Group #20265747 — subgroups

Subgroups — #20265747

Order 6464.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#13108073	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#15692255	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692253	C₂	1	1	yes	maximal, characteristic, complemented
5	#15692242	C₂	1	1	yes	maximal, characteristic, complemented
6	#15692237	C₂	1	4	yes	maximal, complemented
7	#15692232	C₂	1	4	yes	maximal, complemented
8	#15692212	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#13108073	—	2	4	no	—
11	C₂×Dic₂₀₂	C₂²	1	2	yes	—
12	C₂×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	Q₈×D₁₀₁	—	2	4	no	—
15	C₂×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
16	C₂²×Dic₁₀₁	C₂²	1	2	yes	—
17	C₂×C₄×D₁₀₁	C₂²	1	2	yes	—
18	C₂×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₀₁	C₂²	1	4	yes	—
20	#13108053	C₂²	1	2	yes	—
21	#13108053	C₂²	1	4	yes	—
22	C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
23	#13108051	C₂²	1	2	yes	—
24	C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	—
28	C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂³×C₁₀₁	C₂³	1	2	yes	—
30	C₂²×D₁₀₁	C₂³	1	1	yes	characteristic
31	Q₈×C₁₀₁	—	2	2	no	—
32	D₄×C₁₀₁	—	2	2	no	—
33	C₂×C₄×C₁₀₁	C₂³	1	2	yes	—
34	C₂×C₄×C₁₀₁	C₂³	1	2	yes	—
35	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
36	C₂²⋊D₁₀₁	—	2	4	no	—
37	C₂×Dic₁₀₁	—	2	4	no	—
38	C₂×Dic₁₀₁	C₂³	1	2	yes	—
39	C₂×Dic₁₀₁	C₂³	1	2	yes	—
40	C₂×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂×Dic₁₀₁	C₂³	1	1	yes	characteristic
42	C₄×D₁₀₁	—	2	4	no	—
43	C₄×D₁₀₁	D₄	1	4	yes	—
44	Dic₂₀₂	—	2	2	no	—
45	Dic₂₀₂	—	2	2	no	—
46	Dic₂₀₂	—	2	4	no	—
47	C₂²×C₁₀₁	—	2	2	no	—
48	C₂²×C₁₀₁	—	2	4	no	—
49	C₂²×C₁₀₁	C₂⁴	1	1	yes	characteristic
50	C₂×D₁₀₁	—	2	2	no	—
51	C₂×D₁₀₁	C₂×D₄	1	2	yes	—
52	C₄×C₁₀₁	—	2	2	no	—
53	C₄×C₁₀₁	—	2	2	no	—
54	C₄×C₁₀₁	C₂×D₄	1	2	yes	—
55	Dic₁₀₁	—	2	2	no	—
56	Dic₁₀₁	—	2	2	no	—
57	Dic₁₀₁	—	2	2	no	—
58	Dic₁₀₁	C₂×D₄	1	2	yes	—
59	C₂×C₁₀₁	—	4	2	no	—
60	C₂×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
62	D₁₀₁	—	2	2	no	—
63	C₁₀₁	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	101	1	no	maximal, Sylow
65	C₂×#41	—	101	1	no	—
66	C₂²×Q₈	—	101	1	no	—
67	C₄⋊Q₈	—	101	1	no	—
68	C₄⋊Q₈	—	101	1	no	—
69	#124	—	101	1	no	—
70	#124	D₁₀₁	1	1	yes	characteristic, complemented
71	#123	—	101	4	no	—
72	C₂²⋊Q₈	—	101	4	no	—
73	#117	—	101	1	no	—
74	#41	—	202	4	no	—
75	C₂×Q₈	—	101	2	no	—
76	C₂×Q₈	—	101	1	no	—
77	C₂×Q₈	—	101	1	no	—
78	C₂×Q₈	—	202	4	no	—
79	C₂×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
80	C₂×D₄	—	101	2	no	—
81	C₂×D₄	C₂×D₁₀₁	1	1	yes	characteristic
82	C₂²×C₄	—	101	1	no	—
83	C₂²×C₄	—	101	2	no	—
84	C₂²×C₄	—	101	2	no	—
85	C₄⋊C₄	—	101	2	no	—
86	C₄⋊C₄	—	101	4	no	—
87	C₄⋊C₄	—	101	4	no	—
88	C₂²⋊C₄	—	101	4	no	—
89	C₂²⋊C₄	—	101	2	no	—
90	C₂²⋊C₄	C₂×D₁₀₁	1	4	yes	—
91	C₄²	—	101	1	no	—
92	C₄²	C₂×D₁₀₁	1	1	yes	characteristic
93	C₂³	—	101	1	no	—
94	C₂³	C₂²×D₁₀₁	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	202	4	no	—
97	Q₈	—	202	2	no	—
98	Q₈	—	202	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	202	4	no	—
101	C₂×C₄	—	101	1	no	—
102	C₂×C₄	—	101	4	no	—
103	C₂×C₄	—	101	2	no	—
104	C₂×C₄	—	101	2	no	—
105	C₂×C₄	—	101	2	no	—
106	C₂×C₄	—	202	4	no	—
107	C₂×C₄	—	202	4	no	—
108	C₂×C₄	C₂²×D₁₀₁	1	2	yes	—
109	C₂×C₄	C₂²×D₁₀₁	1	2	yes	—
110	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	101	2	no	—
114	C₂²	—	202	2	no	—
115	C₂²	C₂³×D₁₀₁	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	101	2	no	—
119	C₄	—	202	2	no	—
120	C₄	—	202	2	no	—
121	C₄	—	202	2	no	—
122	C₄	D₄×D₁₀₁	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	202	2	no	—
125	C₂	C₂×D₄×D₁₀₁	1	1	yes	characteristic
126	C₂	#15692302	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13108073

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₀₁

C₂

yes

maximal, characteristic, complemented

#15692255

C₂

yes

maximal, characteristic, complemented

#15692253

C₂

yes

maximal, characteristic, complemented

#15692242

C₂

yes

maximal, characteristic, complemented

#15692237

C₂

yes

maximal, complemented

#15692232

C₂

yes

maximal, complemented

#15692212

C₂

yes

maximal, characteristic, complemented

#124×C₁₀₁

C₂

yes

maximal, characteristic, complemented

#13108073

—

C₂×Dic₂₀₂

C₂²

yes

—

C₂×Dic₂₀₂

C₂²

yes

characteristic

C₂×Dic₂₀₂

C₂²

yes

characteristic

Q₈×D₁₀₁

—

C₂×C₂²⋊D₁₀₁

C₂²

yes

—

C₂²×Dic₁₀₁

C₂²

yes

—

C₂×C₄×D₁₀₁

C₂²

yes

—

C₂×C₄×D₁₀₁

C₂²

yes

characteristic

C₄⋊Dic₁₀₁

C₂²

yes

—

#13108053

C₂²

yes

—

#13108053

C₂²

yes

—

C₂²⋊Dic₁₀₁

C₂²

yes

—

#13108051

C₂²

yes

—

C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂×D₄×C₁₀₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₁

C₂²

yes

—

C₄²×C₁₀₁

C₂²

yes

characteristic

C₂³×C₁₀₁

C₂³

yes

—

C₂²×D₁₀₁

C₂³

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

D₄

yes

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂⁴

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

C₂×D₄

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×D₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

C₂×D₄

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂²×D₄

yes

characteristic

C₂×C₁₀₁

2₋¹⁺⁴

yes

—

D₁₀₁

—

C₁₀₁

#536

yes

characteristic, complemented, Sylow

#536

—

101

maximal, Sylow

C₂×#41

—

101

—

C₂²×Q₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

101

—

#124

—

101

—

#124

D₁₀₁

yes

characteristic, complemented

#123

—

101

—

C₂²⋊Q₈

—

101

—

#117

—

101

—

#41

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

101

—

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂×D₁₀₁

yes

characteristic

C₂×D₄

—

101

—

C₂×D₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

C₂×D₁₀₁

yes

—

C₄²

—

101

—

C₄²

C₂×D₁₀₁

yes

characteristic

C₂³

—

101

—

C₂³

C₂²×D₁₀₁

yes

—

Q₈

—

Q₈

—

202

—

Q₈

—

202

—

Q₈

—

202

—

D₄

—

100

D₄

—

202

—

101

C₂×C₄

—

101

—

102

C₂×C₄

—

101

—

103

C₂×C₄

—

101

—

104

C₂×C₄

—

101

—

105

C₂×C₄

—

101

—

106

C₂×C₄

—

202

—

107

C₂×C₄

—

202

—

108

C₂×C₄

C₂²×D₁₀₁

yes

—

109

C₂×C₄

C₂²×D₁₀₁

yes

—

110

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

101

—

114

C₂²

—

202

—

115

C₂²

C₂³×D₁₀₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

101

—

119

C₄

—

202

—

120

C₄

—

202

—

121

C₄

—

202

—

122

C₄

D₄×D₁₀₁

yes

—

123

C₂

—

124

C₂

—

202

—

125

C₂

C₂×D₄×D₁₀₁

yes

characteristic

126

C₂

#15692302

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂⁴
Fitting	#124×C₁₀₁	C₂
hypercenter	#124	D₁₀₁
hypocenter	C₁₀₁	#536
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂⁴

Fitting

#124×C₁₀₁

C₂

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₁₀₁	C₂	semidirect
3. #15692255	C₂	semidirect
4. #15692253	C₂	semidirect
5. #15692242	C₂	semidirect
6. #15692237	C₂	semidirect
7. #15692232	C₂	semidirect
8. #15692212	C₂	semidirect
9. #124×C₁₀₁	C₂	semidirect
63. C₁₀₁	#536	semidirect
70. #124	D₁₀₁	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₁₀₁

C₂

semidirect

3. #15692255

C₂

semidirect

4. #15692253

C₂

semidirect

5. #15692242

C₂

semidirect

6. #15692237

C₂

semidirect

7. #15692232

C₂

semidirect

8. #15692212

C₂

semidirect

9. #124×C₁₀₁

C₂

semidirect

63. C₁₀₁

#536

semidirect

70. #124

D₁₀₁

semidirect

Subgroups — #20265747

Conjugacy classes of subgroups

Complements