristic

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

#117

C₄×C₁₀₉

yes

characteristic

C₂×C₁₆

C₂²×C₁₀₉

yes

—

C₂×C₁₆

C₂²×C₁₀₉

yes

—

#97

C₄×C₁₀₉

yes

characteristic

#97

C₄×C₁₀₉

yes

characteristic

C₄×C₈

C₂²×C₁₀₉

yes

characteristic

C₄×C₈

C₂²×C₁₀₉

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₀₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₀₉

yes

characteristic

C₂×C₈

C₂×C₄×C₁₀₉

yes

characteristic

C₂×C₈

C₂×C₄×C₁₀₉

yes

characteristic

C₂×C₈

C₂³×C₁₀₉

yes

characteristic

C₄⋊C₄

C₈×C₁₀₉

yes

—

C₂²⋊C₄

C₈×C₁₀₉

yes

—

C₄²

C₂×C₄×C₁₀₉

yes

characteristic

C₄²

C₂×C₄×C₁₀₉

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂×C₈×C₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₈×C₁₀₉

yes

—

C₂×C₄

C₂×C₈×C₁₀₉

yes

—

C₂×C₄

C₂×C₈×C₁₀₉

yes

characteristic

C₂×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

C₈

—

C₈

—

C₈

M₁₆×C₁₀₉

yes

—

C₈

#41×C₁₀₉

yes

—

C₂²

—

C₂²

—

C₂²

C₂²×C₈×C₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#117×C₁₀₉

yes

characteristic

C₄

C₂×M₁₆×C₁₀₉

yes

characteristic

C₂

—

C₂

#431×C₁₀₉

yes

characteristic

100

C₂

#504×C₁₀₉

yes

—

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

subgroup

G/H

center

C₂×C₈×C₁₀₉

C₂³

commutator

C₂²

C₂²×C₈×C₁₀₉

Frattini, cosocle

C₂×C₈

C₂³×C₁₀₉

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂³×C₁₀₉

C₂×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. C₄⋊C₁₆×C₁₀₉	C₂	semidirect
4. #346×C₁₀₉	C₂	semidirect
5. C₄×C₁₆×C₁₀₉	C₂	semidirect
50. #2164	C₁₀₉	direct (N × H)
51. C₁₀₉	#2164	direct (N × H)
53. C₄⋊C₁₆	C₂×C₁₀₉	semidirect
55. #346	C₂×C₁₀₉	semidirect
56. C₄×C₁₆	C₂×C₁₀₉	semidirect

H ≅ G/N

type

2. C₄⋊C₁₆×C₁₀₉

C₂

semidirect

4. #346×C₁₀₉

C₂

semidirect

5. C₄×C₁₆×C₁₀₉

C₂

semidirect

50. #2164

C₁₀₉

direct (N × H)

51. C₁₀₉

#2164

direct (N × H)

53. C₄⋊C₁₆

C₂×C₁₀₉

semidirect

55. #346

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
#405×C₁₀₉	2
C₄⋊C₁₆×C₁₀₉	2
C₂²⋊C₁₆×C₁₀₉	2
#346×C₁₀₉	2
C₄×C₁₆×C₁₀₉	2
#2164	109

subgroup

index

#405×C₁₀₉

C₄⋊C₁₆×C₁₀₉

C₂²⋊C₁₆×C₁₀₉

#346×C₁₀₉

C₄×C₁₆×C₁₀₉

#2164

109

Subgroups — C₁₀₉×#2164 #28359418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#2164 #28359418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2164 #28359418