Group #28364195 — subgroups

Subgroups — #28364195

Order 13952.

466 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20294881	C₂	1	3	yes	maximal
2	#20294793	C₂	1	3	yes	maximal, complemented
3	#543×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₁₀₉	C₂²	1	3	yes	—
6	C₂×#13182006	—	2	6	no	—
7	C₂²⋊Q₈×C₁₀₉	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
9	C₂³×D₁₀₉	—	2	3	no	—
10	C₂³×D₁₀₉	—	2	3	no	—
11	C₂³×D₁₀₉	—	4	4	no	—
12	C₂³×D₁₀₉	Q₈	1	2	yes	complemented
13	C₂²×Dic₁₀₉	—	2	3	no	—
14	C₂²⋊Dic₁₀₉	—	2	6	no	—
15	#13182006	—	4	12	no	—
16	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₁₀₉	—	2	2	no	—
18	C₂²×C₄×C₁₀₉	D₄	1	6	yes	—
19	C₄⋊C₄×C₁₀₉	—	2	6	no	—
20	C₂²⋊C₄×C₁₀₉	—	2	6	no	—
21	C₂³×C₁₀₉	—	4	2	no	—
22	C₂³×C₁₀₉	C₂×D₄	1	3	yes	—
23	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
24	C₂³×C₁₀₉	#41	1	3	yes	—
25	C₂²×D₁₀₉	—	2	12	no	—
26	C₂²×D₁₀₉	—	2	6	no	—
27	C₂²×D₁₀₉	—	4	4	no	—
28	C₂²×D₁₀₉	—	4	12	no	—
29	C₂²×D₁₀₉	—	4	6	no	—
30	C₂²×D₁₀₉	—	4	1	no	—
31	C₂²×D₁₀₉	—	4	3	no	—
32	Q₈×C₁₀₉	—	8	2	no	—
33	C₂×C₄×C₁₀₉	—	2	6	no	—
34	C₂×C₄×C₁₀₉	—	4	6	no	—
35	C₂×Dic₁₀₉	—	4	6	no	—
36	C₂×Dic₁₀₉	—	4	3	no	—
37	C₂²×C₁₀₉	—	2	6	no	—
38	C₂²×C₁₀₉	—	4	1	no	—
39	C₂²×C₁₀₉	—	4	3	no	—
40	C₂²×C₁₀₉	#120	1	1	yes	characteristic
41	C₂²×C₁₀₉	C₂²⋊Q₈	1	3	yes	—
42	C₂²×C₁₀₉	#124	1	3	yes	—
43	C₂×D₁₀₉	—	2	12	no	—
44	C₂×D₁₀₉	—	4	12	no	—
45	C₂×D₁₀₉	—	4	6	no	—
46	C₂×D₁₀₉	—	4	2	no	—
47	C₂×D₁₀₉	—	4	4	no	—
48	C₄×C₁₀₉	—	4	6	no	—
49	Dic₁₀₉	—	8	3	no	—
50	C₂×C₁₀₉	—	2	3	no	—
51	C₂×C₁₀₉	—	2	3	no	—
52	C₂×C₁₀₉	#393	1	1	yes	characteristic
53	C₂×C₁₀₉	B(4,𝔽₂)	1	2	yes	—
54	D₁₀₉	—	4	4	no	—
55	#2029	—	109	1	no	maximal, Sylow
56	C₁₀₉	#2029	1	1	yes	characteristic, complemented, Sylow
57	#543	D₁₀₉	1	1	yes	characteristic, complemented
58	#379	—	109	3	no	—
59	#342	—	109	3	no	—
60	C₂⁵	—	109	1	no	—
61	C₂²⋊Q₈	—	2	6	no	—
62	C₂×C₂²⋊C₄	—	109	3	no	—
63	C₂×C₂²⋊C₄	—	218	6	no	—
64	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	3	yes	—
65	C₂⁴	—	109	2	no	—
66	C₂⁴	—	436	4	no	—
67	C₂⁴	—	218	3	no	—
68	C₂⁴	—	218	3	no	—
69	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	218	3	no	—
72	C₂²×C₄	C₂²⋊D₁₀₉	1	6	yes	—
73	C₄⋊C₄	—	2	6	no	—
74	C₂²⋊C₄	—	2	6	no	—
75	C₂²⋊C₄	—	436	12	no	—
76	C₂²⋊C₄	—	218	6	no	—
77	C₂³	—	4	2	no	—
78	C₂³	—	436	6	no	—
79	C₂³	—	436	12	no	—
80	C₂³	—	436	4	no	—
81	C₂³	—	436	1	no	—
82	C₂³	—	436	3	no	—
83	C₂³	—	218	12	no	—
84	C₂³	—	218	6	no	—
85	C₂³	C₂×C₂²⋊D₁₀₉	1	3	yes	—
86	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
87	C₂³	#13182030	1	3	yes	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	4	6	no	—
91	C₂×C₄	—	436	6	no	—
92	C₂×C₄	—	436	3	no	—
93	C₂²	—	2	6	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	3	no	—
96	C₂²	—	436	12	no	—
97	C₂²	—	436	6	no	—
98	C₂²	—	436	2	no	—
99	C₂²	—	436	4	no	—
100	C₂²	—	218	12	no	—
101	C₂²	#16812668	1	1	yes	characteristic
102	C₂²	#16812675	1	3	yes	—
103	C₂²	#16812688	1	3	yes	—
104	C₄	—	4	6	no	—
105	C₄	—	872	3	no	—
106	C₂	—	2	3	no	—
107	C₂	—	2	3	no	—
108	C₂	—	436	4	no	—
109	C₂	#20294937	1	1	yes	characteristic
110	C₂	#20295213	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20294881

C₂

yes

maximal

#20294793

C₂

yes

maximal, complemented

#543×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₀₉

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182006

—

C₂²⋊Q₈×C₁₀₉

—

C₂×C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₂³×D₁₀₉

—

C₂³×D₁₀₉

—

C₂³×D₁₀₉

—

C₂³×D₁₀₉

Q₈

yes

complemented

C₂²×Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

#13182006

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂³×C₁₀₉

#41

yes

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#120

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

#124

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#393

yes

characteristic

C₂×C₁₀₉

B(4,𝔽₂)

yes

—

D₁₀₉

—

#2029

—

109

maximal, Sylow

C₁₀₉

#2029

yes

characteristic, complemented, Sylow

#543

D₁₀₉

yes

characteristic, complemented

#379

—

109

—

#342

—

109

—

C₂⁵

—

109

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

218

—

C₂×C₂²⋊C₄

C₂×D₁₀₉

yes

—

C₂⁴

—

109

—

C₂⁴

—

436

—

C₂⁴

—

218

—

C₂⁴

—

218

—

C₂⁴

C₂²×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

436

—

C₂²⋊C₄

—

218

—

C₂³

—

C₂³

—

436

—

C₂³

—

436

—

C₂³

—

436

—

C₂³

—

436

—

C₂³

—

436

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

—

C₂³

Q₈×D₁₀₉

yes

characteristic

C₂³

#13182030

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

436

—

100

C₂²

—

218

—

101

C₂²

#16812668

yes

characteristic

102

C₂²

#16812675

yes

—

103

C₂²

#16812688

yes

—

104

C₄

—

105

C₄

—

872

—

106

C₂

—

107

C₂

—

108

C₂

—

436

—

109

C₂

#20294937

yes

characteristic

110

C₂

#20295213

yes

—

	subgroup	G/H
center	C₂²	#16812668
commutator, cosocle	C₂⁴×C₁₀₉	C₂³
Frattini	C₂⁴	C₂²×D₁₀₉
Fitting	#543×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#120
prime radical	C₂⁴×D₁₀₉	C₂²
hypercenter	#543	D₁₀₉
hypocenter	C₁₀₉	#2029
LCS	G ▷ C₂⁴×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#16812668

commutator, cosocle

C₂⁴×C₁₀₉

C₂³

Frattini

C₂⁴

C₂²×D₁₀₉

Fitting

#543×C₁₀₉

C₂

socle

C₂²×C₁₀₉

#120

prime radical

C₂⁴×D₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂⁴×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #20294793	C₂	semidirect
3. #543×C₁₀₉	C₂	semidirect
12. C₂³×D₁₀₉	Q₈	semidirect
56. C₁₀₉	#2029	semidirect
57. #543	D₁₀₉	semidirect

H ≅ G/N

type

2. #20294793

C₂

semidirect

3. #543×C₁₀₉

C₂

semidirect

12. C₂³×D₁₀₉

Q₈

semidirect

56. C₁₀₉

#2029

semidirect

57. #543

D₁₀₉

semidirect

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

subgroup	index
#20294881	2
#20294793	2
#543×C₁₀₉	2
#2029	109

subgroup

index

#20294881

#20294793

#543×C₁₀₉

#2029

109

Subgroups — #28364195

Conjugacy classes of subgroups

Complements

Maximal subgroups