Group #28363196 — subgroups

Subgroups — #28363196

Order 13952.

248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812633	C₂	1	2	yes	maximal
2	C₂×#16812631	C₂	1	2	yes	maximal
3	#20294892	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
6	#16812633	—	2	4	no	—
7	#16812631	—	2	4	no	—
8	#16812606	—	2	4	no	—
9	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₀₉	C₄	1	4	yes	complemented
13	C₂²×Dic₁₀₉	—	2	4	no	—
14	C₄⋊Dic₁₀₉	—	2	4	no	—
15	C₄⋊Dic₁₀₉	—	2	4	no	—
16	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
17	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
21	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
22	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
23	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
24	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₄²×C₁₀₉	Q₈	1	2	yes	—
26	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	2	4	no	—
28	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₁₀₉	—	4	4	no	—
36	C₂×Dic₁₀₉	—	4	8	no	—
37	C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
38	C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
39	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₉	#127	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	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
48	Dic₁₀₉	—	8	4	no	—
49	C₂×C₁₀₉	#384	1	1	yes	characteristic
50	C₂×C₁₀₉	#427	1	2	yes	—
51	C₂×C₁₀₉	#495	1	2	yes	—
52	C₂×C₁₀₉	#499	1	2	yes	—
53	#1846	—	109	1	no	maximal, Sylow
54	C₁₀₉	#1846	1	1	yes	characteristic, complemented, Sylow
55	C₂×#107	—	109	2	no	—
56	C₂×#106	—	109	2	no	—
57	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
58	#382	—	109	2	no	—
59	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	109	4	no	—
62	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
63	#107	—	218	4	no	—
64	#106	—	218	4	no	—
65	C₄×C₈	Dic₁₀₉	1	4	yes	complemented
66	#95	—	218	4	no	—
67	C₂²×C₄	—	218	4	no	—
68	C₂²×C₄	D₄₃₆	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
70	C₂×C₈	Dic₂₁₈	1	2	yes	—
71	C₂×C₈	Dic₂₁₈	1	2	yes	—
72	C₂×C₈	D₄₃₆	1	2	yes	—
73	C₂×C₈	D₄₃₆	1	2	yes	—
74	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
75	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
76	C₄⋊C₄	—	218	4	no	—
77	C₄⋊C₄	—	218	4	no	—
78	C₄²	Dic₂₁₈	1	2	yes	—
79	C₄²	C₂×Dic₁₀₉	1	2	yes	—
80	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	436	4	no	—
83	C₂×C₄	—	436	8	no	—
84	C₂×C₄	C₄⋊Dic₁₀₉	1	4	yes	—
85	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
87	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
89	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
90	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
91	C₈	C₄⋊Dic₁₀₉	1	4	yes	—
92	C₈	C₄⋊Dic₁₀₉	1	4	yes	—
93	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
94	C₂²	#16812696	1	1	yes	characteristic
95	C₂²	#16812699	1	1	yes	characteristic
96	C₂²	#16812721	1	4	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	872	4	no	—
99	C₄	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
100	C₄	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
101	C₄	#16812699	1	1	yes	characteristic
102	C₄	#16812699	1	1	yes	characteristic
103	C₂	#20294896	1	1	yes	characteristic
104	C₂	#20295064	1	2	yes	—
105	C₂	#20295410	1	2	yes	—
106	C₂	#20295428	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812633

C₂

yes

maximal

C₂×#16812631

C₂

yes

maximal

#20294892

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812633

—

#16812631

—

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

complemented

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#127

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#135

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

#384

yes

characteristic

C₂×C₁₀₉

#427

yes

—

C₂×C₁₀₉

#495

yes

—

C₂×C₁₀₉

#499

yes

—

#1846

—

109

maximal, Sylow

C₁₀₉

#1846

yes

characteristic, complemented, Sylow

C₂×#107

—

109

—

C₂×#106

—

109

—

C₂×C₄×C₈

D₁₀₉

yes

characteristic

#382

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#107

—

218

—

#106

—

218

—

C₄×C₈

Dic₁₀₉

yes

complemented

#95

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

D₄₃₆

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

D₄₃₆

yes

—

C₂×C₈

D₄₃₆

yes

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

Dic₂₁₈

yes

—

C₄²

C₂×Dic₁₀₉

yes

—

C₂³

C₂×D₄₃₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

C₄⋊Dic₁₀₉

yes

—

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

C₈

C₄⋊Dic₁₀₉

yes

—

C₈

C₄⋊Dic₁₀₉

yes

—

C₂²

C₂×C₄⋊Dic₁₀₉

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

872

—

C₄

C₂×C₄⋊Dic₁₀₉

yes

characteristic

100

C₄

C₂×C₄⋊Dic₁₀₉

yes

characteristic

101

C₄

#16812699

yes

characteristic

102

C₄

#16812699

yes

characteristic

103

C₂

#20294896

yes

characteristic

104

C₂

#20295064

yes

—

105

C₂

#20295410

yes

—

106

C₂

#20295428

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	#1846
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₄₃₆

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×C₄×C₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×C₄×C₈

D₁₀₉

hypocenter

C₁₀₉

#1846

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
12. C₄×C₈×C₁₀₉	C₄	semidirect
54. C₁₀₉	#1846	semidirect
65. C₄×C₈	Dic₁₀₉	semidirect

H ≅ G/N

type

12. C₄×C₈×C₁₀₉

C₄

semidirect

54. C₁₀₉

#1846

semidirect

65. C₄×C₈

Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812633	2
C₂×#16812631	2
#20294892	2
C₂×C₄×C₈×C₁₀₉	2
#1846	109

subgroup

index

C₂×C₄×C₈×C₁₀₉

109

Subgroups — #28363196

Conjugacy classes of subgroups

Complements

Maximal subgroups