Group #28359896 — subgroups

Subgroups — C₁₀₉×#2642 #28359896

Order 13952.

416 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 200 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×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#400×C₁₀₉	C₂	1	8	yes	maximal
4	#391×C₁₀₉	C₂	1	4	yes	maximal
5	#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
8	#125×C₁₀₉	—	2	4	no	—
9	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	#95×C₁₀₉	C₂²	1	4	yes	—
15	#95×C₁₀₉	C₂²	1	16	yes	—
16	C₂×Q₈×C₁₀₉	—	2	4	no	—
17	C₂×Q₈×C₁₀₉	—	2	8	no	—
18	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
19	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
20	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
22	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₀₉	—	2	8	no	—
24	C₄⋊C₄×C₁₀₉	—	2	4	no	—
25	C₄⋊C₄×C₁₀₉	—	2	8	no	—
26	C₄²×C₁₀₉	—	2	4	no	—
27	C₄²×C₁₀₉	D₄	1	4	yes	—
28	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
29	Q₈×C₁₀₉	—	4	8	no	—
30	C₂×C₄×C₁₀₉	—	2	4	no	—
31	C₂×C₄×C₁₀₉	—	2	4	no	—
32	C₂×C₄×C₁₀₉	—	2	4	no	—
33	C₂×C₄×C₁₀₉	—	2	4	no	—
34	C₂×C₄×C₁₀₉	—	2	8	no	—
35	C₂×C₄×C₁₀₉	—	2	8	no	—
36	C₂×C₄×C₁₀₉	—	2	8	no	—
37	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	#41	1	4	yes	—
40	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
41	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
42	C₂²×C₁₀₉	2₋¹⁺⁴	1	4	yes	—
43	C₄×C₁₀₉	—	2	4	no	—
44	C₄×C₁₀₉	—	4	4	no	—
45	C₄×C₁₀₉	—	4	4	no	—
46	C₄×C₁₀₉	—	4	4	no	—
47	C₄×C₁₀₉	#124	1	4	yes	—
48	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
49	C₂×C₁₀₉	#536	1	2	yes	—
50	C₂×C₁₀₉	#541	1	4	yes	—
51	#2642	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₁₀₉	#2642	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic
54	C₂×#125	C₂×C₁₀₉	1	1	yes	characteristic
55	#400	C₂×C₁₀₉	1	8	yes	—
56	#391	C₂×C₁₀₉	1	4	yes	—
57	#376	C₂×C₁₀₉	1	1	yes	characteristic
58	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
59	C₄⋊Q₈	—	2	4	no	—
60	#125	—	2	4	no	—
61	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
62	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
63	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
64	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
65	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
66	#95	C₂²×C₁₀₉	1	4	yes	—
67	#95	C₂²×C₁₀₉	1	16	yes	—
68	C₂×Q₈	—	2	4	no	—
69	C₂×Q₈	—	2	8	no	—
70	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
71	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
72	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
73	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
74	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
75	C₄⋊C₄	—	2	8	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	2	8	no	—
78	C₄²	—	2	4	no	—
79	C₄²	D₄×C₁₀₉	1	4	yes	—
80	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
81	Q₈	—	4	8	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	8	no	—
87	C₂×C₄	—	2	8	no	—
88	C₂×C₄	—	2	8	no	—
89	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
90	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
91	C₂×C₄	#41×C₁₀₉	1	4	yes	—
92	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
93	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
94	C₂²	2₋¹⁺⁴×C₁₀₉	1	4	yes	—
95	C₄	—	2	4	no	—
96	C₄	—	4	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	4	4	no	—
99	C₄	#124×C₁₀₉	1	4	yes	—
100	C₂	C₂×#124×C₁₀₉	1	1	yes	characteristic
101	C₂	#536×C₁₀₉	1	2	yes	—
102	C₂	#541×C₁₀₉	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#125×C₁₀₉

C₂

yes

maximal, characteristic

#400×C₁₀₉

C₂

yes

maximal

#391×C₁₀₉

C₂

yes

maximal

#376×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

—

#125×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

2₋¹⁺⁴

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₂×C₁₀₉

C₂×#124

yes

characteristic

C₂×C₁₀₉

#536

yes

—

C₂×C₁₀₉

#541

yes

—

#2642

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#2642

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₁₀₉

yes

characteristic

C₂×#125

C₂×C₁₀₉

yes

characteristic

#400

C₂×C₁₀₉

yes

—

#391

C₂×C₁₀₉

yes

—

#376

C₂×C₁₀₉

yes

characteristic

C₂²×Q₈

C₂²×C₁₀₉

yes

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

characteristic

#95

C₂²×C₁₀₉

yes

—

#95

C₂²×C₁₀₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂³×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₄×C₁₀₉

yes

—

C₂³

C₂⁴×C₁₀₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

C₂×C₄

#41×C₁₀₉

yes

—

C₂²

C₂²×D₄×C₁₀₉

yes

characteristic

C₂²

C₂×#41×C₁₀₉

yes

—

C₂²

2₋¹⁺⁴×C₁₀₉

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#124×C₁₀₉

yes

—

100

C₂

C₂×#124×C₁₀₉

yes

characteristic

101

C₂

#536×C₁₀₉

yes

—

102

C₂

#541×C₁₀₉

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₉

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₀₉

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ G

N	H ≅ G/N	type
51. #2642	C₁₀₉	direct (N × H)
52. C₁₀₉	#2642	direct (N × H)

H ≅ G/N

type

51. #2642

C₁₀₉

direct (N × H)

52. C₁₀₉

#2642

direct (N × H)

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

subgroup	index
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#125×C₁₀₉	2
#400×C₁₀₉	2
#391×C₁₀₉	2
#376×C₁₀₉	2
#2642	109

subgroup

index

C₂×C₄⋊Q₈×C₁₀₉

C₂×#125×C₁₀₉

#400×C₁₀₉

#391×C₁₀₉

#376×C₁₀₉

#2642

109

Subgroups — C₁₀₉×#2642 #28359896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#2642 #28359896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2642 #28359896