Group #28359245 — subgroups

Subgroups — C₁₀₉×#1991 #28359245

Order 13952.

252 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 128 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×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#340×C₁₀₉	C₂	1	4	yes	maximal
5	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
6	#125×C₁₀₉	—	2	2	no	—
7	#125×C₁₀₉	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₀₉	—	2	2	no	—
13	#95×C₁₀₉	—	2	2	no	—
14	C₂²×C₄×C₁₀₉	—	2	2	no	—
15	C₂²×C₄×C₁₀₉	—	2	2	no	—
16	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
17	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	—	2	2	no	—
19	C₂×C₈×C₁₀₉	—	2	2	no	—
20	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
21	C₄⋊C₄×C₁₀₉	—	2	8	no	—
22	C₄⋊C₄×C₁₀₉	—	2	2	no	—
23	C₄⋊C₄×C₁₀₉	—	2	4	no	—
24	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₄²×C₁₀₉	D₄	1	2	yes	—
27	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	4	no	—
29	C₂×C₄×C₁₀₉	—	2	4	no	—
30	C₂×C₄×C₁₀₉	—	4	4	no	—
31	C₂×C₄×C₁₀₉	—	4	2	no	—
32	C₂×C₄×C₁₀₉	—	4	4	no	—
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₀₉	#41	1	2	yes	—
38	C₈×C₁₀₉	—	2	4	no	—
39	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₉	#124	1	1	yes	characteristic
41	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₀₉	#135	1	4	yes	—
43	C₄×C₁₀₉	—	2	4	no	—
44	C₄×C₁₀₉	—	4	4	no	—
45	C₄×C₁₀₉	—	8	2	no	—
46	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
47	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
48	C₂×C₁₀₉	#391	1	1	yes	characteristic
49	C₂×C₁₀₉	#416	1	2	yes	—
50	C₂×C₁₀₉	#488	1	2	yes	—
51	C₂×C₁₀₉	#499	1	2	yes	—
52	#1991	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₁₀₉	#1991	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×#125	C₂×C₁₀₉	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic
56	#382	C₂×C₁₀₉	1	1	yes	characteristic
57	#340	C₂×C₁₀₉	1	4	yes	—
58	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
59	#125	—	2	2	no	—
60	#125	C₄×C₁₀₉	1	4	yes	complemented
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
63	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
64	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
65	C₄×C₈	—	2	2	no	—
66	#95	—	2	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	D₄×C₁₀₉	1	2	yes	—
70	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	Q₈×C₁₀₉	1	4	yes	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
78	C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
79	C₄²	D₄×C₁₀₉	1	2	yes	—
80	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	4	4	no	—
86	C₂×C₄	C₂²⋊C₄×C₁₀₉	1	4	yes	—
87	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
89	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
90	C₂×C₄	#41×C₁₀₉	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
93	C₂²	#124×C₁₀₉	1	1	yes	characteristic
94	C₂²	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
95	C₂²	#135×C₁₀₉	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	8	2	no	—
99	C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
100	C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
101	C₂	#391×C₁₀₉	1	1	yes	characteristic
102	C₂	#416×C₁₀₉	1	2	yes	—
103	C₂	#488×C₁₀₉	1	2	yes	—
104	C₂	#499×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₀₉

C₂

yes

maximal, characteristic

#382×C₁₀₉

C₂

yes

maximal, characteristic

#340×C₁₀₉

C₂

yes

maximal

C₂²×C₈×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

—

#125×C₁₀₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

—

#95×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

#124

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#135

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₁₀₉

#391

yes

characteristic

C₂×C₁₀₉

#416

yes

—

C₂×C₁₀₉

#488

yes

—

C₂×C₁₀₉

#499

yes

—

#1991

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#1991

yes

characteristic, direct product factor, Sylow

C₂×#125

C₂×C₁₀₉

yes

characteristic

C₂×C₄×C₈

C₂×C₁₀₉

yes

characteristic

#382

C₂×C₁₀₉

yes

characteristic

#340

C₂×C₁₀₉

yes

—

C₂²×C₈

C₂²×C₁₀₉

yes

—

#125

—

#125

C₄×C₁₀₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

characteristic

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₈×C₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₁₀₉

yes

—

C₄²

C₂×C₄×C₁₀₉

yes

—

C₄²

D₄×C₁₀₉

yes

—

C₂³

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₁₀₉

yes

—

C₂×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₄

C₂×Q₈×C₁₀₉

yes

—

C₂×C₄

#41×C₁₀₉

yes

—

C₈

—

C₂²

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

C₂²

#124×C₁₀₉

yes

characteristic

C₂²

C₄⋊Q₈×C₁₀₉

yes

characteristic

C₂²

#135×C₁₀₉

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₄×Q₈×C₁₀₉

yes

—

100

C₄

C₂²⋊Q₈×C₁₀₉

yes

—

101

C₂

#391×C₁₀₉

yes

characteristic

102

C₂

#416×C₁₀₉

yes

—

103

C₂

#488×C₁₀₉

yes

—

104

C₂

#499×C₁₀₉

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₉

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₉

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
7. #125×C₁₀₉	C₄	semidirect
52. #1991	C₁₀₉	direct (N × H)
53. C₁₀₉	#1991	direct (N × H)
60. #125	C₄×C₁₀₉	semidirect

H ≅ G/N

type

7. #125×C₁₀₉

C₄

semidirect

52. #1991

C₁₀₉

direct (N × H)

53. C₁₀₉

#1991

direct (N × H)

60. #125

C₄×C₁₀₉

semidirect

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

subgroup	index
C₂×#125×C₁₀₉	2
C₂×C₄×C₈×C₁₀₉	2
#382×C₁₀₉	2
#340×C₁₀₉	2
#1991	109

subgroup

index

C₂×#125×C₁₀₉

C₂×C₄×C₈×C₁₀₉

#382×C₁₀₉

#340×C₁₀₉

#1991

109

Subgroups — C₁₀₉×#1991 #28359245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#1991 #28359245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#1991 #28359245