Group #21432699 — subgroups

Subgroups — C₂²×#14273663 #21432699

Order 7552.

480 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 309 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×#11745664	C₂	1	2	yes	maximal
2	C₂×#14273663	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745664	—	2	6	no	—
5	C₂×#11745664	C₂²	1	12	yes	—
6	C₂²×C₅₉⋊C₈	—	2	2	no	—
7	#14273663	C₂²	1	16	yes	direct product factor
8	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₅₉	C₄	1	6	yes	—
10	C₂²×C₈×C₅₉	C₄	1	2	yes	—
11	C₂²×C₈×C₅₉	C₂²	1	6	yes	—
12	C₂⁴×C₅₉	Q₈	1	1	yes	characteristic
13	C₂²×C₄×C₅₉	C₂×C₄	1	3	yes	—
14	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	D₄	1	3	yes	—
16	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	Q₈	1	3	yes	—
18	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
19	C₂×C₈×C₅₉	C₂×C₄	1	12	yes	—
20	C₂×C₈×C₅₉	C₂×C₄	1	12	yes	—
21	C₂×C₈×C₅₉	C₂³	1	4	yes	—
22	#11745664	—	2	12	no	—
23	#11745664	C₂³	1	8	yes	—
24	C₂×C₅₉⋊C₈	—	2	12	no	—
25	C₂³×C₅₉	—	2	4	no	—
26	C₂³×C₅₉	C₄⋊C₄	1	3	yes	—
27	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂³×C₅₉	C₂×Q₈	1	3	yes	—
29	C₂×C₄×C₅₉	C₄⋊C₄	1	6	yes	—
30	C₂×C₄×C₅₉	C₄⋊C₄	1	3	yes	—
31	C₂×C₄×C₅₉	C₄⋊C₄	1	3	yes	—
32	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
33	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
34	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
35	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
36	C₂×C₄×C₅₉	C₂×Q₈	1	3	yes	—
37	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
38	C₈×C₅₉	C₂²×C₄	1	8	yes	—
39	C₅₉⋊C₈	—	2	8	no	—
40	C₂²×C₅₉	—	2	6	no	—
41	C₂²×C₅₉	—	2	6	no	—
42	C₂²×C₅₉	#108	1	4	yes	complemented
43	C₂²×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
44	C₂²×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
45	C₂²×C₅₉	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
47	C₄×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
48	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
49	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
50	C₂²×#108	—	59	1	no	maximal, Sylow
51	C₂×C₅₉	—	2	3	no	—
52	C₂×C₅₉	—	2	1	no	—
53	C₂×C₅₉	C₂×#108	1	6	yes	complemented
54	C₂×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×M₁₆	—	59	2	no	—
56	C₂³×C₈	D₅₉	1	1	yes	characteristic
57	C₂×#108	—	59	12	no	—
58	C₅₉	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
59	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
60	C₂×M₁₆	—	59	12	no	—
61	C₂×M₁₆	—	118	6	no	—
62	C₂²×C₈	—	118	2	no	—
63	C₂²×C₈	Dic₅₉	1	2	yes	—
64	C₂²×C₈	Dic₅₉	1	6	yes	—
65	C₂²×C₈	C₂×D₅₉	1	6	yes	—
66	#108	—	59	16	no	—
67	C₂⁴	Dic₁₁₈	1	1	yes	characteristic
68	C₂²×C₄	Dic₁₁₈	1	3	yes	—
69	C₂²×C₄	D₂₃₆	1	3	yes	—
70	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
71	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
72	C₂²×C₄	C₂×Dic₅₉	1	3	yes	—
73	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
74	M₁₆	—	59	8	no	—
75	M₁₆	—	118	12	no	—
76	C₂×C₈	—	118	12	no	—
77	C₂×C₈	C₂×Dic₅₉	1	12	yes	—
78	C₂×C₈	C₂×Dic₅₉	1	12	yes	—
79	C₂×C₈	C₂²×D₅₉	1	4	yes	—
80	C₂³	—	2	4	no	—
81	C₂³	C₄⋊Dic₅₉	1	3	yes	—
82	C₂³	C₄⋊Dic₅₉	1	1	yes	characteristic
83	C₂³	C₂×Dic₁₁₈	1	3	yes	—
84	C₂×C₄	C₄⋊Dic₅₉	1	6	yes	—
85	C₂×C₄	C₄⋊Dic₅₉	1	3	yes	—
86	C₂×C₄	C₄⋊Dic₅₉	1	3	yes	—
87	C₂×C₄	C₂×Dic₁₁₈	1	3	yes	—
88	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
89	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
90	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
91	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
92	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
93	C₈	—	118	8	no	—
94	C₈	C₂²×Dic₅₉	1	8	yes	—
95	C₂²	—	2	6	no	—
96	C₂²	—	2	6	no	—
97	C₂²	#14273663	1	4	yes	direct product factor
98	C₂²	C₂×C₄⋊Dic₅₉	1	3	yes	—
99	C₂²	C₂×C₄⋊Dic₅₉	1	3	yes	—
100	C₂²	C₂²×Dic₁₁₈	1	1	yes	characteristic
101	C₄	C₂×C₄⋊Dic₅₉	1	3	yes	—
102	C₄	C₂×C₄⋊Dic₅₉	1	3	yes	—
103	C₄	C₂³×Dic₅₉	1	1	yes	characteristic
104	C₄	C₂²×D₂₃₆	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	2	1	no	—
107	C₂	C₂×#14273663	1	6	yes	direct product factor
108	C₂	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745664

C₂

yes

maximal

C₂×#14273663

C₂

yes

maximal, direct product factor

C₂³×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

—

#14273663

C₂²

yes

direct product factor

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₄

yes

—

C₂²×C₈×C₅₉

C₄

yes

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂⁴×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

#11745664

—

#11745664

C₂³

yes

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂³×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₈×C₅₉

C₂²×C₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#108

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

—

C₂²×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₄×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

C₂²×D₄

yes

characteristic

C₂²×#108

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×#108

yes

complemented

C₂×C₅₉

C₂²×C₄⋊C₄

yes

characteristic

C₂²×M₁₆

—

C₂³×C₈

D₅₉

yes

characteristic

C₂×#108

—

C₅₉

C₂²×#108

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂²×C₈

—

118

—

C₂²×C₈

Dic₅₉

yes

—

C₂²×C₈

Dic₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

—

#108

—

C₂⁴

Dic₁₁₈

yes

characteristic

C₂²×C₄

Dic₁₁₈

yes

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

—

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂²×D₅₉

yes

—

C₂³

—

C₂³

C₄⋊Dic₅₉

yes

—

C₂³

C₄⋊Dic₅₉

yes

characteristic

C₂³

C₂×Dic₁₁₈

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₂×Dic₁₁₈

yes

—

C₂×C₄

C₂×D₂₃₆

yes

—

C₂×C₄

C₂×D₂₃₆

yes

—

C₂×C₄

C₂²×Dic₅₉

yes

—

C₂×C₄

C₂²×Dic₅₉

yes

—

C₂×C₄

C₂³×D₅₉

yes

characteristic

C₈

—

118

—

C₈

C₂²×Dic₅₉

yes

—

C₂²

—

C₂²

—

C₂²

#14273663

yes

direct product factor

C₂²

C₂×C₄⋊Dic₅₉

yes

—

C₂²

C₂×C₄⋊Dic₅₉

yes

—

100

C₂²

C₂²×Dic₁₁₈

yes

characteristic

101

C₄

C₂×C₄⋊Dic₅₉

yes

—

102

C₄

C₂×C₄⋊Dic₅₉

yes

—

103

C₄

C₂³×Dic₅₉

yes

characteristic

104

C₄

C₂²×D₂₃₆

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

C₂×#14273663

yes

direct product factor

108

C₂

C₂²×C₄⋊Dic₅₉

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₂₃₆
commutator	C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂³×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₄⋊C₄
hypercenter	C₂³×C₈	D₅₉
hypocenter	C₅₉	C₂²×#108
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₂₃₆

commutator

C₄×C₅₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂³×C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₄⋊C₄

hypercenter

C₂³×C₈

D₅₉

hypocenter

C₅₉

C₂²×#108

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. C₂×#14273663	C₂	direct (N × H)
7. #14273663	C₂²	direct (N × H)
42. C₂²×C₅₉	#108	semidirect
53. C₂×C₅₉	C₂×#108	semidirect
58. C₅₉	C₂²×#108	semidirect
97. C₂²	#14273663	direct (N × H)
107. C₂	C₂×#14273663	direct (N × H)

H ≅ G/N

type

2. C₂×#14273663

C₂

direct (N × H)

7. #14273663

C₂²

direct (N × H)

42. C₂²×C₅₉

#108

semidirect

53. C₂×C₅₉

C₂×#108

semidirect

58. C₅₉

C₂²×#108

semidirect

97. C₂²

#14273663

direct (N × H)

107. C₂

C₂×#14273663

direct (N × H)

Subgroups — C₂²×#14273663 #21432699

Conjugacy classes of subgroups

Complements

Subgroups — C22×#14273663 #21432699

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14273663 #21432699