Group #27171276 — subgroups

Subgroups — #27171276

Order 12928.

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

normal subgroups only

This group has 196 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	#20265023	C₂	1	1	yes	maximal, characteristic
2	#20264932	C₂	1	1	yes	maximal, characteristic
3	#20264931	C₂	1	2	yes	maximal
4	#20264931	C₂	1	2	yes	maximal
5	#336×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11717579	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	#15692162	C₄	1	2	yes	—
10	#15692162	C₄	1	2	yes	—
11	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
16	C₂×#11717579	—	2	2	no	—
17	C₂×#11717579	—	2	2	no	—
18	C₂×#11717579	—	2	2	no	—
19	C₄×Dic₁₀₁	—	2	1	no	—
20	C₄×Dic₁₀₁	—	2	2	no	—
21	C₄×Dic₁₀₁	—	2	1	no	—
22	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₀₁	—	2	2	no	—
25	C₂×C₈×C₁₀₁	—	2	2	no	—
26	C₂×C₈×C₁₀₁	—	2	2	no	—
27	C₄²×C₁₀₁	—	2	2	no	—
28	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₁	—	2	2	no	—
30	C₂×C₄×C₁₀₁	—	2	2	no	—
31	C₂×C₄×C₁₀₁	—	2	2	no	—
32	C₂×C₄×C₁₀₁	#41	1	2	yes	—
33	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	#41	1	2	yes	—
35	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
36	C₂×Dic₁₀₁	—	2	1	no	—
37	C₂×Dic₁₀₁	—	2	1	no	—
38	C₂×Dic₁₀₁	—	2	2	no	—
39	C₂×Dic₁₀₁	—	2	2	no	—
40	C₂×Dic₁₀₁	—	2	2	no	—
41	C₂×Dic₁₀₁	—	2	2	no	—
42	C₂×Dic₁₀₁	—	2	2	no	—
43	C₈×C₁₀₁	—	4	2	no	—
44	#11717579	—	4	2	no	—
45	C₂²×C₁₀₁	#117	1	1	yes	characteristic
46	C₂²×C₁₀₁	#117	1	1	yes	characteristic
47	C₂²×C₁₀₁	#117	1	1	yes	characteristic
48	C₂²×C₁₀₁	#131	1	2	yes	—
49	C₂²×C₁₀₁	#131	1	2	yes	—
50	C₄×C₁₀₁	—	4	2	no	—
51	C₄×C₁₀₁	#126	1	2	yes	—
52	C₄×C₁₀₁	#126	1	2	yes	—
53	Dic₁₀₁	—	4	1	no	—
54	Dic₁₀₁	—	4	2	no	—
55	Dic₁₀₁	—	4	1	no	—
56	C₂×C₁₀₁	#383	1	1	yes	characteristic
57	C₂×C₁₀₁	#433	1	2	yes	—
58	C₂×C₁₀₁	#433	1	1	yes	characteristic
59	C₂×C₁₀₁	#433	1	2	yes	—
60	C₂×C₁₀₁	#433	1	1	yes	characteristic
61	#1840	—	101	1	no	maximal, Sylow
62	C₁₀₁	#1840	1	1	yes	characteristic, complemented, Sylow
63	#376	—	101	1	no	—
64	#336	—	101	2	no	—
65	#336	—	101	2	no	—
66	#336	—	101	1	no	—
67	#336	D₁₀₁	1	1	yes	characteristic
68	C₂²×C₈	—	101	2	no	—
69	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
70	C₂×C₄²	—	101	1	no	—
71	C₂×C₄²	—	101	1	no	—
72	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
73	#95	—	101	2	no	—
74	#95	—	101	2	no	—
75	C₂²×C₄	—	101	1	no	—
76	C₂²×C₄	—	101	1	no	—
77	C₂²×C₄	—	101	2	no	—
78	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
79	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	202	2	no	—
84	C₂×C₈	—	202	2	no	—
85	C₂×C₈	—	202	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	202	1	no	—
88	C₄²	—	202	2	no	—
89	C₄²	—	202	1	no	—
90	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	202	1	no	—
95	C₂×C₄	—	202	1	no	—
96	C₂×C₄	—	202	2	no	—
97	C₂×C₄	—	202	2	no	—
98	C₂×C₄	—	202	2	no	—
99	C₂×C₄	—	202	2	no	—
100	C₂×C₄	—	202	2	no	—
101	C₂×C₄	#13108073	1	1	yes	characteristic
102	C₂×C₄	#13108073	1	2	yes	—
103	C₂×C₄	#13108074	1	2	yes	—
104	C₂×C₄	#13108075	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	404	2	no	—
107	C₂²	#15692211	1	1	yes	characteristic
108	C₂²	#15692212	1	1	yes	characteristic
109	C₂²	#15692213	1	1	yes	characteristic
110	C₂²	#15692261	1	2	yes	—
111	C₂²	#15692263	1	2	yes	—
112	C₄	—	4	2	no	—
113	C₄	—	404	1	no	—
114	C₄	—	404	2	no	—
115	C₄	—	404	1	no	—
116	C₄	#15692248	1	2	yes	—
117	C₄	#15692250	1	2	yes	—
118	C₂	#20265044	1	1	yes	characteristic
119	C₂	#20265235	1	2	yes	—
120	C₂	#20265236	1	1	yes	characteristic
121	C₂	#20265237	1	2	yes	—
122	C₂	#20265239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265023

C₂

yes

maximal, characteristic

#20264932

C₂

yes

maximal, characteristic

#20264931

C₂

yes

maximal

#20264931

C₂

yes

maximal

#336×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

#15692162

C₄

yes

—

#15692162

C₄

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#131

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#126

yes

—

C₄×C₁₀₁

#126

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#383

yes

characteristic

C₂×C₁₀₁

#433

yes

—

C₂×C₁₀₁

#433

yes

characteristic

C₂×C₁₀₁

#433

yes

—

C₂×C₁₀₁

#433

yes

characteristic

#1840

—

101

maximal, Sylow

C₁₀₁

#1840

yes

characteristic, complemented, Sylow

#376

—

101

—

#336

—

101

—

#336

—

101

—

#336

—

101

—

#336

D₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#95

—

101

—

#95

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₄×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄²

—

C₄²

—

202

—

C₄²

—

202

—

C₄²

—

202

—

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

100

C₂×C₄

—

202

—

101

C₂×C₄

#13108073

yes

characteristic

102

C₂×C₄

#13108073

yes

—

103

C₂×C₄

#13108074

yes

—

104

C₂×C₄

#13108075

yes

characteristic

105

C₈

—

106

C₈

—

404

—

107

C₂²

#15692211

yes

characteristic

108

C₂²

#15692212

yes

characteristic

109

C₂²

#15692213

yes

characteristic

110

C₂²

#15692261

yes

—

111

C₂²

#15692263

yes

—

112

C₄

—

113

C₄

—

404

—

114

C₄

—

404

—

115

C₄

—

404

—

116

C₄

#15692248

yes

—

117

C₄

#15692250

yes

—

118

C₂

#20265044

yes

characteristic

119

C₂

#20265235

yes

—

120

C₂

#20265236

yes

characteristic

121

C₂

#20265237

yes

—

122

C₂

#20265239

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₁
commutator, socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
Fitting	#336×C₁₀₁	C₂
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#336	D₁₀₁
hypocenter	C₁₀₁	#1840
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₁

commutator, socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

Fitting

#336×C₁₀₁

C₂

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₁₀₁	#1840	semidirect

H ≅ G/N

type

62. C₁₀₁

#1840

semidirect

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

subgroup	index
#20265023	2
#20264932	2
#20264931	2
#336×C₁₀₁	2
#1840	101

subgroup

index

#336×C₁₀₁

101

Subgroups — #27171276

Conjugacy classes of subgroups

Complements

Maximal subgroups