Group #27172098 — subgroups

Subgroups — #27172098

Order 12928.

252 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄×#11717579	C₂	1	1	yes	maximal, characteristic
2	#20265043	C₂	1	1	yes	maximal, characteristic
3	#20264939	C₂	1	4	yes	maximal
4	C₂×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717579	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
7	C₄×#11717579	—	2	2	no	—
8	#15692162	—	2	2	no	—
9	#125×C₁₀₁	—	2	2	no	—
10	#125×C₁₀₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₁	—	2	2	no	—
15	C₂×#11717579	—	2	2	no	—
16	C₂×#11717579	—	2	2	no	—
17	C₂×#11717579	Q₈	1	4	yes	—
18	C₄⋊Dic₁₀₁	—	2	4	no	—
19	C₂²×C₄×C₁₀₁	—	2	2	no	—
20	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
21	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₁	—	2	8	no	—
23	C₄⋊C₄×C₁₀₁	—	2	2	no	—
24	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
25	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
26	C₄²×C₁₀₁	D₄	1	2	yes	—
27	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₁	—	2	4	no	—
29	C₂×C₄×C₁₀₁	—	2	4	no	—
30	C₂×C₄×C₁₀₁	—	4	4	no	—
31	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₀₁	#41	1	2	yes	—
36	C₂×Dic₁₀₁	—	4	2	no	—
37	C₂×Dic₁₀₁	—	4	4	no	—
38	#11717579	—	2	4	no	—
39	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₁	#124	1	1	yes	characteristic
41	C₂²×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₀₁	#135	1	4	yes	—
43	C₄×C₁₀₁	—	2	4	no	—
44	C₄×C₁₀₁	—	4	4	no	—
45	C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
46	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
47	Dic₁₀₁	—	8	2	no	—
48	C₂×C₁₀₁	#391	1	1	yes	characteristic
49	C₂×C₁₀₁	#416	1	2	yes	—
50	C₂×C₁₀₁	#488	1	2	yes	—
51	C₂×C₁₀₁	#499	1	2	yes	—
52	#1991	—	101	1	no	maximal, Sylow
53	C₁₀₁	#1991	1	1	yes	characteristic, complemented, Sylow
54	C₂×#125	D₁₀₁	1	1	yes	characteristic
55	C₂×C₄×C₈	—	101	1	no	—
56	#382	—	101	1	no	—
57	#340	—	101	4	no	—
58	C₂²×C₈	—	101	2	no	—
59	#125	—	2	2	no	—
60	#125	Dic₁₀₁	1	4	yes	complemented
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	101	2	no	—
63	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
64	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
65	C₄×C₈	—	202	2	no	—
66	#95	—	202	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	202	2	no	—
69	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
71	C₂×C₈	—	101	4	no	—
72	C₂×C₈	—	202	2	no	—
73	C₂×C₈	—	202	2	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	202	4	no	—
77	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
78	C₄²	C₂×Dic₁₀₁	1	2	yes	—
79	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	404	2	no	—
85	C₂×C₄	—	404	4	no	—
86	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
87	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
89	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
90	C₂×C₄	#13108075	1	2	yes	—
91	C₈	—	202	4	no	—
92	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
93	C₂²	#15692244	1	1	yes	characteristic
94	C₂²	#15692253	1	1	yes	characteristic
95	C₂²	#15692275	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	808	2	no	—
99	C₄	Q₈×Dic₁₀₁	1	2	yes	—
100	C₄	#15692231	1	2	yes	—
101	C₂	#20265084	1	1	yes	characteristic
102	C₂	#20265173	1	2	yes	—
103	C₂	#20265534	1	2	yes	—
104	C₂	#20265576	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11717579

C₂

yes

maximal, characteristic

#20265043

C₂

yes

maximal, characteristic

#20264939

C₂

yes

maximal

C₂×#125×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₄×#11717579

—

#15692162

—

#125×C₁₀₁

—

#125×C₁₀₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

Q₈

yes

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#124

yes

characteristic

C₂²×C₁₀₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₁

#135

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×Q₈

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#391

yes

characteristic

C₂×C₁₀₁

#416

yes

—

C₂×C₁₀₁

#488

yes

—

C₂×C₁₀₁

#499

yes

—

#1991

—

101

maximal, Sylow

C₁₀₁

#1991

yes

characteristic, complemented, Sylow

C₂×#125

D₁₀₁

yes

characteristic

C₂×C₄×C₈

—

101

—

#382

—

101

—

#340

—

101

—

C₂²×C₈

—

101

—

#125

—

#125

Dic₁₀₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄×C₈

—

202

—

#95

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄⋊C₄

C₂×Dic₁₀₁

yes

—

C₄²

C₂×Dic₁₀₁

yes

—

C₄²

C₂²⋊D₁₀₁

yes

—

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

Q₈×D₁₀₁

yes

—

C₂×C₄

#13108075

yes

—

C₈

—

202

—

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

808

—

C₄

Q₈×Dic₁₀₁

yes

—

100

C₄

#15692231

yes

—

101

C₂

#20265084

yes

characteristic

102

C₂

#20265173

yes

—

103

C₂

#20265534

yes

—

104

C₂

#20265576

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#125×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#125	D₁₀₁
hypocenter	C₁₀₁	#1991
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#125×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₀₁	C₄	semidirect
53. C₁₀₁	#1991	semidirect
60. #125	Dic₁₀₁	semidirect

H ≅ G/N

type

10. #125×C₁₀₁

C₄

semidirect

53. C₁₀₁

#1991

semidirect

60. #125

Dic₁₀₁

semidirect

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

subgroup	index
C₂×C₄×#11717579	2
#20265043	2
#20264939	2
C₂×#125×C₁₀₁	2
#1991	101

subgroup

index

C₂×#125×C₁₀₁

101

Subgroups — #27172098

Conjugacy classes of subgroups

Complements

Maximal subgroups