Group #20265830 — subgroups

Subgroups — #20265830

Order 6464.

292 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 107 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×#13108075	C₂	1	2	yes	maximal, complemented
2	#15692240	C₂	1	4	yes	maximal, complemented
3	#15692231	C₂	1	2	yes	maximal, complemented
4	#15692230	C₂	1	2	yes	maximal, complemented
5	C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×D₁₀₁	C₂	1	2	yes	maximal, complemented
8	C₄⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#13108075	—	2	8	no	—
10	C₂×D₄₀₄	C₂²	1	4	yes	—
11	C₂×D₄₀₄	C₂²	1	1	yes	characteristic
12	C₂×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₀₁	C₂²	1	4	yes	—
14	C₂×C₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄×D₁₀₁	C₂²	1	2	yes	—
16	C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	#13108053	C₂²	1	4	yes	—
19	#13108051	C₂²	1	2	yes	—
20	#13108051	C₂²	1	2	yes	—
21	#13108051	C₂²	1	4	yes	—
22	C₂×Q₈×C₁₀₁	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
25	C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂²×D₁₀₁	C₂³	1	2	yes	—
27	C₂²×D₁₀₁	C₂³	1	2	yes	—
28	Q₈×C₁₀₁	—	2	4	no	—
29	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
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	C₂×Dic₁₀₁	C₂³	1	2	yes	—
34	C₂×Dic₁₀₁	C₂³	1	2	yes	—
35	D₄₀₄	—	2	8	no	—
36	D₄₀₄	—	2	2	no	—
37	D₄₀₄	D₄	1	4	yes	complemented
38	C₄×D₁₀₁	—	2	8	no	—
39	C₄×D₁₀₁	—	2	4	no	—
40	C₄×D₁₀₁	—	2	4	no	—
41	C₂²×C₁₀₁	C₂⁴	1	1	yes	characteristic
42	C₂×D₁₀₁	—	2	2	no	—
43	C₂×D₁₀₁	—	2	2	no	—
44	C₂×D₁₀₁	—	2	2	no	—
45	C₂×D₁₀₁	—	2	2	no	—
46	C₂×D₁₀₁	—	2	2	no	—
47	C₂×D₁₀₁	C₂×D₄	1	4	yes	—
48	C₄×C₁₀₁	—	2	4	no	—
49	C₄×C₁₀₁	—	2	1	no	—
50	C₄×C₁₀₁	C₂×D₄	1	2	yes	—
51	C₄×C₁₀₁	#41	1	2	yes	complemented
52	Dic₁₀₁	—	2	2	no	—
53	Dic₁₀₁	—	2	2	no	—
54	C₂×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₁₀₁	C₂×#41	1	1	yes	characteristic
56	C₂×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
57	D₁₀₁	—	2	4	no	—
58	D₁₀₁	—	4	2	no	—
59	C₁₀₁	#547	1	1	yes	characteristic, complemented, Sylow
60	#547	—	101	1	no	maximal, Sylow
61	C₂×#41	—	101	2	no	—
62	C₄⋊Q₈	D₁₀₁	1	1	yes	characteristic, complemented
63	#123	—	101	4	no	—
64	C₂²⋊Q₈	—	101	2	no	—
65	#121	—	101	2	no	—
66	C₄×D₄	—	101	1	no	—
67	C₄×D₄	—	101	1	no	—
68	C₂×C₄⋊C₄	—	101	2	no	—
69	#41	—	202	8	no	—
70	C₂×Q₈	C₂×D₁₀₁	1	2	yes	—
71	C₂×D₄	—	101	1	no	—
72	C₂×D₄	—	101	1	no	—
73	C₂×D₄	—	101	4	no	—
74	C₂²×C₄	—	101	2	no	—
75	C₂²×C₄	—	101	4	no	—
76	C₂²×C₄	—	101	2	no	—
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	2	no	—
84	C₂²⋊C₄	—	101	4	no	—
85	C₄²	C₂×D₁₀₁	1	1	yes	characteristic
86	C₂³	—	101	2	no	—
87	C₂³	—	101	2	no	—
88	Q₈	—	2	4	no	—
89	D₄	—	101	4	no	—
90	D₄	—	202	8	no	—
91	D₄	—	202	2	no	—
92	C₂×C₄	—	101	2	no	—
93	C₂×C₄	—	101	2	no	—
94	C₂×C₄	—	202	8	no	—
95	C₂×C₄	—	202	4	no	—
96	C₂×C₄	—	202	4	no	—
97	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
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	—	101	4	no	—
102	C₂²	—	202	2	no	—
103	C₂²	—	202	2	no	—
104	C₂²	—	202	2	no	—
105	C₂²	—	202	2	no	—
106	C₂²	—	202	2	no	—
107	C₂²	C₂³×D₁₀₁	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	2	1	no	—
110	C₄	—	202	2	no	—
111	C₄	—	202	2	no	—
112	C₄	D₄×D₁₀₁	1	2	yes	—
113	C₄	#13108075	1	2	yes	complemented
114	C₂	—	404	2	no	—
115	C₂	—	202	4	no	—
116	C₂	C₂×D₄×D₁₀₁	1	1	yes	characteristic
117	C₂	C₂×#13108075	1	1	yes	characteristic
118	C₂	#15692303	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13108075

C₂

yes

maximal, complemented

#15692240

C₂

yes

maximal, complemented

#15692231

C₂

yes

maximal, complemented

#15692230

C₂

yes

maximal, complemented

C₄×D₄₀₄

C₂

yes

maximal, characteristic, complemented

C₄×D₄₀₄

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₀₁

C₂

yes

maximal, complemented

C₄⋊Q₈×C₁₀₁

C₂

yes

maximal, characteristic, complemented

#13108075

—

C₂×D₄₀₄

C₂²

yes

—

C₂×D₄₀₄

C₂²

yes

characteristic

C₂×D₄₀₄

C₂²

yes

characteristic

C₂×C₄×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

—

#13108051

C₂²

yes

—

#13108051

C₂²

yes

—

#13108051

C₂²

yes

—

C₂×Q₈×C₁₀₁

C₂²

yes

—

C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×D₁₀₁

C₂³

yes

—

C₂²×D₁₀₁

C₂³

yes

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

D₄

yes

complemented

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₂²×C₁₀₁

C₂⁴

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

C₂×D₄

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×D₄

yes

—

C₄×C₁₀₁

#41

yes

complemented

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

C₂²×D₄

yes

characteristic

C₂×C₁₀₁

C₂×#41

yes

characteristic

C₂×C₁₀₁

2₋¹⁺⁴

yes

characteristic

D₁₀₁

—

D₁₀₁

—

C₁₀₁

#547

yes

characteristic, complemented, Sylow

#547

—

101

maximal, Sylow

C₂×#41

—

101

—

C₄⋊Q₈

D₁₀₁

yes

characteristic, complemented

#123

—

101

—

C₂²⋊Q₈

—

101

—

#121

—

101

—

C₄×D₄

—

101

—

C₄×D₄

—

101

—

C₂×C₄⋊C₄

—

101

—

#41

—

202

—

C₂×Q₈

C₂×D₁₀₁

yes

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

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

—

C₂²⋊C₄

—

101

—

C₄²

C₂×D₁₀₁

yes

characteristic

C₂³

—

101

—

C₂³

—

101

—

Q₈

—

D₄

—

101

—

D₄

—

202

—

D₄

—

202

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₄

C₂²×D₁₀₁

yes

—

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

100

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

101

C₂²

—

101

—

102

C₂²

—

202

—

103

C₂²

—

202

—

104

C₂²

—

202

—

105

C₂²

—

202

—

106

C₂²

—

202

—

107

C₂²

C₂³×D₁₀₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

202

—

111

C₄

—

202

—

112

C₄

D₄×D₁₀₁

yes

—

113

C₄

#13108075

yes

complemented

114

C₂

—

404

—

115

C₂

—

202

—

116

C₂

C₂×D₄×D₁₀₁

yes

characteristic

117

C₂

C₂×#13108075

yes

characteristic

118

C₂

#15692303

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂⁴
Fitting	C₄⋊Q₈×C₁₀₁	C₂
hypercenter	C₄⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	#547
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂⁴

Fitting

C₄⋊Q₈×C₁₀₁

C₂

hypercenter

C₄⋊Q₈

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#13108075	C₂	semidirect
2. #15692240	C₂	semidirect
3. #15692231	C₂	semidirect
4. #15692230	C₂	semidirect
5. C₄×D₄₀₄	C₂	semidirect
6. C₄×D₄₀₄	C₂	semidirect
7. C₄⋊C₄×D₁₀₁	C₂	semidirect
8. C₄⋊Q₈×C₁₀₁	C₂	semidirect
37. D₄₀₄	D₄	semidirect
51. C₄×C₁₀₁	#41	semidirect
59. C₁₀₁	#547	semidirect
62. C₄⋊Q₈	D₁₀₁	semidirect
113. C₄	#13108075	semidirect

H ≅ G/N

type

1. C₂×#13108075

C₂

semidirect

2. #15692240

C₂

semidirect

3. #15692231

C₂

semidirect

4. #15692230

C₂

semidirect

5. C₄×D₄₀₄

C₂

semidirect

6. C₄×D₄₀₄

C₂

semidirect

7. C₄⋊C₄×D₁₀₁

C₂

semidirect

8. C₄⋊Q₈×C₁₀₁

C₂

semidirect

37. D₄₀₄

D₄

semidirect

51. C₄×C₁₀₁

#41

semidirect

59. C₁₀₁

#547

semidirect

62. C₄⋊Q₈

D₁₀₁

semidirect

113. C₄

#13108075

semidirect

Subgroups — #20265830

Conjugacy classes of subgroups

Complements