Group #20295899 — subgroups

Subgroups — #20295899

Order 6976.

114 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 46 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×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#89137	—	2	1	no	—
5	D₁₀₉⋊C₈	—	2	1	no	—
6	D₁₀₉⋊C₈	C₄	1	2	yes	complemented
7	C₂×#11724786	—	2	1	no	—
8	C₄⋊#89137	—	2	1	no	—
9	C₄⋊#89137	C₄	1	2	yes	complemented
10	C₂×C₄×D₁₀₉	—	2	1	no	—
11	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic, complemented
13	#13182008	—	2	1	no	—
14	C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
16	C₂×#89137	—	2	2	no	—
17	C₂×#89137	—	4	1	no	—
18	C₂×#89137	—	4	1	no	—
19	C₂×C₄×C₁₀₉	—	2	1	no	—
20	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	2	1	no	—
22	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
23	C₄×D₁₀₉	—	4	1	no	—
24	C₄×D₁₀₉	—	4	1	no	—
25	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×D₁₀₉	D₄	1	1	yes	characteristic
28	C₄×D₁₀₉	Q₈	1	1	yes	characteristic
29	#11724786	—	2	2	no	—
30	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₁₀₉	D₈	1	1	yes	characteristic
34	C₂×D₁₀₉	SD₁₆	1	1	yes	characteristic
35	C₂×D₁₀₉	SD₁₆	1	1	yes	characteristic
36	C₂×D₁₀₉	Q₁₆	1	1	yes	characteristic
37	#89137	—	4	2	no	—
38	C₄×C₁₀₉	—	4	1	no	—
39	C₄×C₁₀₉	C₄²	1	1	yes	characteristic
40	C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
41	Dic₁₀₉	—	4	1	no	—
42	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
43	Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₀₉	#95	1	1	yes	characteristic
45	C₂×C₁₀₉	#102	1	1	yes	characteristic
46	C₂×C₁₀₉	#103	1	1	yes	characteristic
47	D₁₀₉	#102	1	1	yes	characteristic
48	D₁₀₉	#103	1	1	yes	characteristic
49	D₁₀₉	#106	1	1	yes	characteristic
50	D₁₀₉	#107	1	1	yes	characteristic
51	C₁₀₉	#340	1	1	yes	characteristic, complemented, Sylow
52	#340	—	109	1	no	maximal, Sylow
53	C₂²×C₈	—	109	1	no	—
54	C₂×C₄⋊C₄	—	109	1	no	—
55	C₂×C₄⋊C₄	—	109	1	no	—
56	C₂²×C₄	—	109	1	no	—
57	C₂²×C₄	—	218	1	no	—
58	C₂²×C₄	—	218	1	no	—
59	C₂×C₈	—	109	2	no	—
60	C₂×C₈	—	218	1	no	—
61	C₂×C₈	—	218	1	no	—
62	C₄⋊C₄	—	109	2	no	—
63	C₄⋊C₄	—	109	1	no	—
64	C₄⋊C₄	—	218	1	no	—
65	C₄⋊C₄	—	218	1	no	—
66	C₄⋊C₄	#89137	1	1	yes	characteristic, complemented
67	C₂³	—	109	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	109	1	no	—
70	C₂×C₄	—	109	1	no	—
71	C₂×C₄	—	109	1	no	—
72	C₂×C₄	—	109	1	no	—
73	C₂×C₄	—	109	1	no	—
74	C₂×C₄	—	436	1	no	—
75	C₂×C₄	—	436	1	no	—
76	C₂×C₄	—	436	1	no	—
77	C₂×C₄	—	436	1	no	—
78	C₂×C₄	—	218	2	no	—
79	C₂×C₄	—	218	1	no	—
80	C₂×C₄	C₂×#89137	1	1	yes	characteristic
81	C₈	—	218	2	no	—
82	C₂²	—	109	1	no	—
83	C₂²	—	109	1	no	—
84	C₂²	—	109	1	no	—
85	C₂²	—	109	1	no	—
86	C₂²	—	109	1	no	—
87	C₂²	—	109	1	no	—
88	C₂²	C₂²⋊#89137	1	1	yes	characteristic
89	C₄	—	4	1	no	—
90	C₄	—	109	1	no	—
91	C₄	—	109	1	no	—
92	C₄	—	436	2	no	—
93	C₄	—	436	1	no	—
94	C₄	C₄×#89137	1	1	yes	characteristic
95	C₄	C₄⋊#89137	1	1	yes	characteristic
96	C₂	—	109	1	no	—
97	C₂	—	109	1	no	—
98	C₂	—	109	1	no	—
99	C₂	—	109	1	no	—
100	C₂	(C₂×#89137)⋊C₄	1	1	yes	characteristic
101	C₂	#16812761	1	1	yes	characteristic
102	C₂	#16812762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊#89137

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₀₉

C₂

yes

maximal, characteristic

C₂²×#89137

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

C₄

yes

—

—

C₄

yes

complemented

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₄

yes

characteristic, complemented

#13182008

—

C₄⋊C₄×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×D₁₀₉

D₄

yes

—

—

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₄×D₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

Q₈

yes

characteristic

#11724786

—

C₂²×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

D₈

yes

characteristic

C₂×D₁₀₉

SD₁₆

yes

characteristic

C₂×D₁₀₉

SD₁₆

yes

characteristic

C₂×D₁₀₉

Q₁₆

yes

characteristic

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄²

yes

characteristic

C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

characteristic

Dic₁₀₉

C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

C₁₀₉

yes

characteristic, complemented, Sylow

#340

—

109

maximal, Sylow

C₂²×C₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

#89137

yes

characteristic, complemented

C₂³

—

109

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×#89137

yes

characteristic

C₈

—

218

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

C₂²⋊#89137

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

109

—

C₄

—

436

—

C₄

—

436

—

C₄

yes

C₄

yes

C₂

—

109

—

C₂

—

109

—

C₂

—

109

—

C₂

—

109

—

100

C₂

(C₂×#89137)⋊C₄

yes

characteristic

101

C₂

#16812761

yes

characteristic

102

C₂

#16812762

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₄²
Frattini	C₂×C₄	C₂×#89137
Fitting	C₄⋊C₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²⋊C₄
hypercenter	C₄⋊C₄	#89137
hypocenter	C₁₀₉	#340
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

subgroup

G/H

center

C₂²

C₂²⋊#89137

commutator

C₄×C₁₀₉

C₄²

Frattini

C₂×C₄

C₂×#89137

Fitting

C₄⋊C₄×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²⋊C₄

C₄⋊C₄

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

N	H ≅ G/N	type
6. D₁₀₉⋊C₈	C₄	semidirect
9. C₄⋊#89137	C₄	semidirect
12. C₄⋊Dic₁₀₉	C₄	semidirect
14. C₄⋊C₄×C₁₀₉	C₄	semidirect
51. C₁₀₉	#340	semidirect
66. C₄⋊C₄	#89137	semidirect

H ≅ G/N

type

6. D₁₀₉⋊C₈

C₄

semidirect

9. C₄⋊#89137

C₄

semidirect

12. C₄⋊Dic₁₀₉

C₄

semidirect

14. C₄⋊C₄×C₁₀₉

C₄

semidirect

51. C₁₀₉

#340

semidirect

66. C₄⋊C₄

#89137

semidirect

Subgroups — #20295899

Conjugacy classes of subgroups

Complements