Group #28364912 — subgroups

Subgroups — #28364912

Order 13952.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20295458	C₂	1	1	yes	maximal, characteristic
2	#20295142	C₂	1	2	yes	maximal, complemented
3	#20294832	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812721	C₂²	1	2	yes	—
7	#16812707	—	2	2	no	—
8	Dic₈₇₂	C₄	1	1	yes	characteristic, complemented
9	#16812644	C₄	1	2	yes	complemented
10	D₁₇₄₄	C₄	1	1	yes	characteristic, complemented
11	C₂×#13182004	C₂²	1	1	yes	characteristic
12	#16812625	—	2	2	no	—
13	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	#108×C₁₀₉	C₂²	1	2	yes	—
16	#13182029	—	2	2	no	—
17	Dic₄₃₆	C₂×C₄	1	2	yes	—
18	#13182018	—	2	2	no	—
19	D₈₇₂	C₂×C₄	1	2	yes	—
20	#13182012	—	4	2	no	—
21	C₂×#11724784	D₄	1	1	yes	characteristic
22	C₈×D₁₀₉	—	4	2	no	—
23	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
24	#13182004	—	2	1	no	—
25	M₁₆×C₁₀₉	—	2	2	no	—
26	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
30	C₂²⋊D₁₀₉	—	4	2	no	—
31	C₂×Dic₁₀₉	—	2	1	no	—
32	D₄₃₆	—	2	2	no	—
33	C₄×D₁₀₉	—	4	2	no	—
34	Dic₂₁₈	—	2	2	no	—
35	C₈×C₁₀₉	—	4	2	no	—
36	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₁₀₉	#41	1	1	yes	characteristic
38	#11724784	D₈	1	1	yes	characteristic
39	#11724784	D₈	1	1	yes	characteristic
40	C₂²×C₁₀₉	#135	1	1	yes	characteristic
41	C₂×D₁₀₉	—	4	2	no	—
42	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	—	4	2	no	—
46	C₂×C₁₀₉	—	2	1	no	—
47	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
48	D₁₀₉	—	8	2	no	—
49	#2178	—	109	1	no	maximal, Sylow
50	C₁₀₉	#2178	1	1	yes	characteristic, complemented, Sylow
51	#508	—	109	1	no	—
52	#443	—	109	2	no	—
53	#368	D₁₀₉	1	1	yes	characteristic, complemented
54	#359	—	109	2	no	—
55	C₄×C₁₆	—	109	1	no	—
56	#135	—	109	2	no	—
57	#131	—	218	2	no	—
58	Q₃₂	—	109	1	no	—
59	SD₃₂	—	109	2	no	—
60	D₁₆	—	109	1	no	—
61	C₂×C₁₆	—	109	1	no	—
62	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
63	#108	C₂×D₁₀₉	1	2	yes	—
64	#104	—	218	2	no	—
65	C₄×C₈	—	109	1	no	—
66	#41	—	218	2	no	—
67	Q₁₆	—	109	2	no	—
68	SD₁₆	—	218	2	no	—
69	D₈	—	109	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	436	2	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	436	2	no	—
74	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
75	C₄²	—	109	1	no	—
76	C₁₆	—	218	1	no	—
77	C₁₆	C₄×D₁₀₉	1	1	yes	characteristic
78	C₁₆	C₄×D₁₀₉	1	1	yes	characteristic
79	Q₈	—	218	2	no	—
80	D₄	—	436	2	no	—
81	D₄	—	218	2	no	—
82	C₂×C₄	—	436	2	no	—
83	C₂×C₄	—	218	1	no	—
84	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
85	C₈	—	4	2	no	—
86	C₈	—	109	1	no	—
87	C₈	—	109	1	no	—
88	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
89	C₈	#13182030	1	1	yes	characteristic
90	C₂²	—	436	2	no	—
91	C₂²	#16812722	1	1	yes	characteristic
92	C₄	—	436	2	no	—
93	C₄	—	218	2	no	—
94	C₄	#16812661	1	1	yes	characteristic
95	C₄	D₈×D₁₀₉	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	872	2	no	—
98	C₂	#20295108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295458

C₂

yes

maximal, characteristic

#20295142

C₂

yes

maximal, complemented

#20294832

C₂

yes

maximal, complemented

C₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

#368×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812721

C₂²

yes

—

#16812707

—

Dic₈₇₂

C₄

yes

characteristic, complemented

#16812644

C₄

yes

complemented

D₁₇₄₄

C₄

yes

characteristic, complemented

C₂×#13182004

C₂²

yes

characteristic

#16812625

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

#108×C₁₀₉

C₂²

yes

—

#13182029

—

Dic₄₃₆

C₂×C₄

yes

—

#13182018

—

D₈₇₂

C₂×C₄

yes

—

#13182012

—

C₂×#11724784

D₄

yes

characteristic

C₈×D₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂²×C₄

yes

C₈×C₁₀₉

yes

D₈

yes

characteristic

#11724784

D₈

yes

characteristic

C₂²×C₁₀₉

#135

yes

characteristic

C₂×D₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₂×D₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₈

yes

characteristic

D₁₀₉

—

#2178

—

109

maximal, Sylow

C₁₀₉

#2178

yes

characteristic, complemented, Sylow

#508

—

109

—

#443

—

109

—

#368

D₁₀₉

yes

characteristic, complemented

#359

—

109

—

C₄×C₁₆

—

109

—

#135

—

109

—

#131

—

218

—

Q₃₂

—

109

—

SD₃₂

—

109

—

D₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

#108

C₂×D₁₀₉

yes

—

#104

—

218

—

C₄×C₈

—

109

—

#41

—

218

—

Q₁₆

—

109

—

SD₁₆

—

218

—

D₈

—

109

—

M₁₆

—

M₁₆

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₁₆

—

218

—

C₁₆

C₄×D₁₀₉

yes

characteristic

C₁₆

C₄×D₁₀₉

yes

characteristic

Q₈

—

218

—

D₄

—

436

—

D₄

—

218

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

D₄×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

C₂×C₄×D₁₀₉

yes

characteristic

C₈

#13182030

yes

characteristic

C₂²

—

436

—

C₂²

#16812722

yes

characteristic

C₄

—

436

—

C₄

—

218

—

C₄

#16812661

yes

characteristic

C₄

D₈×D₁₀₉

yes

characteristic

C₂

—

C₂

—

872

—

C₂

#20295108

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₁₀₉
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#368×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₄×D₈
prime radical	#20295458	C₂
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	#368	D₁₀₉
hypocenter	C₁₀₉	#2178
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×D₁₀₉

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#368×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₄×D₈

prime radical

#20295458

C₂

cosocle

C₂×C₈×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
2. #20295142	C₂	semidirect
3. #20294832	C₂	semidirect
4. C₁₆×Dic₁₀₉	C₂	semidirect
5. #368×C₁₀₉	C₂	semidirect
8. Dic₈₇₂	C₄	semidirect
9. #16812644	C₄	semidirect
10. D₁₇₄₄	C₄	semidirect
50. C₁₀₉	#2178	semidirect
53. #368	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295142

C₂

semidirect

3. #20294832

C₂

semidirect

4. C₁₆×Dic₁₀₉

C₂

semidirect

5. #368×C₁₀₉

C₂

semidirect

8. Dic₈₇₂

C₄

semidirect

9. #16812644

C₄

semidirect

10. D₁₇₄₄

C₄

semidirect

50. C₁₀₉

#2178

semidirect

53. #368

D₁₀₉

semidirect

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

subgroup	index
#20295458	2
#20295142	2
#20294832	2
C₁₆×Dic₁₀₉	2
#368×C₁₀₉	2
#2178	109

subgroup

index

#20295458

#20295142

#20294832

C₁₆×Dic₁₀₉

#368×C₁₀₉

#2178

109

Subgroups — #28364912

Conjugacy classes of subgroups

Complements

Maximal subgroups