Group #27170886 — subgroups

Subgroups — #27170886

Order 12928.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20265135	C₂	1	1	yes	maximal, characteristic
2	#20265133	C₂	1	2	yes	maximal
3	#20264935	C₂	1	1	yes	maximal, characteristic
4	#20264933	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	2	yes	—
7	C₂²×#11717579	C₂²	1	1	yes	characteristic
8	#15692214	C₂²	1	1	yes	characteristic
9	#15692166	C₄	1	4	yes	complemented
10	#15692165	C₄	1	2	yes	complemented
11	C₄×#11717579	C₄	1	4	yes	complemented
12	C₈×Dic₁₀₁	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
14	#117×C₁₀₁	C₂²	1	2	yes	—
15	#13108058	—	2	2	no	—
16	#13108058	—	2	2	no	—
17	C₂×#11717579	C₂×C₄	1	4	yes	—
18	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
19	C₂×#11717579	C₂×C₄	1	2	yes	—
20	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
21	C₂×#11717579	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₀₁	—	2	1	no	—
23	C₂²⋊Dic₁₀₁	—	2	1	no	—
24	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₁	—	2	2	no	—
27	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₀₁	—	2	2	no	—
29	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
30	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
31	C₂³×C₁₀₁	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	—	2	4	no	—
33	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₁	#41	1	2	yes	—
36	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
37	C₂×Dic₁₀₁	—	2	2	no	—
38	C₈×C₁₀₁	—	2	2	no	—
39	#11717579	—	2	4	no	—
40	#11717579	C₄²	1	1	yes	characteristic
41	#11717579	C₄²	1	1	yes	characteristic
42	#11717579	C₄²	1	2	yes	—
43	C₂²×C₁₀₁	—	4	1	no	—
44	C₂²×C₁₀₁	#117	1	2	yes	—
45	C₂²×C₁₀₁	#117	1	1	yes	characteristic
46	C₄×C₁₀₁	—	4	4	no	—
47	C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₁	#117	1	1	yes	characteristic
49	C₄×C₁₀₁	#117	1	2	yes	—
50	Dic₁₀₁	—	4	2	no	—
51	C₂×C₁₀₁	—	2	1	no	—
52	C₂×C₁₀₁	—	2	2	no	—
53	C₂×C₁₀₁	#376	1	1	yes	characteristic
54	#1747	—	101	1	no	maximal, Sylow
55	C₁₀₁	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	101	1	no	—
57	#405	—	101	2	no	—
58	#338	—	101	1	no	—
59	H(3,ℤ/4)	—	101	2	no	—
60	H(3,ℤ/4)	D₁₀₁	1	1	yes	characteristic
61	C₂×M₁₆	—	101	2	no	—
62	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
63	C₂²×C₈	—	101	1	no	—
64	#117	—	101	1	no	—
65	#117	C₂×D₁₀₁	1	2	yes	—
66	#97	—	101	2	no	—
67	#97	—	101	4	no	—
68	C₄×C₈	—	101	2	no	—
69	C₄×C₈	—	101	4	no	—
70	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	202	2	no	—
73	M₁₆	—	202	2	no	—
74	C₂×C₈	—	101	1	no	—
75	C₂×C₈	—	101	2	no	—
76	C₂×C₈	—	101	1	no	—
77	C₂×C₈	—	101	2	no	—
78	C₂×C₈	—	101	4	no	—
79	C₂×C₈	C₄×D₁₀₁	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	202	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	202	1	no	—
84	C₄²	—	101	2	no	—
85	C₄²	C₄×D₁₀₁	1	4	yes	—
86	C₂³	#13108073	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	202	2	no	—
89	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
91	C₂×C₄	#13108073	1	2	yes	—
92	C₂×C₄	#13108075	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	101	1	no	—
95	C₈	—	101	1	no	—
96	C₈	—	101	2	no	—
97	C₈	—	202	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#15692211	1	2	yes	—
100	C₂²	#15692213	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	404	2	no	—
103	C₄	C₄²×D₁₀₁	1	1	yes	characteristic
104	C₄	#15692211	1	1	yes	characteristic
105	C₄	#15692213	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#20265022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265135

C₂

yes

maximal, characteristic

#20265133

C₂

yes

maximal

#20264935

C₂

yes

maximal, characteristic

#20264933

C₂

yes

maximal

H(3,ℤ/4)×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₂²×#11717579

C₂²

yes

characteristic

#15692214

C₂²

yes

characteristic

#15692166

C₄

yes

complemented

#15692165

C₄

yes

complemented

C₄×#11717579

C₄

yes

complemented

C₈×Dic₁₀₁

C₄

yes

complemented

C₂×M₁₆×C₁₀₁

C₂²

yes

characteristic

#117×C₁₀₁

C₂²

yes

—

#13108058

—

#13108058

—

C₂×#11717579

C₂×C₄

yes

—

C₂×#11717579

C₂×C₄

yes

characteristic

C₂×#11717579

C₂×C₄

yes

—

C₂×#11717579

C₂×C₄

yes

characteristic

C₂×#11717579

C₂×C₄

yes

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₂³×C₁₀₁

#41

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

#11717579

C₄²

yes

characteristic

#11717579

C₄²

yes

characteristic

#11717579

C₄²

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#117

yes

—

C₂²×C₁₀₁

#117

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×C₄²

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#376

yes

characteristic

#1747

—

101

maximal, Sylow

C₁₀₁

#1747

yes

characteristic, complemented, Sylow

#405

—

101

—

#405

—

101

—

#338

—

101

—

H(3,ℤ/4)

—

101

—

H(3,ℤ/4)

D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

#117

—

101

—

#117

C₂×D₁₀₁

yes

—

#97

—

101

—

#97

—

101

—

C₄×C₈

—

101

—

C₄×C₈

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

C₄×D₁₀₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

C₂²⋊C₄

—

202

—

C₄²

—

101

—

C₄²

C₄×D₁₀₁

yes

—

C₂³

#13108073

yes

characteristic

C₂×C₄

—

C₂×C₄

—

202

—

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

C₂×C₄

#13108073

yes

—

C₂×C₄

#13108075

yes

characteristic

C₈

—

C₈

—

101

—

C₈

—

101

—

C₈

—

101

—

C₈

—

202

—

C₂²

—

C₂²

#15692211

yes

—

100

C₂²

#15692213

yes

characteristic

101

C₄

—

102

C₄

—

404

—

103

C₄

C₄²×D₁₀₁

yes

characteristic

104

C₄

#15692211

yes

characteristic

105

C₄

#15692213

yes

—

106

C₂

—

107

C₂

—

108

C₂

#20265022

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₀₁
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	H(3,ℤ/4)×C₁₀₁	C₂
socle	C₂×C₁₀₁	#376
prime radical	C₂³×C₁₀₁	#41
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	H(3,ℤ/4)	D₁₀₁
hypocenter	C₁₀₁	#1747
LCS	G ▷ C₄×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₀₁

commutator

C₄×C₁₀₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

H(3,ℤ/4)×C₁₀₁

C₂

C₂×C₁₀₁

C₂³×C₁₀₁

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₄×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #15692166	C₄	semidirect
10. #15692165	C₄	semidirect
11. C₄×#11717579	C₄	semidirect
12. C₈×Dic₁₀₁	C₄	semidirect
55. C₁₀₁	#1747	semidirect

H ≅ G/N

type

9. #15692166

C₄

semidirect

10. #15692165

C₄

semidirect

11. C₄×#11717579

C₄

semidirect

12. C₈×Dic₁₀₁

C₄

semidirect

55. C₁₀₁

#1747

semidirect

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

subgroup	index
#20265135	2
#20265133	2
#20264935	2
#20264933	2
H(3,ℤ/4)×C₁₀₁	2
#1747	101

subgroup

index

H(3,ℤ/4)×C₁₀₁

101

Subgroups — #27170886

Conjugacy classes of subgroups

Complements

Maximal subgroups