Group #21435550 — subgroups

Subgroups — #21435550

Order 7552.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16838014	C₂	1	2	yes	maximal
2	#16837710	C₂	1	4	yes	maximal, complemented
3	#16837660	C₂	1	2	yes	maximal
4	#16837643	C₂	1	2	yes	maximal
5	#16837613	C₂	1	4	yes	maximal, complemented
6	C₅₉×#424	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₂₃₆	C₂²	1	2	yes	—
8	C₂×Dic₂₃₆	C₂²	1	2	yes	—
9	#14273727	C₂²	1	2	yes	—
10	#14273720	—	2	4	no	—
11	#14273719	C₂²	1	2	yes	—
12	#14273717	—	2	4	no	—
13	#14273706	C₂²	1	4	yes	—
14	C₄×Dic₁₁₈	C₂²	1	4	yes	—
15	#14273661	C₂²	1	2	yes	—
16	#14273659	C₂²	1	2	yes	—
17	#14273652	C₂²	1	4	yes	—
18	#14273652	C₂²	1	4	yes	—
19	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	#117×C₅₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
23	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₁₈	C₂³	1	4	yes	—
25	M₁₆×C₅₉	—	4	1	no	—
26	C₂×C₈×C₅₉	—	4	1	no	—
27	C₂×C₈×C₅₉	C₂³	1	2	yes	—
28	C₂×C₈×C₅₉	C₂³	1	2	yes	—
29	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
30	C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
31	C₄²×C₅₉	C₂³	1	2	yes	—
32	C₂²⋊Dic₅₉	—	2	4	no	—
33	C₄⋊Dic₅₉	—	2	4	no	—
34	C₄⋊Dic₅₉	C₂³	1	4	yes	—
35	Dic₅₉⋊C₄	—	2	4	no	—
36	Dic₅₉⋊C₄	—	2	4	no	—
37	Dic₅₉⋊C₄	—	2	4	no	—
38	C₄×Dic₅₉	—	2	4	no	—
39	Dic₂₃₆	—	4	2	no	—
40	Dic₂₃₆	—	4	2	no	—
41	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	2	1	no	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₅₉	—	2	4	no	—
48	C₂×Dic₅₉	—	2	4	no	—
49	Dic₁₁₈	—	2	4	no	—
50	Dic₁₁₈	—	2	4	no	—
51	C₈×C₅₉	—	2	2	no	—
52	C₈×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	—	2	1	no	—
54	C₂²×C₅₉	—	2	2	no	—
55	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₅₉	—	2	1	no	—
57	C₄×C₅₉	—	2	4	no	—
58	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₅₉	—	4	4	no	—
61	Dic₅₉	—	4	4	no	—
62	#3261	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	—	4	1	no	—
64	C₂×C₅₉	#537	1	1	yes	characteristic
65	C₂×C₅₉	#578	1	2	yes	—
66	#541	—	59	2	no	—
67	#479	—	59	4	no	—
68	#470	—	59	2	no	—
69	#467	—	59	2	no	—
70	#462	—	59	4	no	—
71	#424	D₅₉	1	1	yes	characteristic
72	C₅₉	#3261	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₁₆	—	59	2	no	—
74	C₂×Q₁₆	—	59	2	no	—
75	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
77	C₄⋊Q₈	—	59	2	no	—
78	#126	—	118	4	no	—
79	#125	—	59	2	no	—
80	#125	—	118	4	no	—
81	C₂²⋊Q₈	—	59	4	no	—
82	C₄×Q₈	—	59	4	no	—
83	#117	C₂×D₅₉	1	1	yes	characteristic
84	#107	—	59	2	no	—
85	#106	—	59	2	no	—
86	C₄⋊C₈	C₂×D₅₉	1	4	yes	—
87	#103	—	59	4	no	—
88	#103	—	59	4	no	—
89	C₂×Q₈	—	59	4	no	—
90	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
91	Q₁₆	—	236	2	no	—
92	Q₁₆	—	236	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₅₉	1	2	yes	—
96	C₂×C₈	C₂²×D₅₉	1	2	yes	—
97	C₄⋊C₄	—	59	4	no	—
98	C₄⋊C₄	—	118	4	no	—
99	C₄⋊C₄	—	118	4	no	—
100	C₄⋊C₄	—	118	4	no	—
101	C₄⋊C₄	—	118	4	no	—
102	C₄⋊C₄	D₂₃₆	1	2	yes	—
103	C₂²⋊C₄	—	118	4	no	—
104	C₂²⋊C₄	D₂₃₆	1	2	yes	—
105	C₄²	—	118	4	no	—
106	C₄²	C₂²×D₅₉	1	2	yes	—
107	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
108	Q₈	—	118	4	no	—
109	Q₈	—	118	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	118	4	no	—
113	C₂×C₄	—	118	4	no	—
114	C₂×C₄	C₂×D₂₃₆	1	4	yes	—
115	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	236	4	no	—
125	C₄	—	236	4	no	—
126	C₄	#14273772	1	1	yes	characteristic
127	C₄	#14273775	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#16837983	1	1	yes	characteristic
130	C₂	#16838219	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16838014

C₂

yes

maximal

#16837710

C₂

yes

maximal, complemented

#16837660

C₂

yes

maximal

#16837643

C₂

yes

maximal

#16837613

C₂

yes

maximal, complemented

C₅₉×#424

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

—

C₂×Dic₂₃₆

C₂²

yes

—

#14273727

C₂²

yes

—

#14273720

—

#14273719

C₂²

yes

—

#14273717

—

#14273706

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273661

C₂²

yes

—

#14273659

C₂²

yes

—

#14273652

C₂²

yes

—

#14273652

C₂²

yes

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

C₂³

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#3261

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#537

yes

characteristic

C₂×C₅₉

#578

yes

—

—

—

—

—

—

D₅₉

yes

characteristic

C₅₉

#3261

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

#126

—

118

—

#125

—

#125

—

118

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₅₉

yes

characteristic

#107

—

#106

—

C₄⋊C₈

C₂×D₅₉

yes

—

#103

—

#103

—

C₂×Q₈

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

Q₁₆

—

236

—

Q₁₆

—

236

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₉

yes

—

C₂×C₈

C₂²×D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

100

C₄⋊C₄

—

118

—

101

C₄⋊C₄

—

118

—

102

C₄⋊C₄

D₂₃₆

yes

—

103

C₂²⋊C₄

—

118

—

104

C₂²⋊C₄

D₂₃₆

yes

—

105

C₄²

—

118

—

106

C₄²

C₂²×D₅₉

yes

—

107

C₂³

C₂×D₂₃₆

yes

characteristic

108

Q₈

—

118

—

109

Q₈

—

118

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

118

—

113

C₂×C₄

—

118

—

114

C₂×C₄

C₂×D₂₃₆

yes

—

115

C₂×C₄

C₂×D₂₃₆

yes

characteristic

116

C₂×C₄

C₂³×D₅₉

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₂₃₆

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

236

—

125

C₄

—

236

—

126

C₄

#14273772

yes

characteristic

127

C₄

#14273775

yes

characteristic

128

C₂

—

129

C₂

#16837983

yes

characteristic

130

C₂

#16838219

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#424	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#424	D₅₉
hypocenter	C₅₉	#3261
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#424

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #16837710	C₂	semidirect
5. #16837613	C₂	semidirect
72. C₅₉	#3261	semidirect

H ≅ G/N

type

2. #16837710

C₂

semidirect

5. #16837613

C₂

semidirect

72. C₅₉

#3261

semidirect

Subgroups — #21435550

Conjugacy classes of subgroups

Complements