Group #27169000 — subgroups

Subgroups — #27169000

Order 12928.

228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 77 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	#20265037	C₂	1	2	yes	maximal
2	C₂×C₂²⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₁₀₁	—	2	2	no	—
4	C₂×C₂²⋊Dic₁₀₁	—	2	2	no	—
5	#15692162	C₄	1	4	yes	complemented
6	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₁	C₄	1	2	yes	complemented
8	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
9	C₂²×Dic₁₀₁	—	2	4	no	—
10	C₄⋊Dic₁₀₁	—	4	4	no	—
11	C₂²⋊Dic₁₀₁	—	4	2	no	—
12	C₂²⋊Dic₁₀₁	—	4	2	no	—
13	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₁	—	4	2	no	—
15	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
18	C₂²×C₄×C₁₀₁	Q₈	1	1	yes	characteristic
19	C₂×C₈×C₁₀₁	—	2	2	no	—
20	C₂×C₈×C₁₀₁	—	4	2	no	—
21	C₂³×C₁₀₁	—	4	1	no	—
22	C₂³×C₁₀₁	—	4	1	no	—
23	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
24	C₂³×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂³×C₁₀₁	D₈	1	1	yes	characteristic
26	C₂³×C₁₀₁	SD₁₆	1	2	yes	—
27	C₂³×C₁₀₁	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₁₀₁	—	2	2	no	—
29	C₂×C₄×C₁₀₁	—	2	4	no	—
30	C₂×C₄×C₁₀₁	—	4	2	no	—
31	C₂×C₄×C₁₀₁	—	4	1	no	—
32	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₁₀₁	—	4	4	no	—
36	C₂×Dic₁₀₁	—	4	4	no	—
37	C₂×Dic₁₀₁	—	4	4	no	—
38	C₈×C₁₀₁	—	4	2	no	—
39	C₂²×C₁₀₁	—	2	2	no	—
40	C₂²×C₁₀₁	—	2	1	no	—
41	C₂²×C₁₀₁	—	2	2	no	—
42	C₂²×C₁₀₁	—	2	1	no	—
43	C₂²×C₁₀₁	—	4	1	no	—
44	C₂²×C₁₀₁	—	4	2	no	—
45	C₂²×C₁₀₁	—	4	1	no	—
46	C₂²×C₁₀₁	#95	1	1	yes	characteristic
47	C₂²×C₁₀₁	#102	1	2	yes	—
48	C₂²×C₁₀₁	#103	1	2	yes	—
49	C₂²×C₁₀₁	#106	1	1	yes	characteristic
50	C₂²×C₁₀₁	#107	1	1	yes	characteristic
51	C₄×C₁₀₁	—	2	1	no	—
52	C₄×C₁₀₁	—	2	2	no	—
53	C₄×C₁₀₁	#99	1	2	yes	—
54	Dic₁₀₁	—	8	4	no	—
55	C₂×C₁₀₁	—	2	2	no	—
56	C₂×C₁₀₁	—	2	2	no	—
57	C₂×C₁₀₁	—	2	1	no	—
58	C₂×C₁₀₁	—	2	1	no	—
59	C₂×C₁₀₁	H(3,ℤ/4)	1	1	yes	characteristic
60	C₂×C₁₀₁	#340	1	1	yes	characteristic
61	C₂×C₁₀₁	#342	1	1	yes	characteristic
62	#1289	—	101	1	no	maximal, Sylow
63	C₁₀₁	#1289	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₂²⋊C₈	D₁₀₁	1	1	yes	characteristic
65	#380	—	101	2	no	—
66	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
67	C₂²×C₈	Dic₁₀₁	1	2	yes	complemented
68	C₂×C₄⋊C₄	—	202	2	no	—
69	C₂×C₂²⋊C₄	—	202	2	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	#95	—	101	4	no	—
72	C₂⁴	D₄₀₄	1	1	yes	characteristic
73	C₂²×C₄	—	4	2	no	—
74	C₂²×C₄	—	202	4	no	—
75	C₂²×C₄	Dic₂₀₂	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
77	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₄⋊C₄	—	404	4	no	—
82	C₂²⋊C₄	—	404	2	no	—
83	C₂²⋊C₄	—	404	2	no	—
84	C₂³	—	4	1	no	—
85	C₂³	—	4	1	no	—
86	C₂³	#13108051	1	2	yes	—
87	C₂³	C₄⋊Dic₁₀₁	1	1	yes	characteristic
88	C₂³	D₈₀₈	1	1	yes	characteristic
89	C₂³	#13108063	1	2	yes	—
90	C₂³	Dic₄₀₄	1	1	yes	characteristic
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	404	4	no	—
96	C₂×C₄	—	404	4	no	—
97	C₂×C₄	—	404	4	no	—
98	C₂×C₄	C₄×Dic₁₀₁	1	1	yes	characteristic
99	C₂×C₄	C₂²⋊Dic₁₀₁	1	1	yes	characteristic
100	C₂×C₄	#13108053	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	4	1	no	—
107	C₂²	—	4	2	no	—
108	C₂²	—	4	1	no	—
109	C₂²	#15692162	1	1	yes	characteristic
110	C₂²	#15692177	1	2	yes	—
111	C₂²	#15692180	1	2	yes	—
112	C₂²	#15692187	1	1	yes	characteristic
113	C₂²	#15692189	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	808	4	no	—
117	C₄	#15692170	1	2	yes	—
118	C₂	—	2	2	no	—
119	C₂	—	2	2	no	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#20264934	1	1	yes	characteristic
123	C₂	#20264940	1	1	yes	characteristic
124	C₂	#20264944	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265037

C₂

yes

maximal

C₂×C₂²⋊C₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₁

—

C₂×C₂²⋊Dic₁₀₁

—

#15692162

C₄

yes

complemented

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₄

yes

complemented

C₂²⋊C₈×C₁₀₁

—

C₂²×Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₂⁴×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

Q₈

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂³×C₁₀₁

D₈

yes

characteristic

C₂³×C₁₀₁

SD₁₆

yes

—

C₂³×C₁₀₁

Q₁₆

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄²

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#95

yes

characteristic

C₂²×C₁₀₁

#102

yes

—

C₂²×C₁₀₁

#103

yes

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#99

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1289

yes

characteristic, complemented, Sylow

C₂×C₂²⋊C₈

D₁₀₁

yes

characteristic

#380

—

101

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

Dic₁₀₁

yes

complemented

C₂×C₄⋊C₄

—

202

—

C₂×C₂²⋊C₄

—

202

—

C₂²⋊C₈

—

#95

—

101

—

C₂⁴

D₄₀₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

Dic₂₀₂

yes

characteristic

C₂²×C₄

C₄×D₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

404

—

C₂²⋊C₄

—

404

—

C₂²⋊C₄

—

404

—

C₂³

—

C₂³

—

C₂³

#13108051

yes

—

C₂³

C₄⋊Dic₁₀₁

yes

characteristic

C₂³

D₈₀₈

yes

characteristic

C₂³

#13108063

yes

—

C₂³

Dic₄₀₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

C₄×Dic₁₀₁

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₀₁

yes

characteristic

100

C₂×C₄

#13108053

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#15692162

yes

characteristic

110

C₂²

#15692177

yes

—

111

C₂²

#15692180

yes

—

112

C₂²

#15692187

yes

characteristic

113

C₂²

#15692189

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

808

—

117

C₄

#15692170

yes

—

118

C₂

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#20264934

yes

characteristic

123

C₂

#20264940

yes

characteristic

124

C₂

#20264944

yes

characteristic

	subgroup	G/H
center	C₂²	#15692162
commutator	C₂×C₄×C₁₀₁	C₄²
Frattini	C₂³×C₄	C₂×D₁₀₁
Fitting	C₂×C₂²⋊C₈×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#95
prime radical	C₂⁴×C₁₀₁	D₄
cosocle	C₂³×C₄×C₁₀₁	C₂²
hypercenter	C₂×C₂²⋊C₈	D₁₀₁
hypocenter	C₁₀₁	#1289
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂²

#15692162

commutator

C₂×C₄×C₁₀₁

C₄²

Frattini

C₂³×C₄

C₂×D₁₀₁

Fitting

C₂×C₂²⋊C₈×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#95

prime radical

C₂⁴×C₁₀₁

D₄

cosocle

C₂³×C₄×C₁₀₁

C₂²

hypercenter

C₂×C₂²⋊C₈

D₁₀₁

hypocenter

C₁₀₁

#1289

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
5. #15692162	C₄	semidirect
7. C₂²×C₈×C₁₀₁	C₄	semidirect
63. C₁₀₁	#1289	semidirect
67. C₂²×C₈	Dic₁₀₁	semidirect

H ≅ G/N

type

5. #15692162

C₄

semidirect

7. C₂²×C₈×C₁₀₁

C₄

semidirect

63. C₁₀₁

#1289

semidirect

67. C₂²×C₈

Dic₁₀₁

semidirect

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

subgroup	index
#20265037	2
C₂×C₂²⋊C₈×C₁₀₁	2
#1289	101

subgroup

index

#20265037

C₂×C₂²⋊C₈×C₁₀₁

#1289

101

Subgroups — #27169000

Conjugacy classes of subgroups

Complements

Maximal subgroups