Group #20265658 — subgroups

Subgroups — #20265658

Order 6464.

294 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 151 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
2	#15692220	C₂	1	2	yes	maximal, complemented
3	#15692217	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊D₁₀₁	C₂	1	4	yes	maximal, complemented
5	#15692214	C₂	1	1	yes	maximal, characteristic, complemented
6	#15692213	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₄×D₁₀₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#13108074	C₄	1	8	yes	complemented
10	C₂×Dic₂₀₂	C₂²	1	1	yes	characteristic
11	C₂×D₄₀₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
13	C₂×C₄×D₁₀₁	C₂²	1	4	yes	—
14	C₂×C₄×D₁₀₁	C₂²	1	2	yes	—
15	C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
17	#13108053	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
19	#13108051	C₂²	1	4	yes	—
20	C₄×Dic₁₀₁	C₂²	1	4	yes	—
21	C₄×Dic₁₀₁	C₂²	1	2	yes	—
22	C₂²×C₄×C₁₀₁	C₂²	1	2	yes	—
23	C₂²×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
25	C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
26	C₂³×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₁	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₁	—	4	2	no	—
29	C₂×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
30	C₂×C₄×C₁₀₁	C₂³	1	4	yes	—
31	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	complemented
34	C₂×Dic₁₀₁	C₂³	1	2	yes	—
35	C₂×Dic₁₀₁	C₂³	1	2	yes	—
36	C₂×Dic₁₀₁	C₂³	1	2	yes	—
37	D₄₀₄	C₂×C₄	1	4	yes	complemented
38	C₄×D₁₀₁	—	4	4	no	—
39	C₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
40	Dic₂₀₂	C₂×C₄	1	4	yes	complemented
41	C₂²×C₁₀₁	—	2	1	no	—
42	C₂²×C₁₀₁	—	2	1	no	—
43	C₂²×C₁₀₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₁₀₁	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₀₁	—	2	2	no	—
46	C₂×D₁₀₁	—	2	2	no	—
47	C₂×D₁₀₁	C₂²×C₄	1	4	yes	—
48	C₄×C₁₀₁	—	2	4	no	—
49	C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
50	C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
51	Dic₁₀₁	—	2	2	no	—
52	Dic₁₀₁	—	2	2	no	—
53	Dic₁₀₁	C₂²×C₄	1	4	yes	—
54	C₂×C₁₀₁	—	2	2	no	—
55	C₂×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
56	C₂×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₀₁	—	2	4	no	—
59	C₁₀₁	#520	1	1	yes	characteristic, complemented, Sylow
60	#520	—	101	1	no	maximal, Sylow
61	C₂×#41	—	101	1	no	—
62	C₄×Q₈	—	101	2	no	—
63	C₄×D₄	—	101	4	no	—
64	C₄×D₄	—	101	2	no	—
65	#117	—	101	1	no	—
66	#117	—	101	2	no	—
67	C₂×C₄⋊C₄	—	101	2	no	—
68	C₂×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
69	#41	—	101	8	no	—
70	C₂×Q₈	—	101	1	no	—
71	C₂×D₄	—	101	2	no	—
72	C₂×D₄	—	101	1	no	—
73	C₂²×C₄	—	101	2	no	—
74	C₂²×C₄	—	101	4	no	—
75	C₂²×C₄	C₂×D₁₀₁	1	2	yes	—
76	C₂²×C₄	C₂×D₁₀₁	1	1	yes	characteristic
77	C₄⋊C₄	—	101	4	no	—
78	C₄⋊C₄	—	101	1	no	—
79	C₄⋊C₄	—	101	1	no	—
80	C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
81	C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
82	C₂²⋊C₄	—	101	2	no	—
83	C₂²⋊C₄	—	101	4	no	—
84	C₄²	—	101	4	no	—
85	C₄²	—	101	2	no	—
86	C₂³	—	101	2	no	—
87	C₂³	C₂²×D₁₀₁	1	1	yes	characteristic
88	Q₈	—	101	4	no	—
89	D₄	—	101	8	no	—
90	D₄	—	101	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	101	2	no	—
93	C₂×C₄	—	101	8	no	—
94	C₂×C₄	—	101	2	no	—
95	C₂×C₄	—	101	2	no	—
96	C₂×C₄	—	404	4	no	—
97	C₂×C₄	C₄×D₁₀₁	1	4	yes	complemented
98	C₂×C₄	C₂²×D₁₀₁	1	4	yes	—
99	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	101	4	no	—
104	C₂²	—	202	2	no	—
105	C₂²	—	202	2	no	—
106	C₂²	C₂×C₄×D₁₀₁	1	2	yes	—
107	C₂²	C₂³×D₁₀₁	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	101	4	no	—
110	C₄	—	202	2	no	—
111	C₄	—	202	2	no	—
112	C₄	C₂×C₄×D₁₀₁	1	2	yes	—
113	C₄	C₂×C₄×D₁₀₁	1	2	yes	—
114	C₂	—	2	2	no	—
115	C₂	—	202	4	no	—
116	C₂	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
117	C₂	#15692300	1	1	yes	characteristic
118	C₂	#15692303	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13108074

C₂

yes

maximal, characteristic

#15692220

C₂

yes

maximal, complemented

#15692217

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₀₁

C₂

yes

maximal, complemented

#15692214

C₂

yes

maximal, characteristic, complemented

#15692213

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₀₁

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₁₀₁

C₂

yes

maximal, characteristic, complemented

#13108074

C₄

yes

complemented

C₂×Dic₂₀₂

C₂²

yes

characteristic

C₂×D₄₀₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₁

C₂²

yes

—

C₂×C₄×D₁₀₁

C₂²

yes

—

C₂×C₄×D₁₀₁

C₂²

yes

—

C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₄⋊Dic₁₀₁

C₂²

yes

characteristic

#13108053

C₂²

yes

—

C₂²⋊Dic₁₀₁

C₂²

yes

—

#13108051

C₂²

yes

—

C₄×Dic₁₀₁

C₂²

yes

—

C₄×Dic₁₀₁

C₂²

yes

—

C₂²×C₄×C₁₀₁

C₂²

yes

—

C₂²×C₄×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂³×C₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊D₁₀₁

C₂×C₄

yes

complemented

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

D₄₀₄

C₂×C₄

yes

complemented

C₄×D₁₀₁

—

C₄×D₁₀₁

C₂×C₄

yes

complemented

Dic₂₀₂

C₂×C₄

yes

complemented

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²×C₄

yes

—

C₂²×C₁₀₁

C₂⁴

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

C₂²×C₄

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²×C₄

yes

—

C₄×C₁₀₁

C₂²×C₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

C₂²×C₄

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂³×C₄

yes

characteristic

C₂×C₁₀₁

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₁

2₋¹⁺⁴

yes

characteristic

D₁₀₁

—

C₁₀₁

#520

yes

characteristic, complemented, Sylow

#520

—

101

maximal, Sylow

C₂×#41

—

101

—

C₄×Q₈

—

101

—

C₄×D₄

—

101

—

C₄×D₄

—

101

—

#117

—

101

—

#117

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

D₁₀₁

yes

characteristic, complemented

#41

—

101

—

C₂×Q₈

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×D₁₀₁

yes

—

C₂²×C₄

C₂×D₁₀₁

yes

characteristic

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

C₂×D₁₀₁

yes

—

C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₄²

—

101

—

C₄²

—

101

—

C₂³

—

101

—

C₂³

C₂²×D₁₀₁

yes

characteristic

Q₈

—

101

—

D₄

—

101

—

D₄

—

101

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

C₄×D₁₀₁

yes

complemented

C₂×C₄

C₂²×D₁₀₁

yes

—

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

100

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

101

—

104

C₂²

—

202

—

105

C₂²

—

202

—

106

C₂²

C₂×C₄×D₁₀₁

yes

—

107

C₂²

C₂³×D₁₀₁

yes

characteristic

108

C₄

—

109

C₄

—

101

—

110

C₄

—

202

—

111

C₄

—

202

—

112

C₄

C₂×C₄×D₁₀₁

yes

—

113

C₄

C₂×C₄×D₁₀₁

yes

—

114

C₂

—

115

C₂

—

202

—

116

C₂

C₂²×C₄×D₁₀₁

yes

characteristic

117

C₂

#15692300

yes

characteristic

118

C₂

#15692303

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₁
commutator	C₂×C₁₀₁	C₂³×C₄
Fitting	C₂×C₄⋊C₄×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂⁴
hypercenter	C₂×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#520
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₁

commutator

C₂×C₁₀₁

C₂³×C₄

Fitting

C₂×C₄⋊C₄×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂⁴

hypercenter

C₂×C₄⋊C₄

D₁₀₁

C₁₀₁

G ▷ C₂×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. #15692220	C₂	semidirect
3. #15692217	C₂	semidirect
4. C₄×C₂²⋊D₁₀₁	C₂	semidirect
5. #15692214	C₂	semidirect
6. #15692213	C₂	semidirect
7. C₄⋊C₄×D₁₀₁	C₂	semidirect
8. C₂×C₄⋊C₄×C₁₀₁	C₂	semidirect
9. #13108074	C₄	semidirect
29. C₂×C₄×C₁₀₁	C₂×C₄	semidirect
33. C₂²⋊D₁₀₁	C₂×C₄	semidirect
37. D₄₀₄	C₂×C₄	semidirect
39. C₄×D₁₀₁	C₂×C₄	semidirect
40. Dic₂₀₂	C₂×C₄	semidirect
59. C₁₀₁	#520	semidirect
68. C₂×C₄⋊C₄	D₁₀₁	semidirect
97. C₂×C₄	C₄×D₁₀₁	semidirect

H ≅ G/N

type

2. #15692220

C₂

semidirect

3. #15692217

C₂

semidirect

4. C₄×C₂²⋊D₁₀₁

C₂

semidirect

5. #15692214

C₂

semidirect

6. #15692213

C₂

semidirect

7. C₄⋊C₄×D₁₀₁

C₂

semidirect

8. C₂×C₄⋊C₄×C₁₀₁

C₂

semidirect

9. #13108074

C₄

semidirect

29. C₂×C₄×C₁₀₁

C₂×C₄

semidirect

33. C₂²⋊D₁₀₁

C₂×C₄

semidirect

37. D₄₀₄

C₂×C₄

semidirect

39. C₄×D₁₀₁

C₂×C₄

semidirect

40. Dic₂₀₂

C₂×C₄

semidirect

59. C₁₀₁

#520

semidirect

68. C₂×C₄⋊C₄

D₁₀₁

semidirect

97. C₂×C₄

C₄×D₁₀₁

semidirect

Subgroups — #20265658

Conjugacy classes of subgroups

Complements