Group #28336625 — subgroups

Subgroups — #28336625

Order 13696.

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×#16812043	C₂	1	2	yes	maximal
2	C₂×#16812041	C₂	1	2	yes	maximal
3	#20292493	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₀₇	C₂²	1	4	yes	—
6	#16812043	—	2	4	no	—
7	#16812041	—	2	4	no	—
8	C₁₀₇⋊#95	—	2	4	no	—
9	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₀₇	C₄	1	4	yes	complemented
13	C₂²×Dic₁₀₇	—	2	4	no	—
14	C₄⋊Dic₁₀₇	—	2	4	no	—
15	C₄⋊Dic₁₀₇	—	2	4	no	—
16	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
17	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
21	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
22	C₂×C₈×C₁₀₇	Q₈	1	2	yes	—
23	C₂×C₈×C₁₀₇	Q₈	1	2	yes	—
24	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₀₇	Q₈	1	2	yes	—
26	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₇	—	2	4	no	—
28	C₂×C₄×C₁₀₇	C₄⋊C₄	1	4	yes	—
29	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₁₀₇	—	4	4	no	—
36	C₂×Dic₁₀₇	—	4	8	no	—
37	C₈×C₁₀₇	C₄⋊C₄	1	4	yes	—
38	C₈×C₁₀₇	C₄⋊C₄	1	4	yes	—
39	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₇	#127	1	1	yes	characteristic
41	C₂²×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₀₇	#135	1	4	yes	—
43	C₄×C₁₀₇	—	2	4	no	—
44	C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
48	Dic₁₀₇	—	8	4	no	—
49	C₂×C₁₀₇	#384	1	1	yes	characteristic
50	C₂×C₁₀₇	#427	1	2	yes	—
51	C₂×C₁₀₇	#495	1	2	yes	—
52	C₂×C₁₀₇	#499	1	2	yes	—
53	#1846	—	107	1	no	maximal, Sylow
54	C₁₀₇	#1846	1	1	yes	characteristic, complemented, Sylow
55	C₂×#107	—	107	2	no	—
56	C₂×#106	—	107	2	no	—
57	C₂×C₄×C₈	D₁₀₇	1	1	yes	characteristic
58	#382	—	107	2	no	—
59	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	107	4	no	—
62	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
63	#107	—	214	4	no	—
64	#106	—	214	4	no	—
65	C₄×C₈	Dic₁₀₇	1	4	yes	complemented
66	#95	—	214	4	no	—
67	C₂²×C₄	—	214	4	no	—
68	C₂²×C₄	D₄₂₈	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
70	C₂×C₈	Dic₂₁₄	1	2	yes	—
71	C₂×C₈	Dic₂₁₄	1	2	yes	—
72	C₂×C₈	D₄₂₈	1	2	yes	—
73	C₂×C₈	D₄₂₈	1	2	yes	—
74	C₂×C₈	C₂×Dic₁₀₇	1	2	yes	—
75	C₂×C₈	C₂×Dic₁₀₇	1	2	yes	—
76	C₄⋊C₄	—	214	4	no	—
77	C₄⋊C₄	—	214	4	no	—
78	C₄²	Dic₂₁₄	1	2	yes	—
79	C₄²	C₂×Dic₁₀₇	1	2	yes	—
80	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	428	4	no	—
83	C₂×C₄	—	428	8	no	—
84	C₂×C₄	C₄⋊Dic₁₀₇	1	4	yes	—
85	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
87	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
89	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
90	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
91	C₈	C₄⋊Dic₁₀₇	1	4	yes	—
92	C₈	C₄⋊Dic₁₀₇	1	4	yes	—
93	C₂²	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
94	C₂²	#16812106	1	1	yes	characteristic
95	C₂²	#16812109	1	1	yes	characteristic
96	C₂²	#16812131	1	4	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	856	4	no	—
99	C₄	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
100	C₄	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
101	C₄	#16812109	1	1	yes	characteristic
102	C₄	#16812109	1	1	yes	characteristic
103	C₂	#20292497	1	1	yes	characteristic
104	C₂	#20292665	1	2	yes	—
105	C₂	#20293011	1	2	yes	—
106	C₂	#20293029	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812043

C₂

yes

maximal

C₂×#16812041

C₂

yes

maximal

#20292493

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

#16812043

—

#16812041

—

C₁₀₇⋊#95

—

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄×C₈×C₁₀₇

C₄

yes

complemented

C₂²×Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₂×C₈×C₁₀₇

Q₈

yes

—

C₂×C₈×C₁₀₇

Q₈

yes

—

C₄²×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

Q₈

yes

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₈×C₁₀₇

C₄⋊C₄

yes

—

C₈×C₁₀₇

C₄⋊C₄

yes

—

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₇

#127

yes

characteristic

C₂²×C₁₀₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₇

#135

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

Dic₁₀₇

—

C₂×C₁₀₇

#384

yes

characteristic

C₂×C₁₀₇

#427

yes

—

C₂×C₁₀₇

#495

yes

—

C₂×C₁₀₇

#499

yes

—

#1846

—

107

maximal, Sylow

C₁₀₇

#1846

yes

characteristic, complemented, Sylow

C₂×#107

—

107

—

C₂×#106

—

107

—

C₂×C₄×C₈

D₁₀₇

yes

characteristic

#382

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

C₂×C₄⋊C₄

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#107

—

214

—

#106

—

214

—

C₄×C₈

Dic₁₀₇

yes

complemented

#95

—

214

—

C₂²×C₄

—

214

—

C₂²×C₄

D₄₂₈

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

Dic₂₁₄

yes

—

C₂×C₈

Dic₂₁₄

yes

—

C₂×C₈

D₄₂₈

yes

—

C₂×C₈

D₄₂₈

yes

—

C₂×C₈

C₂×Dic₁₀₇

yes

—

C₂×C₈

C₂×Dic₁₀₇

yes

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

Dic₂₁₄

yes

—

C₄²

C₂×Dic₁₀₇

yes

—

C₂³

C₂×D₄₂₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

428

—

C₂×C₄

—

428

—

C₂×C₄

C₄⋊Dic₁₀₇

yes

—

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

C₂×C₄

C₂×D₄₂₈

yes

characteristic

C₂×C₄

C₂×D₄₂₈

yes

characteristic

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

C₈

C₄⋊Dic₁₀₇

yes

—

C₈

C₄⋊Dic₁₀₇

yes

—

C₂²

C₂×C₄⋊Dic₁₀₇

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

856

—

C₄

C₂×C₄⋊Dic₁₀₇

yes

characteristic

100

C₄

C₂×C₄⋊Dic₁₀₇

yes

characteristic

101

C₄

#16812109

yes

characteristic

102

C₄

#16812109

yes

characteristic

103

C₂

#20292497

yes

characteristic

104

C₂

#20292665

yes

—

105

C₂

#20293011

yes

—

106

C₂

#20293029

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₂₈
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×C₄×C₈×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₇
hypocenter	C₁₀₇	#1846
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₄₂₈

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂×C₄×C₈×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₀₇

hypocenter

C₁₀₇

#1846

LCS

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
12. C₄×C₈×C₁₀₇	C₄	semidirect
54. C₁₀₇	#1846	semidirect
65. C₄×C₈	Dic₁₀₇	semidirect

H ≅ G/N

type

12. C₄×C₈×C₁₀₇

C₄

semidirect

54. C₁₀₇

#1846

semidirect

65. C₄×C₈

Dic₁₀₇

semidirect

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

subgroup	index
C₂×#16812043	2
C₂×#16812041	2
#20292493	2
C₂×C₄×C₈×C₁₀₇	2
#1846	107

subgroup

index

C₂×C₄×C₈×C₁₀₇

107

Subgroups — #28336625

Conjugacy classes of subgroups

Complements

Maximal subgroups