Group #28362786 — subgroups

Subgroups — #28362786

Order 13952.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20294984	C₂	1	1	yes	maximal, characteristic
2	#20294982	C₂	1	2	yes	maximal
3	#20294784	C₂	1	1	yes	maximal, characteristic
4	#20294782	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	#16812658	C₂²	1	1	yes	characteristic
9	#16812610	C₄	1	4	yes	complemented
10	#16812609	C₄	1	2	yes	complemented
11	C₄×#11724784	C₄	1	4	yes	complemented
12	C₈×Dic₁₀₉	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	#117×C₁₀₉	C₂²	1	2	yes	—
15	#13182013	—	2	2	no	—
16	#13182013	—	2	2	no	—
17	C₂×#11724784	C₂×C₄	1	4	yes	—
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂×C₄	1	2	yes	—
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₂×#11724784	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₀₉	—	2	1	no	—
23	C₂²⋊Dic₁₀₉	—	2	1	no	—
24	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₉	—	2	2	no	—
27	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₀₉	—	2	2	no	—
29	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₂³×C₁₀₉	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	4	no	—
33	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	#41	1	2	yes	—
36	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
37	C₂×Dic₁₀₉	—	2	2	no	—
38	C₈×C₁₀₉	—	2	2	no	—
39	#11724784	—	2	4	no	—
40	#11724784	C₄²	1	1	yes	characteristic
41	#11724784	C₄²	1	1	yes	characteristic
42	#11724784	C₄²	1	2	yes	—
43	C₂²×C₁₀₉	—	4	1	no	—
44	C₂²×C₁₀₉	#117	1	2	yes	—
45	C₂²×C₁₀₉	#117	1	1	yes	characteristic
46	C₄×C₁₀₉	—	4	4	no	—
47	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₉	#117	1	1	yes	characteristic
49	C₄×C₁₀₉	#117	1	2	yes	—
50	Dic₁₀₉	—	4	2	no	—
51	C₂×C₁₀₉	—	2	1	no	—
52	C₂×C₁₀₉	—	2	2	no	—
53	C₂×C₁₀₉	#376	1	1	yes	characteristic
54	#1747	—	109	1	no	maximal, Sylow
55	C₁₀₉	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	109	1	no	—
57	#405	—	109	2	no	—
58	#338	—	109	1	no	—
59	H(3,ℤ/4)	—	109	2	no	—
60	H(3,ℤ/4)	D₁₀₉	1	1	yes	characteristic
61	C₂×M₁₆	—	109	2	no	—
62	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
63	C₂²×C₈	—	109	1	no	—
64	#117	—	109	1	no	—
65	#117	C₂×D₁₀₉	1	2	yes	—
66	#97	—	109	2	no	—
67	#97	—	109	4	no	—
68	C₄×C₈	—	109	2	no	—
69	C₄×C₈	—	109	4	no	—
70	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	218	2	no	—
73	M₁₆	—	218	2	no	—
74	C₂×C₈	—	109	1	no	—
75	C₂×C₈	—	109	2	no	—
76	C₂×C₈	—	109	1	no	—
77	C₂×C₈	—	109	2	no	—
78	C₂×C₈	—	109	4	no	—
79	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	218	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	218	1	no	—
84	C₄²	—	109	2	no	—
85	C₄²	C₄×D₁₀₉	1	4	yes	—
86	C₂³	#13182028	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	218	2	no	—
89	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
91	C₂×C₄	#13182028	1	2	yes	—
92	C₂×C₄	#13182030	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	109	1	no	—
95	C₈	—	109	1	no	—
96	C₈	—	109	2	no	—
97	C₈	—	218	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#16812655	1	2	yes	—
100	C₂²	#16812657	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	436	2	no	—
103	C₄	C₄²×D₁₀₉	1	1	yes	characteristic
104	C₄	#16812655	1	1	yes	characteristic
105	C₄	#16812657	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#20294871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20294984

C₂

yes

maximal, characteristic

#20294982

C₂

yes

maximal

#20294784

C₂

yes

maximal, characteristic

#20294782

C₂

yes

maximal

H(3,ℤ/4)×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

#16812658

C₂²

yes

characteristic

#16812610

C₄

yes

complemented

#16812609

C₄

yes

complemented

C₄×#11724784

C₄

yes

complemented

C₈×Dic₁₀₉

C₄

yes

complemented

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

#13182013

—

#13182013

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

C₄²

yes

characteristic

#11724784

C₄²

yes

characteristic

#11724784

C₄²

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#117

yes

—

C₂²×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#376

yes

characteristic

#1747

—

109

maximal, Sylow

C₁₀₉

#1747

yes

characteristic, complemented, Sylow

#405

—

109

—

#405

—

109

—

#338

—

109

—

H(3,ℤ/4)

—

109

—

H(3,ℤ/4)

D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

#117

—

109

—

#117

C₂×D₁₀₉

yes

—

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

C₄×D₁₀₉

yes

—

C₂³

#13182028

yes

characteristic

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

C₂×C₄

#13182028

yes

—

C₂×C₄

#13182030

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₂²

—

C₂²

#16812655

yes

—

100

C₂²

#16812657

yes

characteristic

101

C₄

—

102

C₄

—

436

—

103

C₄

C₄²×D₁₀₉

yes

characteristic

104

C₄

#16812655

yes

characteristic

105

C₄

#16812657

yes

—

106

C₂

—

107

C₂

—

108

C₂

#20294871

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	H(3,ℤ/4)×C₁₀₉	C₂
socle	C₂×C₁₀₉	#376
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	H(3,ℤ/4)	D₁₀₉
hypocenter	C₁₀₉	#1747
LCS	G ▷ C₄×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₀₉

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

H(3,ℤ/4)×C₁₀₉

C₂

C₂×C₁₀₉

C₂³×C₁₀₉

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #16812610	C₄	semidirect
10. #16812609	C₄	semidirect
11. C₄×#11724784	C₄	semidirect
12. C₈×Dic₁₀₉	C₄	semidirect
55. C₁₀₉	#1747	semidirect

H ≅ G/N

type

9. #16812610

C₄

semidirect

10. #16812609

C₄

semidirect

11. C₄×#11724784

C₄

semidirect

12. C₈×Dic₁₀₉

C₄

semidirect

55. C₁₀₉

#1747

semidirect

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

subgroup	index
#20294984	2
#20294982	2
#20294784	2
#20294782	2
H(3,ℤ/4)×C₁₀₉	2
#1747	109

subgroup

index

H(3,ℤ/4)×C₁₀₉

109

Subgroups — #28362786

Conjugacy classes of subgroups

Complements

Maximal subgroups