Group #28362983 — subgroups

Subgroups — #28362983

Order 13952.

548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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	#20295010	C₂	1	3	yes	maximal, complemented
2	#20294993	C₂	1	3	yes	maximal, complemented
3	#585×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182013	C₂²	1	3	yes	complemented
5	C₂²×#11724784	C₂²	1	1	yes	characteristic, complemented
6	#16812618	—	2	3	no	—
7	#16812616	—	2	3	no	—
8	C₂²⋊#11724784	—	2	6	no	—
9	2₋¹⁺⁴×C₁₀₉	—	2	3	no	—
10	H(5,𝔽₂)×C₁₀₉	—	2	3	no	—
11	H(5,𝔽₂)×C₁₀₉	C₄	1	4	yes	—
12	C₂×#41×C₁₀₉	—	2	6	no	—
13	C₂×#41×C₁₀₉	C₂²	1	3	yes	—
14	#13182013	—	2	6	no	—
15	C₂×#11724784	—	2	3	no	—
16	C₂×#11724784	—	2	3	no	—
17	#41×C₁₀₉	—	2	4	no	—
18	#41×C₁₀₉	—	2	12	no	—
19	#41×C₁₀₉	—	2	12	no	—
20	#41×C₁₀₉	—	2	6	no	—
21	#41×C₁₀₉	—	2	6	no	—
22	C₂×Q₈×C₁₀₉	—	2	6	no	—
23	C₂×Q₈×C₁₀₉	D₄	1	3	yes	—
24	C₂×D₄×C₁₀₉	—	2	6	no	—
25	C₂×D₄×C₁₀₉	—	2	12	no	—
26	C₂×D₄×C₁₀₉	C₂×C₄	1	6	yes	—
27	C₂×D₄×C₁₀₉	D₄	1	3	yes	—
28	C₂²×C₄×C₁₀₉	—	2	4	no	—
29	C₂²×C₄×C₁₀₉	D₄	1	6	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂³×C₁₀₉	—	2	4	no	—
32	C₂³×C₁₀₉	C₂²⋊C₄	1	6	yes	—
33	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	Q₈×C₁₀₉	—	2	4	no	—
35	Q₈×C₁₀₉	—	2	6	no	—
36	D₄×C₁₀₉	—	2	12	no	—
37	D₄×C₁₀₉	—	2	12	no	—
38	D₄×C₁₀₉	—	2	6	no	—
39	C₂×C₄×C₁₀₉	—	2	12	no	—
40	C₂×C₄×C₁₀₉	—	2	6	no	—
41	C₂×C₄×C₁₀₉	—	2	6	no	—
42	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	6	yes	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
45	#11724784	—	4	1	no	—
46	#11724784	—	4	3	no	—
47	C₂²×C₁₀₉	—	2	6	no	—
48	C₂²×C₁₀₉	—	4	1	no	—
49	C₂²×C₁₀₉	—	4	6	no	—
50	C₂²×C₁₀₉	—	8	4	no	—
51	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	3	yes	—
52	C₄×C₁₀₉	—	2	6	no	—
53	C₄×C₁₀₉	#120	1	3	yes	—
54	C₄×C₁₀₉	#120	1	1	yes	characteristic
55	C₂×C₁₀₉	—	2	3	no	—
56	C₂×C₁₀₉	—	4	6	no	—
57	C₂×C₁₀₉	#379	1	1	yes	characteristic
58	#1791	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1791	1	1	yes	characteristic, complemented, Sylow
60	#585	D₁₀₉	1	1	yes	characteristic
61	#413	—	109	3	no	—
62	#408	—	109	3	no	—
63	2₋¹⁺⁴	—	2	3	no	—
64	H(5,𝔽₂)	—	2	3	no	—
65	H(5,𝔽₂)	Dic₁₀₉	1	4	yes	—
66	C₂×#41	—	2	6	no	—
67	C₂×#41	C₂×D₁₀₉	1	3	yes	—
68	C₂×M₁₆	—	109	3	no	—
69	C₂²×C₈	—	109	1	no	—
70	#101	—	218	3	no	—
71	#100	—	218	3	no	—
72	C₂²⋊C₈	—	218	6	no	—
73	#41	—	2	4	no	—
74	#41	—	2	12	no	—
75	#41	—	2	12	no	—
76	#41	—	2	6	no	—
77	#41	—	2	6	no	—
78	C₂×Q₈	—	2	6	no	—
79	C₂×Q₈	C₂²⋊D₁₀₉	1	3	yes	—
80	C₂×D₄	—	2	6	no	—
81	C₂×D₄	—	2	12	no	—
82	C₂×D₄	C₂×Dic₁₀₉	1	6	yes	—
83	C₂×D₄	C₂²⋊D₁₀₉	1	3	yes	—
84	C₂²×C₄	—	2	4	no	—
85	C₂²×C₄	C₂²⋊D₁₀₉	1	6	yes	—
86	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
87	M₁₆	—	218	6	no	—
88	C₂×C₈	—	218	3	no	—
89	C₂×C₈	—	218	3	no	—
90	C₂³	—	2	4	no	—
91	C₂³	C₂²⋊Dic₁₀₉	1	6	yes	—
92	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	2	6	no	—
95	D₄	—	2	12	no	—
96	D₄	—	2	12	no	—
97	D₄	—	2	6	no	—
98	C₂×C₄	—	2	12	no	—
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	C₂²⋊Dic₁₀₉	1	6	yes	—
102	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
103	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
104	C₈	—	436	3	no	—
105	C₈	—	436	1	no	—
106	C₂²	—	2	6	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	6	no	—
109	C₂²	—	8	4	no	—
110	C₂²	C₂×C₂²⋊Dic₁₀₉	1	3	yes	—
111	C₄	—	2	6	no	—
112	C₄	#16812668	1	3	yes	—
113	C₄	#16812668	1	1	yes	characteristic
114	C₂	—	2	3	no	—
115	C₂	—	4	6	no	—
116	C₂	#20294882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295010

C₂

yes

maximal, complemented

#20294993

C₂

yes

maximal, complemented

#585×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

complemented

C₂²×#11724784

C₂²

yes

characteristic, complemented

#16812618

—

#16812616

—

C₂²⋊#11724784

—

2₋¹⁺⁴×C₁₀₉

—

H(5,𝔽₂)×C₁₀₉

—

H(5,𝔽₂)×C₁₀₉

C₄

yes

—

C₂×#41×C₁₀₉

—

C₂×#41×C₁₀₉

C₂²

yes

—

#13182013

—

C₂×#11724784

—

C₂×#11724784

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

—

C₄×C₁₀₉

#120

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#379

yes

characteristic

#1791

—

109

maximal, Sylow

C₁₀₉

#1791

yes

characteristic, complemented, Sylow

#585

D₁₀₉

yes

characteristic

#413

—

109

—

#408

—

109

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

Dic₁₀₉

yes

—

C₂×#41

—

C₂×#41

C₂×D₁₀₉

yes

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#101

—

218

—

#100

—

218

—

C₂²⋊C₈

—

218

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₀₉

yes

—

C₂×D₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂³

—

C₂³

C₂²⋊Dic₁₀₉

yes

—

C₂³

C₂²×Dic₁₀₉

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

102

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

103

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

104

C₈

—

436

—

105

C₈

—

436

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

—

111

C₄

—

112

C₄

#16812668

yes

—

113

C₄

#16812668

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#20294882

yes

characteristic

	subgroup	G/H
center	C₄	#16812668
commutator	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting, prime radical	#585×C₁₀₉	C₂
socle	C₂×C₁₀₉	#379
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#585	D₁₀₉
hypocenter	C₁₀₉	#1791
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₄

#16812668

commutator

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting, prime radical

#585×C₁₀₉

C₂

socle

C₂×C₁₀₉

#379

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #20295010	C₂	semidirect
2. #20294993	C₂	semidirect
4. C₂×#13182013	C₂²	semidirect
5. C₂²×#11724784	C₂²	semidirect
59. C₁₀₉	#1791	semidirect

H ≅ G/N

type

1. #20295010

C₂

semidirect

2. #20294993

C₂

semidirect

4. C₂×#13182013

C₂²

semidirect

5. C₂²×#11724784

C₂²

semidirect

59. C₁₀₉

#1791

semidirect

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

subgroup	index
#20295010	2
#20294993	2
#585×C₁₀₉	2
#1791	109

subgroup

index

#20295010

#20294993

#585×C₁₀₉

#1791

109

Subgroups — #28362983

Conjugacy classes of subgroups

Complements

Maximal subgroups