Group #28364625 — subgroups

Subgroups — #28364625

Order 13952.

196 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#16812638	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294802	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
5	#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₄²×D₁₀₉	C₄	1	1	yes	characteristic
9	#16812638	C₄	1	8	yes	complemented
10	C₂×#13182004	C₂²	1	2	yes	—
11	C₄×#11724784	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₈	1	2	yes	—
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₀₉	C₈	1	4	yes	—
21	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
22	C₄×Dic₁₀₉	C₈	1	2	yes	—
23	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
24	#13182004	C₂×C₄	1	4	yes	complemented
25	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
29	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
34	C₄×D₁₀₉	—	2	4	no	—
35	C₄×D₁₀₉	C₄²	1	2	yes	—
36	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
37	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
38	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
39	#11724784	C₄²	1	2	yes	—
40	#11724784	C₂×C₈	1	2	yes	—
41	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
42	C₂×D₁₀₉	—	2	2	no	—
43	C₂×D₁₀₉	C₄×C₈	1	2	yes	—
44	C₄×C₁₀₉	—	2	2	no	—
45	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
47	Dic₁₀₉	C₄×C₈	1	2	yes	—
48	Dic₁₀₉	M₃₂	1	4	yes	complemented
49	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
50	C₂×C₁₀₉	C₂×M₃₂	1	2	yes	—
51	D₁₀₉	—	2	2	no	—
52	C₄×M₃₂	—	109	1	no	maximal, Sylow
53	C₁₀₉	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
54	C₂×M₃₂	—	109	2	no	—
55	C₂×C₄×C₈	—	109	1	no	—
56	#346	—	109	1	no	—
57	#346	D₁₀₉	1	1	yes	characteristic, complemented
58	C₄×C₁₆	—	109	2	no	—
59	C₂²×C₈	—	109	1	no	—
60	C₂²×C₈	—	109	1	no	—
61	C₂×C₄²	—	109	1	no	—
62	M₃₂	—	109	8	no	—
63	C₂×C₁₆	—	109	2	no	—
64	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
65	C₄×C₈	—	109	1	no	—
66	C₄×C₈	—	109	1	no	—
67	C₄×C₈	—	109	1	no	—
68	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
69	C₂²×C₄	—	109	1	no	—
70	C₂²×C₄	—	109	2	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	109	4	no	—
74	C₂×C₈	—	109	4	no	—
75	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
76	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
77	C₄²	—	109	2	no	—
78	C₄²	—	109	1	no	—
79	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
80	C₁₆	—	109	4	no	—
81	C₁₆	C₄×D₁₀₉	1	4	yes	complemented
82	C₂³	—	109	1	no	—
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	109	1	no	—
85	C₂×C₄	—	109	2	no	—
86	C₂×C₄	—	109	2	no	—
87	C₂×C₄	—	218	4	no	—
88	C₂×C₄	C₈×D₁₀₉	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
90	C₈	—	109	2	no	—
91	C₈	—	109	2	no	—
92	C₈	C₈×D₁₀₉	1	2	yes	—
93	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
94	C₂²	—	109	2	no	—
95	C₂²	—	218	2	no	—
96	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	109	4	no	—
99	C₄	—	109	2	no	—
100	C₄	C₄²×D₁₀₉	1	1	yes	characteristic
101	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
102	C₂	—	218	2	no	—
103	C₂	C₄×C₈×D₁₀₉	1	1	yes	characteristic
104	C₂	M₃₂×D₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812638

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#20294802

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₁₀₉

C₂

yes

maximal, complemented

#346×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₄

yes

characteristic

#16812638

C₄

yes

complemented

C₂×#13182004

C₂²

yes

—

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₈

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₈

yes

—

C₈×D₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₈

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#13182004

C₂×C₄

yes

complemented

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

complemented

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₄²

yes

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

C₂²×C₄

yes

—

#11724784

C₄²

yes

—

#11724784

C₂×C₈

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄×C₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

Dic₁₀₉

C₄×C₈

yes

—

Dic₁₀₉

M₃₂

yes

complemented

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

—

D₁₀₉

—

C₄×M₃₂

—

109

maximal, Sylow

C₁₀₉

C₄×M₃₂

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×C₄×C₈

—

109

—

#346

—

109

—

#346

D₁₀₉

yes

characteristic, complemented

C₄×C₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

M₃₂

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₁₆

—

109

—

C₁₆

C₄×D₁₀₉

yes

complemented

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₈×D₁₀₉

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

C₈×D₁₀₉

yes

—

C₈

C₂×C₄×D₁₀₉

yes

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

109

—

100

C₄

C₄²×D₁₀₉

yes

characteristic

101

C₄

C₂×C₈×D₁₀₉

yes

characteristic

102

C₂

—

218

—

103

C₂

C₄×C₈×D₁₀₉

yes

characteristic

104

C₂

M₃₂×D₁₀₉

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Fitting	#346×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	#346	D₁₀₉
hypocenter	C₁₀₉	C₄×M₃₂
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Fitting

#346×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂²×D₁₀₉

C₂×C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

#346

D₁₀₉

hypocenter

C₁₀₉

C₄×M₃₂

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
3. #20294802	C₂	semidirect
4. C₁₆×Dic₁₀₉	C₂	semidirect
5. #346×C₁₀₉	C₂	semidirect
9. #16812638	C₄	semidirect
24. #13182004	C₂×C₄	semidirect
28. C₁₆×C₁₀₉	C₂×C₄	semidirect
48. Dic₁₀₉	M₃₂	semidirect
53. C₁₀₉	C₄×M₃₂	semidirect
57. #346	D₁₀₉	semidirect
81. C₁₆	C₄×D₁₀₉	semidirect

H ≅ G/N

type

3. #20294802

C₂

semidirect

4. C₁₆×Dic₁₀₉

C₂

semidirect

5. #346×C₁₀₉

C₂

semidirect

9. #16812638

C₄

semidirect

24. #13182004

C₂×C₄

semidirect

28. C₁₆×C₁₀₉

C₂×C₄

semidirect

48. Dic₁₀₉

M₃₂

semidirect

53. C₁₀₉

C₄×M₃₂

semidirect

57. #346

D₁₀₉

semidirect

81. C₁₆

C₄×D₁₀₉

semidirect

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

subgroup	index
C₂×#16812638	2
C₄×C₈×D₁₀₉	2
#20294802	2
C₁₆×Dic₁₀₉	2
#346×C₁₀₉	2
C₄×M₃₂	109

subgroup

index

C₂×#16812638

C₄×C₈×D₁₀₉

#20294802

C₁₆×Dic₁₀₉

#346×C₁₀₉

C₄×M₃₂

109

Subgroups — #28364625

Conjugacy classes of subgroups

Complements

Maximal subgroups