Group #20296069 — subgroups

Subgroups — #20296069

Order 6976.

106 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 40 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	#16812787	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812786	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812769	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812764	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812763	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#89137	C₂	1	1	yes	maximal, characteristic, complemented
7	#16812720	C₂	1	1	yes	maximal, characteristic
8	#13182039	—	2	1	no	—
9	#13182039	C₂²	1	1	yes	characteristic
10	#13182039	C₂²	1	1	yes	characteristic
11	#13182038	—	2	1	no	—
12	D₁₀₉⋊C₈	—	2	1	no	—
13	D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#11724786	—	2	1	no	—
15	C₄×#89137	C₂²	1	1	yes	characteristic
16	#13182030	C₂²	1	1	yes	characteristic
17	#13182028	C₂²	1	1	yes	characteristic
18	#13182019	C₄	1	1	yes	characteristic, complemented
19	#13182018	C₄	1	1	yes	characteristic, complemented
20	#13182014	C₄	1	1	yes	characteristic, complemented
21	C₈×D₁₀₉	C₂²	1	1	yes	characteristic
22	SD₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
23	C₂×#89137	—	2	1	no	—
24	Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²⋊D₁₀₉	—	2	1	no	—
27	C₂×Dic₁₀₉	—	2	1	no	—
28	D₄₃₆	—	2	1	no	—
29	D₄₃₆	C₂×C₄	1	1	yes	characteristic
30	C₄×D₁₀₉	—	2	1	no	—
31	C₄×D₁₀₉	C₂³	1	1	yes	characteristic
32	Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
33	#11724786	—	2	1	no	—
34	#11724786	—	2	1	no	—
35	#11724786	D₄	1	1	yes	characteristic
36	#11724786	D₄	1	1	yes	characteristic
37	C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	#11724784	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₀₉	—	2	1	no	—
40	C₂×D₁₀₉	—	2	1	no	—
41	C₂×D₁₀₉	#41	1	1	yes	characteristic
42	#89137	—	2	2	no	—
43	C₄×C₁₀₉	—	2	1	no	—
44	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	Dic₁₀₉	—	2	1	no	—
46	Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₁₀₉	—	4	1	no	—
48	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
49	D₁₀₉	—	2	1	no	—
50	D₁₀₉	—	4	1	no	—
51	C₁₀₉	#443	1	1	yes	characteristic, complemented, Sylow
52	#443	—	109	1	no	maximal, Sylow
53	#135	—	109	1	no	—
54	#131	—	109	1	no	—
55	#131	—	109	1	no	—
56	#108	—	109	1	no	—
57	#104	—	109	1	no	—
58	#104	—	109	1	no	—
59	C₄×C₈	—	109	1	no	—
60	#41	—	109	1	no	—
61	#41	—	109	1	no	—
62	Q₁₆	—	109	1	no	—
63	SD₁₆	—	109	1	no	—
64	SD₁₆	#89137	1	1	yes	characteristic, complemented
65	D₈	—	109	1	no	—
66	M₁₆	—	109	1	no	—
67	M₁₆	—	109	1	no	—
68	M₁₆	—	218	1	no	—
69	M₁₆	—	218	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	218	1	no	—
73	C₂×C₈	—	218	1	no	—
74	C₄²	—	109	1	no	—
75	Q₈	—	109	1	no	—
76	Q₈	C₂×#89137	1	1	yes	characteristic
77	D₄	—	109	1	no	—
78	D₄	—	218	1	no	—
79	D₄	—	218	1	no	—
80	D₄	C₂×#89137	1	1	yes	characteristic
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	218	1	no	—
83	C₂×C₄	—	218	1	no	—
84	C₂×C₄	—	218	1	no	—
85	C₈	—	109	1	no	—
86	C₈	—	109	1	no	—
87	C₈	—	109	1	no	—
88	C₈	—	218	1	no	—
89	C₈	—	218	1	no	—
90	C₈	C₂×#89137	1	1	yes	characteristic
91	C₂²	—	2	1	no	—
92	C₂²	—	109	1	no	—
93	C₂²	—	218	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	109	1	no	—
96	C₄	—	218	2	no	—
97	C₄	—	218	1	no	—
98	C₄	C₂²×#89137	1	1	yes	characteristic
99	C₂	—	4	1	no	—
100	C₂	—	436	1	no	—
101	C₂	—	218	1	no	—
102	C₂	D₄×#89137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812787

C₂

yes

maximal, characteristic, complemented

#16812786

C₂

yes

maximal, characteristic, complemented

#16812769

C₂

yes

maximal, characteristic, complemented

#16812764

C₂

yes

maximal, characteristic, complemented

#16812763

C₂

yes

maximal, characteristic, complemented

C₈×#89137

C₂

yes

maximal, characteristic, complemented

#16812720

C₂

yes

maximal, characteristic

#13182039

—

#13182039

C₂²

yes

characteristic

#13182039

C₂²

yes

characteristic

#13182038

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

C₂²

yes

characteristic

C₂×#11724786

—

C₄×#89137

C₂²

yes

characteristic

#13182030

C₂²

yes

characteristic

#13182028

C₂²

yes

characteristic

#13182019

C₄

yes

characteristic, complemented

#13182018

C₄

yes

characteristic, complemented

#13182014

C₄

yes

characteristic, complemented

C₈×D₁₀₉

C₂²

yes

characteristic

SD₁₆×C₁₀₉

C₄

yes

characteristic, complemented

C₂×#89137

—

Q₈×C₁₀₉

C₂×C₄

yes

characteristic

D₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

C₂×C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂³

yes

characteristic

Dic₂₁₈

C₂×C₄

yes

—

—

D₄

yes

characteristic

#11724786

D₄

yes

characteristic

C₈×C₁₀₉

C₂×C₄

yes

characteristic

#11724784

C₂×C₄

yes

characteristic

C₂²×C₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#41

yes

characteristic

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

characteristic

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#443

yes

characteristic, complemented, Sylow

#443

—

109

maximal, Sylow

#135

—

109

—

#131

—

109

—

#131

—

109

—

#108

—

109

—

#104

—

109

—

#104

—

109

—

C₄×C₈

—

109

—

#41

—

109

—

#41

—

109

—

Q₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

#89137

yes

characteristic, complemented

D₈

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

Q₈

—

109

—

Q₈

C₂×#89137

yes

characteristic

D₄

—

109

—

D₄

—

218

—

D₄

—

218

—

D₄

C₂×#89137

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₈

—

218

—

C₈

C₂×#89137

yes

characteristic

C₂²

—

C₂²

—

109

—

C₂²

—

218

—

C₄

—

C₄

—

109

—

C₄

—

218

—

C₄

—

218

—

C₄

C₂²×#89137

yes

characteristic

C₂

—

100

C₂

—

436

—

101

C₂

—

218

—

102

C₂

D₄×#89137

yes

characteristic

	subgroup	G/H
center	C₂	D₄×#89137
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₄	C₂²×#89137
Fitting	SD₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₄×D₄
hypercenter	SD₁₆	#89137
hypocenter	C₁₀₉	#443
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

subgroup

G/H

center

C₂

D₄×#89137

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₄

C₂²×#89137

Fitting

SD₁₆×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₄×D₄

SD₁₆

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

N	H ≅ G/N	type
1. #16812787	C₂	semidirect
2. #16812786	C₂	semidirect
3. #16812769	C₂	semidirect
4. #16812764	C₂	semidirect
5. #16812763	C₂	semidirect
6. C₈×#89137	C₂	semidirect
18. #13182019	C₄	semidirect
19. #13182018	C₄	semidirect
20. #13182014	C₄	semidirect
22. SD₁₆×C₁₀₉	C₄	semidirect
51. C₁₀₉	#443	semidirect
64. SD₁₆	#89137	semidirect

H ≅ G/N

type

1. #16812787

C₂

semidirect

2. #16812786

C₂

semidirect

3. #16812769

C₂

semidirect

4. #16812764

C₂

semidirect

5. #16812763

C₂

semidirect

6. C₈×#89137

C₂

semidirect

18. #13182019

C₄

semidirect

19. #13182018

C₄

semidirect

20. #13182014

C₄

semidirect

22. SD₁₆×C₁₀₉

C₄

semidirect

51. C₁₀₉

#443

semidirect

64. SD₁₆

#89137

semidirect

Subgroups — #20296069

Conjugacy classes of subgroups

Complements