Group #20265085 — subgroups

Subgroups — #20265085

Order 6464.

322 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×C₂²⋊D₁₀₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊Dic₁₀₁	C₂	1	2	yes	maximal, complemented
3	C₂×#13108051	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₀₁	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
8	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
9	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
10	C₂²×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₀₁	C₂²	1	2	yes	—
12	C₂²⋊Dic₁₀₁	—	2	4	no	—
13	#13108051	—	2	2	no	—
14	C₂⁴×C₁₀₁	C₂²	1	2	yes	—
15	C₂×D₄×C₁₀₁	—	2	4	no	—
16	C₂×D₄×C₁₀₁	—	2	2	no	—
17	C₂²×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂³×C₁₀₁	—	2	2	no	—
19	C₂³×C₁₀₁	—	2	2	no	—
20	C₂³×C₁₀₁	—	2	2	no	—
21	C₂³×C₁₀₁	—	2	2	no	—
22	C₂³×C₁₀₁	—	2	2	no	—
23	C₂³×C₁₀₁	—	2	2	no	—
24	C₂³×C₁₀₁	D₄	1	4	yes	complemented
25	C₂³×C₁₀₁	C₂³	1	1	yes	characteristic
26	C₂²×D₁₀₁	—	2	1	no	—
27	C₂²×D₁₀₁	—	2	2	no	—
28	C₂²×D₁₀₁	—	2	2	no	—
29	C₂²×D₁₀₁	—	2	1	no	—
30	C₂²×D₁₀₁	D₄	1	2	yes	complemented
31	D₄×C₁₀₁	—	2	8	no	—
32	C₂×C₄×C₁₀₁	—	2	2	no	—
33	C₂×C₄×C₁₀₁	D₄	1	2	yes	complemented
34	C₂²⋊D₁₀₁	—	2	16	no	—
35	C₂×Dic₁₀₁	—	2	2	no	—
36	C₂×Dic₁₀₁	—	2	2	no	—
37	C₂×Dic₁₀₁	—	2	1	no	—
38	C₂×Dic₁₀₁	—	2	2	no	—
39	C₂×Dic₁₀₁	D₄	1	4	yes	complemented
40	C₂²×C₁₀₁	—	2	4	no	—
41	C₂²×C₁₀₁	—	2	4	no	—
42	C₂²×C₁₀₁	—	2	4	no	—
43	C₂²×C₁₀₁	—	4	2	no	—
44	C₂²×C₁₀₁	—	4	2	no	—
45	C₂²×C₁₀₁	—	4	2	no	—
46	C₂²×C₁₀₁	—	4	2	no	—
47	C₂²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₁₀₁	C₂×D₄	1	2	yes	—
49	C₂²×C₁₀₁	C₂×D₄	1	2	yes	—
50	C₂²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₁₀₁	#41	1	1	yes	characteristic
52	C₂×D₁₀₁	—	2	4	no	—
53	C₂×D₁₀₁	—	2	2	no	—
54	C₂×D₁₀₁	—	4	1	no	—
55	C₂×D₁₀₁	—	4	2	no	—
56	C₂×D₁₀₁	—	4	1	no	—
57	C₄×C₁₀₁	—	2	2	no	—
58	Dic₁₀₁	—	2	4	no	—
59	Dic₁₀₁	—	4	1	no	—
60	C₂×C₁₀₁	—	4	4	no	—
61	C₂×C₁₀₁	#120	1	2	yes	—
62	C₂×C₁₀₁	#120	1	1	yes	characteristic
63	C₂×C₁₀₁	#121	1	1	yes	characteristic
64	C₂×C₁₀₁	#121	1	2	yes	—
65	C₂×C₁₀₁	#127	1	1	yes	characteristic
66	D₁₀₁	—	4	2	no	—
67	C₁₀₁	#392	1	1	yes	characteristic, complemented, Sylow
68	#392	—	101	1	no	maximal, Sylow
69	C₂²×D₄	—	101	2	no	—
70	C₂²×D₄	D₁₀₁	1	1	yes	characteristic, complemented
71	C₂×C₂²⋊C₄	—	101	2	no	—
72	C₂×C₂²⋊C₄	—	101	1	no	—
73	#95	—	101	1	no	—
74	C₂⁴	—	101	1	no	—
75	C₂⁴	C₂×D₁₀₁	1	2	yes	—
76	C₂×D₄	—	2	4	no	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	—	202	4	no	—
79	C₂×D₄	—	202	4	no	—
80	C₂×D₄	—	202	4	no	—
81	C₂²×C₄	—	101	2	no	—
82	C₂²×C₄	—	101	1	no	—
83	C₂²×C₄	C₂×D₁₀₁	1	1	yes	characteristic
84	C₂²⋊C₄	—	202	4	no	—
85	C₂²⋊C₄	—	202	2	no	—
86	C₂³	—	2	2	no	—
87	C₂³	—	2	2	no	—
88	C₂³	—	2	2	no	—
89	C₂³	—	2	2	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	2	2	no	—
92	C₂³	—	101	2	no	—
93	C₂³	—	202	2	no	—
94	C₂³	—	202	1	no	—
95	C₂³	—	202	2	no	—
96	C₂³	—	202	1	no	—
97	C₂³	C₂²⋊D₁₀₁	1	4	yes	complemented
98	C₂³	C₂²×D₁₀₁	1	1	yes	characteristic
99	D₄	—	2	8	no	—
100	D₄	—	202	16	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	101	4	no	—
103	C₂×C₄	—	202	2	no	—
104	C₂×C₄	—	202	1	no	—
105	C₂×C₄	—	202	2	no	—
106	C₂×C₄	—	202	2	no	—
107	C₂×C₄	C₂²⋊D₁₀₁	1	2	yes	complemented
108	C₂²	—	2	4	no	—
109	C₂²	—	2	4	no	—
110	C₂²	—	2	4	no	—
111	C₂²	—	4	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	2	no	—
114	C₂²	—	4	2	no	—
115	C₂²	—	404	1	no	—
116	C₂²	—	404	2	no	—
117	C₂²	—	404	1	no	—
118	C₂²	—	202	4	no	—
119	C₂²	—	202	2	no	—
120	C₂²	D₄×D₁₀₁	1	1	yes	characteristic
121	C₂²	D₄×D₁₀₁	1	2	yes	—
122	C₂²	C₂×C₂²⋊D₁₀₁	1	2	yes	—
123	C₂²	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
124	C₂²	#13108073	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	404	1	no	—
127	C₄	—	202	4	no	—
128	C₂	—	4	4	no	—
129	C₂	—	404	2	no	—
130	C₂	#15692223	1	2	yes	—
131	C₂	#15692224	1	1	yes	characteristic
132	C₂	#15692226	1	1	yes	characteristic
133	C₂	#15692227	1	2	yes	—
134	C₂	#15692251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊D₁₀₁

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₀₁

C₂

yes

maximal, complemented

C₂×#13108051

C₂

yes

maximal, characteristic, complemented

#15692162

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₁

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₁

—

C₂×C₂²⋊D₁₀₁

—

C₂×C₂²⋊D₁₀₁

—

C₂²×Dic₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

C₂²

yes

—

C₂²⋊Dic₁₀₁

—

#13108051

—

C₂⁴×C₁₀₁

C₂²

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂²

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

D₄

yes

complemented

C₂³×C₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

D₄

yes

complemented

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

D₄

yes

complemented

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

D₄

yes

complemented

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×C₁₀₁

C₂×D₄

yes

—

C₂²×C₁₀₁

C₂×D₄

yes

—

C₂²×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×C₁₀₁

#41

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#120

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#127

yes

characteristic

D₁₀₁

—

C₁₀₁

#392

yes

characteristic, complemented, Sylow

#392

—

101

maximal, Sylow

C₂²×D₄

—

101

—

C₂²×D₄

D₁₀₁

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

#95

—

101

—

C₂⁴

—

101

—

C₂⁴

C₂×D₁₀₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×D₁₀₁

yes

characteristic

C₂²⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

101

—

C₂³

—

202

—

C₂³

—

202

—

C₂³

—

202

—

C₂³

—

202

—

C₂³

C₂²⋊D₁₀₁

yes

complemented

C₂³

C₂²×D₁₀₁

yes

characteristic

D₄

—

100

D₄

—

202

—

101

C₂×C₄

—

102

C₂×C₄

—

101

—

103

C₂×C₄

—

202

—

104

C₂×C₄

—

202

—

105

C₂×C₄

—

202

—

106

C₂×C₄

—

202

—

107

C₂×C₄

C₂²⋊D₁₀₁

yes

complemented

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

404

—

116

C₂²

—

404

—

117

C₂²

—

404

—

118

C₂²

—

202

—

119

C₂²

—

202

—

120

C₂²

D₄×D₁₀₁

yes

characteristic

121

C₂²

D₄×D₁₀₁

yes

—

122

C₂²

C₂×C₂²⋊D₁₀₁

yes

—

123

C₂²

C₂×C₂²⋊D₁₀₁

yes

characteristic

124

C₂²

#13108073

yes

characteristic

125

C₄

—

126

C₄

—

404

—

127

C₄

—

202

—

128

C₂

—

129

C₂

—

404

—

130

C₂

#15692223

yes

—

131

C₂

#15692224

yes

characteristic

132

C₂

#15692226

yes

characteristic

133

C₂

#15692227

yes

—

134

C₂

#15692251

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₁
commutator, socle	C₂³×C₁₀₁	C₂³
Fitting	C₂²×D₄×C₁₀₁	C₂
hypercenter	C₂²×D₄	D₁₀₁
hypocenter	C₁₀₁	#392
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₁

commutator, socle

C₂³×C₁₀₁

C₂³

Fitting

C₂²×D₄×C₁₀₁

C₂

hypercenter

C₂²×D₄

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₁₀₁	C₂	semidirect
2. C₂×C₂²⋊Dic₁₀₁	C₂	semidirect
3. C₂×#13108051	C₂	semidirect
4. #15692162	C₂	semidirect
5. C₂²×D₄×C₁₀₁	C₂	semidirect
24. C₂³×C₁₀₁	D₄	semidirect
30. C₂²×D₁₀₁	D₄	semidirect
33. C₂×C₄×C₁₀₁	D₄	semidirect
39. C₂×Dic₁₀₁	D₄	semidirect
67. C₁₀₁	#392	semidirect
70. C₂²×D₄	D₁₀₁	semidirect
97. C₂³	C₂²⋊D₁₀₁	semidirect
107. C₂×C₄	C₂²⋊D₁₀₁	semidirect

H ≅ G/N

type

1. C₂²×C₂²⋊D₁₀₁

C₂

semidirect

2. C₂×C₂²⋊Dic₁₀₁

C₂

semidirect

3. C₂×#13108051

C₂

semidirect

4. #15692162

C₂

semidirect

5. C₂²×D₄×C₁₀₁

C₂

semidirect

24. C₂³×C₁₀₁

D₄

semidirect

30. C₂²×D₁₀₁

D₄

semidirect

33. C₂×C₄×C₁₀₁

D₄

semidirect

39. C₂×Dic₁₀₁

D₄

semidirect

67. C₁₀₁

#392

semidirect

70. C₂²×D₄

D₁₀₁

semidirect

97. C₂³

C₂²⋊D₁₀₁

semidirect

107. C₂×C₄

C₂²⋊D₁₀₁

semidirect

Subgroups — #20265085

Conjugacy classes of subgroups

Complements