Group #21425650 — subgroups

Subgroups — #21425650

Order 7552.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16837451	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837429	C₂	1	2	yes	maximal, complemented
3	#16837420	C₂	1	2	yes	maximal
4	#16837370	C₂	1	1	yes	maximal, characteristic
5	C₅₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745685	C₂²	1	1	yes	characteristic
7	C₂×#11745664	C₂²	1	2	yes	—
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	#14273663	—	2	2	no	—
10	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
11	#14273655	—	2	4	no	—
12	(C₂×D₅₉)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	#117×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₅₉	—	2	1	no	—
17	#11745685	—	2	2	no	—
18	#11745685	—	2	2	no	—
19	C₂×D₂₃₆	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₅₉	—	2	1	no	—
21	C₂×Dic₁₁₈	Q₈	1	1	yes	characteristic
22	M₁₆×C₅₉	—	2	2	no	—
23	C₂×C₈×C₅₉	D₄	1	2	yes	—
24	C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₂²⋊C₄×C₅₉	—	2	2	no	—
26	C₄²×C₅₉	D₄	1	4	yes	—
27	#11745664	—	2	2	no	—
28	#11745664	—	2	2	no	—
29	C₂×C₅₉⋊C₈	—	2	1	no	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂³×C₅₉	#41	1	1	yes	characteristic
33	C₂²×D₅₉	—	2	1	no	—
34	C₂×C₄×C₅₉	—	2	4	no	—
35	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₉	#41	1	2	yes	—
39	C₂²⋊D₅₉	—	4	2	no	—
40	C₂×Dic₅₉	—	2	1	no	—
41	D₂₃₆	—	2	2	no	—
42	C₄×D₅₉	—	4	1	no	—
43	C₄×D₅₉	—	4	1	no	—
44	Dic₁₁₈	—	2	2	no	—
45	C₈×C₅₉	—	2	2	no	—
46	C₅₉⋊C₈	—	4	1	no	—
47	C₅₉⋊C₈	—	4	2	no	—
48	C₂²×C₅₉	—	4	1	no	—
49	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₅₉	#123	1	2	yes	—
51	C₂×D₅₉	—	4	2	no	—
52	C₂×D₅₉	—	4	1	no	—
53	C₄×C₅₉	—	4	4	no	—
54	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₅₉	#123	1	1	yes	characteristic
56	C₄×C₅₉	#127	1	1	yes	characteristic
57	Dic₅₉	—	4	1	no	—
58	#2098	—	59	1	no	maximal, Sylow
59	C₂×C₅₉	—	2	2	no	—
60	C₂×C₅₉	—	2	1	no	—
61	C₂×C₅₉	#399	1	1	yes	characteristic
62	D₅₉	—	8	1	no	—
63	#430	—	59	1	no	—
64	#424	—	59	2	no	—
65	#421	—	59	2	no	—
66	#408	—	59	1	no	—
67	H(3,ℤ/4)	D₅₉	1	1	yes	characteristic, complemented
68	C₅₉	#2098	1	1	yes	characteristic, complemented, Sylow
69	C₂×#41	—	59	1	no	—
70	C₂×M₁₆	—	59	2	no	—
71	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
72	C₂²×C₈	—	59	1	no	—
73	#117	C₂×D₅₉	1	2	yes	—
74	#108	—	118	2	no	—
75	C₄⋊C₈	—	118	4	no	—
76	#104	—	118	4	no	—
77	C₂²⋊C₈	—	118	2	no	—
78	#41	—	118	2	no	—
79	#41	—	118	2	no	—
80	C₂×Q₈	—	59	1	no	—
81	C₂×D₄	—	59	1	no	—
82	C₂×D₄	—	118	1	no	—
83	C₂²×C₄	—	118	1	no	—
84	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	118	2	no	—
87	M₁₆	—	118	2	no	—
88	C₂×C₈	—	118	1	no	—
89	C₂×C₈	—	118	2	no	—
90	C₂×C₈	—	118	2	no	—
91	C₂×C₈	D₂₃₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₂₃₆	1	4	yes	—
95	C₂³	—	118	1	no	—
96	C₂³	#11745687	1	1	yes	characteristic
97	Q₈	—	118	2	no	—
98	D₄	—	118	2	no	—
99	D₄	—	236	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	118	1	no	—
102	C₂×C₄	—	236	1	no	—
103	C₂×C₄	—	236	1	no	—
104	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
105	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
106	C₂×C₄	#11745687	1	2	yes	—
107	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	236	1	no	—
110	C₈	—	236	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	236	2	no	—
113	C₂²	—	236	1	no	—
114	C₂²	#14273707	1	1	yes	characteristic
115	C₂²	#14273711	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	236	1	no	—
118	C₄	#14273707	1	2	yes	—
119	C₄	#14273711	1	1	yes	characteristic
120	C₄	#14273724	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	472	1	no	—
124	C₂	#16837339	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837451

C₂

yes

maximal, characteristic, complemented

#16837429

C₂

yes

maximal, complemented

#16837420

C₂

yes

maximal

#16837370

C₂

yes

maximal, characteristic

C₅₉×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

#14273655

—

(C₂×D₅₉)⋊C₈

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

Q₈

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

Q₈

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

D₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

#123

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2098

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#399

yes

characteristic

D₅₉

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#2098

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

#117

C₂×D₅₉

yes

—

#108

—

118

—

C₄⋊C₈

—

118

—

#104

—

118

—

C₂²⋊C₈

—

118

—

#41

—

118

—

#41

—

118

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

D₂₃₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₂₃₆

yes

—

C₂³

—

118

—

C₂³

#11745687

yes

characteristic

Q₈

—

118

—

D₄

—

118

—

D₄

—

236

—

100

C₂×C₄

—

101

C₂×C₄

—

118

—

102

C₂×C₄

—

236

—

103

C₂×C₄

—

236

—

104

C₂×C₄

C₂×D₂₃₆

yes

—

105

C₂×C₄

C₂×D₂₃₆

yes

characteristic

106

C₂×C₄

#11745687

yes

—

107

C₂×C₄

Q₈×D₅₉

yes

characteristic

108

C₈

—

109

C₈

—

236

—

110

C₈

—

236

—

111

C₂²

—

112

C₂²

—

236

—

113

C₂²

—

236

—

114

C₂²

#14273707

yes

characteristic

115

C₂²

#14273711

yes

—

116

C₄

—

117

C₄

—

236

—

118

C₄

#14273707

yes

—

119

C₄

#14273711

yes

characteristic

120

C₄

#14273724

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

472

—

124

C₂

#16837339

yes

characteristic

	subgroup	G/H
center	C₄	#14273724
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×H(3,ℤ/4)	C₂
socle	C₂×C₅₉	#399
hypercenter	H(3,ℤ/4)	D₅₉
hypocenter	C₅₉	#2098
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#14273724

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×H(3,ℤ/4)

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #16837451	C₂	semidirect
2. #16837429	C₂	semidirect
5. C₅₉×H(3,ℤ/4)	C₂	semidirect
67. H(3,ℤ/4)	D₅₉	semidirect
68. C₅₉	#2098	semidirect

H ≅ G/N

type

1. #16837451

C₂

semidirect

2. #16837429

C₂

semidirect

5. C₅₉×H(3,ℤ/4)

C₂

semidirect

67. H(3,ℤ/4)

D₅₉

semidirect

68. C₅₉

#2098

semidirect

Subgroups — #21425650

Conjugacy classes of subgroups

Complements