Group #20265346 — subgroups

Subgroups — #20265346

Order 6464.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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	#15692301	C₂	1	1	yes	maximal, characteristic, complemented
2	#15692283	C₂	1	1	yes	maximal, characteristic, complemented
3	#15692280	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692252	C₂	1	1	yes	maximal, characteristic, complemented
5	#15692183	C₂	1	1	yes	maximal, characteristic, complemented
6	#15692171	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#13108075	—	2	1	no	—
9	#13108074	—	2	1	no	—
10	#13108074	C₂²	1	1	yes	characteristic
11	C₂×D₄₀₄	—	2	1	no	—
12	C₂×D₄₀₄	—	2	2	no	—
13	C₂×D₄₀₄	C₂²	1	1	yes	characteristic
14	C₂×D₄₀₄	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₁₀₁	—	2	2	no	—
16	D₄×D₁₀₁	—	2	1	no	—
17	#13108063	—	2	1	no	—
18	#13108061	—	2	1	no	—
19	D₈₀₈	—	2	1	no	—
20	#13108059	—	2	1	no	—
21	#13108058	C₂²	1	1	yes	characteristic, complemented
22	#41×C₁₀₁	C₂²	1	1	yes	characteristic
23	M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₁₀₁	—	2	2	no	—
26	C₂²×D₁₀₁	—	2	1	no	—
27	C₂²×D₁₀₁	D₄	1	2	yes	—
28	Q₈×C₁₀₁	D₄	1	1	yes	characteristic, complemented
29	D₄×C₁₀₁	—	2	1	no	—
30	D₄×C₁₀₁	D₄	1	1	yes	characteristic, complemented
31	C₂×C₄×C₁₀₁	—	2	1	no	—
32	C₂×C₄×C₁₀₁	—	2	1	no	—
33	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₂²⋊D₁₀₁	—	2	2	no	—
35	C₂²⋊D₁₀₁	—	2	1	no	—
36	D₄₀₄	—	2	2	no	—
37	D₄₀₄	—	2	1	no	—
38	D₄₀₄	—	2	1	no	—
39	D₄₀₄	—	2	1	no	—
40	D₄₀₄	—	2	1	no	—
41	D₄₀₄	—	4	4	no	—
42	D₄₀₄	D₄	1	1	yes	characteristic, complemented
43	C₄×D₁₀₁	—	2	2	no	—
44	C₄×D₁₀₁	—	2	1	no	—
45	Dic₂₀₂	D₄	1	1	yes	characteristic, complemented
46	C₈×C₁₀₁	—	2	1	no	—
47	#11717579	—	2	1	no	—
48	C₂²×C₁₀₁	—	2	1	no	—
49	C₂²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
50	C₂×D₁₀₁	—	2	2	no	—
51	C₂×D₁₀₁	—	2	1	no	—
52	C₂×D₁₀₁	—	4	2	no	—
53	C₂×D₁₀₁	—	4	2	no	—
54	C₂×D₁₀₁	—	4	1	no	—
55	C₂×D₁₀₁	—	8	2	no	—
56	C₄×C₁₀₁	—	2	2	no	—
57	C₄×C₁₀₁	—	2	1	no	—
58	C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
59	C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
60	Dic₁₀₁	—	2	1	no	—
61	C₂×C₁₀₁	—	2	1	no	—
62	C₂×C₁₀₁	—	4	1	no	—
63	C₂×C₁₀₁	#120	1	1	yes	characteristic
64	D₁₀₁	—	4	2	no	—
65	D₁₀₁	—	4	1	no	—
66	D₁₀₁	—	8	1	no	—
67	C₁₀₁	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	101	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	101	1	no	—
70	AGL(1,ℤ/8)	—	101	1	no	—
71	AGL(1,ℤ/8)	—	101	1	no	—
72	#127	—	101	1	no	—
73	#104	—	101	1	no	—
74	#104	D₁₀₁	1	1	yes	characteristic, complemented
75	#100	—	101	1	no	—
76	#41	—	101	1	no	—
77	#41	—	202	1	no	—
78	#41	—	202	1	no	—
79	#41	C₂×D₁₀₁	1	1	yes	characteristic
80	C₂×D₄	—	101	1	no	—
81	C₂×D₄	—	101	1	no	—
82	C₂×D₄	—	202	1	no	—
83	C₂×D₄	—	202	2	no	—
84	C₂×D₄	—	202	2	no	—
85	C₂×D₄	—	202	1	no	—
86	SD₁₆	—	202	1	no	—
87	SD₁₆	—	202	1	no	—
88	D₈	—	202	1	no	—
89	D₈	—	202	1	no	—
90	M₁₆	—	101	1	no	—
91	M₁₆	C₂×D₁₀₁	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₁₀₁	1	1	yes	characteristic, complemented
93	C₂³	—	101	2	no	—
94	C₂³	—	202	2	no	—
95	C₂³	—	202	1	no	—
96	Q₈	—	101	1	no	—
97	Q₈	D₄₀₄	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	101	1	no	—
100	D₄	—	404	4	no	—
101	D₄	—	202	2	no	—
102	D₄	—	202	2	no	—
103	D₄	—	202	1	no	—
104	D₄	—	202	1	no	—
105	D₄	—	202	1	no	—
106	D₄	—	202	1	no	—
107	D₄	—	202	1	no	—
108	D₄	D₄₀₄	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	202	2	no	—
112	C₂×C₄	—	202	1	no	—
113	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	202	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	808	2	no	—
118	C₂²	—	404	2	no	—
119	C₂²	—	404	2	no	—
120	C₂²	—	404	1	no	—
121	C₂²	—	202	1	no	—
122	C₂²	—	202	2	no	—
123	C₂²	D₄×D₁₀₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	202	1	no	—
127	C₄	D₄×D₁₀₁	1	1	yes	characteristic
128	C₄	C₂×D₄₀₄	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	808	1	no	—
132	C₂	—	404	1	no	—
133	C₂	—	404	2	no	—
134	C₂	#15692225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15692301

C₂

yes

maximal, characteristic, complemented

#15692283

C₂

yes

maximal, characteristic, complemented

#15692280

C₂

yes

maximal, characteristic, complemented

#15692252

C₂

yes

maximal, characteristic, complemented

#15692183

C₂

yes

maximal, characteristic, complemented

#15692171

C₂

yes

maximal, characteristic, complemented

#104×C₁₀₁

C₂

yes

maximal, characteristic, complemented

#13108075

—

#13108074

—

#13108074

C₂²

yes

characteristic

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

C₂²

yes

characteristic

C₂×D₄₀₄

C₂²

yes

characteristic, complemented

D₄×D₁₀₁

—

D₄×D₁₀₁

—

#13108063

—

#13108061

—

D₈₀₈

—

#13108059

—

#13108058

C₂²

yes

characteristic, complemented

#41×C₁₀₁

C₂²

yes

characteristic

M₁₆×C₁₀₁

C₂²

yes

characteristic, complemented

C₄²×C₁₀₁

C₂²

yes

characteristic, complemented

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

D₄

yes

—

Q₈×C₁₀₁

D₄

yes

characteristic, complemented

D₄×C₁₀₁

—

D₄×C₁₀₁

D₄

yes

characteristic, complemented

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊D₁₀₁

—

C₂²⋊D₁₀₁

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

D₄

yes

characteristic, complemented

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

D₄

yes

characteristic, complemented

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×D₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₄×C₁₀₁

C₂×D₄

yes

characteristic

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#120

yes

characteristic

D₁₀₁

—

D₁₀₁

—

D₁₀₁

—

C₁₀₁

#453

yes

characteristic, complemented, Sylow

#453

—

101

maximal, Sylow

H(5,𝔽₂)

—

101

—

AGL(1,ℤ/8)

—

101

—

AGL(1,ℤ/8)

—

101

—

#127

—

101

—

#104

—

101

—

#104

D₁₀₁

yes

characteristic, complemented

#100

—

101

—

#41

—

101

—

#41

—

202

—

#41

—

202

—

#41

C₂×D₁₀₁

yes

characteristic

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

SD₁₆

—

202

—

SD₁₆

—

202

—

D₈

—

202

—

D₈

—

202

—

M₁₆

—

101

—

M₁₆

C₂×D₁₀₁

yes

characteristic, complemented

C₄²

C₂×D₁₀₁

yes

characteristic, complemented

C₂³

—

101

—

C₂³

—

202

—

C₂³

—

202

—

Q₈

—

101

—

Q₈

D₄₀₄

yes

characteristic, complemented

D₄

—

D₄

—

101

—

100

D₄

—

404

—

101

D₄

—

202

—

102

D₄

—

202

—

103

D₄

—

202

—

104

D₄

—

202

—

105

D₄

—

202

—

106

D₄

—

202

—

107

D₄

—

202

—

108

D₄

D₄₀₄

yes

characteristic, complemented

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

202

—

112

C₂×C₄

—

202

—

113

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

114

C₈

—

115

C₈

—

202

—

116

C₂²

—

117

C₂²

—

808

—

118

C₂²

—

404

—

119

C₂²

—

404

—

120

C₂²

—

404

—

121

C₂²

—

202

—

122

C₂²

—

202

—

123

C₂²

D₄×D₁₀₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

202

—

127

C₄

D₄×D₁₀₁

yes

characteristic

128

C₄

C₂×D₄₀₄

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

808

—

132

C₂

—

404

—

133

C₂

—

404

—

134

C₂

#15692225

yes

characteristic

	subgroup	G/H
center	C₂	#15692225
commutator	C₂×C₄×C₁₀₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₁
Fitting	#104×C₁₀₁	C₂
socle	C₂×C₁₀₁	#120
hypercenter	#104	D₁₀₁
hypocenter	C₁₀₁	#453
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

subgroup

G/H

center

C₂

#15692225

commutator

C₂×C₄×C₁₀₁

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₁

Fitting

#104×C₁₀₁

C₂

C₂×C₁₀₁

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

N	H ≅ G/N	type
1. #15692301	C₂	semidirect
2. #15692283	C₂	semidirect
3. #15692280	C₂	semidirect
4. #15692252	C₂	semidirect
5. #15692183	C₂	semidirect
6. #15692171	C₂	semidirect
7. #104×C₁₀₁	C₂	semidirect
14. C₂×D₄₀₄	C₂²	semidirect
21. #13108058	C₂²	semidirect
23. M₁₆×C₁₀₁	C₂²	semidirect
24. C₄²×C₁₀₁	C₂²	semidirect
28. Q₈×C₁₀₁	D₄	semidirect
30. D₄×C₁₀₁	D₄	semidirect
42. D₄₀₄	D₄	semidirect
45. Dic₂₀₂	D₄	semidirect
67. C₁₀₁	#453	semidirect
74. #104	D₁₀₁	semidirect
91. M₁₆	C₂×D₁₀₁	semidirect
92. C₄²	C₂×D₁₀₁	semidirect
97. Q₈	D₄₀₄	semidirect
108. D₄	D₄₀₄	semidirect

H ≅ G/N

type

1. #15692301

C₂

semidirect

2. #15692283

C₂

semidirect

3. #15692280

C₂

semidirect

4. #15692252

C₂

semidirect

5. #15692183

C₂

semidirect

6. #15692171

C₂

semidirect

7. #104×C₁₀₁

C₂

semidirect

14. C₂×D₄₀₄

C₂²

semidirect

21. #13108058

C₂²

semidirect

23. M₁₆×C₁₀₁

C₂²

semidirect

24. C₄²×C₁₀₁

C₂²

semidirect

28. Q₈×C₁₀₁

D₄

semidirect

30. D₄×C₁₀₁

D₄

semidirect

42. D₄₀₄

D₄

semidirect

45. Dic₂₀₂

D₄

semidirect

67. C₁₀₁

#453

semidirect

74. #104

D₁₀₁

semidirect

91. M₁₆

C₂×D₁₀₁

semidirect

92. C₄²

C₂×D₁₀₁

semidirect

97. Q₈

D₄₀₄

semidirect

108. D₄

D₄₀₄

semidirect

Subgroups — #20265346

Conjugacy classes of subgroups

Complements