Group #27171180 — subgroups

Subgroups — #27171180

Order 12928.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#15692262	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692190	C₂	1	2	yes	maximal, complemented
3	#20265161	C₂	1	1	yes	maximal, characteristic
4	#20264935	C₂	1	2	yes	maximal, complemented
5	#413×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	#15692262	—	2	4	no	—
7	C₂×#13108058	C₄	1	2	yes	—
8	C₂×#13108058	C₂²	1	2	yes	—
9	C₂×#13108058	C₂²	1	1	yes	characteristic
10	C₂²×#11717579	C₄	1	2	yes	—
11	C₂²×#11717579	C₂²	1	1	yes	characteristic
12	#15692190	—	2	4	no	—
13	#15692174	—	2	1	no	—
14	#15692172	—	2	1	no	—
15	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
17	#101×C₁₀₁	—	2	1	no	—
18	#100×C₁₀₁	—	2	1	no	—
19	#13108058	—	2	2	no	—
20	#13108058	—	2	4	no	—
21	#13108058	—	2	2	no	—
22	#13108058	—	2	2	no	—
23	C₂×#11717579	—	2	4	no	—
24	C₂×#11717579	—	2	2	no	—
25	C₂×#11717579	C₂×C₄	1	2	yes	—
26	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
27	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
28	C₂×#11717579	D₄	1	4	yes	—
29	#41×C₁₀₁	—	2	2	no	—
30	#41×C₁₀₁	—	2	2	no	—
31	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
32	C₂×D₄×C₁₀₁	D₄	1	1	yes	characteristic
33	C₂×D₄×C₁₀₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₀₁	—	2	2	no	—
37	M₁₆×C₁₀₁	—	2	2	no	—
38	C₂×C₈×C₁₀₁	—	2	2	no	—
39	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
41	Q₈×C₁₀₁	—	2	2	no	—
42	D₄×C₁₀₁	—	2	4	no	—
43	D₄×C₁₀₁	—	2	2	no	—
44	C₂×C₄×C₁₀₁	—	2	2	no	—
45	C₂×C₄×C₁₀₁	—	2	2	no	—
46	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	#41	1	2	yes	—
51	C₈×C₁₀₁	—	4	2	no	—
52	#11717579	—	2	2	no	—
53	#11717579	—	4	2	no	—
54	#11717579	C₂²⋊C₄	1	2	yes	—
55	#11717579	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₁₀₁	—	2	2	no	—
57	C₂²×C₁₀₁	—	4	1	no	—
58	C₂²×C₁₀₁	—	4	2	no	—
59	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₁₀₁	—	2	2	no	—
62	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₁₀₁	#117	1	1	yes	characteristic
64	C₄×C₁₀₁	#121	1	2	yes	—
65	C₂×C₁₀₁	—	2	1	no	—
66	C₂×C₁₀₁	—	2	2	no	—
67	C₂×C₁₀₁	—	4	2	no	—
68	C₂×C₁₀₁	#380	1	1	yes	characteristic
69	#1813	—	101	1	no	maximal, Sylow
70	C₁₀₁	#1813	1	1	yes	characteristic, complemented, Sylow
71	C₂×#131	—	101	1	no	—
72	C₂×#108	—	101	2	no	—
73	#413	—	101	1	no	—
74	#413	D₁₀₁	1	1	yes	characteristic
75	#338	—	101	2	no	—
76	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
77	#131	—	202	4	no	—
78	C₂×M₁₆	—	101	2	no	—
79	C₂×M₁₆	—	101	1	no	—
80	C₂×M₁₆	—	101	2	no	—
81	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
82	C₂²×C₈	—	101	1	no	—
83	C₂²×C₈	—	101	2	no	—
84	#108	—	202	4	no	—
85	#101	—	2	1	no	—
86	#101	—	202	1	no	—
87	#100	—	2	1	no	—
88	#100	—	202	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
92	C₂×D₄	Dic₂₀₂	1	2	yes	—
93	C₂×D₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	202	4	no	—
99	M₁₆	—	202	2	no	—
100	M₁₆	—	202	2	no	—
101	M₁₆	—	202	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	101	1	no	—
104	C₂×C₈	—	101	1	no	—
105	C₂×C₈	—	101	4	no	—
106	C₂×C₈	—	101	2	no	—
107	C₂×C₈	—	202	4	no	—
108	C₂×C₈	—	202	2	no	—
109	C₂³	#13108053	1	2	yes	—
110	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	#13108053	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
118	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
120	C₂×C₄	#13108073	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₈	—	101	2	no	—
123	C₈	—	101	2	no	—
124	C₈	—	404	2	no	—
125	C₈	—	202	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×#13108053	1	1	yes	characteristic
130	C₂²	#15692232	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
133	C₄	#15692211	1	1	yes	characteristic
134	C₄	#15692227	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#20265035	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692262

C₂

yes

maximal, characteristic

C₂×#15692190

C₂

yes

maximal, complemented

#20265161

C₂

yes

maximal, characteristic

#20264935

C₂

yes

maximal, complemented

#413×C₁₀₁

C₂

yes

maximal, characteristic

#15692262

—

C₂×#13108058

C₄

yes

—

C₂×#13108058

C₂²

yes

—

C₂×#13108058

C₂²

yes

characteristic

C₂²×#11717579

C₄

yes

—

C₂²×#11717579

C₂²

yes

—

—

—

C₂×#41×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

C₂²

yes

—

#101×C₁₀₁

—

#100×C₁₀₁

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11717579

C₂×C₄

yes

characteristic

C₂×#11717579

C₂×C₄

yes

characteristic

C₂×#11717579

D₄

yes

—

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

characteristic

C₂×D₄×C₁₀₁

D₄

yes

characteristic

C₂×D₄×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

C₄⋊C₄

yes

—

C₂³×C₁₀₁

C₂×Q₈

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₈×C₁₀₁

—

#11717579

—

#11717579

—

#11717579

C₂²⋊C₄

yes

—

#11717579

C₂²⋊C₄

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#380

yes

characteristic

#1813

—

101

maximal, Sylow

C₁₀₁

#1813

yes

characteristic, complemented, Sylow

C₂×#131

—

101

—

C₂×#108

—

101

—

#413

—

101

—

#413

D₁₀₁

yes

characteristic

#338

—

101

—

C₂×#41

C₂×D₁₀₁

yes

characteristic

#131

—

202

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

#108

—

202

—

#101

—

#101

—

202

—

#100

—

#100

—

202

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₁₀₁

yes

characteristic

C₂×D₄

Dic₂₀₂

yes

—

C₂×D₄

C₂²⋊D₁₀₁

yes

characteristic

C₂²×C₄

C₄×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

202

—

M₁₆

—

202

—

100

M₁₆

—

202

—

101

M₁₆

—

202

—

102

C₂×C₈

—

103

C₂×C₈

—

101

—

104

C₂×C₈

—

101

—

105

C₂×C₈

—

101

—

106

C₂×C₈

—

101

—

107

C₂×C₈

—

202

—

108

C₂×C₈

—

202

—

109

C₂³

#13108053

yes

—

110

C₂³

C₂×Dic₂₀₂

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#13108053

yes

—

117

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

118

C₂×C₄

D₄×D₁₀₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

120

C₂×C₄

#13108073

yes

—

121

C₈

—

122

C₈

—

101

—

123

C₈

—

101

—

124

C₈

—

404

—

125

C₈

—

202

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×#13108053

yes

characteristic

130

C₂²

#15692232

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₁₀₁

yes

characteristic

133

C₄

#15692211

yes

characteristic

134

C₄

#15692227

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#20265035

yes

characteristic

	subgroup	G/H
center	C₄	#15692211
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#413×C₁₀₁	C₂
socle	C₂×C₁₀₁	#380
prime radical	C₂×#41×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#413	D₁₀₁
hypocenter	C₁₀₁	#1813
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#15692211

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#413×C₁₀₁

C₂

socle

C₂×C₁₀₁

#380

prime radical

C₂×#41×C₁₀₁

C₂²

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. C₂×#15692190	C₂	semidirect
4. #20264935	C₂	semidirect
70. C₁₀₁	#1813	semidirect

H ≅ G/N

type

2. C₂×#15692190

C₂

semidirect

4. #20264935

C₂

semidirect

70. C₁₀₁

#1813

semidirect

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

subgroup	index
C₂×#15692262	2
C₂×#15692190	2
#20265161	2
#20264935	2
#413×C₁₀₁	2
#1813	101

subgroup

index

#413×C₁₀₁

101

Subgroups — #27171180

Conjugacy classes of subgroups

Complements

Maximal subgroups