Group #28379433 — subgroups

Subgroups — #28379433

Order 13952.

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

normal subgroups only

This group has 146 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	#20296070	C₂	1	2	yes	maximal, complemented
2	#20295954	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295941	C₂	1	2	yes	maximal, complemented
4	C₁₆×#89137	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295459	C₂	1	1	yes	maximal, characteristic
6	#16812787	—	2	2	no	—
7	D₁₀₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16812770	C₂²	1	2	yes	—
9	#16812764	—	2	2	no	—
10	C₈×#89137	C₂²	1	1	yes	characteristic
11	#16812722	C₂²	1	2	yes	—
12	#16812642	C₄	1	2	yes	complemented
13	D₁₇₄₄	C₄	1	1	yes	characteristic, complemented
14	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	#13182039	—	2	2	no	—
17	#13182039	—	4	2	no	—
18	D₁₀₉⋊C₈	—	4	2	no	—
19	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
20	C₄×#89137	D₄	1	1	yes	characteristic
21	#13182032	—	2	1	no	—
22	#13182030	—	2	2	no	—
23	#13182016	—	2	2	no	—
24	D₈₇₂	C₂×C₄	1	2	yes	—
25	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
26	#13182004	C₂×C₄	1	1	yes	characteristic
27	Q₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×#89137	—	2	1	no	—
30	Q₈×C₁₀₉	—	2	2	no	—
31	D₄₃₆	—	2	2	no	—
32	D₄₃₆	—	4	2	no	—
33	C₄×D₁₀₉	—	4	2	no	—
34	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
35	#11724786	—	4	2	no	—
36	#11724786	D₈	1	1	yes	characteristic
37	#11724786	D₈	1	1	yes	characteristic
38	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	#11724784	#41	1	1	yes	characteristic
40	C₂×D₁₀₉	—	4	2	no	—
41	C₂×D₁₀₉	#135	1	1	yes	characteristic
42	#89137	—	2	2	no	—
43	C₄×C₁₀₉	—	4	2	no	—
44	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
45	Dic₁₀₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
47	D₁₀₉	—	2	1	no	—
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	—	109	1	no	—
54	#359	—	109	2	no	—
55	C₄×C₁₆	—	109	1	no	—
56	#135	—	109	2	no	—
57	#131	—	218	2	no	—
58	Q₃₂	#89137	1	1	yes	characteristic, complemented
59	SD₃₂	—	109	2	no	—
60	D₁₆	—	109	1	no	—
61	C₂×C₁₆	—	109	1	no	—
62	C₂×C₁₆	—	109	1	no	—
63	#108	—	109	2	no	—
64	#104	—	218	2	no	—
65	C₄×C₈	—	109	1	no	—
66	#41	—	218	2	no	—
67	Q₁₆	C₂×#89137	1	2	yes	—
68	SD₁₆	—	218	2	no	—
69	D₈	—	109	2	no	—
70	M₁₆	—	436	2	no	—
71	M₁₆	—	218	2	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	109	1	no	—
74	C₂×C₈	—	436	2	no	—
75	C₄²	—	109	1	no	—
76	C₁₆	—	109	1	no	—
77	C₁₆	—	218	1	no	—
78	C₁₆	C₂×#89137	1	1	yes	characteristic
79	Q₈	—	2	2	no	—
80	D₄	—	436	2	no	—
81	D₄	—	218	2	no	—
82	C₂×C₄	—	109	1	no	—
83	C₂×C₄	—	436	2	no	—
84	C₂×C₄	—	218	1	no	—
85	C₈	—	109	1	no	—
86	C₈	—	109	1	no	—
87	C₈	—	109	1	no	—
88	C₈	—	436	2	no	—
89	C₈	C₂²×#89137	1	1	yes	characteristic
90	C₂²	—	109	1	no	—
91	C₂²	—	436	2	no	—
92	C₄	—	4	2	no	—
93	C₄	—	109	1	no	—
94	C₄	—	218	2	no	—
95	C₄	D₄×#89137	1	1	yes	characteristic
96	C₂	—	872	2	no	—
97	C₂	—	218	1	no	—
98	C₂	D₈×#89137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296070

C₂

yes

maximal, complemented

#20295954

C₂

yes

maximal, characteristic, complemented

#20295941

C₂

yes

maximal, complemented

C₁₆×#89137

C₂

yes

maximal, characteristic, complemented

#20295459

C₂

yes

maximal, characteristic

#16812787

—

D₁₀₉⋊C₁₆

C₂²

yes

characteristic

#16812770

C₂²

yes

—

#16812764

—

C₈×#89137

C₂²

yes

characteristic

#16812722

C₂²

yes

—

#16812642

C₄

yes

complemented

D₁₇₄₄

C₄

yes

characteristic, complemented

C₁₆×D₁₀₉

C₂²

yes

characteristic

Q₃₂×C₁₀₉

C₄

yes

characteristic, complemented

#13182039

—

#13182039

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

D₄

yes

characteristic

C₄×#89137

D₄

yes

—

—

—

D₈₇₂

C₂×C₄

yes

—

C₈×D₁₀₉

C₂³

yes

characteristic

#13182004

C₂×C₄

yes

characteristic

Q₁₆×C₁₀₉

C₂×C₄

yes

—

C₁₆×C₁₀₉

C₂×C₄

yes

characteristic

C₂×#89137

—

Q₈×C₁₀₉

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

characteristic

#11724786

—

#11724786

D₈

yes

characteristic

#11724786

D₈

yes

characteristic

C₈×C₁₀₉

C₂²×C₄

yes

yes

C₂×D₁₀₉

—

C₂×D₁₀₉

#135

yes

characteristic

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

characteristic

Dic₁₀₉

C₂×D₈

yes

characteristic

C₂×C₁₀₉

C₄×D₈

yes

characteristic

D₁₀₉

—

D₁₀₉

—

#2178

—

109

maximal, Sylow

C₁₀₉

#2178

yes

characteristic, complemented, Sylow

#508

—

109

—

#443

—

109

—

#368

—

109

—

#359

—

109

—

C₄×C₁₆

—

109

—

#135

—

109

—

#131

—

218

—

Q₃₂

#89137

yes

characteristic, complemented

SD₃₂

—

109

—

D₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

#108

—

109

—

#104

—

218

—

C₄×C₈

—

109

—

#41

—

218

—

Q₁₆

C₂×#89137

yes

—

SD₁₆

—

218

—

D₈

—

109

—

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₄²

—

109

—

C₁₆

—

109

—

C₁₆

—

218

—

C₁₆

C₂×#89137

yes

characteristic

Q₈

—

D₄

—

436

—

D₄

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

436

—

C₈

C₂²×#89137

yes

characteristic

C₂²

—

109

—

C₂²

—

436

—

C₄

—

C₄

—

109

—

C₄

—

218

—

C₄

D₄×#89137

yes

characteristic

C₂

—

872

—

C₂

—

218

—

C₂

D₈×#89137

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#89137
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	Q₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₄×D₈
prime radical	#20295459	C₂
cosocle	C₈×D₁₀₉	C₂³
hypercenter	Q₃₂	#89137
hypocenter	C₁₀₉	#2178
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

D₈×#89137

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₈

C₂²×#89137

Fitting

Q₃₂×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₄×D₈

prime radical

#20295459

C₂

cosocle

C₈×D₁₀₉

C₂³

Q₃₂

C₁₀₉

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

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
1. #20296070	C₂	semidirect
2. #20295954	C₂	semidirect
3. #20295941	C₂	semidirect
4. C₁₆×#89137	C₂	semidirect
12. #16812642	C₄	semidirect
13. D₁₇₄₄	C₄	semidirect
15. Q₃₂×C₁₀₉	C₄	semidirect
50. C₁₀₉	#2178	semidirect
58. Q₃₂	#89137	semidirect

H ≅ G/N

type

1. #20296070

C₂

semidirect

2. #20295954

C₂

semidirect

3. #20295941

C₂

semidirect

4. C₁₆×#89137

C₂

semidirect

12. #16812642

C₄

semidirect

13. D₁₇₄₄

C₄

semidirect

15. Q₃₂×C₁₀₉

C₄

semidirect

50. C₁₀₉

#2178

semidirect

58. Q₃₂

#89137

semidirect

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

subgroup	index
#20296070	2
#20295954	2
#20295941	2
C₁₆×#89137	2
#20295459	2
#2178	109

subgroup

index

C₁₆×#89137

109

Subgroups — #28379433

Conjugacy classes of subgroups

Complements

Maximal subgroups