Group #27187058 — subgroups

Subgroups — #27187058

Order 12928.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20266182	C₂	1	1	yes	maximal, characteristic
2	#20266181	C₂	1	1	yes	maximal, characteristic
3	#20266053	C₂	1	4	yes	maximal
4	#97×D₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108084	C₂²	1	2	yes	—
6	C₂×#13108084	C₂²	1	2	yes	—
7	#15692322	—	2	2	no	—
8	C₄⋊#11717581	—	2	2	no	—
9	C₂×C₈×D₁₀₁	C₂²	1	2	yes	—
10	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
11	#15692166	C₄	1	1	yes	characteristic
12	#15692165	C₄	1	2	yes	—
13	#97×C₁₀₁	C₄	1	1	yes	characteristic
14	#13108084	—	2	4	no	—
15	#13108084	—	2	4	no	—
16	C₂×#11717581	—	2	2	no	—
17	C₂×#11717581	—	2	2	no	—
18	C₂×C₄×D₁₀₁	Q₈	1	2	yes	—
19	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
20	C₂×#11717579	C₂×C₄	1	2	yes	—
21	C₈×D₁₀₁	—	2	2	no	—
22	C₈×D₁₀₁	—	2	2	no	—
23	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₀₁	D₄	1	2	yes	—
25	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
26	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₁₀₁	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
32	C₄×D₁₀₁	—	4	2	no	—
33	C₄×D₁₀₁	#41	1	2	yes	—
34	C₄×D₁₀₁	#41	1	2	yes	—
35	#11717581	—	4	2	no	—
36	#11717581	—	4	2	no	—
37	C₈×C₁₀₁	—	2	2	no	—
38	#11717579	—	2	2	no	—
39	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×D₁₀₁	—	2	2	no	—
41	C₂×D₁₀₁	#125	1	2	yes	—
42	C₄×C₁₀₁	—	2	2	no	—
43	C₄×C₁₀₁	#117	1	2	yes	—
44	Dic₁₀₁	—	2	2	no	—
45	Dic₁₀₁	#124	1	1	yes	characteristic
46	Dic₁₀₁	#124	1	1	yes	characteristic
47	C₂×C₁₀₁	#382	1	1	yes	characteristic
48	C₂×C₁₀₁	#430	1	2	yes	—
49	D₁₀₁	—	2	2	no	—
50	#1838	—	101	1	no	maximal, Sylow
51	C₁₀₁	#1838	1	1	yes	characteristic, complemented, Sylow
52	#423	—	101	1	no	—
53	#423	—	101	1	no	—
54	C₂×#97	—	101	1	no	—
55	#341	—	101	4	no	—
56	C₂×M₁₆	—	101	2	no	—
57	C₂×M₁₆	—	101	2	no	—
58	C₂²×C₈	—	101	2	no	—
59	C₂×C₄²	—	101	1	no	—
60	C₄⋊C₈	—	202	2	no	—
61	C₄⋊C₈	—	202	2	no	—
62	#97	—	101	2	no	—
63	#97	—	101	1	no	—
64	#97	#87938	1	1	yes	characteristic
65	C₂²×C₄	—	101	1	no	—
66	C₂²×C₄	—	101	2	no	—
67	M₁₆	—	202	4	no	—
68	M₁₆	—	202	4	no	—
69	C₂×C₈	—	101	2	no	—
70	C₂×C₈	—	202	2	no	—
71	C₂×C₈	—	202	2	no	—
72	C₂×C₈	—	202	2	no	—
73	C₂×C₈	—	202	2	no	—
74	C₂×C₈	C₂×#87938	1	2	yes	—
75	C₄²	—	101	2	no	—
76	C₄²	—	101	1	no	—
77	C₄²	C₂×#87938	1	1	yes	characteristic
78	C₂³	—	101	1	no	—
79	C₂×C₄	—	101	1	no	—
80	C₂×C₄	—	101	2	no	—
81	C₂×C₄	—	101	2	no	—
82	C₂×C₄	—	101	2	no	—
83	C₂×C₄	—	404	2	no	—
84	C₂×C₄	C₄⋊#87938	1	2	yes	—
85	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	404	2	no	—
88	C₈	—	404	2	no	—
89	C₈	—	202	2	no	—
90	C₂²	—	101	2	no	—
91	C₂²	—	202	2	no	—
92	C₂²	C₂×C₄⋊#87938	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	101	1	no	—
95	C₄	—	101	1	no	—
96	C₄	—	202	2	no	—
97	C₄	#15692334	1	2	yes	—
98	C₂	—	202	2	no	—
99	C₂	#20266121	1	1	yes	characteristic
100	C₂	#20266192	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20266182

C₂

yes

maximal, characteristic

#20266181

C₂

yes

maximal, characteristic

#20266053

C₂

yes

maximal

#97×D₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108084

C₂²

yes

—

C₂×#13108084

C₂²

yes

—

#15692322

—

C₄⋊#11717581

—

C₂×C₈×D₁₀₁

C₂²

yes

—

C₄²×D₁₀₁

C₂²

yes

characteristic

#15692166

C₄

yes

characteristic

#15692165

C₄

yes

—

#97×C₁₀₁

C₄

yes

—

—

—

—

C₂×C₄×D₁₀₁

Q₈

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

characteristic

C₂×#11717579

C₂×C₄

yes

—

C₈×D₁₀₁

—

C₈×D₁₀₁

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×D₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

#41

yes

—

C₄×D₁₀₁

#41

yes

—

#11717581

—

#11717581

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

#125

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#117

yes

—

Dic₁₀₁

—

Dic₁₀₁

yes

Dic₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

D₁₀₁

—

#1838

—

101

maximal, Sylow

C₁₀₁

#1838

yes

characteristic, complemented, Sylow

#423

—

101

—

#423

—

101

—

C₂×#97

—

101

—

#341

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂²×C₈

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

C₄⋊C₈

—

202

—

#97

—

101

—

#97

—

101

—

#97

#87938

yes

characteristic

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

C₂×#87938

yes

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

C₂×#87938

yes

characteristic

C₂³

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

C₄⋊#87938

yes

—

C₂×C₄

C₂²×#87938

yes

characteristic

C₈

—

C₈

—

404

—

C₈

—

404

—

C₈

—

202

—

C₂²

—

101

—

C₂²

—

202

—

C₂²

C₂×C₄⋊#87938

yes

characteristic

C₄

—

C₄

—

101

—

C₄

—

101

—

C₄

—

202

—

C₄

#15692334

yes

—

C₂

—

202

—

C₂

#20266121

yes

characteristic

100

C₂

#20266192

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	#97×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₁	C₂×Q₈
cosocle	C₂×C₄×D₁₀₁	C₂³
hypercenter	#97	#87938
hypocenter	C₁₀₁	#1838
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

#97×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₁

C₂×Q₈

cosocle

C₂×C₄×D₁₀₁

C₂³

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
51. C₁₀₁	#1838	semidirect

H ≅ G/N

type

51. C₁₀₁

#1838

semidirect

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

subgroup	index
#20266182	2
#20266181	2
#20266053	2
#97×D₁₀₁	2
#1838	101

subgroup

index

#97×D₁₀₁

101

Subgroups — #27187058

Conjugacy classes of subgroups

Complements

Maximal subgroups