Group #27180538 — subgroups

Subgroups — #27180538

Order 12928.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#20265748	C₂	1	3	yes	maximal, complemented
2	#20265367	C₂	1	8	yes	maximal, complemented
3	#20265151	C₂	1	3	yes	maximal, complemented
4	#585×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic, complemented
6	#15692253	—	2	3	no	—
7	#15692242	—	2	3	no	—
8	#15692236	C₂²	1	12	yes	complemented
9	#15692234	—	2	6	no	—
10	#15692214	C₂²	1	3	yes	complemented
11	#15692170	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₁₀₁	—	2	3	no	—
13	H(5,𝔽₂)×C₁₀₁	—	2	3	no	—
14	H(5,𝔽₂)×C₁₀₁	C₂²	1	4	yes	—
15	C₂×#41×C₁₀₁	—	2	6	no	—
16	C₂×#41×C₁₀₁	C₂²	1	3	yes	—
17	C₂×Dic₂₀₂	—	2	3	no	—
18	C₂×Dic₂₀₂	—	2	3	no	—
19	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
20	C₄⋊Dic₁₀₁	—	2	3	no	—
21	#13108053	—	2	12	no	—
22	C₂²⋊Dic₁₀₁	—	2	12	no	—
23	C₂²⋊Dic₁₀₁	C₂³	1	6	yes	—
24	C₄×Dic₁₀₁	—	2	3	no	—
25	#41×C₁₀₁	—	2	4	no	—
26	#41×C₁₀₁	—	2	12	no	—
27	#41×C₁₀₁	—	2	6	no	—
28	#41×C₁₀₁	—	2	12	no	—
29	#41×C₁₀₁	—	2	6	no	—
30	C₂×Q₈×C₁₀₁	—	2	6	no	—
31	C₂×Q₈×C₁₀₁	D₄	1	3	yes	—
32	C₂×D₄×C₁₀₁	—	2	6	no	—
33	C₂×D₄×C₁₀₁	—	2	12	no	—
34	C₂×D₄×C₁₀₁	D₄	1	3	yes	—
35	C₂×D₄×C₁₀₁	C₂³	1	6	yes	—
36	C₂²×C₄×C₁₀₁	—	2	4	no	—
37	C₂²×C₄×C₁₀₁	D₄	1	6	yes	—
38	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
39	C₂³×C₁₀₁	—	2	4	no	—
40	C₂³×C₁₀₁	C₂×D₄	1	6	yes	—
41	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
42	Q₈×C₁₀₁	—	2	4	no	—
43	Q₈×C₁₀₁	—	2	6	no	—
44	D₄×C₁₀₁	—	2	12	no	—
45	D₄×C₁₀₁	—	2	12	no	—
46	D₄×C₁₀₁	—	2	6	no	—
47	C₂×C₄×C₁₀₁	—	2	12	no	—
48	C₂×C₄×C₁₀₁	—	2	6	no	—
49	C₂×C₄×C₁₀₁	—	2	6	no	—
50	C₂×C₄×C₁₀₁	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
53	C₂×Dic₁₀₁	—	2	6	no	—
54	C₂×Dic₁₀₁	—	2	6	no	—
55	Dic₂₀₂	—	4	1	no	—
56	Dic₂₀₂	—	4	3	no	—
57	C₂²×C₁₀₁	—	2	6	no	—
58	C₂²×C₁₀₁	—	4	1	no	—
59	C₂²×C₁₀₁	—	4	6	no	—
60	C₂²×C₁₀₁	—	8	4	no	—
61	C₂²×C₁₀₁	C₂²×D₄	1	3	yes	—
62	C₄×C₁₀₁	—	2	6	no	—
63	C₄×C₁₀₁	#120	1	1	yes	characteristic
64	C₄×C₁₀₁	#120	1	3	yes	—
65	Dic₁₀₁	—	4	6	no	—
66	Dic₁₀₁	—	4	2	no	—
67	C₂×C₁₀₁	—	2	3	no	—
68	C₂×C₁₀₁	—	4	6	no	—
69	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
70	#3028	—	101	1	no	maximal, Sylow
71	C₁₀₁	#3028	1	1	yes	characteristic, complemented, Sylow
72	#585	D₁₀₁	1	1	yes	characteristic
73	#536	—	101	3	no	—
74	#458	—	101	8	no	—
75	#410	—	101	3	no	—
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₁₀₁	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₁₀₁	1	3	yes	—
81	C₂²×Q₈	—	101	1	no	—
82	C₄⋊Q₈	—	202	3	no	—
83	#124	—	202	3	no	—
84	#123	—	101	12	no	—
85	C₂²⋊Q₈	—	202	6	no	—
86	#117	—	101	3	no	—
87	#99	—	101	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	202	3	no	—
95	C₂×Q₈	—	202	3	no	—
96	C₂×Q₈	C₂²⋊D₁₀₁	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₁₀₁	1	3	yes	—
100	C₂×D₄	C₂²×D₁₀₁	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	101	2	no	—
103	C₂²×C₄	C₂²⋊D₁₀₁	1	6	yes	—
104	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
105	C₄⋊C₄	—	202	12	no	—
106	C₄⋊C₄	—	202	3	no	—
107	C₂²⋊C₄	—	101	6	no	—
108	C₂²⋊C₄	—	202	12	no	—
109	C₄²	—	202	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₁₀₁	1	6	yes	—
112	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	404	1	no	—
116	Q₈	—	404	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	202	6	no	—
124	C₂×C₄	—	202	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₁₀₁	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	404	6	no	—
135	C₄	—	404	2	no	—
136	C₄	#15692224	1	1	yes	characteristic
137	C₄	#15692224	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#15692224	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265748

C₂

yes

maximal, complemented

#20265367

C₂

yes

maximal, complemented

#20265151

C₂

yes

maximal, complemented

#585×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

characteristic, complemented

#15692253

—

#15692242

—

#15692236

C₂²

yes

complemented

#15692234

—

#15692214

C₂²

yes

complemented

#15692170

C₂²

yes

complemented

2₋¹⁺⁴×C₁₀₁

—

H(5,𝔽₂)×C₁₀₁

—

H(5,𝔽₂)×C₁₀₁

C₂²

yes

—

C₂×#41×C₁₀₁

—

C₂×#41×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₄⋊Dic₁₀₁

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

C₂³

yes

—

C₄×Dic₁₀₁

—

#41×C₁₀₁

—

#41×C₁₀₁

—

#41×C₁₀₁

—

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²×D₄

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

#120

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#120

yes

characteristic

#3028

—

101

maximal, Sylow

C₁₀₁

#3028

yes

characteristic, complemented, Sylow

#585

D₁₀₁

yes

characteristic

#536

—

101

—

#458

—

101

—

#410

—

101

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₁₀₁

yes

—

C₂×#41

—

C₂×#41

C₂×D₁₀₁

yes

—

C₂²×Q₈

—

101

—

C₄⋊Q₈

—

202

—

#124

—

202

—

#123

—

101

—

C₂²⋊Q₈

—

202

—

#117

—

101

—

#99

—

101

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂²⋊D₁₀₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₁

yes

—

100

C₂×D₄

C₂²×D₁₀₁

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

101

—

103

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

104

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

105

C₄⋊C₄

—

202

—

106

C₄⋊C₄

—

202

—

107

C₂²⋊C₄

—

101

—

108

C₂²⋊C₄

—

202

—

109

C₄²

—

202

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₁₀₁

yes

—

112

C₂³

C₂³×D₁₀₁

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

404

—

116

Q₈

—

404

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

202

—

124

C₂×C₄

—

202

—

125

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

126

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

127

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₁₀₁

yes

—

133

C₄

—

134

C₄

—

404

—

135

C₄

—

404

—

136

C₄

#15692224

yes

characteristic

137

C₄

#15692224

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#15692224

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#15692224
commutator, cosocle	C₂³×C₁₀₁	C₂⁴
Frattini	C₂³	C₂³×D₁₀₁
Fitting, prime radical	#585×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂×#120
hypercenter	#585	D₁₀₁
hypocenter	C₁₀₁	#3028
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#15692224

commutator, cosocle

C₂³×C₁₀₁

C₂⁴

Frattini

C₂³

C₂³×D₁₀₁

Fitting, prime radical

#585×C₁₀₁

C₂

C₂×C₁₀₁

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #20265748	C₂	semidirect
2. #20265367	C₂	semidirect
3. #20265151	C₂	semidirect
5. C₂²×Dic₂₀₂	C₂²	semidirect
8. #15692236	C₂²	semidirect
10. #15692214	C₂²	semidirect
11. #15692170	C₂²	semidirect
71. C₁₀₁	#3028	semidirect

H ≅ G/N

type

1. #20265748

C₂

semidirect

2. #20265367

C₂

semidirect

3. #20265151

C₂

semidirect

5. C₂²×Dic₂₀₂

C₂²

semidirect

8. #15692236

C₂²

semidirect

10. #15692214

C₂²

semidirect

11. #15692170

C₂²

semidirect

71. C₁₀₁

#3028

semidirect

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

subgroup	index
#20265748	2
#20265367	2
#20265151	2
#585×C₁₀₁	2
#3028	101

subgroup

index

#585×C₁₀₁

101

Subgroups — #27180538

Conjugacy classes of subgroups

Complements

Maximal subgroups