Group #20295229 — subgroups

Subgroups — #20295229

Order 6976.

120 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 43 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×#13182017	C₂	1	1	yes	maximal, characteristic
2	#16812697	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812628	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812624	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812623	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	—	2	1	no	—
10	#13182017	—	2	2	no	—
11	C₂×#11724784	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
13	#13182008	—	2	1	no	—
14	#13182008	—	2	1	no	—
15	C₂²⋊Dic₁₀₉	—	2	1	no	—
16	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₀₉	—	2	2	no	—
20	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₉	—	2	1	no	—
22	Q₈×C₁₀₉	—	2	1	no	—
23	Q₈×C₁₀₉	—	2	1	no	—
24	D₄×C₁₀₉	—	2	1	no	—
25	D₄×C₁₀₉	D₄	1	2	yes	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	1	no	—
30	C₂×Dic₁₀₉	—	4	2	no	—
31	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
33	Dic₂₁₈	—	2	1	no	—
34	Dic₂₁₈	—	2	1	no	—
35	C₈×C₁₀₉	—	2	1	no	—
36	#11724784	—	2	1	no	—
37	C₂²×C₁₀₉	—	2	2	no	—
38	C₂²×C₁₀₉	—	4	1	no	—
39	C₂²×C₁₀₉	—	4	1	no	—
40	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₄×C₁₀₉	—	4	1	no	—
42	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₀₉	#41	1	1	yes	characteristic
44	Dic₁₀₉	—	2	1	no	—
45	Dic₁₀₉	—	4	1	no	—
46	Dic₁₀₉	—	4	1	no	—
47	Dic₁₀₉	SD₁₆	1	2	yes	complemented
48	C₂×C₁₀₉	—	4	2	no	—
49	C₂×C₁₀₉	#121	1	1	yes	characteristic
50	C₂×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
51	C₂×C₁₀₉	#137	1	1	yes	characteristic
52	C₁₀₉	#461	1	1	yes	characteristic, complemented, Sylow
53	#461	—	109	1	no	maximal, Sylow
54	C₂×SD₁₆	—	109	1	no	—
55	C₂×SD₁₆	D₁₀₉	1	1	yes	characteristic
56	C₄⋊Q₈	—	109	1	no	—
57	C₄×D₄	—	109	1	no	—
58	C₄⋊C₈	—	109	1	no	—
59	#103	—	109	1	no	—
60	#103	—	109	1	no	—
61	C₂×Q₈	—	109	1	no	—
62	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
63	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
64	C₂²×C₄	—	218	1	no	—
65	SD₁₆	—	2	2	no	—
66	SD₁₆	—	218	2	no	—
67	C₂×C₈	—	109	1	no	—
68	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
69	C₄⋊C₄	—	109	1	no	—
70	C₄⋊C₄	—	218	1	no	—
71	C₄⋊C₄	—	218	1	no	—
72	C₂²⋊C₄	—	218	1	no	—
73	C₄²	—	109	1	no	—
74	C₂³	—	2	1	no	—
75	Q₈	—	2	1	no	—
76	Q₈	—	2	1	no	—
77	Q₈	—	218	1	no	—
78	Q₈	—	218	1	no	—
79	D₄	—	2	1	no	—
80	D₄	C₂²⋊D₁₀₉	1	2	yes	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	109	1	no	—
83	C₂×C₄	—	109	1	no	—
84	C₂×C₄	—	436	2	no	—
85	C₂×C₄	—	218	1	no	—
86	C₂×C₄	—	218	1	no	—
87	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	218	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	4	1	no	—
93	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
94	C₄	—	4	1	no	—
95	C₄	—	109	2	no	—
96	C₄	—	436	1	no	—
97	C₄	—	436	1	no	—
98	C₄	—	218	1	no	—
99	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
100	C₄	#13182028	1	1	yes	characteristic
101	C₂	—	4	2	no	—
102	C₂	#16812671	1	1	yes	characteristic
103	C₂	SD₁₆×D₁₀₉	1	1	yes	characteristic
104	C₂	#16812728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182017

C₂

yes

maximal, characteristic

#16812697

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₀₉

C₂

yes

maximal, characteristic

#16812628

C₂

yes

maximal, characteristic, complemented

#16812624

C₂

yes

maximal, characteristic, complemented

#16812623

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

#13182017

—

C₂×#11724784

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182008

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

SD₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₂×Dic₁₀₉

D₄

yes

characteristic

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

SD₁₆

yes

complemented

C₂×C₁₀₉

—

C₂×C₁₀₉

#121

yes

characteristic

C₂×C₁₀₉

C₂×SD₁₆

yes

C₂×C₁₀₉

yes

C₁₀₉

yes

characteristic, complemented, Sylow

#461

—

109

maximal, Sylow

C₂×SD₁₆

—

109

—

C₂×SD₁₆

D₁₀₉

yes

characteristic

C₄⋊Q₈

—

109

—

C₄×D₄

—

109

—

C₄⋊C₈

—

109

—

#103

—

109

—

#103

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

218

—

SD₁₆

—

SD₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

218

—

Q₈

—

218

—

D₄

—

D₄

C₂²⋊D₁₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₀₉

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

436

—

C₄

—

436

—

C₄

—

218

—

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

100

C₄

#13182028

yes

characteristic

101

C₂

—

102

C₂

#16812671

yes

characteristic

103

C₂

SD₁₆×D₁₀₉

yes

characteristic

104

C₂

#16812728

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×SD₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	#461
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

C₂×SD₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₀₉

hypocenter

C₁₀₉

#461

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
2. #16812697	C₂	semidirect
4. #16812628	C₂	semidirect
5. #16812624	C₂	semidirect
6. #16812623	C₂	semidirect
47. Dic₁₀₉	SD₁₆	semidirect
52. C₁₀₉	#461	semidirect

H ≅ G/N

type

2. #16812697

C₂

semidirect

4. #16812628

C₂

semidirect

5. #16812624

C₂

semidirect

6. #16812623

C₂

semidirect

47. Dic₁₀₉

SD₁₆

semidirect

52. C₁₀₉

#461

semidirect

Subgroups — #20295229

Conjugacy classes of subgroups

Complements