Group #42130507 — subgroups

Subgroups — C₁₀₉×#12513 #42130507

Order 27904.

290 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₁₀₉×#2240	C₂	1	4	yes	maximal, complemented
2	C₁₀₉×#1712	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#500×C₁₀₉	C₂²	1	2	yes	—
6	#493×C₁₀₉	—	2	2	no	—
7	#493×C₁₀₉	C₂²	1	2	yes	—
8	#487×C₁₀₉	—	2	2	no	—
9	#357×C₁₀₉	—	2	8	no	—
10	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
11	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₁₀₉	—	2	4	no	—
13	C₂×D₈×C₁₀₉	—	4	4	no	—
14	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
15	#127×C₁₀₉	—	2	2	no	—
16	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
17	#107×C₁₀₉	—	2	4	no	—
18	#103×C₁₀₉	—	4	4	no	—
19	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
20	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₁₀₉	—	4	2	no	—
22	C₂×D₄×C₁₀₉	—	4	2	no	—
23	C₂×D₄×C₁₀₉	—	4	4	no	—
24	D₈×C₁₀₉	—	4	8	no	—
25	D₈×C₁₀₉	—	8	8	no	—
26	C₂×C₈×C₁₀₉	D₈	1	4	yes	—
27	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
28	C₄⋊C₄×C₁₀₉	—	4	2	no	—
29	C₄⋊C₄×C₁₀₉	—	4	2	no	—
30	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₁₆×C₁₀₉	—	2	4	no	—
32	C₂³×C₁₀₉	—	8	2	no	—
33	Q₈×C₁₀₉	—	8	4	no	—
34	D₄×C₁₀₉	—	8	4	no	—
35	D₄×C₁₀₉	—	8	8	no	—
36	C₂×C₄×C₁₀₉	—	8	2	no	—
37	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
39	C₈×C₁₀₉	—	2	4	no	—
40	C₈×C₁₀₉	#135	1	4	yes	—
41	C₂²×C₁₀₉	—	16	2	no	—
42	C₂²×C₁₀₉	—	16	4	no	—
43	C₂²×C₁₀₉	#493	1	1	yes	characteristic
44	C₄×C₁₀₉	—	16	2	no	—
45	C₄×C₁₀₉	#495	1	2	yes	—
46	C₄×C₁₀₉	#509	1	4	yes	—
47	#12513	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₁₀₉	—	32	2	no	—
49	C₂×C₁₀₉	#1709	1	1	yes	characteristic
50	C₂×C₁₀₉	#2250	1	2	yes	—
51	#2240	C₂×C₁₀₉	1	4	yes	complemented
52	#1712	C₂×C₁₀₉	1	1	yes	characteristic
53	#1710	C₂×C₁₀₉	1	1	yes	characteristic, complemented
54	#1311	C₂×C₁₀₉	1	1	yes	characteristic, complemented
55	C₁₀₉	#12513	1	1	yes	characteristic, direct product factor, Sylow
56	#500	C₂²×C₁₀₉	1	2	yes	—
57	#493	—	2	2	no	—
58	#493	C₂²×C₁₀₉	1	2	yes	—
59	#487	—	2	2	no	—
60	#357	—	2	8	no	—
61	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
62	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
63	C₂×D₈	—	2	4	no	—
64	C₂×D₈	—	4	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	2	2	no	—
67	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
71	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	4	2	no	—
74	C₂×D₄	—	4	4	no	—
75	D₈	—	4	8	no	—
76	D₈	—	8	8	no	—
77	C₂×C₈	D₈×C₁₀₉	1	4	yes	—
78	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	8	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	8	8	no	—
86	D₄	—	8	4	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
89	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#135×C₁₀₉	1	4	yes	—
92	C₂²	—	16	2	no	—
93	C₂²	—	16	4	no	—
94	C₂²	#493×C₁₀₉	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#495×C₁₀₉	1	2	yes	—
97	C₄	#509×C₁₀₉	1	4	yes	—
98	C₂	—	32	2	no	—
99	C₂	C₁₀₉×#1709	1	1	yes	characteristic
100	C₂	C₁₀₉×#2250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2240

C₂

yes

maximal, complemented

C₁₀₉×#1712

C₂

yes

maximal, characteristic

C₁₀₉×#1710

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1311

C₂

yes

maximal, characteristic, complemented

#500×C₁₀₉

C₂²

yes

—

#493×C₁₀₉

—

#493×C₁₀₉

C₂²

yes

—

#487×C₁₀₉

—

#357×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

—

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

#127×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#107×C₁₀₉

—

#103×C₁₀₉

—

C₄×C₈×C₁₀₉

D₄

yes

—

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#127

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#493

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#495

yes

—

C₄×C₁₀₉

#509

yes

—

#12513

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1709

yes

characteristic

C₂×C₁₀₉

#2250

yes

—

#2240

C₂×C₁₀₉

yes

complemented

#1712

C₂×C₁₀₉

yes

characteristic

#1710

C₂×C₁₀₉

yes

characteristic, complemented

#1311

C₂×C₁₀₉

yes

characteristic, complemented

C₁₀₉

#12513

yes

characteristic, direct product factor, Sylow

#500

C₂²×C₁₀₉

yes

—

#493

—

#493

C₂²×C₁₀₉

yes

—

#487

—

#357

—

C₄×C₁₆

C₂²×C₁₀₉

yes

—

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

#107

—

#103

—

C₄×C₈

D₄×C₁₀₉

yes

—

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

D₈×C₁₀₉

yes

—

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#127×C₁₀₉

yes

characteristic

C₂×C₄

C₂×D₈×C₁₀₉

yes

—

C₈

—

C₈

#135×C₁₀₉

yes

—

C₂²

—

C₂²

—

C₂²

#493×C₁₀₉

yes

characteristic

C₄

—

C₄

#495×C₁₀₉

yes

—

C₄

#509×C₁₀₉

yes

—

C₂

—

C₂

C₁₀₉×#1709

yes

characteristic

100

C₂

C₁₀₉×#2250

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
prime radical	C₁₀₉×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

prime radical

C₁₀₉×#1712

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₁₀₉×#2240	C₂	semidirect
3. C₁₀₉×#1710	C₂	semidirect
4. C₁₀₉×#1311	C₂	semidirect
47. #12513	C₁₀₉	direct (N × H)
51. #2240	C₂×C₁₀₉	semidirect
53. #1710	C₂×C₁₀₉	semidirect
54. #1311	C₂×C₁₀₉	semidirect
55. C₁₀₉	#12513	direct (N × H)

H ≅ G/N

type

1. C₁₀₉×#2240

C₂

semidirect

3. C₁₀₉×#1710

C₂

semidirect

4. C₁₀₉×#1311

C₂

semidirect

47. #12513

C₁₀₉

direct (N × H)

51. #2240

C₂×C₁₀₉

semidirect

53. #1710

C₂×C₁₀₉

semidirect

54. #1311

C₂×C₁₀₉

semidirect

55. C₁₀₉

#12513

direct (N × H)

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

subgroup	index
C₁₀₉×#2240	2
C₁₀₉×#1712	2
C₁₀₉×#1710	2
C₁₀₉×#1311	2
#12513	109

subgroup

index

C₁₀₉×#2240

C₁₀₉×#1712

C₁₀₉×#1710

C₁₀₉×#1311

#12513

109

Subgroups — C₁₀₉×#12513 #42130507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12513 #42130507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12513 #42130507