Group #27170849 — subgroups

Subgroups — #27170849

Order 12928.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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×#15692214	C₂	1	2	yes	maximal
2	#20264940	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692214	—	2	4	no	—
5	#15692214	C₄	1	8	yes	complemented
6	C₂×C₄⋊Dic₁₀₁	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₁₀₁	—	2	2	no	—
8	C₂×C₄×Dic₁₀₁	—	2	2	no	—
9	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₁	—	2	4	no	—
11	C₂²×C₈×C₁₀₁	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
13	C₂²×Dic₁₀₁	—	2	4	no	—
14	C₄⋊Dic₁₀₁	—	2	4	no	—
15	C₄⋊Dic₁₀₁	C₂×C₄	1	8	yes	—
16	C₂²⋊Dic₁₀₁	—	2	8	no	—
17	C₄×Dic₁₀₁	—	2	8	no	—
18	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
23	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
24	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₀₁	—	2	8	no	—
26	C₂×C₈×C₁₀₁	—	2	4	no	—
27	C₂×C₈×C₁₀₁	C₂×C₄	1	4	yes	—
28	C₂³×C₁₀₁	—	2	4	no	—
29	C₂³×C₁₀₁	C₂²⋊C₄	1	4	yes	—
30	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
31	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	C₄²	1	4	yes	—
33	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₀₁	C₄⋊C₄	1	8	yes	—
36	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₀₁	—	2	8	no	—
42	C₂×Dic₁₀₁	—	4	8	no	—
43	C₈×C₁₀₁	—	2	4	no	—
44	C₂²×C₁₀₁	—	2	8	no	—
45	C₂²×C₁₀₁	—	2	4	no	—
46	C₂²×C₁₀₁	#95	1	4	yes	—
47	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
48	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₁	#135	1	4	yes	—
50	C₄×C₁₀₁	#95	1	4	yes	—
51	C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
54	Dic₁₀₁	—	4	8	no	—
55	C₂×C₁₀₁	—	2	4	no	—
56	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
57	C₂×C₁₀₁	#416	1	4	yes	—
58	C₂×C₁₀₁	#427	1	2	yes	—
59	#1735	—	101	1	no	maximal, Sylow
60	C₁₀₁	#1735	1	1	yes	characteristic, complemented, Sylow
61	C₂³×C₈	D₁₀₁	1	1	yes	characteristic
62	C₂×#117	—	101	2	no	—
63	#340	—	101	4	no	—
64	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
65	C₂²×C₈	—	2	4	no	—
66	C₂²×C₈	Dic₁₀₁	1	4	yes	complemented
67	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
68	#117	—	101	8	no	—
69	#117	—	202	4	no	—
70	C₂×C₄⋊C₄	—	101	4	no	—
71	C₂×C₂²⋊C₄	—	202	2	no	—
72	C₂×C₄²	—	202	2	no	—
73	C₂⁴	D₄₀₄	1	1	yes	characteristic
74	C₂²×C₄	—	202	4	no	—
75	C₂²×C₄	Dic₂₀₂	1	2	yes	—
76	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
77	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
79	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
80	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×Dic₁₀₁	1	4	yes	—
84	C₄⋊C₄	—	101	8	no	—
85	C₄⋊C₄	—	202	4	no	—
86	C₂²⋊C₄	—	202	8	no	—
87	C₄²	—	202	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	#13108051	1	4	yes	—
90	C₂³	C₄⋊Dic₁₀₁	1	2	yes	—
91	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
92	C₂×C₄	—	404	8	no	—
93	C₂×C₄	—	202	8	no	—
94	C₂×C₄	C₄×Dic₁₀₁	1	4	yes	—
95	C₂×C₄	#13108051	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
97	C₂×C₄	#13108053	1	8	yes	—
98	C₂×C₄	C₄⋊Dic₁₀₁	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
100	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
102	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	#15692162	1	4	yes	—
107	C₂²	C₂×#13108051	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
109	C₂²	#15692277	1	4	yes	—
110	C₄	—	404	8	no	—
111	C₄	#15692162	1	4	yes	—
112	C₄	C₂×C₄×Dic₁₀₁	1	1	yes	characteristic
113	C₄	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
114	C₄	C₂×#13108053	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	C₂×#15692162	1	1	yes	characteristic
117	C₂	#20265174	1	4	yes	—
118	C₂	#20265215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692214

C₂

yes

maximal

#20264940

C₂

yes

maximal, complemented

C₂³×C₈×C₁₀₁

C₂

yes

maximal, characteristic

#15692214

—

#15692214

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

—

C₂×C₄×Dic₁₀₁

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

—

C₂²×C₈×C₁₀₁

C₄

yes

complemented

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂²×Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₄⋊C₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₄²

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#95

yes

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

#135

yes

—

C₄×C₁₀₁

#95

yes

—

C₄×C₁₀₁

C₂×C₄²

yes

characteristic

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₁

C₂×C₄⋊C₄

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#95

yes

characteristic

C₂×C₁₀₁

#416

yes

—

C₂×C₁₀₁

#427

yes

—

#1735

—

101

maximal, Sylow

C₁₀₁

#1735

yes

characteristic, complemented, Sylow

C₂³×C₈

D₁₀₁

yes

characteristic

C₂×#117

—

101

—

#340

—

101

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₁₀₁

yes

complemented

C₂²×C₈

C₂×D₁₀₁

yes

—

#117

—

101

—

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

202

—

C₂×C₄²

—

202

—

C₂⁴

D₄₀₄

yes

characteristic

C₂²×C₄

—

202

—

C₂²×C₄

Dic₂₀₂

yes

—

C₂²×C₄

C₄×D₁₀₁

yes

—

C₂²×C₄

D₄₀₄

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₀₁

yes

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

202

—

C₂³

—

C₂³

#13108051

yes

—

C₂³

C₄⋊Dic₁₀₁

yes

—

C₂³

C₂×D₄₀₄

yes

characteristic

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

C₄×Dic₁₀₁

yes

—

C₂×C₄

#13108051

yes

—

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

C₂×C₄

#13108053

yes

—

C₂×C₄

C₄⋊Dic₁₀₁

yes

—

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

100

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

102

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#15692162

yes

—

107

C₂²

C₂×#13108051

yes

—

108

C₂²

C₂×C₄⋊Dic₁₀₁

yes

characteristic

109

C₂²

#15692277

yes

—

110

C₄

—

404

—

111

C₄

#15692162

yes

—

112

C₄

C₂×C₄×Dic₁₀₁

yes

characteristic

113

C₄

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

114

C₄

C₂×#13108053

yes

—

115

C₂

—

116

C₂

C₂×#15692162

yes

characteristic

117

C₂

#20265174

yes

—

118

C₂

#20265215

yes

—

	subgroup	G/H
center	C₂²×C₄	D₄₀₄
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂³×C₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
prime radical	C₂⁴×C₁₀₁	D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂³×C₈	D₁₀₁
hypocenter	C₁₀₁	#1735
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₄₀₄

commutator

C₄×C₁₀₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂³×C₈×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

prime radical

C₂⁴×C₁₀₁

D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂³×C₈

D₁₀₁

hypocenter

C₁₀₁

#1735

LCS

G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #20264940	C₂	semidirect
5. #15692214	C₄	semidirect
11. C₂²×C₈×C₁₀₁	C₄	semidirect
60. C₁₀₁	#1735	semidirect
66. C₂²×C₈	Dic₁₀₁	semidirect

H ≅ G/N

type

2. #20264940

C₂

semidirect

5. #15692214

C₄

semidirect

11. C₂²×C₈×C₁₀₁

C₄

semidirect

60. C₁₀₁

#1735

semidirect

66. C₂²×C₈

Dic₁₀₁

semidirect

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

subgroup	index
C₂×#15692214	2
#20264940	2
C₂³×C₈×C₁₀₁	2
#1735	101

subgroup

index

C₂×#15692214

#20264940

C₂³×C₈×C₁₀₁

#1735

101

Subgroups — #27170849

Conjugacy classes of subgroups

Complements

Maximal subgroups