Group #27187319 — subgroups

Subgroups — #27187319

Order 12928.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20266181	C₂	1	1	yes	maximal, characteristic
2	#20266113	C₂	1	1	yes	maximal, characteristic
3	#20266047	C₂	1	4	yes	maximal
4	#20265733	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108084	C₂²	1	2	yes	—
6	C₂×C₄×#87938	C₂²	1	2	yes	—
7	C₄⋊#11717581	—	2	2	no	—
8	(C₂×#87938)⋊C₄	—	2	2	no	—
9	#15692254	C₄	1	2	yes	complemented
10	#15692247	C₄	1	1	yes	characteristic, complemented
11	#15692233	—	2	2	no	—
12	#15692220	—	2	2	no	—
13	#15692213	C₂²	1	2	yes	—
14	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
15	#125×C₁₀₁	C₄	1	1	yes	characteristic, complemented
16	C₂²×#87938	—	2	2	no	—
17	#13108084	—	2	4	no	—
18	C₂×#11717581	—	2	2	no	—
19	C₄×#87938	—	2	2	no	—
20	C₄×#87938	Q₈	1	4	yes	complemented
21	C₂×Dic₂₀₂	—	2	2	no	—
22	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
23	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
25	#13108053	—	2	2	no	—
26	#13108053	—	2	2	no	—
27	#13108051	—	2	2	no	—
28	C₄×Dic₁₀₁	—	2	2	no	—
29	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₀₁	D₄	1	2	yes	—
31	C₄⋊C₄×C₁₀₁	—	2	2	no	—
32	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
33	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
35	C₂×#87938	—	2	4	no	—
36	C₂×#87938	—	4	4	no	—
37	C₂×C₄×C₁₀₁	—	2	2	no	—
38	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₁	—	2	2	no	—
41	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
43	C₄×D₁₀₁	—	4	2	no	—
44	C₄×D₁₀₁	C₂×Q₈	1	2	yes	—
45	C₄×D₁₀₁	#41	1	2	yes	—
46	Dic₂₀₂	—	4	4	no	—
47	#11717581	—	4	2	no	—
48	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₀₁	—	2	2	no	—
50	C₂×D₁₀₁	C₂²⋊Q₈	1	2	yes	—
51	#87938	—	4	4	no	—
52	C₄×C₁₀₁	—	2	2	no	—
53	C₄×C₁₀₁	—	4	2	no	—
54	C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
55	Dic₁₀₁	—	2	2	no	—
56	Dic₁₀₁	—	4	2	no	—
57	Dic₁₀₁	#124	1	1	yes	characteristic
58	Dic₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₁₀₁	#391	1	1	yes	characteristic
60	C₂×C₁₀₁	#421	1	2	yes	—
61	D₁₀₁	—	2	2	no	—
62	#1995	—	101	1	no	maximal, Sylow
63	C₁₀₁	#1995	1	1	yes	characteristic, complemented, Sylow
64	#533	—	101	1	no	—
65	#423	—	101	1	no	—
66	#376	—	101	1	no	—
67	#339	—	101	4	no	—
68	C₂×M₁₆	—	101	2	no	—
69	C₄⋊Q₈	—	101	2	no	—
70	#125	—	101	1	no	—
71	#125	#87938	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	202	2	no	—
73	C₄×Q₈	—	202	2	no	—
74	#117	—	101	2	no	—
75	C₂×C₄²	—	101	1	no	—
76	C₂×C₄²	—	101	2	no	—
77	C₄⋊C₈	—	202	2	no	—
78	#95	—	202	2	no	—
79	C₂×Q₈	—	202	2	no	—
80	C₂²×C₄	—	101	1	no	—
81	C₂²×C₄	—	101	2	no	—
82	C₂²×C₄	—	202	2	no	—
83	M₁₆	—	202	4	no	—
84	C₂×C₈	—	202	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	101	2	no	—
87	C₄⋊C₄	—	202	2	no	—
88	C₄⋊C₄	—	202	2	no	—
89	C₄⋊C₄	C₂×#87938	1	2	yes	—
90	C₂²⋊C₄	—	202	2	no	—
91	C₄²	—	101	2	no	—
92	C₄²	—	101	1	no	—
93	C₄²	—	101	4	no	—
94	C₄²	—	202	2	no	—
95	C₄²	—	202	2	no	—
96	C₄²	C₂×#87938	1	1	yes	characteristic
97	C₂³	—	101	1	no	—
98	Q₈	—	404	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	101	1	no	—
101	C₂×C₄	—	101	2	no	—
102	C₂×C₄	—	101	2	no	—
103	C₂×C₄	—	101	2	no	—
104	C₂×C₄	—	404	2	no	—
105	C₂×C₄	—	404	4	no	—
106	C₂×C₄	—	202	4	no	—
107	C₂×C₄	—	202	2	no	—
108	C₂×C₄	C₂²⋊#87938	1	2	yes	—
109	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
110	C₈	—	404	2	no	—
111	C₂²	—	101	2	no	—
112	C₂²	—	202	2	no	—
113	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	101	1	no	—
117	C₄	—	101	1	no	—
118	C₄	—	404	4	no	—
119	C₄	—	404	2	no	—
120	C₄	—	202	2	no	—
121	C₄	Q₈×#87938	1	2	yes	—
122	C₂	—	202	2	no	—
123	C₂	(C₂×#87938)⋊Q₈	1	1	yes	characteristic
124	C₂	#20266174	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20266181

C₂

yes

maximal, characteristic

#20266113

C₂

yes

maximal, characteristic

#20266047

C₂

yes

maximal

#20265733

C₂

yes

maximal, characteristic

C₂×#13108084

C₂²

yes

—

C₂×C₄×#87938

C₂²

yes

—

C₄⋊#11717581

—

(C₂×#87938)⋊C₄

—

#15692254

C₄

yes

complemented

#15692247

C₄

yes

characteristic, complemented

#15692233

—

#15692220

—

#15692213

C₂²

yes

—

C₄²×D₁₀₁

C₂²

yes

characteristic

#125×C₁₀₁

C₄

yes

characteristic, complemented

C₂²×#87938

—

—

—

—

Q₈

yes

complemented

C₂×Dic₂₀₂

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

characteristic

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

#13108053

—

#13108053

—

#13108051

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×D₁₀₁

C₂×D₄

yes

characteristic

C₂×#87938

—

C₂×#87938

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

C₂×Q₈

yes

—

C₄×D₁₀₁

#41

yes

—

Dic₂₀₂

—

#11717581

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

C₂²⋊Q₈

yes

—

#87938

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#124

yes

characteristic

Dic₁₀₁

C₄⋊Q₈

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

D₁₀₁

—

#1995

—

101

maximal, Sylow

C₁₀₁

#1995

yes

characteristic, complemented, Sylow

#533

—

101

—

#423

—

101

—

#376

—

101

—

#339

—

101

—

C₂×M₁₆

—

101

—

C₄⋊Q₈

—

101

—

#125

—

101

—

#125

#87938

yes

characteristic, complemented

C₂²⋊Q₈

—

202

—

C₄×Q₈

—

202

—

#117

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

#95

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

M₁₆

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄⋊C₄

C₂×#87938

yes

—

C₂²⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

202

—

C₄²

—

202

—

C₄²

C₂×#87938

yes

characteristic

C₂³

—

101

—

Q₈

—

404

—

C₂×C₄

—

100

C₂×C₄

—

101

—

101

C₂×C₄

—

101

—

102

C₂×C₄

—

101

—

103

C₂×C₄

—

101

—

104

C₂×C₄

—

404

—

105

C₂×C₄

—

404

—

106

C₂×C₄

—

202

—

107

C₂×C₄

—

202

—

108

C₂×C₄

C₂²⋊#87938

yes

—

109

C₂×C₄

C₂²×#87938

yes

characteristic

110

C₈

—

404

—

111

C₂²

—

101

—

112

C₂²

—

202

—

113

C₂²

C₂×C₂²⋊#87938

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

101

—

117

C₄

—

101

—

118

C₄

—

404

—

119

C₄

—

404

—

120

C₄

—

202

—

121

C₄

Q₈×#87938

yes

—

122

C₂

—

202

—

123

C₂

(C₂×#87938)⋊Q₈

yes

characteristic

124

C₂

#20266174

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	#125×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₁	C₂×D₄
cosocle	C₂×C₄×D₁₀₁	C₂³
hypercenter	#125	#87938
hypocenter	C₁₀₁	#1995
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

#125×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₁

C₂×D₄

cosocle

C₂×C₄×D₁₀₁

C₂³

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #15692254	C₄	semidirect
10. #15692247	C₄	semidirect
15. #125×C₁₀₁	C₄	semidirect
20. C₄×#87938	Q₈	semidirect
63. C₁₀₁	#1995	semidirect
71. #125	#87938	semidirect

H ≅ G/N

type

9. #15692254

C₄

semidirect

10. #15692247

C₄

semidirect

15. #125×C₁₀₁

C₄

semidirect

20. C₄×#87938

Q₈

semidirect

63. C₁₀₁

#1995

semidirect

71. #125

#87938

semidirect

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

subgroup	index
#20266181	2
#20266113	2
#20266047	2
#20265733	2
#1995	101

subgroup

index

101

Subgroups — #27187319

Conjugacy classes of subgroups

Complements

Maximal subgroups