Group #28379254 — subgroups

Subgroups — #28379254

Order 13952.

160 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 78 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	D₁₀₉⋊M₃₂	C₂	1	2	yes	maximal
2	#20295919	C₂	1	2	yes	maximal, complemented
3	#20295918	C₂	1	2	yes	maximal, complemented
4	#20294980	C₂	1	1	yes	maximal, characteristic
5	#16812774	C₄	1	4	yes	complemented
6	#16812773	C₂²	1	4	yes	—
7	D₁₀₉⋊C₁₆	C₄	1	4	yes	complemented
8	C₂×#13182012	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
10	#16812656	C₄	1	1	yes	characteristic
11	#16812610	C₂²	1	1	yes	characteristic
12	#16812609	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
14	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	#13182032	C₂×C₄	1	8	yes	—
16	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
17	#13182012	—	2	2	no	—
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂³	1	1	yes	characteristic
20	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
21	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
22	#13182008	—	2	1	no	—
23	#13182006	—	2	1	no	—
24	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₀₉	M₁₆	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	2	1	no	—
32	C₂×Dic₁₀₉	M₁₆	1	1	yes	characteristic
33	C₄×D₁₀₉	C₄²	1	2	yes	—
34	C₄×D₁₀₉	M₁₆	1	2	yes	—
35	C₈×C₁₀₉	—	2	1	no	—
36	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
37	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
38	#11724784	—	2	1	no	—
39	#11724784	C₂²×C₄	1	2	yes	—
40	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
41	C₂×D₁₀₉	—	4	1	no	—
42	C₂×D₁₀₉	#97	1	2	yes	—
43	C₄×C₁₀₉	—	4	1	no	—
44	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
45	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
46	Dic₁₀₉	—	4	1	no	—
47	Dic₁₀₉	#97	1	2	yes	—
48	C₂×C₁₀₉	—	2	1	no	—
49	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
50	D₁₀₉	—	2	2	no	—
51	#2110	—	109	1	no	maximal, Sylow
52	C₁₀₉	#2110	1	1	yes	characteristic, complemented, Sylow
53	C₂×M₃₂	—	109	2	no	—
54	#405	—	109	1	no	—
55	#347	—	109	2	no	—
56	#347	—	109	2	no	—
57	C₂×M₁₆	—	109	1	no	—
58	C₂²×C₈	—	109	1	no	—
59	#117	—	109	1	no	—
60	M₃₂	—	109	4	no	—
61	M₃₂	—	109	4	no	—
62	C₂×C₁₆	—	109	4	no	—
63	#97	—	109	1	no	—
64	#97	—	109	1	no	—
65	C₄×C₈	—	109	1	no	—
66	C₄×C₈	#89137	1	1	yes	characteristic
67	C₂²×C₄	—	109	1	no	—
68	M₁₆	—	218	2	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	2	no	—
71	C₂×C₈	—	109	2	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	C₂×#89137	1	1	yes	characteristic
74	C₂×C₈	C₂×#89137	1	1	yes	characteristic
75	C₄⋊C₄	—	218	1	no	—
76	C₂²⋊C₄	—	218	1	no	—
77	C₄²	—	109	1	no	—
78	C₄²	C₂×#89137	1	1	yes	characteristic
79	C₁₆	—	109	8	no	—
80	C₂³	—	109	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	109	1	no	—
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	218	1	no	—
86	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
87	C₈	—	2	1	no	—
88	C₈	—	109	2	no	—
89	C₈	—	218	1	no	—
90	C₈	C₄×#89137	1	1	yes	characteristic
91	C₈	C₄×#89137	1	1	yes	characteristic
92	C₂²	—	109	2	no	—
93	C₂²	—	436	1	no	—
94	C₂²	D₁₀₉⋊M₁₆	1	1	yes	characteristic
95	C₄	—	4	1	no	—
96	C₄	—	109	2	no	—
97	C₄	—	436	1	no	—
98	C₄	C₂×C₄×#89137	1	1	yes	characteristic
99	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	218	2	no	—
102	C₂	#20295985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₉⋊M₃₂

C₂

yes

maximal

#20295919

C₂

yes

maximal, complemented

#20295918

C₂

yes

maximal, complemented

#20294980

C₂

yes

maximal, characteristic

#16812774

C₄

yes

complemented

#16812773

C₂²

yes

—

D₁₀₉⋊C₁₆

C₄

yes

complemented

C₂×#13182012

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812656

C₄

yes

characteristic

#16812610

C₂²

yes

characteristic

#16812609

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

characteristic

#13182032

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

#13182012

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

—

C₈×D₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

M₁₆

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

M₁₆

yes

characteristic

C₄×D₁₀₉

C₄²

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄²

yes

characteristic

C₈×C₁₀₉

C₄²

yes

characteristic

#11724784

—

#11724784

C₂²×C₄

yes

—

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#97

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

#97

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#97

yes

characteristic

D₁₀₉

—

#2110

—

109

maximal, Sylow

C₁₀₉

#2110

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

#405

—

109

—

#347

—

109

—

#347

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

M₃₂

—

109

—

M₃₂

—

109

—

C₂×C₁₆

—

109

—

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₁₆

—

109

—

C₂³

—

109

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

218

—

C₈

yes

C₈

yes

C₂²

—

109

—

C₂²

—

436

—

C₂²

D₁₀₉⋊M₁₆

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

436

—

C₄

C₂×C₄×#89137

yes

characteristic

C₄

D₁₀₉⋊M₁₆

yes

characteristic

100

C₂

—

101

C₂

—

218

—

102

C₂

#20295985

yes

characteristic

	subgroup	G/H
center	C₈	C₄×#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×#97
prime radical	C₂²×D₁₀₉	M₁₆
cosocle	C₂×#11724784	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#2110
LCS	G ▷ C₄×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄×#89137

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₂×#97

prime radical

C₂²×D₁₀₉

M₁₆

C₂³

C₄×C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #20295919	C₂	semidirect
3. #20295918	C₂	semidirect
5. #16812774	C₄	semidirect
7. D₁₀₉⋊C₁₆	C₄	semidirect
52. C₁₀₉	#2110	semidirect

H ≅ G/N

type

2. #20295919

C₂

semidirect

3. #20295918

C₂

semidirect

5. #16812774

C₄

semidirect

7. D₁₀₉⋊C₁₆

C₄

semidirect

52. C₁₀₉

#2110

semidirect

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

subgroup	index
D₁₀₉⋊M₃₂	2
#20295919	2
#20295918	2
#20294980	2
#2110	109

subgroup

index

D₁₀₉⋊M₃₂

109

Subgroups — #28379254

Conjugacy classes of subgroups

Complements

Maximal subgroups