Group #20265688 — subgroups

Subgroups — #20265688

Order 6464.

330 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 119 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×#13108074	C₂	1	1	yes	maximal, characteristic, complemented
2	#15692238	C₂	1	2	yes	maximal, complemented
3	#15692234	C₂	1	2	yes	maximal, complemented
4	#15692229	C₂	1	2	yes	maximal, complemented
5	#15692224	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊D₁₀₁	C₂	1	4	yes	maximal, complemented
7	#15692214	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#13108074	—	2	4	no	—
10	C₂×Dic₂₀₂	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄₀₄	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
14	C₂×C₄×D₁₀₁	C₂²	1	2	yes	complemented
15	C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
16	#13108053	C₂²	1	4	yes	complemented
17	C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
18	C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
19	#13108051	C₂²	1	4	yes	complemented
20	C₄×Dic₁₀₁	C₂²	1	2	yes	complemented
21	C₂⁴×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₁	—	2	4	no	—
23	C₂²×C₄×C₁₀₁	C₂²	1	2	yes	—
24	C₂²×C₄×C₁₀₁	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂³×C₁₀₁	—	2	2	no	—
28	C₂³×C₁₀₁	—	2	4	no	—
29	C₂³×C₁₀₁	C₂³	1	1	yes	characteristic
30	C₂³×C₁₀₁	C₂³	1	2	yes	—
31	C₂²×D₁₀₁	C₂³	1	2	yes	—
32	C₂×C₄×C₁₀₁	—	2	2	no	—
33	C₂×C₄×C₁₀₁	—	2	4	no	—
34	C₂×C₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄×C₁₀₁	D₄	1	4	yes	complemented
36	C₂×C₄×C₁₀₁	C₂³	1	2	yes	—
37	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
39	C₂²⋊D₁₀₁	—	2	4	no	—
40	C₂²⋊D₁₀₁	—	2	8	no	—
41	C₂×Dic₁₀₁	C₂³	1	4	yes	—
42	C₂×Dic₁₀₁	C₂³	1	2	yes	—
43	D₄₀₄	—	2	2	no	—
44	C₄×D₁₀₁	—	2	4	no	—
45	Dic₂₀₂	—	2	2	no	—
46	C₂²×C₁₀₁	—	2	8	no	—
47	C₂²×C₁₀₁	—	2	2	no	—
48	C₂²×C₁₀₁	—	2	2	no	—
49	C₂²×C₁₀₁	—	2	2	no	—
50	C₂²×C₁₀₁	C₂×D₄	1	2	yes	—
51	C₂²×C₁₀₁	#41	1	4	yes	complemented
52	C₂²×C₁₀₁	C₂⁴	1	1	yes	characteristic
53	C₂×D₁₀₁	—	2	2	no	—
54	C₂×D₁₀₁	—	2	4	no	—
55	C₄×C₁₀₁	—	2	2	no	—
56	C₄×C₁₀₁	C₂×D₄	1	2	yes	—
57	C₄×C₁₀₁	C₂×D₄	1	2	yes	—
58	Dic₁₀₁	—	2	2	no	—
59	Dic₁₀₁	—	2	4	no	—
60	C₂×C₁₀₁	—	2	2	no	—
61	C₂×C₁₀₁	—	2	4	no	—
62	C₂×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₁₀₁	C₂×#41	1	2	yes	—
64	D₁₀₁	—	4	2	no	—
65	C₁₀₁	#525	1	1	yes	characteristic, complemented, Sylow
66	#525	—	101	1	no	maximal, Sylow
67	C₂×#41	—	101	1	no	—
68	C₂³×C₄	D₁₀₁	1	1	yes	characteristic, complemented
69	#123	—	101	2	no	—
70	C₂²⋊Q₈	—	101	2	no	—
71	#121	—	101	2	no	—
72	#120	—	101	2	no	—
73	C₄×D₄	—	101	4	no	—
74	#117	—	101	1	no	—
75	C₂⁴	C₂×D₁₀₁	1	1	yes	characteristic
76	#41	—	202	4	no	—
77	C₂×Q₈	—	101	1	no	—
78	C₂×D₄	—	101	4	no	—
79	C₂×D₄	—	101	2	no	—
80	C₂×D₄	—	101	1	no	—
81	C₂²×C₄	—	2	4	no	—
82	C₂²×C₄	—	101	2	no	—
83	C₂²×C₄	C₂×D₁₀₁	1	2	yes	—
84	C₂²×C₄	C₂×D₁₀₁	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₁₀₁	1	2	yes	—
86	C₂²×C₄	C₂×D₁₀₁	1	1	yes	characteristic
87	C₄⋊C₄	—	101	4	no	—
88	C₄⋊C₄	—	101	2	no	—
89	C₂²⋊C₄	—	101	2	no	—
90	C₂²⋊C₄	—	101	4	no	—
91	C₂²⋊C₄	—	101	4	no	—
92	C₄²	—	101	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	101	2	no	—
96	C₂³	C₂²×D₁₀₁	1	1	yes	characteristic
97	C₂³	C₂²×D₁₀₁	1	2	yes	—
98	Q₈	—	202	2	no	—
99	D₄	—	202	4	no	—
100	D₄	—	202	8	no	—
101	D₄	—	202	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	101	4	no	—
106	C₂×C₄	—	101	2	no	—
107	C₂×C₄	—	202	4	no	—
108	C₂×C₄	C₂²⋊D₁₀₁	1	4	yes	complemented
109	C₂×C₄	C₂²×D₁₀₁	1	2	yes	—
110	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
112	C₂²	—	2	8	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	202	2	no	—
117	C₂²	—	202	4	no	—
118	C₂²	C₂×C₂²⋊D₁₀₁	1	2	yes	—
119	C₂²	#13108074	1	4	yes	complemented
120	C₂²	C₂³×D₁₀₁	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	202	2	no	—
123	C₄	—	202	4	no	—
124	C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
125	C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	2	4	no	—
128	C₂	—	404	2	no	—
129	C₂	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
130	C₂	C₂×#13108074	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13108074

C₂

yes

maximal, characteristic, complemented

#15692238

C₂

yes

maximal, complemented

#15692234

C₂

yes

maximal, complemented

#15692229

C₂

yes

maximal, complemented

#15692224

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₀₁

C₂

yes

maximal, complemented

#15692214

C₂

yes

maximal, characteristic, complemented

C₂³×C₄×C₁₀₁

C₂

yes

maximal, characteristic, complemented

#13108074

—

C₂×Dic₂₀₂

C₂²

yes

characteristic, complemented

C₂×D₄₀₄

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₁₀₁

C₂²

yes

—

C₂×C₂²⋊D₁₀₁

C₂²

yes

—

C₂×C₄×D₁₀₁

C₂²

yes

complemented

C₄⋊Dic₁₀₁

C₂²

yes

complemented

#13108053

C₂²

yes

complemented

C₂²⋊Dic₁₀₁

C₂²

yes

—

C₂²⋊Dic₁₀₁

C₂²

yes

—

#13108051

C₂²

yes

complemented

C₄×Dic₁₀₁

C₂²

yes

complemented

C₂⁴×C₁₀₁

C₂²

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂²

yes

—

C₂²×C₄×C₁₀₁

C₂²

yes

—

C₂²×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₄×C₁₀₁

C₂²

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

C₂³

yes

—

C₂²×D₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

D₄

yes

complemented

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊D₁₀₁

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

D₄₀₄

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×D₄

yes

—

C₂²×C₁₀₁

#41

yes

complemented

C₂²×C₁₀₁

C₂⁴

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×D₄

yes

—

C₄×C₁₀₁

C₂×D₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂²×D₄

yes

characteristic

C₂×C₁₀₁

C₂×#41

yes

—

D₁₀₁

—

C₁₀₁

#525

yes

characteristic, complemented, Sylow

#525

—

101

maximal, Sylow

C₂×#41

—

101

—

C₂³×C₄

D₁₀₁

yes

characteristic, complemented

#123

—

101

—

C₂²⋊Q₈

—

101

—

#121

—

101

—

#120

—

101

—

C₄×D₄

—

101

—

#117

—

101

—

C₂⁴

C₂×D₁₀₁

yes

characteristic

#41

—

202

—

C₂×Q₈

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×D₁₀₁

yes

—

C₂²×C₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₄

C₂×D₁₀₁

yes

—

C₂²×C₄

C₂×D₁₀₁

yes

characteristic

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₄²

—

101

—

C₂³

—

C₂³

—

C₂³

—

101

—

C₂³

C₂²×D₁₀₁

yes

characteristic

C₂³

C₂²×D₁₀₁

yes

—

Q₈

—

202

—

D₄

—

202

—

100

D₄

—

202

—

101

D₄

—

202

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

101

—

106

C₂×C₄

—

101

—

107

C₂×C₄

—

202

—

108

C₂×C₄

C₂²⋊D₁₀₁

yes

complemented

109

C₂×C₄

C₂²×D₁₀₁

yes

—

110

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

111

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

202

—

117

C₂²

—

202

—

118

C₂²

C₂×C₂²⋊D₁₀₁

yes

—

119

C₂²

#13108074

yes

complemented

120

C₂²

C₂³×D₁₀₁

yes

characteristic

121

C₄

—

122

C₄

—

202

—

123

C₄

—

202

—

124

C₄

C₂×C₂²⋊D₁₀₁

yes

—

125

C₄

C₂×C₂²⋊D₁₀₁

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

404

—

129

C₂

C₂²×C₂²⋊D₁₀₁

yes

characteristic

130

C₂

C₂×#13108074

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂⁴
Frattini	C₂²	C₂³×D₁₀₁
Fitting	C₂³×C₄×C₁₀₁	C₂
hypercenter	C₂³×C₄	D₁₀₁
hypocenter	C₁₀₁	#525
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

subgroup

G/H

center

C₂×C₄

C₂²×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂⁴

Frattini

C₂²

C₂³×D₁₀₁

Fitting

C₂³×C₄×C₁₀₁

C₂

hypercenter

C₂³×C₄

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂×C₄ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂×#13108074	C₂	semidirect
2. #15692238	C₂	semidirect
3. #15692234	C₂	semidirect
4. #15692229	C₂	semidirect
5. #15692224	C₂	semidirect
6. C₄×C₂²⋊D₁₀₁	C₂	semidirect
7. #15692214	C₂	semidirect
8. C₂³×C₄×C₁₀₁	C₂	semidirect
10. C₂×Dic₂₀₂	C₂²	semidirect
11. C₂×D₄₀₄	C₂²	semidirect
14. C₂×C₄×D₁₀₁	C₂²	semidirect
15. C₄⋊Dic₁₀₁	C₂²	semidirect
16. #13108053	C₂²	semidirect
19. #13108051	C₂²	semidirect
20. C₄×Dic₁₀₁	C₂²	semidirect
35. C₂×C₄×C₁₀₁	D₄	semidirect
51. C₂²×C₁₀₁	#41	semidirect
65. C₁₀₁	#525	semidirect
68. C₂³×C₄	D₁₀₁	semidirect
108. C₂×C₄	C₂²⋊D₁₀₁	semidirect
119. C₂²	#13108074	semidirect

H ≅ G/N

type

1. C₂×#13108074

C₂

semidirect

2. #15692238

C₂

semidirect

3. #15692234

C₂

semidirect

4. #15692229

C₂

semidirect

5. #15692224

C₂

semidirect

6. C₄×C₂²⋊D₁₀₁

C₂

semidirect

7. #15692214

C₂

semidirect

8. C₂³×C₄×C₁₀₁

C₂

semidirect

10. C₂×Dic₂₀₂

C₂²

semidirect

11. C₂×D₄₀₄

C₂²

semidirect

14. C₂×C₄×D₁₀₁

C₂²

semidirect

15. C₄⋊Dic₁₀₁

C₂²

semidirect

16. #13108053

C₂²

semidirect

19. #13108051

C₂²

semidirect

20. C₄×Dic₁₀₁

C₂²

semidirect

35. C₂×C₄×C₁₀₁

D₄

semidirect

51. C₂²×C₁₀₁

#41

semidirect

65. C₁₀₁

#525

semidirect

68. C₂³×C₄

D₁₀₁

semidirect

108. C₂×C₄

C₂²⋊D₁₀₁

semidirect

119. C₂²

#13108074

semidirect

Subgroups — #20265688

Conjugacy classes of subgroups

Complements