Group #28363998 — subgroups

Subgroups — #28363998

Order 13952.

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

normal subgroups only

This group has 252 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×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
2	#20294892	C₂	1	1	yes	maximal, characteristic
3	#20294788	C₂	1	4	yes	maximal
4	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
7	C₄×#11724784	—	2	2	no	—
8	#16812606	—	2	2	no	—
9	#125×C₁₀₉	—	2	2	no	—
10	#125×C₁₀₉	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₉	—	2	2	no	—
15	C₂×#11724784	—	2	2	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	Q₈	1	4	yes	—
18	C₄⋊Dic₁₀₉	—	2	4	no	—
19	C₂²×C₄×C₁₀₉	—	2	2	no	—
20	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
21	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₉	—	2	8	no	—
23	C₄⋊C₄×C₁₀₉	—	2	2	no	—
24	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₄²×C₁₀₉	D₄	1	2	yes	—
27	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	4	no	—
29	C₂×C₄×C₁₀₉	—	2	4	no	—
30	C₂×C₄×C₁₀₉	—	4	4	no	—
31	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₀₉	#41	1	2	yes	—
36	C₂×Dic₁₀₉	—	4	2	no	—
37	C₂×Dic₁₀₉	—	4	4	no	—
38	#11724784	—	2	4	no	—
39	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₉	#124	1	1	yes	characteristic
41	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₀₉	#135	1	4	yes	—
43	C₄×C₁₀₉	—	2	4	no	—
44	C₄×C₁₀₉	—	4	4	no	—
45	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
46	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
47	Dic₁₀₉	—	8	2	no	—
48	C₂×C₁₀₉	#391	1	1	yes	characteristic
49	C₂×C₁₀₉	#416	1	2	yes	—
50	C₂×C₁₀₉	#488	1	2	yes	—
51	C₂×C₁₀₉	#499	1	2	yes	—
52	#1991	—	109	1	no	maximal, Sylow
53	C₁₀₉	#1991	1	1	yes	characteristic, complemented, Sylow
54	C₂×#125	D₁₀₉	1	1	yes	characteristic
55	C₂×C₄×C₈	—	109	1	no	—
56	#382	—	109	1	no	—
57	#340	—	109	4	no	—
58	C₂²×C₈	—	109	2	no	—
59	#125	—	2	2	no	—
60	#125	Dic₁₀₉	1	4	yes	complemented
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	109	2	no	—
63	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
64	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
65	C₄×C₈	—	218	2	no	—
66	#95	—	218	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	218	2	no	—
69	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
71	C₂×C₈	—	109	4	no	—
72	C₂×C₈	—	218	2	no	—
73	C₂×C₈	—	218	2	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	218	4	no	—
77	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
78	C₄²	C₂×Dic₁₀₉	1	2	yes	—
79	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	436	2	no	—
85	C₂×C₄	—	436	4	no	—
86	C₂×C₄	C₂²⋊Dic₁₀₉	1	4	yes	—
87	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
89	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
90	C₂×C₄	#13182030	1	2	yes	—
91	C₈	—	218	4	no	—
92	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
93	C₂²	#16812688	1	1	yes	characteristic
94	C₂²	#16812697	1	1	yes	characteristic
95	C₂²	#16812719	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	872	2	no	—
99	C₄	Q₈×Dic₁₀₉	1	2	yes	—
100	C₄	#16812675	1	2	yes	—
101	C₂	#20294933	1	1	yes	characteristic
102	C₂	#20295022	1	2	yes	—
103	C₂	#20295383	1	2	yes	—
104	C₂	#20295425	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

#20294892

C₂

yes

maximal, characteristic

#20294788

C₂

yes

maximal

C₂×#125×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₄×#11724784

—

#16812606

—

#125×C₁₀₉

—

#125×C₁₀₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

Q₈

yes

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

#124

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#135

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

#391

yes

characteristic

C₂×C₁₀₉

#416

yes

—

C₂×C₁₀₉

#488

yes

—

C₂×C₁₀₉

#499

yes

—

#1991

—

109

maximal, Sylow

C₁₀₉

#1991

yes

characteristic, complemented, Sylow

C₂×#125

D₁₀₉

yes

characteristic

C₂×C₄×C₈

—

109

—

#382

—

109

—

#340

—

109

—

C₂²×C₈

—

109

—

#125

—

#125

Dic₁₀₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄×C₈

—

218

—

#95

—

218

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

C₄²

C₂×Dic₁₀₉

yes

—

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

Q₈×D₁₀₉

yes

—

C₂×C₄

#13182030

yes

—

C₈

—

218

—

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

872

—

C₄

Q₈×Dic₁₀₉

yes

—

100

C₄

#16812675

yes

—

101

C₂

#20294933

yes

characteristic

102

C₂

#20295022

yes

—

103

C₂

#20295383

yes

—

104

C₂

#20295425

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#125×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#125	D₁₀₉
hypocenter	C₁₀₉	#1991
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#125×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₀₉	C₄	semidirect
53. C₁₀₉	#1991	semidirect
60. #125	Dic₁₀₉	semidirect

H ≅ G/N

type

10. #125×C₁₀₉

C₄

semidirect

53. C₁₀₉

#1991

semidirect

60. #125

Dic₁₀₉

semidirect

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

subgroup	index
C₂×C₄×#11724784	2
#20294892	2
#20294788	2
C₂×#125×C₁₀₉	2
#1991	109

subgroup

index

C₂×#125×C₁₀₉

109

Subgroups — #28363998

Conjugacy classes of subgroups

Complements

Maximal subgroups