Group #27171258 — subgroups

Subgroups — #27171258

Order 12928.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₈×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#20265046	C₂	1	2	yes	maximal
3	#20264940	C₂	1	1	yes	maximal, characteristic
4	#20264939	C₂	1	2	yes	maximal
5	#340×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11717579	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
8	C₂×#13108053	—	2	2	no	—
9	C₂×#13108053	—	2	2	no	—
10	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₀₁	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
14	C₂²×Dic₁₀₁	—	2	2	no	—
15	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
16	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
17	C₂×#11717579	—	2	2	no	—
18	C₂×#11717579	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₁₀₁	—	2	2	no	—
20	C₄⋊Dic₁₀₁	—	2	2	no	—
21	#13108053	—	4	4	no	—
22	#13108053	—	4	4	no	—
23	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₀₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₁	—	2	2	no	—
26	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₀₁	—	2	2	no	—
28	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₁₀₁	—	2	2	no	—
30	C₄⋊C₄×C₁₀₁	—	2	2	no	—
31	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	—	4	2	no	—
33	C₂×C₄×C₁₀₁	—	4	2	no	—
34	C₂×C₄×C₁₀₁	—	4	2	no	—
35	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₀₁	#41	1	2	yes	—
38	C₂×C₄×C₁₀₁	#41	1	2	yes	—
39	C₂×Dic₁₀₁	—	4	2	no	—
40	C₂×Dic₁₀₁	—	4	2	no	—
41	C₂×Dic₁₀₁	—	4	2	no	—
42	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
43	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₁₀₁	D₈	1	2	yes	—
45	C₂×Dic₁₀₁	SD₁₆	1	4	yes	—
46	C₂×Dic₁₀₁	Q₁₆	1	2	yes	—
47	C₈×C₁₀₁	—	2	2	no	—
48	#11717579	—	2	2	no	—
49	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₀₁	#124	1	2	yes	—
51	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₀₁	C₂×SD₁₆	1	2	yes	—
53	C₂²×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
54	C₄×C₁₀₁	—	8	2	no	—
55	C₄×C₁₀₁	#117	1	1	yes	characteristic
56	C₄×C₁₀₁	#117	1	1	yes	characteristic
57	C₄×C₁₀₁	#125	1	2	yes	—
58	Dic₁₀₁	—	8	2	no	—
59	Dic₁₀₁	#106	1	4	yes	—
60	Dic₁₀₁	#107	1	4	yes	—
61	C₂×C₁₀₁	#382	1	1	yes	characteristic
62	C₂×C₁₀₁	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₀₁	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₀₁	#486	1	2	yes	—
65	C₂×C₁₀₁	#487	1	2	yes	—
66	#1834	—	101	1	no	maximal, Sylow
67	C₁₀₁	#1834	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×C₈	—	101	1	no	—
69	#384	—	101	2	no	—
70	#340	—	101	1	no	—
71	#340	—	101	2	no	—
72	#340	D₁₀₁	1	1	yes	characteristic
73	C₂²×C₈	—	101	1	no	—
74	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	101	2	no	—
76	C₂×C₄⋊C₄	—	202	2	no	—
77	C₂×C₄⋊C₄	—	202	2	no	—
78	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
79	C₂×C₄²	—	101	1	no	—
80	C₄×C₈	—	101	4	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	101	1	no	—
83	C₂²×C₄	—	101	1	no	—
84	C₂²×C₄	—	202	2	no	—
85	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	101	2	no	—
88	C₂×C₈	—	202	2	no	—
89	C₂×C₈	C₄×D₁₀₁	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	404	4	no	—
93	C₄⋊C₄	—	404	4	no	—
94	C₄⋊C₄	—	202	2	no	—
95	C₄⋊C₄	—	202	2	no	—
96	C₄²	—	101	2	no	—
97	C₄²	—	101	2	no	—
98	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	101	2	no	—
103	C₂×C₄	—	101	2	no	—
104	C₂×C₄	—	101	2	no	—
105	C₂×C₄	—	101	4	no	—
106	C₂×C₄	—	101	2	no	—
107	C₂×C₄	—	404	2	no	—
108	C₂×C₄	—	404	2	no	—
109	C₂×C₄	—	404	2	no	—
110	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₀₁	1	1	yes	characteristic
112	C₂×C₄	#13108073	1	2	yes	—
113	C₂×C₄	#13108074	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	202	2	no	—
116	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
117	C₂²	#15692241	1	2	yes	—
118	C₂²	D₈×D₁₀₁	1	1	yes	characteristic
119	C₂²	SD₁₆×D₁₀₁	1	2	yes	—
120	C₂²	Q₁₆×D₁₀₁	1	1	yes	characteristic
121	C₄	—	8	2	no	—
122	C₄	—	808	2	no	—
123	C₄	—	101	4	no	—
124	C₄	—	101	4	no	—
125	C₄	#15692211	1	1	yes	characteristic
126	C₄	#15692212	1	1	yes	characteristic
127	C₄	#15692245	1	2	yes	—
128	C₂	#20265041	1	1	yes	characteristic
129	C₂	#106×D₁₀₁	1	1	yes	characteristic
130	C₂	#107×D₁₀₁	1	1	yes	characteristic
131	C₂	#20265522	1	2	yes	—
132	C₂	#20265526	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₀₁

C₂

yes

maximal, characteristic

#20265046

C₂

yes

maximal

#20264940

C₂

yes

maximal, characteristic

#20264939

C₂

yes

maximal

#340×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×#13108053

—

C₂×#13108053

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₈×Dic₁₀₁

C₄

yes

complemented

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

D₄

yes

characteristic

C₂²×Dic₁₀₁

D₄

yes

characteristic

C₂×#11717579

—

C₂×#11717579

C₂×C₄

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

#13108053

—

#13108053

—

C₄×Dic₁₀₁

C₂×C₄

yes

—

C₄×Dic₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

D₈

yes

—

C₂×Dic₁₀₁

SD₁₆

yes

—

C₂×Dic₁₀₁

Q₁₆

yes

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

#124

yes

—

C₂²×C₁₀₁

C₂×D₈

yes

characteristic

C₂²×C₁₀₁

C₂×SD₁₆

yes

—

C₂²×C₁₀₁

C₂×Q₁₆

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

Dic₁₀₁

—

Dic₁₀₁

#106

yes

—

Dic₁₀₁

#107

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#487

yes

—

#1834

—

101

maximal, Sylow

C₁₀₁

#1834

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

101

—

#384

—

101

—

#340

—

101

—

#340

—

101

—

#340

D₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

—

101

—

C₄×C₈

—

101

—

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

C₄×D₁₀₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₂³

D₄×D₁₀₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

101

—

103

C₂×C₄

—

101

—

104

C₂×C₄

—

101

—

105

C₂×C₄

—

101

—

106

C₂×C₄

—

101

—

107

C₂×C₄

—

404

—

108

C₂×C₄

—

404

—

109

C₂×C₄

—

404

—

110

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

111

C₂×C₄

Q₈×D₁₀₁

yes

characteristic

112

C₂×C₄

#13108073

yes

—

113

C₂×C₄

#13108074

yes

—

114

C₈

—

115

C₈

—

202

—

116

C₂²

C₄⋊C₄×D₁₀₁

yes

characteristic

117

C₂²

#15692241

yes

—

118

C₂²

D₈×D₁₀₁

yes

characteristic

119

C₂²

SD₁₆×D₁₀₁

yes

—

120

C₂²

Q₁₆×D₁₀₁

yes

characteristic

121

C₄

—

122

C₄

—

808

—

123

C₄

—

101

—

124

C₄

—

101

—

125

C₄

#15692211

yes

characteristic

126

C₄

#15692212

yes

characteristic

127

C₄

#15692245

yes

—

128

C₂

#20265041

yes

characteristic

129

C₂

#106×D₁₀₁

yes

characteristic

130

C₂

#107×D₁₀₁

yes

characteristic

131

C₂

#20265522

yes

—

132

C₂

#20265526

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#340×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#340	D₁₀₁
hypocenter	C₁₀₁	#1834
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#340×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
11. C₈×Dic₁₀₁	C₄	semidirect
67. C₁₀₁	#1834	semidirect

H ≅ G/N

type

11. C₈×Dic₁₀₁

C₄

semidirect

67. C₁₀₁

#1834

semidirect

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

subgroup	index
C₂×C₈×Dic₁₀₁	2
#20265046	2
#20264940	2
#20264939	2
#340×C₁₀₁	2
#1834	101

subgroup

index

C₂×C₈×Dic₁₀₁

#340×C₁₀₁

101

Subgroups — #27171258

Conjugacy classes of subgroups

Complements

Maximal subgroups