Group #27186063 — subgroups

Subgroups — #27186063

Order 12928.

134 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 34 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	#20266022	C₂	1	2	yes	maximal
2	#20265083	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11717581	—	2	2	no	—
4	C₂²×Dic₂₀₂	C₄	1	1	yes	characteristic
5	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
6	#15692162	—	2	2	no	—
7	C₂×C₄²×C₁₀₁	C₄	1	1	yes	characteristic
8	C₂×#11717581	—	4	2	no	—
9	C₂×Dic₂₀₂	—	2	1	no	—
10	C₂×Dic₂₀₂	—	4	1	no	—
11	C₂×Dic₂₀₂	—	4	1	no	—
12	C₂×Dic₂₀₂	C₈	1	2	yes	complemented
13	C₂²×Dic₁₀₁	—	2	1	no	—
14	C₂²×Dic₁₀₁	D₄	1	2	yes	—
15	C₄⋊Dic₁₀₁	—	4	1	no	—
16	C₂²×C₄×C₁₀₁	—	2	1	no	—
17	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₁₀₁	—	4	1	no	—
19	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₁	—	2	1	no	—
21	C₂×C₄×C₁₀₁	—	2	1	no	—
22	C₂×C₄×C₁₀₁	—	2	2	no	—
23	C₂×C₄×C₁₀₁	—	4	1	no	—
24	C₂×C₄×C₁₀₁	—	4	1	no	—
25	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₀₁	M₁₆	1	1	yes	characteristic
27	C₂×Dic₁₀₁	—	2	2	no	—
28	C₂×Dic₁₀₁	—	2	2	no	—
29	C₂×Dic₁₀₁	—	4	1	no	—
30	C₂×Dic₁₀₁	—	4	1	no	—
31	C₂×Dic₁₀₁	—	4	2	no	—
32	Dic₂₀₂	—	4	2	no	—
33	Dic₂₀₂	—	4	2	no	—
34	#11717581	—	8	2	no	—
35	C₂²×C₁₀₁	—	2	1	no	—
36	C₂²×C₁₀₁	—	2	1	no	—
37	C₂²×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
38	C₂²×C₁₀₁	#99	1	1	yes	characteristic
39	C₂²×C₁₀₁	#100	1	1	yes	characteristic
40	C₄×C₁₀₁	—	2	1	no	—
41	C₄×C₁₀₁	—	2	1	no	—
42	C₄×C₁₀₁	—	4	2	no	—
43	Dic₁₀₁	—	4	2	no	—
44	Dic₁₀₁	—	4	2	no	—
45	C₂×C₁₀₁	—	2	1	no	—
46	C₂×C₁₀₁	—	2	1	no	—
47	C₂×C₁₀₁	#323	1	1	yes	characteristic
48	C₂×C₁₀₁	#354	1	1	yes	characteristic
49	C₂×C₁₀₁	#356	1	1	yes	characteristic
50	#1319	—	101	1	no	maximal, Sylow
51	C₁₀₁	#1319	1	1	yes	characteristic, complemented, Sylow
52	#391	—	101	1	no	—
53	#324	—	101	2	no	—
54	C₂²×Q₈	—	101	1	no	—
55	C₂×C₄⋊C₄	—	101	1	no	—
56	C₂×C₄²	#87938	1	1	yes	characteristic
57	C₂²⋊C₈	—	202	2	no	—
58	#95	—	202	2	no	—
59	C₂×Q₈	—	101	2	no	—
60	C₂×Q₈	—	404	1	no	—
61	C₂×Q₈	—	404	1	no	—
62	C₂×Q₈	—	202	1	no	—
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	101	2	no	—
65	C₂²×C₄	—	202	1	no	—
66	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
67	C₂×C₈	—	404	2	no	—
68	C₄⋊C₄	—	404	1	no	—
69	C₄²	—	4	1	no	—
70	C₂³	C₂²⋊#87938	1	1	yes	characteristic
71	Q₈	—	404	2	no	—
72	Q₈	—	404	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	404	1	no	—
79	C₂×C₄	—	404	1	no	—
80	C₂×C₄	—	404	2	no	—
81	C₂×C₄	—	202	2	no	—
82	C₂×C₄	—	202	2	no	—
83	C₂×C₄	D₁₀₁⋊C₈	1	1	yes	characteristic
84	C₂×C₄	#13108084	1	1	yes	characteristic
85	C₈	—	808	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	#15692312	1	1	yes	characteristic
89	C₂²	#15692314	1	1	yes	characteristic
90	C₂²	#15692315	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	2	1	no	—
93	C₄	—	4	2	no	—
94	C₄	—	404	2	no	—
95	C₄	—	404	2	no	—
96	C₂	—	2	1	no	—
97	C₂	—	2	1	no	—
98	C₂	#20266021	1	1	yes	characteristic
99	C₂	#20266084	1	1	yes	characteristic
100	C₂	#20266088	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20266022

C₂

yes

maximal

#20265083

C₂

yes

maximal, characteristic

C₂²⋊#11717581

—

C₂²×Dic₂₀₂

C₄

yes

characteristic

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

#15692162

—

C₂×C₄²×C₁₀₁

C₄

yes

characteristic

C₂×#11717581

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₈

yes

complemented

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

D₄

yes

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₁

M₁₆

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717581

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²⋊C₈

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1319

yes

characteristic, complemented, Sylow

#391

—

101

—

#324

—

101

—

C₂²×Q₈

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

#87938

yes

characteristic

C₂²⋊C₈

—

202

—

#95

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

404

—

C₂×Q₈

—

404

—

C₂×Q₈

—

202

—

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂×#87938

yes

characteristic

C₂×C₈

—

404

—

C₄⋊C₄

—

404

—

C₄²

—

C₂³

C₂²⋊#87938

yes

characteristic

Q₈

—

404

—

Q₈

—

404

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

D₁₀₁⋊C₈

yes

characteristic

C₂×C₄

#13108084

yes

characteristic

C₈

—

808

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

404

—

C₄

—

404

—

C₂

—

C₂

—

C₂

yes

C₂

yes

100

C₂

#20266088

yes

characteristic

	subgroup	G/H
center	C₂²	#15692314
commutator	C₂×C₄×C₁₀₁	C₂×C₈
Frattini	C₂²×C₄	C₂×#87938
Fitting	C₂×C₄²×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#99
prime radical	C₂³×C₁₀₁	C₂²⋊C₄
cosocle	C₂×C₄⋊Dic₁₀₁	C₂²
hypercenter	C₂×C₄²	#87938
hypocenter	C₁₀₁	#1319
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²

#15692314

commutator

C₂×C₄×C₁₀₁

C₂×C₈

Frattini

C₂²×C₄

C₂×#87938

Fitting

C₂×C₄²×C₁₀₁

C₄

socle

C₂²×C₁₀₁

#99

prime radical

C₂³×C₁₀₁

C₂²⋊C₄

cosocle

C₂×C₄⋊Dic₁₀₁

C₂²

C₂×C₄²

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
12. C₂×Dic₂₀₂	C₈	semidirect
51. C₁₀₁	#1319	semidirect

H ≅ G/N

type

12. C₂×Dic₂₀₂

C₈

semidirect

51. C₁₀₁

#1319

semidirect

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

subgroup	index
#20266022	2
#20265083	2
#1319	101

subgroup

index

#20266022

#20265083

#1319

101

Subgroups — #27186063

Conjugacy classes of subgroups

Complements

Maximal subgroups