Group #20267960 — subgroups

Subgroups — #20267960

Order 6592.

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	#15693067	C₂	1	1	yes	maximal, characteristic, complemented
2	#15693049	C₂	1	1	yes	maximal, characteristic, complemented
3	#15693046	C₂	1	1	yes	maximal, characteristic, complemented
4	#15693018	C₂	1	1	yes	maximal, characteristic, complemented
5	#15692949	C₂	1	1	yes	maximal, characteristic, complemented
6	#15692937	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#13110106	—	2	1	no	—
9	#13110105	—	2	1	no	—
10	#13110105	C₂²	1	1	yes	characteristic
11	C₂×D₄₁₂	—	2	1	no	—
12	C₂×D₄₁₂	—	2	2	no	—
13	C₂×D₄₁₂	C₂²	1	1	yes	characteristic
14	C₂×D₄₁₂	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₁₀₃	—	2	2	no	—
16	D₄×D₁₀₃	—	2	1	no	—
17	#13110094	—	2	1	no	—
18	#13110092	—	2	1	no	—
19	D₈₂₄	—	2	1	no	—
20	#13110090	—	2	1	no	—
21	#13110089	C₂²	1	1	yes	characteristic, complemented
22	#41×C₁₀₃	C₂²	1	1	yes	characteristic
23	M₁₆×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₁₀₃	—	2	2	no	—
26	C₂²×D₁₀₃	—	2	1	no	—
27	C₂²×D₁₀₃	D₄	1	2	yes	—
28	Q₈×C₁₀₃	D₄	1	1	yes	characteristic, complemented
29	D₄×C₁₀₃	—	2	1	no	—
30	D₄×C₁₀₃	D₄	1	1	yes	characteristic, complemented
31	C₂×C₄×C₁₀₃	—	2	1	no	—
32	C₂×C₄×C₁₀₃	—	2	1	no	—
33	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
34	C₂²⋊D₁₀₃	—	2	2	no	—
35	C₂²⋊D₁₀₃	—	2	1	no	—
36	D₄₁₂	—	2	2	no	—
37	D₄₁₂	—	2	1	no	—
38	D₄₁₂	—	2	1	no	—
39	D₄₁₂	—	2	1	no	—
40	D₄₁₂	—	2	1	no	—
41	D₄₁₂	—	4	4	no	—
42	D₄₁₂	D₄	1	1	yes	characteristic, complemented
43	C₄×D₁₀₃	—	2	2	no	—
44	C₄×D₁₀₃	—	2	1	no	—
45	Dic₂₀₆	D₄	1	1	yes	characteristic, complemented
46	C₈×C₁₀₃	—	2	1	no	—
47	C₁₀₃⋊C₈	—	2	1	no	—
48	C₂²×C₁₀₃	—	2	1	no	—
49	C₂²×C₁₀₃	C₂×D₄	1	1	yes	characteristic
50	C₂×D₁₀₃	—	2	2	no	—
51	C₂×D₁₀₃	—	2	1	no	—
52	C₂×D₁₀₃	—	4	2	no	—
53	C₂×D₁₀₃	—	4	2	no	—
54	C₂×D₁₀₃	—	4	1	no	—
55	C₂×D₁₀₃	—	8	2	no	—
56	C₄×C₁₀₃	—	2	2	no	—
57	C₄×C₁₀₃	—	2	1	no	—
58	C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
59	C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
60	Dic₁₀₃	—	2	1	no	—
61	C₂×C₁₀₃	—	2	1	no	—
62	C₂×C₁₀₃	—	4	1	no	—
63	C₂×C₁₀₃	#120	1	1	yes	characteristic
64	D₁₀₃	—	4	2	no	—
65	D₁₀₃	—	4	1	no	—
66	D₁₀₃	—	8	1	no	—
67	C₁₀₃	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	103	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	103	1	no	—
70	AGL(1,ℤ/8)	—	103	1	no	—
71	AGL(1,ℤ/8)	—	103	1	no	—
72	#127	—	103	1	no	—
73	#104	—	103	1	no	—
74	#104	D₁₀₃	1	1	yes	characteristic, complemented
75	#100	—	103	1	no	—
76	#41	—	103	1	no	—
77	#41	—	206	1	no	—
78	#41	—	206	1	no	—
79	#41	C₂×D₁₀₃	1	1	yes	characteristic
80	C₂×D₄	—	103	1	no	—
81	C₂×D₄	—	103	1	no	—
82	C₂×D₄	—	206	1	no	—
83	C₂×D₄	—	206	2	no	—
84	C₂×D₄	—	206	2	no	—
85	C₂×D₄	—	206	1	no	—
86	SD₁₆	—	206	1	no	—
87	SD₁₆	—	206	1	no	—
88	D₈	—	206	1	no	—
89	D₈	—	206	1	no	—
90	M₁₆	—	103	1	no	—
91	M₁₆	C₂×D₁₀₃	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₁₀₃	1	1	yes	characteristic, complemented
93	C₂³	—	103	2	no	—
94	C₂³	—	206	2	no	—
95	C₂³	—	206	1	no	—
96	Q₈	—	103	1	no	—
97	Q₈	D₄₁₂	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	103	1	no	—
100	D₄	—	412	4	no	—
101	D₄	—	206	2	no	—
102	D₄	—	206	2	no	—
103	D₄	—	206	1	no	—
104	D₄	—	206	1	no	—
105	D₄	—	206	1	no	—
106	D₄	—	206	1	no	—
107	D₄	—	206	1	no	—
108	D₄	D₄₁₂	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	206	2	no	—
112	C₂×C₄	—	206	1	no	—
113	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	206	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	824	2	no	—
118	C₂²	—	412	2	no	—
119	C₂²	—	412	2	no	—
120	C₂²	—	412	1	no	—
121	C₂²	—	206	1	no	—
122	C₂²	—	206	2	no	—
123	C₂²	D₄×D₁₀₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	206	1	no	—
127	C₄	D₄×D₁₀₃	1	1	yes	characteristic
128	C₄	C₂×D₄₁₂	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	824	1	no	—
132	C₂	—	412	1	no	—
133	C₂	—	412	2	no	—
134	C₂	#15692991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15693067

C₂

yes

maximal, characteristic, complemented

#15693049

C₂

yes

maximal, characteristic, complemented

#15693046

C₂

yes

maximal, characteristic, complemented

#15693018

C₂

yes

maximal, characteristic, complemented

#15692949

C₂

yes

maximal, characteristic, complemented

#15692937

C₂

yes

maximal, characteristic, complemented

#104×C₁₀₃

C₂

yes

maximal, characteristic, complemented

#13110106

—

#13110105

—

#13110105

C₂²

yes

characteristic

C₂×D₄₁₂

—

C₂×D₄₁₂

—

C₂×D₄₁₂

C₂²

yes

characteristic

C₂×D₄₁₂

C₂²

yes

characteristic, complemented

D₄×D₁₀₃

—

D₄×D₁₀₃

—

#13110094

—

#13110092

—

D₈₂₄

—

#13110090

—

#13110089

C₂²

yes

characteristic, complemented

#41×C₁₀₃

C₂²

yes

characteristic

M₁₆×C₁₀₃

C₂²

yes

characteristic, complemented

C₄²×C₁₀₃

C₂²

yes

characteristic, complemented

C₂²×D₁₀₃

—

C₂²×D₁₀₃

—

C₂²×D₁₀₃

D₄

yes

—

Q₈×C₁₀₃

D₄

yes

characteristic, complemented

D₄×C₁₀₃

—

D₄×C₁₀₃

D₄

yes

characteristic, complemented

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂²⋊D₁₀₃

—

C₂²⋊D₁₀₃

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

D₄

yes

characteristic, complemented

C₄×D₁₀₃

—

C₄×D₁₀₃

—

Dic₂₀₆

D₄

yes

characteristic, complemented

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂×D₄

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₄×C₁₀₃

C₂×D₄

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#120

yes

characteristic

D₁₀₃

—

D₁₀₃

—

D₁₀₃

—

C₁₀₃

#453

yes

characteristic, complemented, Sylow

#453

—

103

maximal, Sylow

H(5,𝔽₂)

—

103

—

AGL(1,ℤ/8)

—

103

—

AGL(1,ℤ/8)

—

103

—

#127

—

103

—

#104

—

103

—

#104

D₁₀₃

yes

characteristic, complemented

#100

—

103

—

#41

—

103

—

#41

—

206

—

#41

—

206

—

#41

C₂×D₁₀₃

yes

characteristic

C₂×D₄

—

103

—

C₂×D₄

—

103

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

SD₁₆

—

206

—

SD₁₆

—

206

—

D₈

—

206

—

D₈

—

206

—

M₁₆

—

103

—

M₁₆

C₂×D₁₀₃

yes

characteristic, complemented

C₄²

C₂×D₁₀₃

yes

characteristic, complemented

C₂³

—

103

—

C₂³

—

206

—

C₂³

—

206

—

Q₈

—

103

—

Q₈

D₄₁₂

yes

characteristic, complemented

D₄

—

D₄

—

103

—

100

D₄

—

412

—

101

D₄

—

206

—

102

D₄

—

206

—

103

D₄

—

206

—

104

D₄

—

206

—

105

D₄

—

206

—

106

D₄

—

206

—

107

D₄

—

206

—

108

D₄

D₄₁₂

yes

characteristic, complemented

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

206

—

112

C₂×C₄

—

206

—

113

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

114

C₈

—

115

C₈

—

206

—

116

C₂²

—

117

C₂²

—

824

—

118

C₂²

—

412

—

119

C₂²

—

412

—

120

C₂²

—

412

—

121

C₂²

—

206

—

122

C₂²

—

206

—

123

C₂²

D₄×D₁₀₃

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

206

—

127

C₄

D₄×D₁₀₃

yes

characteristic

128

C₄

C₂×D₄₁₂

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

824

—

132

C₂

—

412

—

133

C₂

—

412

—

134

C₂

#15692991

yes

characteristic

	subgroup	G/H
center	C₂	#15692991
commutator	C₂×C₄×C₁₀₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₃
Fitting	#104×C₁₀₃	C₂
socle	C₂×C₁₀₃	#120
hypercenter	#104	D₁₀₃
hypocenter	C₁₀₃	#453
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

subgroup

G/H

center

C₂

#15692991

commutator

C₂×C₄×C₁₀₃

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₃

Fitting

#104×C₁₀₃

C₂

C₂×C₁₀₃

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

N	H ≅ G/N	type
1. #15693067	C₂	semidirect
2. #15693049	C₂	semidirect
3. #15693046	C₂	semidirect
4. #15693018	C₂	semidirect
5. #15692949	C₂	semidirect
6. #15692937	C₂	semidirect
7. #104×C₁₀₃	C₂	semidirect
14. C₂×D₄₁₂	C₂²	semidirect
21. #13110089	C₂²	semidirect
23. M₁₆×C₁₀₃	C₂²	semidirect
24. C₄²×C₁₀₃	C₂²	semidirect
28. Q₈×C₁₀₃	D₄	semidirect
30. D₄×C₁₀₃	D₄	semidirect
42. D₄₁₂	D₄	semidirect
45. Dic₂₀₆	D₄	semidirect
67. C₁₀₃	#453	semidirect
74. #104	D₁₀₃	semidirect
91. M₁₆	C₂×D₁₀₃	semidirect
92. C₄²	C₂×D₁₀₃	semidirect
97. Q₈	D₄₁₂	semidirect
108. D₄	D₄₁₂	semidirect

H ≅ G/N

type

1. #15693067

C₂

semidirect

2. #15693049

C₂

semidirect

3. #15693046

C₂

semidirect

4. #15693018

C₂

semidirect

5. #15692949

C₂

semidirect

6. #15692937

C₂

semidirect

7. #104×C₁₀₃

C₂

semidirect

14. C₂×D₄₁₂

C₂²

semidirect

21. #13110089

C₂²

semidirect

23. M₁₆×C₁₀₃

C₂²

semidirect

24. C₄²×C₁₀₃

C₂²

semidirect

28. Q₈×C₁₀₃

D₄

semidirect

30. D₄×C₁₀₃

D₄

semidirect

42. D₄₁₂

D₄

semidirect

45. Dic₂₀₆

D₄

semidirect

67. C₁₀₃

#453

semidirect

74. #104

D₁₀₃

semidirect

91. M₁₆

C₂×D₁₀₃

semidirect

92. C₄²

C₂×D₁₀₃

semidirect

97. Q₈

D₄₁₂

semidirect

108. D₄

D₄₁₂

semidirect

Subgroups — #20267960

Conjugacy classes of subgroups

Complements