Group #27187049 — subgroups

Subgroups — #27187049

Order 12928.

260 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 114 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⋊#11717581	C₂	1	2	yes	maximal
2	#20266043	C₂	1	4	yes	maximal
3	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11717581	C₂²	1	4	yes	—
5	C₄⋊#11717581	—	2	4	no	—
6	C₂×C₄×Dic₁₀₁	C₄	1	2	yes	—
7	C₂×C₄×Dic₁₀₁	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₀₁	C₄	1	1	yes	characteristic
11	C₂×#11717581	—	2	4	no	—
12	C₂×#11717581	—	2	8	no	—
13	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
15	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
16	C₄×Dic₁₀₁	—	2	4	no	—
17	C₄×Dic₁₀₁	—	2	4	no	—
18	C₄×Dic₁₀₁	C₈	1	4	yes	complemented
19	C₄×Dic₁₀₁	D₄	1	2	yes	—
20	C₄×Dic₁₀₁	Q₈	1	2	yes	—
21	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₀₁	—	2	2	no	—
24	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₁	—	2	2	no	—
26	C₂×C₄×C₁₀₁	—	2	2	no	—
27	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₀₁	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₁₀₁	M₁₆	1	2	yes	—
30	C₂×C₄×C₁₀₁	M₁₆	1	4	yes	—
31	C₂×Dic₁₀₁	—	2	2	no	—
32	C₂×Dic₁₀₁	—	2	2	no	—
33	C₂×Dic₁₀₁	—	2	2	no	—
34	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₁₀₁	C₂×C₈	1	4	yes	—
36	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₁₀₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×Dic₁₀₁	#41	1	4	yes	—
39	#11717581	—	4	4	no	—
40	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₁₀₁	#117	1	2	yes	—
42	C₂²×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
43	C₂²×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
44	C₂²×C₁₀₁	C₂×M₁₆	1	2	yes	—
45	C₄×C₁₀₁	—	2	4	no	—
46	C₄×C₁₀₁	C₄⋊C₈	1	4	yes	—
47	Dic₁₀₁	—	2	2	no	—
48	Dic₁₀₁	—	2	4	no	—
49	Dic₁₀₁	#124	1	2	yes	—
50	Dic₁₀₁	#125	1	2	yes	—
51	C₂×C₁₀₁	#382	1	1	yes	characteristic
52	C₂×C₁₀₁	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₁₀₁	#423	1	1	yes	characteristic
54	C₂×C₁₀₁	#431	1	2	yes	—
55	C₂×C₁₀₁	#432	1	2	yes	—
56	#1831	—	101	1	no	maximal, Sylow
57	C₁₀₁	#1831	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄⋊C₈	—	101	2	no	—
59	C₄³	—	101	1	no	—
60	#336	—	101	4	no	—
61	C₂²×C₈	—	101	4	no	—
62	C₂×C₄²	—	101	1	no	—
63	C₂×C₄²	—	101	1	no	—
64	C₂×C₄²	—	101	2	no	—
65	C₂×C₄²	—	101	2	no	—
66	C₂×C₄²	#87938	1	1	yes	characteristic
67	C₄⋊C₈	—	202	4	no	—
68	C₂²×C₄	—	101	1	no	—
69	C₂²×C₄	—	101	1	no	—
70	C₂²×C₄	—	101	2	no	—
71	C₂²×C₄	C₂×#87938	1	2	yes	—
72	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
73	C₂×C₈	—	202	4	no	—
74	C₂×C₈	—	202	8	no	—
75	C₄²	—	2	2	no	—
76	C₄²	—	101	2	no	—
77	C₄²	—	101	2	no	—
78	C₄²	—	101	4	no	—
79	C₄²	—	202	4	no	—
80	C₄²	—	202	4	no	—
81	C₂³	C₂²×#87938	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	101	1	no	—
85	C₂×C₄	—	101	1	no	—
86	C₂×C₄	—	101	4	no	—
87	C₂×C₄	—	101	2	no	—
88	C₂×C₄	—	101	4	no	—
89	C₂×C₄	—	202	2	no	—
90	C₂×C₄	—	202	2	no	—
91	C₂×C₄	—	202	2	no	—
92	C₂×C₄	C₄⋊#87938	1	2	yes	—
93	C₂×C₄	D₁₀₁⋊C₈	1	2	yes	—
94	C₂×C₄	#13108083	1	2	yes	—
95	C₂×C₄	#13108084	1	4	yes	—
96	C₈	—	404	4	no	—
97	C₂²	C₂×C₄⋊#87938	1	1	yes	characteristic
98	C₂²	#15692334	1	2	yes	—
99	C₂²	C₂×D₁₀₁⋊C₈	1	1	yes	characteristic
100	C₂²	C₂×#13108083	1	1	yes	characteristic
101	C₂²	C₂×#13108084	1	2	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	101	2	no	—
104	C₄	—	101	2	no	—
105	C₄	—	202	2	no	—
106	C₄	—	202	4	no	—
107	C₄	#15692322	1	4	yes	—
108	C₂	#20266121	1	1	yes	characteristic
109	C₂	C₂×#15692322	1	1	yes	characteristic
110	C₂	#20266181	1	1	yes	characteristic
111	C₂	#20266196	1	2	yes	—
112	C₂	#20266198	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11717581

C₂

yes

maximal

#20266043

C₂

yes

maximal

C₄²×Dic₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717581

C₂²

yes

—

C₄⋊#11717581

—

C₂×C₄×Dic₁₀₁

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₄

yes

characteristic

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄²×C₁₀₁

C₄

yes

characteristic

C₂×#11717581

—

C₂×#11717581

—

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂³

yes

characteristic

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

C₈

yes

complemented

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

—

C₂×C₄×C₁₀₁

M₁₆

yes

—

C₂×C₄×C₁₀₁

M₁₆

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×C₈

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₁

#41

yes

—

#11717581

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

#117

yes

—

C₂²×C₁₀₁

C₂²×C₈

yes

characteristic

C₂²×C₁₀₁

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₁

C₂×M₁₆

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄⋊C₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#124

yes

—

Dic₁₀₁

#125

yes

—

C₂×C₁₀₁

#382

yes

characteristic

C₂×C₁₀₁

C₂×C₄⋊C₈

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#432

yes

—

#1831

—

101

maximal, Sylow

C₁₀₁

#1831

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

101

—

C₄³

—

101

—

#336

—

101

—

C₂²×C₈

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

#87938

yes

characteristic

C₄⋊C₈

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×#87938

yes

—

C₂²×C₄

C₂×#87938

yes

characteristic

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄²

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

202

—

C₄²

—

202

—

C₂³

C₂²×#87938

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

C₄⋊#87938

yes

—

C₂×C₄

D₁₀₁⋊C₈

yes

—

C₂×C₄

#13108083

yes

—

C₂×C₄

#13108084

yes

—

C₈

—

404

—

C₂²

C₂×C₄⋊#87938

yes

characteristic

C₂²

#15692334

yes

—

C₂²

C₂×D₁₀₁⋊C₈

yes

characteristic

100

C₂²

C₂×#13108083

yes

characteristic

101

C₂²

C₂×#13108084

yes

—

102

C₄

—

103

C₄

—

101

—

104

C₄

—

101

—

105

C₄

—

202

—

106

C₄

—

202

—

107

C₄

#15692322

yes

—

108

C₂

#20266121

yes

characteristic

109

C₂

C₂×#15692322

yes

characteristic

110

C₂

#20266181

yes

characteristic

111

C₂

#20266196

yes

—

112

C₂

#20266198

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87938
commutator	C₂²×C₁₀₁	C₂²×C₈
Fitting	C₂×C₄²×C₁₀₁	C₄
socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	C₂×C₄²	#87938
hypocenter	C₁₀₁	#1831
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#87938

commutator

C₂²×C₁₀₁

C₂²×C₈

Fitting

C₂×C₄²×C₁₀₁

C₄

socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

cosocle

C₂²×Dic₁₀₁

C₂³

C₂×C₄²

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
18. C₄×Dic₁₀₁	C₈	semidirect
57. C₁₀₁	#1831	semidirect

H ≅ G/N

type

18. C₄×Dic₁₀₁

C₈

semidirect

57. C₁₀₁

#1831

semidirect

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

subgroup	index
C₂×C₄⋊#11717581	2
#20266043	2
C₄²×Dic₁₀₁	2
#1831	101

subgroup

index

C₂×C₄⋊#11717581

#20266043

C₄²×Dic₁₀₁

#1831

101

Subgroups — #27187049

Conjugacy classes of subgroups

Complements

Maximal subgroups