Group #28361625 — subgroups

Subgroups — #28361625

Order 13952.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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	#20295582	C₂	1	1	yes	maximal, characteristic
2	#20294978	C₂	1	2	yes	maximal
3	#20294768	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294767	C₂	1	2	yes	maximal, complemented
5	#330×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182012	—	2	2	no	—
7	#16812698	C₄	1	2	yes	—
8	#16812691	C₂²	1	1	yes	characteristic
9	#16812677	—	2	2	no	—
10	#16812664	—	2	2	no	—
11	#16812657	C₄	1	2	yes	—
12	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
13	#16812610	C₂²	1	2	yes	—
14	C₈×Dic₁₀₉	—	2	2	no	—
15	#125×C₁₀₉	C₂²	1	1	yes	characteristic
16	#97×C₁₀₉	C₂²	1	2	yes	—
17	C₂×Dic₂₁₈	—	2	2	no	—
18	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
20	#13182012	—	2	8	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
23	#13182008	—	2	2	no	—
24	#13182008	—	2	2	no	—
25	#13182006	—	2	2	no	—
26	C₄×Dic₁₀₉	—	2	2	no	—
27	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
37	C₂×Dic₁₀₉	—	2	2	no	—
38	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
40	C₄×D₁₀₉	—	2	2	no	—
41	C₄×D₁₀₉	—	2	4	no	—
42	Dic₂₁₈	—	4	4	no	—
43	C₈×C₁₀₉	—	2	4	no	—
44	#11724784	—	2	4	no	—
45	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×D₁₀₉	—	2	1	no	—
47	C₂×D₁₀₉	—	2	2	no	—
48	C₄×C₁₀₉	—	2	1	no	—
49	C₄×C₁₀₉	—	2	2	no	—
50	C₄×C₁₀₉	—	4	2	no	—
51	Dic₁₀₉	—	4	2	no	—
52	Dic₁₀₉	#101	1	2	yes	—
53	Dic₁₀₉	#104	1	4	yes	—
54	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
55	C₂×C₁₀₉	C₂×#104	1	2	yes	—
56	D₁₀₉	—	4	1	no	—
57	#1526	—	109	1	no	maximal, Sylow
58	C₁₀₉	#1526	1	1	yes	characteristic, complemented, Sylow
59	#533	—	109	1	no	—
60	C₄×M₁₆	—	109	2	no	—
61	#330	—	109	1	no	—
62	#330	—	109	2	no	—
63	#330	D₁₀₉	1	1	yes	characteristic, complemented
64	C₂×M₁₆	—	218	2	no	—
65	C₄⋊Q₈	—	109	2	no	—
66	#125	—	109	1	no	—
67	#125	C₂×D₁₀₉	1	1	yes	characteristic
68	C₂²⋊Q₈	—	218	2	no	—
69	C₄×Q₈	—	218	2	no	—
70	#117	—	109	2	no	—
71	C₂×C₄²	—	109	1	no	—
72	#97	—	109	2	no	—
73	#97	C₂×D₁₀₉	1	2	yes	—
74	C₄×C₈	—	218	2	no	—
75	C₂×Q₈	—	218	2	no	—
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	—	109	2	no	—
78	M₁₆	—	218	8	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	218	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	109	2	no	—
83	C₄⋊C₄	—	218	2	no	—
84	C₄⋊C₄	—	218	2	no	—
85	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	—
86	C₂²⋊C₄	—	218	2	no	—
87	C₄²	—	109	1	no	—
88	C₄²	—	109	2	no	—
89	C₄²	—	218	2	no	—
90	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₂³	—	109	1	no	—
92	Q₈	—	436	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	109	1	no	—
95	C₂×C₄	—	109	2	no	—
96	C₂×C₄	—	218	4	no	—
97	C₂×C₄	—	218	2	no	—
98	C₂×C₄	—	218	2	no	—
99	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	—	218	4	no	—
103	C₂²	—	218	1	no	—
104	C₂²	—	218	2	no	—
105	C₂²	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	4	2	no	—
109	C₄	—	109	2	no	—
110	C₄	—	109	4	no	—
111	C₄	—	436	2	no	—
112	C₂	—	436	1	no	—
113	C₂	#101×D₁₀₉	1	1	yes	characteristic
114	C₂	#104×D₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295582

C₂

yes

maximal, characteristic

#20294978

C₂

yes

maximal

#20294768

C₂

yes

maximal, characteristic, complemented

#20294767

C₂

yes

maximal, complemented

#330×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182012

—

#16812698

C₄

yes

—

#16812691

C₂²

yes

—

—

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812610

C₂²

yes

—

C₈×Dic₁₀₉

—

#125×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

—

#13182012

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#101

yes

—

Dic₁₀₉

#104

yes

—

C₂×C₁₀₉

C₂×#101

yes

characteristic

C₂×C₁₀₉

C₂×#104

yes

—

D₁₀₉

—

#1526

—

109

maximal, Sylow

C₁₀₉

#1526

yes

characteristic, complemented, Sylow

#533

—

109

—

C₄×M₁₆

—

109

—

#330

—

109

—

#330

—

109

—

#330

D₁₀₉

yes

characteristic, complemented

C₂×M₁₆

—

218

—

C₄⋊Q₈

—

109

—

#125

—

109

—

#125

C₂×D₁₀₉

yes

characteristic

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

C₂×C₄²

—

109

—

#97

—

109

—

#97

C₂×D₁₀₉

yes

—

C₄×C₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₄×D₁₀₉

yes

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

109

—

Q₈

—

436

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

100

C₂×C₄

D₄×D₁₀₉

yes

—

101

C₈

—

102

C₈

—

218

—

103

C₂²

—

218

—

104

C₂²

—

218

—

105

C₂²

C₂²⋊C₄×D₁₀₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

109

—

110

C₄

—

109

—

111

C₄

—

436

—

112

C₂

—

436

—

113

C₂

#101×D₁₀₉

yes

characteristic

114

C₂

#104×D₁₀₉

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#330×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₉	C₂²⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#330	D₁₀₉
hypocenter	C₁₀₉	#1526
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#330×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₉

C₂²⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
3. #20294768	C₂	semidirect
4. #20294767	C₂	semidirect
5. #330×C₁₀₉	C₂	semidirect
58. C₁₀₉	#1526	semidirect
63. #330	D₁₀₉	semidirect

H ≅ G/N

type

3. #20294768

C₂

semidirect

4. #20294767

C₂

semidirect

5. #330×C₁₀₉

C₂

semidirect

58. C₁₀₉

#1526

semidirect

63. #330

D₁₀₉

semidirect

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

subgroup	index
#20295582	2
#20294978	2
#20294768	2
#20294767	2
#330×C₁₀₉	2
#1526	109

subgroup

index

#330×C₁₀₉

109

Subgroups — #28361625

Conjugacy classes of subgroups

Complements

Maximal subgroups