Group #28363168 — subgroups

Subgroups — #28363168

Order 13952.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#20295049	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294791	C₂	1	1	yes	maximal, characteristic
4	#20294790	C₂	1	2	yes	maximal
5	#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812628	—	2	4	no	—
10	C₈×Dic₁₀₉	C₄	1	2	yes	—
11	C₈×Dic₁₀₉	C₄	1	2	yes	—
12	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₉	Q₈	1	2	yes	—
15	#13182013	—	2	2	no	—
16	#13182013	—	2	2	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	—	2	2	no	—
19	C₂×#11724784	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₁₀₉	D₄	1	2	yes	—
23	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₀₉	—	2	2	no	—
25	M₁₆×C₁₀₉	—	2	2	no	—
26	C₂×C₈×C₁₀₉	—	2	2	no	—
27	C₂×C₈×C₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	#41	1	2	yes	—
33	C₂×C₄×C₁₀₉	#41	1	2	yes	—
34	C₂×Dic₁₀₉	—	2	4	no	—
35	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
37	C₈×C₁₀₉	—	2	2	no	—
38	C₈×C₁₀₉	—	4	2	no	—
39	#11724784	—	2	2	no	—
40	#11724784	—	4	2	no	—
41	C₂²×C₁₀₉	—	2	2	no	—
42	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₀₉	#125	1	2	yes	—
44	C₄×C₁₀₉	#117	1	1	yes	characteristic
45	C₄×C₁₀₉	#117	1	1	yes	characteristic
46	C₄×C₁₀₉	#124	1	2	yes	—
47	Dic₁₀₉	—	2	2	no	—
48	Dic₁₀₉	#108	1	4	yes	—
49	C₂×C₁₀₉	—	2	2	no	—
50	C₂×C₁₀₉	#382	1	1	yes	characteristic
51	C₂×C₁₀₉	C₂×#108	1	2	yes	—
52	#1837	—	109	1	no	maximal, Sylow
53	C₁₀₉	#1837	1	1	yes	characteristic, complemented, Sylow
54	#423	—	109	2	no	—
55	C₂×C₄×C₈	—	109	1	no	—
56	#341	—	109	2	no	—
57	#341	—	109	1	no	—
58	#341	D₁₀₉	1	1	yes	characteristic
59	C₂×M₁₆	—	109	2	no	—
60	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
61	C₂²×C₈	—	109	1	no	—
62	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
63	C₂×C₄²	—	109	1	no	—
64	C₄⋊C₈	—	218	4	no	—
65	C₄×C₈	—	109	2	no	—
66	C₄×C₈	—	109	2	no	—
67	C₂²×C₄	—	109	2	no	—
68	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	218	2	no	—
72	M₁₆	—	218	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	109	2	no	—
76	C₂×C₈	—	218	2	no	—
77	C₂×C₈	—	218	2	no	—
78	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
79	C₄²	—	109	1	no	—
80	C₄²	—	109	1	no	—
81	C₄²	—	109	2	no	—
82	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	218	4	no	—
86	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
87	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
88	C₂×C₄	#13182028	1	2	yes	—
89	C₂×C₄	#13182029	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	436	2	no	—
93	C₈	—	218	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
96	C₂²	#16812689	1	2	yes	—
97	C₄	—	109	4	no	—
98	C₄	—	218	2	no	—
99	C₄	#16812655	1	1	yes	characteristic
100	C₄	#16812656	1	1	yes	characteristic
101	C₄	#16812685	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#20294890	1	1	yes	characteristic
104	C₂	#108×D₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295049

C₂

yes

maximal

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294791

C₂

yes

maximal, characteristic

#20294790

C₂

yes

maximal

#341×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812628

—

C₈×Dic₁₀₉

C₄

yes

—

C₈×Dic₁₀₉

C₄

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#125

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

Dic₁₀₉

#108

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#382

yes

characteristic

C₂×C₁₀₉

C₂×#108

yes

—

#1837

—

109

maximal, Sylow

C₁₀₉

#1837

yes

characteristic, complemented, Sylow

#423

—

109

—

C₂×C₄×C₈

—

109

—

#341

—

109

—

#341

—

109

—

#341

D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

Q₈×D₁₀₉

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

D₄×D₁₀₉

yes

characteristic

C₂×C₄

#13182028

yes

—

C₂×C₄

#13182029

yes

—

C₈

—

C₈

—

C₈

—

436

—

C₈

—

218

—

C₂²

—

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

C₂²

#16812689

yes

—

C₄

—

109

—

C₄

—

218

—

C₄

#16812655

yes

characteristic

100

C₄

#16812656

yes

characteristic

101

C₄

#16812685

yes

—

102

C₂

—

103

C₂

#20294890

yes

characteristic

104

C₂

#108×D₁₀₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#341×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#341	D₁₀₉
hypocenter	C₁₀₉	#1837
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#341×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
53. C₁₀₉	#1837	semidirect

H ≅ G/N

type

53. C₁₀₉

#1837

semidirect

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

subgroup	index
#20295049	2
C₂×C₈×Dic₁₀₉	2
#20294791	2
#20294790	2
#341×C₁₀₉	2
#1837	109

subgroup

index

C₂×C₈×Dic₁₀₉

#341×C₁₀₉

109

Subgroups — #28363168

Conjugacy classes of subgroups

Complements

Maximal subgroups