Group #28371641 — subgroups

Subgroups — #28371641

Order 13952.

440 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 249 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	#20295793	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812706	C₂	1	1	yes	maximal, characteristic
3	#20295074	C₂	1	6	yes	maximal, complemented
4	C₂×#16812634	C₂	1	6	yes	maximal, complemented
5	#568×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812706	—	2	4	no	—
7	#16812706	C₄	1	8	yes	—
8	C₂×#13182013	C₂²	1	6	yes	—
9	C₂×#13182013	C₂²	1	3	yes	—
10	C₂²×#11724784	C₂²	1	3	yes	—
11	#16812634	C₂²	1	4	yes	—
12	#16812634	C₂²	1	12	yes	—
13	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
14	#131×C₁₀₉	—	2	4	no	—
15	C₂×M₁₆×C₁₀₉	C₂²	1	6	yes	—
16	#13182013	—	2	6	no	—
17	#13182013	C₂×C₄	1	12	yes	—
18	#13182013	C₂³	1	4	yes	—
19	C₂×#11724784	—	2	6	no	—
20	C₂×#11724784	C₂×C₄	1	12	yes	—
21	C₂×#11724784	C₂³	1	1	yes	characteristic
22	C₂×#11724784	C₂³	1	3	yes	—
23	#41×C₁₀₉	C₂×C₄	1	4	yes	—
24	#41×C₁₀₉	D₄	1	4	yes	—
25	C₂×Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₁₀₉	Q₈	1	3	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
28	M₁₆×C₁₀₉	—	2	6	no	—
29	M₁₆×C₁₀₉	C₂³	1	4	yes	—
30	C₂×C₈×C₁₀₉	—	2	6	no	—
31	C₂³×C₁₀₉	C₂×Q₈	1	3	yes	—
32	Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
33	D₄×C₁₀₉	C₄⋊C₄	1	12	yes	—
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₁₀₉	C₂×Q₈	1	3	yes	—
37	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
38	C₈×C₁₀₉	—	2	4	no	—
39	#11724784	—	2	4	no	—
40	#11724784	C₂²×C₄	1	2	yes	—
41	#11724784	C₂²×C₄	1	6	yes	—
42	C₂²×C₁₀₉	—	4	3	no	—
43	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
44	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₀₉	—	2	1	no	—
49	C₂×C₁₀₉	—	2	6	no	—
50	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
51	#2916	—	109	1	no	maximal, Sylow
52	C₁₀₉	#2916	1	1	yes	characteristic, complemented, Sylow
53	#568	—	109	1	no	—
54	#568	D₁₀₉	1	1	yes	characteristic
55	C₂×#131	—	109	1	no	—
56	#430	—	109	6	no	—
57	C₂×#108	—	109	6	no	—
58	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
59	#131	—	2	4	no	—
60	#131	—	109	8	no	—
61	#131	—	218	4	no	—
62	C₂×M₁₆	—	109	3	no	—
63	C₂×M₁₆	—	109	6	no	—
64	C₂×M₁₆	C₂×D₁₀₉	1	6	yes	—
65	C₂²×C₈	—	109	3	no	—
66	#108	—	109	4	no	—
67	#108	—	109	12	no	—
68	#41	C₄×D₁₀₉	1	4	yes	—
69	#41	C₂²⋊D₁₀₉	1	4	yes	—
70	C₂×Q₈	Dic₂₁₈	1	1	yes	characteristic
71	C₂×D₄	Dic₂₁₈	1	3	yes	—
72	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
73	M₁₆	—	2	6	no	—
74	M₁₆	—	109	12	no	—
75	M₁₆	—	109	4	no	—
76	M₁₆	—	218	6	no	—
77	M₁₆	C₂²×D₁₀₉	1	4	yes	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	109	12	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	109	3	no	—
82	C₂×C₈	—	218	6	no	—
83	C₂³	C₂×Dic₂₁₈	1	3	yes	—
84	Q₈	#13182008	1	4	yes	—
85	D₄	#13182008	1	12	yes	—
86	C₂×C₄	C₂×C₄×D₁₀₉	1	6	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	6	yes	—
88	C₂×C₄	C₂×Dic₂₁₈	1	3	yes	—
89	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	109	2	no	—
92	C₈	—	109	6	no	—
93	C₈	—	218	4	no	—
94	C₂²	—	4	3	no	—
95	C₂²	C₂×#13182008	1	6	yes	—
96	C₂²	C₂²×Dic₂₁₈	1	1	yes	characteristic
97	C₄	C₂×#13182008	1	6	yes	—
98	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
99	C₄	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	6	no	—
102	C₂	C₂²×#13182008	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295793

C₂

yes

maximal, characteristic

C₂×#16812706

C₂

yes

maximal, characteristic

#20295074

C₂

yes

maximal, complemented

C₂×#16812634

C₂

yes

maximal, complemented

#568×C₁₀₉

C₂

yes

maximal, characteristic

#16812706

—

#16812706

C₄

yes

—

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

#16812634

C₂²

yes

—

#16812634

C₂²

yes

—

C₂×#41×C₁₀₉

C₂²

yes

characteristic

#131×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

#13182013

—

#13182013

C₂×C₄

yes

—

#13182013

C₂³

yes

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

#41×C₁₀₉

C₂×C₄

yes

—

#41×C₁₀₉

D₄

yes

—

C₂×Q₈×C₁₀₉

Q₈

yes

characteristic

C₂×D₄×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

—

C₂³×C₁₀₉

C₂×Q₈

yes

—

Q₈×C₁₀₉

C₄⋊C₄

yes

—

D₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₈×C₁₀₉

—

#11724784

—

#11724784

C₂²×C₄

yes

—

#11724784

C₂²×C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×C₄⋊C₄

yes

characteristic

#2916

—

109

maximal, Sylow

C₁₀₉

#2916

yes

characteristic, complemented, Sylow

#568

—

109

—

#568

D₁₀₉

yes

characteristic

C₂×#131

—

109

—

#430

—

109

—

C₂×#108

—

109

—

C₂×#41

C₂×D₁₀₉

yes

characteristic

#131

—

#131

—

109

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

#108

—

109

—

#108

—

109

—

#41

C₄×D₁₀₉

yes

—

#41

C₂²⋊D₁₀₉

yes

—

C₂×Q₈

Dic₂₁₈

yes

characteristic

C₂×D₄

Dic₂₁₈

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

M₁₆

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

C₂²×D₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂³

C₂×Dic₂₁₈

yes

—

Q₈

#13182008

yes

—

D₄

#13182008

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

C₂×C₄

C₂×Dic₂₁₈

yes

—

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₂²

—

C₂²

C₂×#13182008

yes

—

C₂²

C₂²×Dic₂₁₈

yes

characteristic

C₄

C₂×#13182008

yes

—

C₄

C₂²×C₄×D₁₀₉

yes

characteristic

C₄

C₂²×C₂²⋊D₁₀₉

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

C₂²×#13182008

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#568×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#568	D₁₀₉
hypocenter	C₁₀₉	#2916
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₄

C₂²×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#568×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂²×C₄⋊C₄

prime radical

C₂×#41×C₁₀₉

C₂²

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
3. #20295074	C₂	semidirect
4. C₂×#16812634	C₂	semidirect
52. C₁₀₉	#2916	semidirect

H ≅ G/N

type

3. #20295074

C₂

semidirect

4. C₂×#16812634

C₂

semidirect

52. C₁₀₉

#2916

semidirect

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

subgroup	index
#20295793	2
C₂×#16812706	2
#20295074	2
C₂×#16812634	2
#568×C₁₀₉	2
#2916	109

subgroup

index

#568×C₁₀₉

109

Subgroups — #28371641

Conjugacy classes of subgroups

Complements

Maximal subgroups