Group #27170251 — subgroups

Subgroups — #27170251

Order 12928.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20265913	C₂	1	1	yes	maximal, characteristic
2	#20265469	C₂	1	2	yes	maximal, complemented
3	#20264921	C₂	1	1	yes	maximal, characteristic
4	#20264909	C₂	1	2	yes	maximal, complemented
5	#492×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	#15692242	—	2	2	no	—
7	#15692236	—	2	2	no	—
8	C₄×Dic₂₀₂	C₂²	1	2	yes	—
9	D₄×Dic₁₀₁	—	2	2	no	—
10	#15692188	—	2	2	no	—
11	C₄⋊#11717579	C₂²	1	2	yes	—
12	#15692178	—	2	2	no	—
13	C₂×SD₁₆×C₁₀₁	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	#127×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₀₂	D₄	1	2	yes	—
18	C₂²×Dic₁₀₁	—	2	2	no	—
19	C₂×#11717579	—	2	2	no	—
20	C₄⋊Dic₁₀₁	D₄	1	2	yes	—
21	#13108053	—	2	2	no	—
22	C₂²⋊Dic₁₀₁	—	2	4	no	—
23	C₂²⋊Dic₁₀₁	—	2	2	no	—
24	C₄×Dic₁₀₁	—	2	2	no	—
25	C₂×Q₈×C₁₀₁	—	2	1	no	—
26	C₂×D₄×C₁₀₁	—	2	2	no	—
27	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
28	SD₁₆×C₁₀₁	—	4	4	no	—
29	C₂×C₈×C₁₀₁	—	2	1	no	—
30	C₄⋊C₄×C₁₀₁	—	2	2	no	—
31	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
32	C₂³×C₁₀₁	—	2	2	no	—
33	Q₈×C₁₀₁	—	4	2	no	—
34	D₄×C₁₀₁	—	4	2	no	—
35	D₄×C₁₀₁	—	4	2	no	—
36	D₄×C₁₀₁	—	4	2	no	—
37	C₂×C₄×C₁₀₁	—	4	1	no	—
38	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₁₀₁	—	2	2	no	—
41	C₂×Dic₁₀₁	—	2	2	no	—
42	C₂×Dic₁₀₁	—	8	2	no	—
43	Dic₂₀₂	—	2	2	no	—
44	Dic₂₀₂	—	2	2	no	—
45	C₈×C₁₀₁	—	2	2	no	—
46	#11717579	—	4	2	no	—
47	C₂²×C₁₀₁	—	4	4	no	—
48	C₂²×C₁₀₁	—	4	2	no	—
49	C₂²×C₁₀₁	#120	1	1	yes	characteristic
50	C₄×C₁₀₁	—	2	1	no	—
51	C₄×C₁₀₁	—	8	1	no	—
52	C₄×C₁₀₁	#135	1	2	yes	—
53	C₄×C₁₀₁	#137	1	2	yes	—
54	Dic₁₀₁	—	4	2	no	—
55	Dic₁₀₁	—	4	2	no	—
56	C₂×C₁₀₁	—	8	2	no	—
57	C₂×C₁₀₁	#452	1	2	yes	—
58	C₂×C₁₀₁	#453	1	1	yes	characteristic
59	#1630	—	101	1	no	maximal, Sylow
60	C₁₀₁	#1630	1	1	yes	characteristic, complemented, Sylow
61	#559	—	101	1	no	—
62	#492	D₁₀₁	1	1	yes	characteristic
63	#477	—	101	2	no	—
64	#331	—	101	1	no	—
65	Q₈⋊C₈	—	101	2	no	—
66	C₂×SD₁₆	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
68	#127	C₂×D₁₀₁	1	1	yes	characteristic
69	#124	—	202	2	no	—
70	#123	—	202	2	no	—
71	C₄×Q₈	—	101	2	no	—
72	C₄×D₄	—	202	2	no	—
73	#107	—	202	2	no	—
74	C₄⋊C₈	—	101	2	no	—
75	#103	—	202	2	no	—
76	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	101	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
81	C₂²×C₄	—	202	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	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	C₂²⋊C₄	—	202	4	no	—
89	C₂²⋊C₄	—	202	2	no	—
90	C₄²	—	202	2	no	—
91	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	202	2	no	—
95	Q₈	—	202	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	808	2	no	—
101	C₂×C₄	—	202	2	no	—
102	C₂×C₄	—	202	2	no	—
103	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
104	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	404	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#15692223	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	8	1	no	—
112	C₄	—	404	2	no	—
113	C₄	—	404	2	no	—
114	C₄	#15692276	1	2	yes	—
115	C₄	#15692284	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#20265336	1	2	yes	—
118	C₂	#20265344	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265913

C₂

yes

maximal, characteristic

#20265469

C₂

yes

maximal, complemented

#20264921

C₂

yes

maximal, characteristic

#20264909

C₂

yes

maximal, complemented

#492×C₁₀₁

C₂

yes

maximal, characteristic

#15692242

—

#15692236

—

C₄×Dic₂₀₂

C₂²

yes

—

D₄×Dic₁₀₁

—

#15692188

—

C₄⋊#11717579

C₂²

yes

—

#15692178

—

C₂×SD₁₆×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

characteristic

#127×C₁₀₁

C₂²

yes

characteristic

C₄×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

D₄

yes

—

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

D₄

yes

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

SD₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#135

yes

—

C₄×C₁₀₁

#137

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#452

yes

—

C₂×C₁₀₁

#453

yes

characteristic

#1630

—

101

maximal, Sylow

C₁₀₁

#1630

yes

characteristic, complemented, Sylow

#559

—

101

—

#492

D₁₀₁

yes

characteristic

#477

—

101

—

#331

—

101

—

Q₈⋊C₈

—

101

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₁₀₁

yes

characteristic

#127

C₂×D₁₀₁

yes

characteristic

#124

—

202

—

#123

—

202

—

C₄×Q₈

—

101

—

C₄×D₄

—

202

—

#107

—

202

—

C₄⋊C₈

—

101

—

#103

—

202

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

101

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

—

202

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

202

—

C₄²

C₂²×D₁₀₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

202

—

Q₈

—

202

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

808

—

101

C₂×C₄

—

202

—

102

C₂×C₄

—

202

—

103

C₂×C₄

D₄×D₁₀₁

yes

—

104

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

105

C₈

—

106

C₈

—

404

—

107

C₂²

—

108

C₂²

—

109

C₂²

#15692223

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

404

—

113

C₄

—

404

—

114

C₄

#15692276

yes

—

115

C₄

#15692284

yes

—

116

C₂

—

117

C₂

#20265336

yes

—

118

C₂

#20265344

yes

characteristic

	subgroup	G/H
center	C₂²	#15692223
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#492×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
prime radical	#127×C₁₀₁	C₂²
hypercenter	#492	D₁₀₁
hypocenter	C₁₀₁	#1630
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#15692223

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#492×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#120

prime radical

#127×C₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #20265469	C₂	semidirect
4. #20264909	C₂	semidirect
60. C₁₀₁	#1630	semidirect

H ≅ G/N

type

2. #20265469

C₂

semidirect

4. #20264909

C₂

semidirect

60. C₁₀₁

#1630

semidirect

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

subgroup	index
#20265913	2
#20265469	2
#20264921	2
#20264909	2
#492×C₁₀₁	2
#1630	101

subgroup

index

#492×C₁₀₁

101

Subgroups — #27170251

Conjugacy classes of subgroups

Complements

Maximal subgroups