Group #20294961 — subgroups

Subgroups — #20294961

Order 6976.

186 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 61 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×#13182008	C₂	1	2	yes	maximal, complemented
2	C₂×#13182006	C₂	1	2	yes	maximal
3	C₂×#13182006	C₂	1	1	yes	maximal, characteristic
4	#16812606	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
9	#13182008	—	2	4	no	—
10	#13182006	—	2	2	no	—
11	#13182006	—	2	4	no	—
12	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₀₉	—	2	2	no	—
15	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
16	C₂²×D₁₀₉	—	2	2	no	—
17	C₂²×D₁₀₉	—	2	1	no	—
18	C₂²×D₁₀₉	—	2	2	no	—
19	C₂²×D₁₀₉	—	2	1	no	—
20	C₂²×D₁₀₉	Q₈	1	2	yes	—
21	C₂×C₄×C₁₀₉	—	2	2	no	—
22	C₂×C₄×C₁₀₉	—	2	2	no	—
23	C₂×C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
26	C₂×Dic₁₀₉	—	2	2	no	—
27	C₂×Dic₁₀₉	—	2	2	no	—
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	2	no	—
30	C₂×Dic₁₀₉	D₄	1	4	yes	—
31	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
32	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₁₀₉	#41	1	2	yes	—
35	C₂²×C₁₀₉	#41	1	1	yes	characteristic
36	C₂×D₁₀₉	—	2	2	no	—
37	C₂×D₁₀₉	—	2	4	no	—
38	C₂×D₁₀₉	—	4	1	no	—
39	C₂×D₁₀₉	—	4	2	no	—
40	C₂×D₁₀₉	—	4	1	no	—
41	C₄×C₁₀₉	—	2	2	no	—
42	C₄×C₁₀₉	—	4	2	no	—
43	Dic₁₀₉	—	2	4	no	—
44	Dic₁₀₉	—	4	1	no	—
45	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
47	C₂×C₁₀₉	#123	1	1	yes	characteristic
48	C₂×C₁₀₉	#123	1	2	yes	—
49	C₂×C₁₀₉	#127	1	1	yes	characteristic
50	D₁₀₉	—	4	2	no	—
51	C₁₀₉	#399	1	1	yes	characteristic, complemented, Sylow
52	#399	—	109	1	no	maximal, Sylow
53	C₂×C₄⋊C₄	—	109	2	no	—
54	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic, complemented
55	C₂×C₂²⋊C₄	—	109	2	no	—
56	C₂×C₂²⋊C₄	—	109	1	no	—
57	#95	—	109	1	no	—
58	C₂⁴	—	109	1	no	—
59	C₂²×C₄	—	109	2	no	—
60	C₂²×C₄	—	109	1	no	—
61	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
62	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	218	4	no	—
65	C₂²⋊C₄	—	218	4	no	—
66	C₂²⋊C₄	—	218	2	no	—
67	C₂³	—	109	2	no	—
68	C₂³	—	218	2	no	—
69	C₂³	—	218	1	no	—
70	C₂³	—	218	2	no	—
71	C₂³	—	218	1	no	—
72	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	109	4	no	—
78	C₂×C₄	—	218	2	no	—
79	C₂×C₄	—	218	1	no	—
80	C₂×C₄	—	218	2	no	—
81	C₂×C₄	—	218	2	no	—
82	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
83	C₂²	—	436	1	no	—
84	C₂²	—	436	2	no	—
85	C₂²	—	436	1	no	—
86	C₂²	—	218	2	no	—
87	C₂²	—	218	4	no	—
88	C₂²	D₄×D₁₀₉	1	2	yes	—
89	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
90	C₂²	Q₈×D₁₀₉	1	1	yes	characteristic
91	C₂²	#13182029	1	2	yes	—
92	C₂²	#13182030	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	4	2	no	—
95	C₄	—	436	1	no	—
96	C₄	—	218	4	no	—
97	C₂	—	436	2	no	—
98	C₂	#16812675	1	1	yes	characteristic
99	C₂	#16812677	1	2	yes	—
100	C₂	#16812682	1	1	yes	characteristic
101	C₂	#16812684	1	2	yes	—
102	C₂	#16812695	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182008

C₂

yes

maximal, complemented

C₂×#13182006

C₂

yes

maximal

C₂×#13182006

C₂

yes

maximal, characteristic

#16812606

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

#13182008

—

#13182006

—

#13182006

—

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

Q₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×Q₈

yes

characteristic

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

#41

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₁₀₉

#123

yes

characteristic

C₂×C₁₀₉

#123

yes

—

C₂×C₁₀₉

#127

yes

characteristic

D₁₀₉

—

C₁₀₉

#399

yes

characteristic, complemented, Sylow

#399

—

109

maximal, Sylow

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

D₁₀₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

#95

—

109

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²⋊D₁₀₉

yes

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

D₄×D₁₀₉

yes

—

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂²

Q₈×D₁₀₉

yes

characteristic

C₂²

#13182029

yes

—

C₂²

#13182030

yes

characteristic

C₄

—

C₄

—

C₄

—

436

—

C₄

—

218

—

C₂

—

436

—

C₂

#16812675

yes

characteristic

C₂

#16812677

yes

—

100

C₂

#16812682

yes

characteristic

101

C₂

#16812684

yes

—

102

C₂

#16812695

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₉
commutator, socle	C₂³×C₁₀₉	C₂³
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#399
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₉

commutator, socle

C₂³×C₁₀₉

C₂³

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₂

hypercenter

C₂×C₄⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×#13182008	C₂	semidirect
4. #16812606	C₂	semidirect
5. C₂×C₄⋊C₄×C₁₀₉	C₂	semidirect
51. C₁₀₉	#399	semidirect
54. C₂×C₄⋊C₄	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#13182008

C₂

semidirect

4. #16812606

C₂

semidirect

5. C₂×C₄⋊C₄×C₁₀₉

C₂

semidirect

51. C₁₀₉

#399

semidirect

54. C₂×C₄⋊C₄

D₁₀₉

semidirect

Subgroups — #20294961

Conjugacy classes of subgroups

Complements