Group #27171213 — subgroups

Subgroups — #27171213

Order 12928.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	1	yes	maximal, characteristic
2	#20265141	C₂	1	1	yes	maximal, characteristic
3	#20264945	C₂	1	4	yes	maximal, complemented
4	#407×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108058	C₂²	1	2	yes	—
6	#15692214	—	2	2	no	—
7	#15692214	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₁₀₁	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₀₁	C₄	1	2	yes	—
11	C₂²⋊#11717579	—	2	2	no	—
12	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
15	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
16	#13108058	—	2	4	no	—
17	C₂×#11717579	—	2	2	no	—
18	C₄⋊Dic₁₀₁	—	2	2	no	—
19	C₄⋊Dic₁₀₁	—	2	2	no	—
20	C₂²⋊Dic₁₀₁	—	2	4	no	—
21	C₄×Dic₁₀₁	—	2	4	no	—
22	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₁	—	2	4	no	—
24	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₁	—	2	4	no	—
29	C₂×C₈×C₁₀₁	—	2	2	no	—
30	C₂³×C₁₀₁	—	2	2	no	—
31	C₂³×C₁₀₁	—	4	2	no	—
32	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	—	2	2	no	—
35	C₂×C₄×C₁₀₁	—	2	4	no	—
36	C₂×C₄×C₁₀₁	—	2	4	no	—
37	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	#41	1	4	yes	—
41	C₂×Dic₁₀₁	—	2	4	no	—
42	C₂×Dic₁₀₁	—	4	4	no	—
43	C₈×C₁₀₁	—	4	2	no	—
44	#11717579	—	4	2	no	—
45	C₂²×C₁₀₁	—	2	2	no	—
46	C₂²×C₁₀₁	—	2	2	no	—
47	C₂²×C₁₀₁	—	2	4	no	—
48	C₂²×C₁₀₁	—	4	4	no	—
49	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₀₁	#117	1	2	yes	—
51	C₄×C₁₀₁	—	2	2	no	—
52	C₄×C₁₀₁	#123	1	2	yes	—
53	C₄×C₁₀₁	#123	1	2	yes	—
54	Dic₁₀₁	—	4	4	no	—
55	C₂×C₁₀₁	—	2	2	no	—
56	C₂×C₁₀₁	—	4	2	no	—
57	C₂×C₁₀₁	#381	1	1	yes	characteristic
58	C₂×C₁₀₁	#413	1	2	yes	—
59	#1822	—	101	1	no	maximal, Sylow
60	C₁₀₁	#1822	1	1	yes	characteristic, complemented, Sylow
61	C₂×#117	—	101	1	no	—
62	#407	—	101	1	no	—
63	#407	D₁₀₁	1	1	yes	characteristic
64	#343	—	101	4	no	—
65	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
66	C₂×M₁₆	—	101	2	no	—
67	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
68	#117	—	202	2	no	—
69	#117	—	202	2	no	—
70	C₂×C₄⋊C₄	—	101	2	no	—
71	C₂×C₂²⋊C₄	—	101	2	no	—
72	C₂×C₄²	—	101	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	202	2	no	—
75	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	101	4	no	—
78	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
79	C₂²×C₄	D₄₀₄	1	2	yes	—
80	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	202	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	202	2	no	—
86	C₄⋊C₄	—	202	2	no	—
87	C₄⋊C₄	—	202	2	no	—
88	C₂²⋊C₄	—	202	4	no	—
89	C₄²	—	202	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#13108051	1	2	yes	—
93	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	404	4	no	—
98	C₂×C₄	—	202	4	no	—
99	C₂×C₄	#13108051	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
102	C₂×C₄	#13108073	1	4	yes	—
103	C₈	—	4	2	no	—
104	C₈	—	404	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#13108051	1	1	yes	characteristic
110	C₂²	#15692211	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	404	4	no	—
113	C₄	#15692236	1	2	yes	—
114	C₄	#15692239	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#20265038	1	1	yes	characteristic
118	C₂	#20265159	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692214

C₂

yes

maximal, characteristic

#20265141

C₂

yes

maximal, characteristic

#20264945

C₂

yes

maximal, complemented

#407×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

#15692214

—

#15692214

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₄

yes

—

C₂²⋊#11717579

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂²⋊C₈×C₁₀₁

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

#13108058

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#117

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#123

yes

—

C₄×C₁₀₁

#123

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#381

yes

characteristic

C₂×C₁₀₁

#413

yes

—

#1822

—

101

maximal, Sylow

C₁₀₁

#1822

yes

characteristic, complemented, Sylow

C₂×#117

—

101

—

#407

—

101

—

#407

D₁₀₁

yes

characteristic

#343

—

101

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

#117

—

202

—

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂²⋊C₈

—

C₂²⋊C₈

—

202

—

C₂⁴

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂²×C₄

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

D₄₀₄

yes

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

202

—

C₂×C₈

—

C₂×C₈

—

202

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

202

—

C₂³

—

C₂³

—

C₂³

#13108051

yes

—

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

#13108051

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

101

C₂×C₄

C₂×D₄₀₄

yes

characteristic

102

C₂×C₄

#13108073

yes

—

103

C₈

—

104

C₈

—

404

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#13108051

yes

characteristic

110

C₂²

#15692211

yes

—

111

C₄

—

112

C₄

—

404

—

113

C₄

#15692236

yes

—

114

C₄

#15692239

yes

—

115

C₂

—

116

C₂

—

117

C₂

#20265038

yes

characteristic

118

C₂

#20265159

yes

—

	subgroup	G/H
center	C₂²	C₂×#13108051
commutator	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#407×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#407	D₁₀₁
hypocenter	C₁₀₁	#1822
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#13108051

commutator

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#407×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₀₁

C₂×C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #20264945	C₂	semidirect
60. C₁₀₁	#1822	semidirect

H ≅ G/N

type

3. #20264945

C₂

semidirect

60. C₁₀₁

#1822

semidirect

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

subgroup	index
C₂×#15692214	2
#20265141	2
#20264945	2
#407×C₁₀₁	2
#1822	101

subgroup

index

#407×C₁₀₁

101

Subgroups — #27171213

Conjugacy classes of subgroups

Complements

Maximal subgroups