Group #20265094 — subgroups

Subgroups — #20265094

Order 6464.

238 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 63 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×D₄₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#13108051	C₂	1	2	yes	maximal, complemented
3	C₂×#13108051	C₂	1	2	yes	maximal, complemented
4	#15692162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₀₁	C₂²	1	2	yes	—
7	C₂×D₄₀₄	—	2	4	no	—
8	C₂×D₄₀₄	—	2	2	no	—
9	C₂²×Dic₁₀₁	C₂²	1	2	yes	—
10	#13108051	—	2	4	no	—
11	#13108051	—	2	4	no	—
12	C₂²×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₁₀₁	—	2	2	no	—
16	C₂³×C₁₀₁	C₂³	1	1	yes	characteristic
17	C₂²×D₁₀₁	—	2	2	no	—
18	C₂²×D₁₀₁	—	2	2	no	—
19	C₂²×D₁₀₁	—	2	2	no	—
20	C₂²×D₁₀₁	—	2	2	no	—
21	C₂²×D₁₀₁	—	2	2	no	—
22	C₂²×D₁₀₁	—	2	2	no	—
23	C₂²×D₁₀₁	D₄	1	4	yes	—
24	C₂×C₄×C₁₀₁	—	2	1	no	—
25	C₂×C₄×C₁₀₁	—	2	2	no	—
26	C₂×C₄×C₁₀₁	—	2	1	no	—
27	C₂×C₄×C₁₀₁	—	2	1	no	—
28	C₂×C₄×C₁₀₁	—	2	2	no	—
29	C₂×C₄×C₁₀₁	D₄	1	2	yes	—
30	C₂×C₄×C₁₀₁	D₄	1	2	yes	complemented
31	C₂×Dic₁₀₁	—	2	2	no	—
32	C₂×Dic₁₀₁	—	2	2	no	—
33	C₂×Dic₁₀₁	—	2	2	no	—
34	D₄₀₄	—	2	8	no	—
35	C₂²×C₁₀₁	C₂×D₄	1	2	yes	—
36	C₂²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₁₀₁	#41	1	1	yes	characteristic
39	C₂²×C₁₀₁	#41	1	2	yes	—
40	C₂×D₁₀₁	—	2	4	no	—
41	C₂×D₁₀₁	—	2	4	no	—
42	C₂×D₁₀₁	—	2	4	no	—
43	C₂×D₁₀₁	—	4	2	no	—
44	C₂×D₁₀₁	—	4	2	no	—
45	C₂×D₁₀₁	—	4	2	no	—
46	C₂×D₁₀₁	—	4	2	no	—
47	C₄×C₁₀₁	—	2	2	no	—
48	C₄×C₁₀₁	—	2	2	no	—
49	C₄×C₁₀₁	—	4	1	no	—
50	Dic₁₀₁	—	4	2	no	—
51	C₂×C₁₀₁	#120	1	1	yes	characteristic
52	C₂×C₁₀₁	#121	1	1	yes	characteristic
53	C₂×C₁₀₁	#121	1	2	yes	—
54	C₂×C₁₀₁	#123	1	2	yes	—
55	C₂×C₁₀₁	#124	1	1	yes	characteristic
56	D₁₀₁	—	4	4	no	—
57	C₁₀₁	#394	1	1	yes	characteristic, complemented, Sylow
58	#394	—	101	1	no	maximal, Sylow
59	C₂²×D₄	—	101	1	no	—
60	C₂×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
61	C₂×C₂²⋊C₄	—	101	2	no	—
62	C₂×C₂²⋊C₄	—	101	2	no	—
63	#95	—	101	1	no	—
64	C₂⁴	—	101	2	no	—
65	C₂×D₄	—	202	4	no	—
66	C₂×D₄	—	202	2	no	—
67	C₂²×C₄	—	101	2	no	—
68	C₂²×C₄	C₂×D₁₀₁	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₁₀₁	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₁₀₁	1	1	yes	characteristic
71	C₄⋊C₄	—	2	2	no	—
72	C₂²⋊C₄	—	202	4	no	—
73	C₂²⋊C₄	—	202	4	no	—
74	C₂³	—	101	4	no	—
75	C₂³	—	202	2	no	—
76	C₂³	—	202	2	no	—
77	C₂³	—	202	2	no	—
78	C₂³	—	202	2	no	—
79	C₂³	—	202	2	no	—
80	C₂³	—	202	2	no	—
81	C₂³	C₂²×D₁₀₁	1	1	yes	characteristic
82	D₄	—	202	8	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	202	2	no	—
89	C₂×C₄	—	202	2	no	—
90	C₂×C₄	—	202	2	no	—
91	C₂×C₄	D₄₀₄	1	2	yes	complemented
92	C₂×C₄	C₂²⋊D₁₀₁	1	2	yes	—
93	C₂²	—	404	2	no	—
94	C₂²	—	404	2	no	—
95	C₂²	—	404	2	no	—
96	C₂²	—	404	2	no	—
97	C₂²	—	202	4	no	—
98	C₂²	—	202	4	no	—
99	C₂²	—	202	4	no	—
100	C₂²	D₄×D₁₀₁	1	2	yes	—
101	C₂²	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
102	C₂²	C₂×D₄₀₄	1	1	yes	characteristic
103	C₂²	#13108074	1	1	yes	characteristic
104	C₂²	#13108075	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	1	no	—
108	C₄	—	404	2	no	—
109	C₂	—	404	4	no	—
110	C₂	#15692223	1	1	yes	characteristic
111	C₂	#15692229	1	1	yes	characteristic
112	C₂	#15692230	1	2	yes	—
113	C₂	#15692240	1	2	yes	—
114	C₂	#15692244	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₀₄

C₂

yes

maximal, characteristic

C₂×#13108051

C₂

yes

maximal, complemented

C₂×#13108051

C₂

yes

maximal, complemented

#15692162

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₁

C₂²

yes

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂²×Dic₁₀₁

C₂²

yes

—

#13108051

—

#13108051

—

C₂²×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₄×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

D₄

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

D₄

yes

—

C₂×C₄×C₁₀₁

D₄

yes

complemented

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

D₄₀₄

—

C₂²×C₁₀₁

C₂×D₄

yes

—

C₂²×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×C₁₀₁

C₂×D₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#123

yes

—

C₂×C₁₀₁

#124

yes

characteristic

D₁₀₁

—

C₁₀₁

#394

yes

characteristic, complemented, Sylow

#394

—

101

maximal, Sylow

C₂²×D₄

—

101

—

C₂×C₄⋊C₄

D₁₀₁

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

#95

—

101

—

C₂⁴

—

101

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₄

C₂×D₁₀₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₂³

—

101

—

C₂³

—

202

—

C₂³

—

202

—

C₂³

—

202

—

C₂³

—

202

—

C₂³

—

202

—

C₂³

—

202

—

C₂³

C₂²×D₁₀₁

yes

characteristic

D₄

—

202

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

D₄₀₄

yes

complemented

C₂×C₄

C₂²⋊D₁₀₁

yes

—

C₂²

—

404

—

C₂²

—

404

—

C₂²

—

404

—

C₂²

—

404

—

C₂²

—

202

—

C₂²

—

202

—

C₂²

—

202

—

100

C₂²

D₄×D₁₀₁

yes

—

101

C₂²

C₂×C₂²⋊D₁₀₁

yes

characteristic

102

C₂²

C₂×D₄₀₄

yes

characteristic

103

C₂²

#13108074

yes

characteristic

104

C₂²

#13108075

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

404

—

109

C₂

—

404

—

110

C₂

#15692223

yes

characteristic

111

C₂

#15692229

yes

characteristic

112

C₂

#15692230

yes

—

113

C₂

#15692240

yes

—

114

C₂

#15692244

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₁
commutator, socle	C₂³×C₁₀₁	C₂³
Fitting	C₂×C₄⋊C₄×C₁₀₁	C₂
hypercenter	C₂×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#394
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₁

commutator, socle

C₂³×C₁₀₁

C₂³

Fitting

C₂×C₄⋊C₄×C₁₀₁

C₂

hypercenter

C₂×C₄⋊C₄

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×#13108051	C₂	semidirect
3. C₂×#13108051	C₂	semidirect
4. #15692162	C₂	semidirect
5. C₂×C₄⋊C₄×C₁₀₁	C₂	semidirect
30. C₂×C₄×C₁₀₁	D₄	semidirect
57. C₁₀₁	#394	semidirect
60. C₂×C₄⋊C₄	D₁₀₁	semidirect
91. C₂×C₄	D₄₀₄	semidirect

H ≅ G/N

type

2. C₂×#13108051

C₂

semidirect

3. C₂×#13108051

C₂

semidirect

4. #15692162

C₂

semidirect

5. C₂×C₄⋊C₄×C₁₀₁

C₂

semidirect

30. C₂×C₄×C₁₀₁

D₄

semidirect

57. C₁₀₁

#394

semidirect

60. C₂×C₄⋊C₄

D₁₀₁

semidirect

91. C₂×C₄

D₄₀₄

semidirect

Subgroups — #20265094

Conjugacy classes of subgroups

Complements