Group #28360412 — subgroups

Subgroups — C₂×#499×C₁₀₉ #28360412

Order 13952.

360 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 232 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	2	yes	maximal
2	#499×C₁₀₉	C₂	1	8	yes	maximal, direct product factor
3	C₂×#107×C₁₀₉	C₂	1	2	yes	maximal
4	C₂×#106×C₁₀₉	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	#125×C₁₀₉	—	2	4	no	—
9	#125×C₁₀₉	C₂²	1	8	yes	—
10	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
11	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
13	#107×C₁₀₉	C₂²	1	8	yes	—
14	#106×C₁₀₉	C₂²	1	8	yes	—
15	C₄×C₈×C₁₀₉	C₂²	1	4	yes	—
16	C₂²×C₄×C₁₀₉	—	2	4	no	—
17	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
18	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
20	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
21	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
22	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
23	C₄⋊C₄×C₁₀₉	—	2	16	no	—
24	C₄⋊C₄×C₁₀₉	—	2	4	no	—
25	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
26	C₄²×C₁₀₉	D₄	1	2	yes	—
27	C₄²×C₁₀₉	C₂³	1	2	yes	—
28	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	2	4	no	—
30	C₂×C₄×C₁₀₉	—	2	8	no	—
31	C₂×C₄×C₁₀₉	—	4	8	no	—
32	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
37	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
38	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
39	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₁₀₉	#135	1	4	yes	—
41	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₁₀₉	—	2	4	no	—
43	C₄×C₁₀₉	—	4	8	no	—
44	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
45	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
47	C₂×C₁₀₉	#499	1	4	yes	direct product factor
48	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂×#135	1	2	yes	—
50	C₂×#499	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₁₀₉	C₂×#499	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×#125	C₂×C₁₀₉	1	2	yes	—
53	#499	C₂×C₁₀₉	1	8	yes	direct product factor
54	C₂×#107	C₂×C₁₀₉	1	2	yes	—
55	C₂×#106	C₂×C₁₀₉	1	2	yes	—
56	C₂×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
59	#125	—	2	4	no	—
60	#125	C₂²×C₁₀₉	1	8	yes	—
61	C₂×C₄⋊C₄	—	2	4	no	—
62	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
63	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
64	#107	C₂²×C₁₀₉	1	8	yes	—
65	#106	C₂²×C₁₀₉	1	8	yes	—
66	C₄×C₈	C₂²×C₁₀₉	1	4	yes	—
67	C₂²×C₄	—	2	4	no	—
68	C₂²×C₄	D₄×C₁₀₉	1	2	yes	—
69	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
70	C₂×C₈	Q₈×C₁₀₉	1	4	yes	—
71	C₂×C₈	Q₈×C₁₀₉	1	4	yes	—
72	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
73	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
74	C₄⋊C₄	—	2	16	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
77	C₄²	D₄×C₁₀₉	1	2	yes	—
78	C₄²	C₂³×C₁₀₉	1	2	yes	—
79	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	8	no	—
82	C₂×C₄	—	4	8	no	—
83	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
84	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
85	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
86	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
87	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
88	C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
89	C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
90	C₂²	C₄⋊Q₈×C₁₀₉	1	2	yes	—
91	C₂²	#135×C₁₀₉	1	4	yes	—
92	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	4	8	no	—
95	C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
96	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
97	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
98	C₂	#499×C₁₀₉	1	4	yes	direct product factor
99	C₂	C₂×C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
100	C₂	C₂×#135×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₁₀₉

C₂

yes

maximal

#499×C₁₀₉

C₂

yes

maximal, direct product factor

C₂×#107×C₁₀₉

C₂

yes

maximal

C₂×#106×C₁₀₉

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#125×C₁₀₉

—

#125×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₈×C₁₀₉

C₂×Q₈

yes

—

C₈×C₁₀₉

C₂×Q₈

yes

—

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

#499

yes

direct product factor

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#135

yes

—

C₂×#499

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

C₂×#499

yes

characteristic, direct product factor, Sylow

C₂×#125

C₂×C₁₀₉

yes

—

#499

C₂×C₁₀₉

yes

direct product factor

C₂×#107

C₂×C₁₀₉

yes

—

C₂×#106

C₂×C₁₀₉

yes

—

C₂×C₄×C₈

C₂×C₁₀₉

yes

characteristic

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

#125

—

#125

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

characteristic

#107

C₂²×C₁₀₉

yes

—

#106

C₂²×C₁₀₉

yes

—

C₄×C₈

C₂²×C₁₀₉

yes

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

C₂×C₈

Q₈×C₁₀₉

yes

—

C₂×C₈

Q₈×C₁₀₉

yes

—

C₂×C₈

C₂³×C₁₀₉

yes

—

C₂×C₈

C₂³×C₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³×C₁₀₉

yes

—

C₄²

D₄×C₁₀₉

yes

—

C₄²

C₂³×C₁₀₉

yes

—

C₂³

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₄

C₂×Q₈×C₁₀₉

yes

—

C₂×C₄

C₂×Q₈×C₁₀₉

yes

—

C₂×C₄

C₂⁴×C₁₀₉

yes

characteristic

C₈

C₂×Q₈×C₁₀₉

yes

—

C₈

C₂×Q₈×C₁₀₉

yes

—

C₂²

C₄⋊Q₈×C₁₀₉

yes

—

C₂²

#135×C₁₀₉

yes

—

C₂²

C₂²×D₄×C₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

C₄⋊Q₈×C₁₀₉

yes

—

C₄

C₂²×Q₈×C₁₀₉

yes

characteristic

C₄

C₂²×Q₈×C₁₀₉

yes

characteristic

C₂

#499×C₁₀₉

yes

direct product factor

C₂

C₂×C₄⋊Q₈×C₁₀₉

yes

characteristic

100

C₂

C₂×#135×C₁₀₉

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×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, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₀₉

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. #499×C₁₀₉	C₂	direct (N × H)
47. C₂×C₁₀₉	#499	direct (N × H)
50. C₂×#499	C₁₀₉	direct (N × H)
51. C₁₀₉	C₂×#499	direct (N × H)
53. #499	C₂×C₁₀₉	direct (N × H)
98. C₂	#499×C₁₀₉	direct (N × H)

H ≅ G/N

type

2. #499×C₁₀₉

C₂

direct (N × H)

47. C₂×C₁₀₉

#499

direct (N × H)

50. C₂×#499

C₁₀₉

direct (N × H)

51. C₁₀₉

C₂×#499

direct (N × H)

53. #499

C₂×C₁₀₉

direct (N × H)

98. C₂

#499×C₁₀₉

direct (N × H)

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

subgroup	index
C₂×#125×C₁₀₉	2
#499×C₁₀₉	2
C₂×#107×C₁₀₉	2
C₂×#106×C₁₀₉	2
C₂×C₄×C₈×C₁₀₉	2
C₂×#499	109

subgroup

index

C₂×#125×C₁₀₉

#499×C₁₀₉

C₂×#107×C₁₀₉

C₂×#106×C₁₀₉

C₂×C₄×C₈×C₁₀₉

C₂×#499

109

Subgroups — C₂×#499×C₁₀₉ #28360412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#499×C109 #28360412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#499×C₁₀₉ #28360412