Group #28364635 — subgroups

Subgroups — #28364635

Order 13952.

160 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 85 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	M₃₂×D₁₀₉	C₂	1	2	yes	maximal
2	#20294983	C₂	1	1	yes	maximal, characteristic
3	#20294804	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294803	C₂	1	2	yes	maximal, complemented
5	#347×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182012	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
8	#16812656	C₄	1	1	yes	characteristic
9	#16812639	C₂²	1	2	yes	—
10	#16812638	C₄	1	4	yes	complemented
11	C₁₆×D₁₀₉	C₄	1	4	yes	complemented
12	#16812609	C₂²	1	1	yes	characteristic
13	C₄×#11724784	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
15	M₃₂×C₁₀₉	C₂²	1	2	yes	—
16	#97×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	#13182012	—	2	2	no	—
19	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
22	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
23	#13182008	—	2	1	no	—
24	#13182006	—	2	1	no	—
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	#13182004	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₂²×D₁₀₉	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	—	2	1	no	—
35	C₂×Dic₁₀₉	M₁₆	1	1	yes	characteristic
36	C₄×D₁₀₉	C₄²	1	2	yes	—
37	C₄×D₁₀₉	M₁₆	1	2	yes	—
38	C₈×C₁₀₉	—	2	1	no	—
39	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
40	#11724784	—	2	1	no	—
41	#11724784	C₄²	1	1	yes	characteristic
42	#11724784	C₄²	1	1	yes	characteristic
43	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂×D₁₀₉	—	4	1	no	—
45	C₂×D₁₀₉	#97	1	2	yes	—
46	C₄×C₁₀₉	—	4	1	no	—
47	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
49	Dic₁₀₉	—	4	1	no	—
50	Dic₁₀₉	#97	1	2	yes	—
51	C₂×C₁₀₉	—	2	1	no	—
52	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
53	D₁₀₉	—	2	2	no	—
54	#2110	—	109	1	no	maximal, Sylow
55	C₁₀₉	#2110	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₃₂	—	109	2	no	—
57	#405	—	109	1	no	—
58	#347	—	109	2	no	—
59	#347	—	109	1	no	—
60	#347	D₁₀₉	1	1	yes	characteristic, complemented
61	C₂×M₁₆	—	109	1	no	—
62	C₂²×C₈	—	109	1	no	—
63	#117	—	109	1	no	—
64	M₃₂	—	109	2	no	—
65	M₃₂	—	109	4	no	—
66	M₃₂	C₂×D₁₀₉	1	2	yes	—
67	C₂×C₁₆	—	109	4	no	—
68	#97	—	109	1	no	—
69	#97	C₂×D₁₀₉	1	1	yes	characteristic
70	C₄×C₈	—	109	1	no	—
71	C₄×C₈	—	109	1	no	—
72	C₂²×C₄	—	109	1	no	—
73	M₁₆	—	218	2	no	—
74	C₂×C₈	—	109	1	no	—
75	C₂×C₈	—	109	2	no	—
76	C₂×C₈	—	109	2	no	—
77	C₂×C₈	—	109	1	no	—
78	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
80	C₄⋊C₄	—	218	1	no	—
81	C₂²⋊C₄	—	218	1	no	—
82	C₄²	—	109	1	no	—
83	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
84	C₁₆	—	109	4	no	—
85	C₁₆	C₄×D₁₀₉	1	4	yes	—
86	C₂³	—	109	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	109	2	no	—
90	C₂×C₄	—	109	2	no	—
91	C₂×C₄	—	218	1	no	—
92	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	109	1	no	—
95	C₈	—	109	1	no	—
96	C₈	—	218	1	no	—
97	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
98	C₂²	—	109	2	no	—
99	C₂²	—	436	1	no	—
100	C₂²	M₁₆×D₁₀₉	1	1	yes	characteristic
101	C₄	—	4	1	no	—
102	C₄	—	109	2	no	—
103	C₄	—	436	1	no	—
104	C₄	C₄²×D₁₀₉	1	1	yes	characteristic
105	C₄	M₁₆×D₁₀₉	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	218	2	no	—
108	C₂	#97×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₁₀₉

C₂

yes

maximal

#20294983

C₂

yes

maximal, characteristic

#20294804

C₂

yes

maximal, characteristic, complemented

#20294803

C₂

yes

maximal, complemented

#347×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182012

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812656

C₄

yes

characteristic

#16812639

C₂²

yes

—

#16812638

C₄

yes

complemented

C₁₆×D₁₀₉

C₄

yes

complemented

#16812609

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

M₃₂×C₁₀₉

C₂²

yes

—

#97×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

#13182012

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

—

C₈×D₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#13182004

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

—

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

C₂²×C₄

yes

—

#11724784

—

#11724784

C₄²

yes

characteristic

#11724784

C₄²

yes

characteristic

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

—

#347

D₁₀₉

yes

characteristic, complemented

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

M₃₂

—

109

—

M₃₂

—

109

—

M₃₂

C₂×D₁₀₉

yes

—

C₂×C₁₆

—

109

—

#97

—

109

—

#97

C₂×D₁₀₉

yes

characteristic

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₁₆

—

109

—

C₁₆

C₄×D₁₀₉

yes

—

C₂³

—

109

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₈

C₂×C₄×D₁₀₉

yes

—

C₂²

—

109

—

C₂²

—

436

—

100

C₂²

M₁₆×D₁₀₉

yes

characteristic

101

C₄

—

102

C₄

—

109

—

103

C₄

—

436

—

104

C₄

C₄²×D₁₀₉

yes

characteristic

105

C₄

M₁₆×D₁₀₉

yes

characteristic

106

C₂

—

107

C₂

—

218

—

108

C₂

#97×D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#347×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×#97
prime radical	C₂²×D₁₀₉	M₁₆
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	#347	D₁₀₉
hypocenter	C₁₀₉	#2110
LCS	G ▷ C₄×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ #347

subgroup

G/H

center

C₄

C₄²×D₁₀₉

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#347×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂×#97

prime radical

C₂²×D₁₀₉

M₁₆

cosocle

C₂×C₈×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₄ ◁ #347

N	H ≅ G/N	type
3. #20294804	C₂	semidirect
4. #20294803	C₂	semidirect
5. #347×C₁₀₉	C₂	semidirect
10. #16812638	C₄	semidirect
11. C₁₆×D₁₀₉	C₄	semidirect
55. C₁₀₉	#2110	semidirect
60. #347	D₁₀₉	semidirect

H ≅ G/N

type

3. #20294804

C₂

semidirect

4. #20294803

C₂

semidirect

5. #347×C₁₀₉

C₂

semidirect

10. #16812638

C₄

semidirect

11. C₁₆×D₁₀₉

C₄

semidirect

55. C₁₀₉

#2110

semidirect

60. #347

D₁₀₉

semidirect

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

subgroup	index
M₃₂×D₁₀₉	2
#20294983	2
#20294804	2
#20294803	2
#347×C₁₀₉	2
#2110	109

subgroup

index

M₃₂×D₁₀₉

#347×C₁₀₉

109

Subgroups — #28364635

Conjugacy classes of subgroups

Complements

Maximal subgroups