Group #27172853 — subgroups

Subgroups — #27172853

Order 12928.

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

normal subgroups only

This group has 220 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	C₂²×Dic₄₀₄	C₂	1	1	yes	maximal, characteristic
2	#20265215	C₂	1	1	yes	maximal, characteristic
3	#20264982	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀₂	—	2	1	no	—
6	C₂×Dic₄₀₄	—	2	1	no	—
7	C₂×Dic₄₀₄	—	2	2	no	—
8	C₂×Dic₄₀₄	C₄	1	4	yes	complemented
9	C₂×Dic₄₀₄	C₂²	1	2	yes	—
10	#15692214	—	2	1	no	—
11	#15692189	C₂²	1	2	yes	—
12	#15692187	—	2	1	no	—
13	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
14	M₃₂×C₁₀₁	—	2	2	no	—
15	C₂×C₁₆×C₁₀₁	C₂²	1	2	yes	—
16	C₂×Dic₂₀₂	—	2	4	no	—
17	C₂×Dic₂₀₂	—	2	2	no	—
18	C₂×Dic₂₀₂	—	4	1	no	—
19	C₂×Dic₂₀₂	—	4	2	no	—
20	C₂²×Dic₁₀₁	—	4	1	no	—
21	Dic₄₀₄	—	2	4	no	—
22	Dic₄₀₄	—	2	2	no	—
23	Dic₄₀₄	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₁₀₁	—	2	2	no	—
25	C₂²⋊Dic₁₀₁	—	4	1	no	—
26	C₄×Dic₁₀₁	—	4	1	no	—
27	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
31	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
32	C₁₆×C₁₀₁	—	2	2	no	—
33	C₂³×C₁₀₁	D₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	SD₁₆	1	2	yes	—
37	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₁₀₁	—	4	4	no	—
39	C₂×Dic₁₀₁	—	4	2	no	—
40	C₂×Dic₁₀₁	—	4	2	no	—
41	Dic₂₀₂	—	2	4	no	—
42	Dic₂₀₂	—	4	4	no	—
43	Dic₂₀₂	—	4	2	no	—
44	C₈×C₁₀₁	C₂²⋊C₄	1	2	yes	—
45	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₁₀₁	—	2	2	no	—
48	C₂²×C₁₀₁	#102	1	2	yes	—
49	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
50	C₄×C₁₀₁	#102	1	2	yes	—
51	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
53	Dic₁₀₁	—	4	4	no	—
54	Dic₁₀₁	—	8	2	no	—
55	C₂×C₁₀₁	—	2	2	no	—
56	C₂×C₁₀₁	C₂×#102	1	1	yes	characteristic
57	C₂×C₁₀₁	#510	1	2	yes	—
58	#2141	—	101	1	no	maximal, Sylow
59	C₁₀₁	#2141	1	1	yes	characteristic, complemented, Sylow
60	C₂²×Q₁₆	—	101	1	no	—
61	C₂×M₃₂	D₁₀₁	1	1	yes	characteristic
62	#427	—	101	1	no	—
63	#358	—	101	4	no	—
64	C₂²×Q₈	—	202	1	no	—
65	C₂×Q₁₆	—	101	4	no	—
66	C₂×Q₁₆	—	101	2	no	—
67	C₂×Q₁₆	—	202	1	no	—
68	C₂×Q₁₆	—	202	2	no	—
69	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
70	#117	—	202	1	no	—
71	M₃₂	—	2	2	no	—
72	C₂×C₁₆	C₂×D₁₀₁	1	2	yes	—
73	#107	—	101	2	no	—
74	#106	—	202	1	no	—
75	C₂×Q₈	—	404	1	no	—
76	C₂×Q₈	—	404	2	no	—
77	C₂×Q₈	—	202	4	no	—
78	C₂×Q₈	—	202	2	no	—
79	C₂²×C₄	—	404	1	no	—
80	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
81	Q₁₆	—	101	4	no	—
82	Q₁₆	—	202	4	no	—
83	Q₁₆	—	202	2	no	—
84	C₂×C₈	C₄×D₁₀₁	1	2	yes	—
85	C₂×C₈	D₄₀₄	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
87	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
88	C₄⋊C₄	—	202	2	no	—
89	C₂²⋊C₄	—	404	1	no	—
90	C₄²	—	404	1	no	—
91	C₁₆	—	2	2	no	—
92	C₂³	D₈₀₈	1	1	yes	characteristic
93	Q₈	—	404	4	no	—
94	Q₈	—	404	2	no	—
95	Q₈	—	202	4	no	—
96	C₂×C₄	—	404	2	no	—
97	C₂×C₄	—	404	4	no	—
98	C₂×C₄	—	404	2	no	—
99	C₂×C₄	#13108051	1	2	yes	—
100	C₂×C₄	D₈₀₈	1	1	yes	characteristic
101	C₂×C₄	#13108063	1	2	yes	—
102	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
103	C₈	#13108051	1	2	yes	—
104	C₈	C₂×C₄×D₁₀₁	1	1	yes	characteristic
105	C₈	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#15692177	1	2	yes	—
108	C₂²	C₂×D₈₀₈	1	1	yes	characteristic
109	C₄	—	808	2	no	—
110	C₄	—	404	4	no	—
111	C₄	#15692177	1	2	yes	—
112	C₄	C₂×#13108051	1	1	yes	characteristic
113	C₄	C₂×#13108063	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#15692177	1	1	yes	characteristic
116	C₂	#20265620	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₀₄

C₂

yes

maximal, characteristic

#20265215

C₂

yes

maximal, characteristic

#20264982

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

—

C₂×Dic₄₀₄

—

C₂×Dic₄₀₄

—

C₂×Dic₄₀₄

C₄

yes

complemented

C₂×Dic₄₀₄

C₂²

yes

—

#15692214

—

#15692189

C₂²

yes

—

#15692187

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

M₃₂×C₁₀₁

—

C₂×C₁₆×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

—

Dic₄₀₄

—

Dic₄₀₄

—

Dic₄₀₄

C₂×C₄

yes

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

D₄

yes

characteristic

C₂×C₈×C₁₀₁

D₄

yes

—

C₂×C₈×C₁₀₁

C₂³

yes

characteristic

C₁₆×C₁₀₁

—

C₂³×C₁₀₁

D₈

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

D₈

yes

characteristic

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

C₂²⋊C₄

yes

—

C₈×C₁₀₁

C₂²×C₄

yes

characteristic

C₈×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#102

yes

—

C₂²×C₁₀₁

C₂×D₈

yes

characteristic

C₄×C₁₀₁

#102

yes

—

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₁

C₂×SD₁₆

yes

characteristic

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#102

yes

characteristic

C₂×C₁₀₁

#510

yes

—

#2141

—

101

maximal, Sylow

C₁₀₁

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

101

—

C₂×M₃₂

D₁₀₁

yes

characteristic

#427

—

101

—

#358

—

101

—

C₂²×Q₈

—

202

—

C₂×Q₁₆

—

101

—

C₂×Q₁₆

—

101

—

C₂×Q₁₆

—

202

—

C₂×Q₁₆

—

202

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

#117

—

202

—

M₃₂

—

C₂×C₁₆

C₂×D₁₀₁

yes

—

#107

—

101

—

#106

—

202

—

C₂×Q₈

—

404

—

C₂×Q₈

—

404

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

404

—

C₂²×C₄

D₄₀₄

yes

characteristic

Q₁₆

—

101

—

Q₁₆

—

202

—

Q₁₆

—

202

—

C₂×C₈

C₄×D₁₀₁

yes

—

C₂×C₈

D₄₀₄

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₁

yes

—

C₂×C₈

C₂²×D₁₀₁

yes

characteristic

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

404

—

C₄²

—

404

—

C₁₆

—

C₂³

D₈₀₈

yes

characteristic

Q₈

—

404

—

Q₈

—

404

—

Q₈

—

202

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

#13108051

yes

—

100

C₂×C₄

D₈₀₈

yes

characteristic

101

C₂×C₄

#13108063

yes

—

102

C₂×C₄

C₂×D₄₀₄

yes

characteristic

103

C₈

#13108051

yes

—

104

C₈

C₂×C₄×D₁₀₁

yes

characteristic

105

C₈

C₂×C₂²⋊D₁₀₁

yes

characteristic

106

C₂²

—

107

C₂²

#15692177

yes

—

108

C₂²

C₂×D₈₀₈

yes

characteristic

109

C₄

—

808

—

110

C₄

—

404

—

111

C₄

#15692177

yes

—

112

C₄

C₂×#13108051

yes

characteristic

113

C₄

C₂×#13108063

yes

characteristic

114

C₂

—

115

C₂

C₂×#15692177

yes

characteristic

116

C₂

#20265620

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈₀₈
commutator	C₈×C₁₀₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	C₂×M₃₂×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×D₈
prime radical	C₂³×C₁₀₁	D₈
cosocle	C₂×C₈×C₁₀₁	C₂³
hypercenter	C₂×M₃₂	D₁₀₁
hypocenter	C₁₀₁	#2141
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₈₀₈

commutator

C₈×C₁₀₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₀₁

Fitting

C₂×M₃₂×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×D₈

prime radical

C₂³×C₁₀₁

D₈

cosocle

C₂×C₈×C₁₀₁

C₂³

hypercenter

C₂×M₃₂

D₁₀₁

hypocenter

C₁₀₁

#2141

LCS

G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #20264982	C₂	semidirect
8. C₂×Dic₄₀₄	C₄	semidirect
59. C₁₀₁	#2141	semidirect

H ≅ G/N

type

3. #20264982

C₂

semidirect

8. C₂×Dic₄₀₄

C₄

semidirect

59. C₁₀₁

#2141

semidirect

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

subgroup	index
C₂²×Dic₄₀₄	2
#20265215	2
#20264982	2
C₂×M₃₂×C₁₀₁	2
#2141	101

subgroup

index

C₂²×Dic₄₀₄

#20265215

#20264982

C₂×M₃₂×C₁₀₁

#2141

101

Subgroups — #27172853

Conjugacy classes of subgroups

Complements

Maximal subgroups