Group #21435490 — subgroups

Subgroups — #21435490

Order 7552.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16837959	C₂	1	2	yes	maximal
2	#16837688	C₂	1	4	yes	maximal, complemented
3	#16837660	C₂	1	4	yes	maximal
4	#16837613	C₂	1	4	yes	maximal, complemented
5	C₅₉×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	C₂²	1	4	yes	—
7	#14273727	C₂²	1	2	yes	—
8	#14273727	C₂²	1	2	yes	—
9	#14273719	—	2	2	no	—
10	#14273706	—	2	2	no	—
11	#14273706	C₂²	1	4	yes	—
12	C₄×Dic₁₁₈	—	2	2	no	—
13	C₄×Dic₁₁₈	C₂²	1	4	yes	—
14	#14273686	—	2	2	no	—
15	#14273659	C₂²	1	4	yes	—
16	#14273652	C₂²	1	8	yes	—
17	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
20	C₂²×C₄×C₅₉	D₄	1	2	yes	—
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₁₈	—	2	2	no	—
23	C₂×Dic₁₁₈	C₂³	1	4	yes	—
24	M₁₆×C₅₉	—	4	2	no	—
25	C₂×C₈×C₅₉	C₂³	1	4	yes	—
26	C₄²×C₅₉	D₄	1	1	yes	characteristic
27	C₄²×C₅₉	D₄	1	1	yes	characteristic
28	C₄²×C₅₉	C₂³	1	2	yes	—
29	C₂²⋊Dic₅₉	—	2	4	no	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	—	2	2	no	—
32	C₄⋊Dic₅₉	—	2	2	no	—
33	C₄⋊Dic₅₉	C₂³	1	4	yes	—
34	Dic₅₉⋊C₄	—	2	4	no	—
35	Dic₅₉⋊C₄	—	2	4	no	—
36	C₄×Dic₅₉	—	2	4	no	—
37	Dic₂₃₆	—	4	4	no	—
38	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	—	2	4	no	—
40	C₂×C₄×C₅₉	—	2	1	no	—
41	C₂×C₄×C₅₉	—	2	1	no	—
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₅₉	—	2	4	no	—
47	C₂×Dic₅₉	—	2	4	no	—
48	Dic₁₁₈	—	2	4	no	—
49	Dic₁₁₈	—	2	4	no	—
50	Dic₁₁₈	—	4	4	no	—
51	C₈×C₅₉	—	2	4	no	—
52	C₂²×C₅₉	—	2	1	no	—
53	C₂²×C₅₉	—	2	2	no	—
54	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₅₉	—	2	1	no	—
56	C₄×C₅₉	—	2	2	no	—
57	C₄×C₅₉	#137	1	4	yes	—
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	#3255	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	—	4	1	no	—
64	C₂×C₅₉	#537	1	1	yes	characteristic
65	C₂×C₅₉	C₂×#137	1	2	yes	—
66	#533	—	59	2	no	—
67	#475	—	59	4	no	—
68	#470	—	59	4	no	—
69	#462	—	59	4	no	—
70	#423	D₅₉	1	1	yes	characteristic
71	C₅₉	#3255	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₁₆	—	59	4	no	—
73	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
74	C₄⋊Q₈	—	59	2	no	—
75	C₄⋊Q₈	—	59	2	no	—
76	#125	—	118	2	no	—
77	C₂²⋊Q₈	—	59	4	no	—
78	C₂²⋊Q₈	—	118	2	no	—
79	C₄×Q₈	—	59	4	no	—
80	C₄×Q₈	—	118	2	no	—
81	#117	—	118	2	no	—
82	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
83	#106	—	59	4	no	—
84	C₄⋊C₈	C₂×D₅₉	1	4	yes	—
85	#103	—	59	8	no	—
86	C₂×Q₈	—	59	4	no	—
87	C₂×Q₈	—	118	2	no	—
88	C₂²×C₄	D₂₃₆	1	2	yes	—
89	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
90	Q₁₆	—	236	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₅₉	1	4	yes	—
93	C₄⋊C₄	—	59	4	no	—
94	C₄⋊C₄	—	118	2	no	—
95	C₄⋊C₄	—	118	2	no	—
96	C₄⋊C₄	—	118	4	no	—
97	C₄⋊C₄	—	118	4	no	—
98	C₄⋊C₄	—	118	2	no	—
99	C₂²⋊C₄	—	118	4	no	—
100	C₄²	—	118	4	no	—
101	C₄²	D₂₃₆	1	1	yes	characteristic
102	C₄²	D₂₃₆	1	1	yes	characteristic
103	C₄²	C₂²×D₅₉	1	2	yes	—
104	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
105	Q₈	—	118	4	no	—
106	Q₈	—	118	4	no	—
107	Q₈	—	236	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	118	4	no	—
112	C₂×C₄	—	118	4	no	—
113	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
114	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
115	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
116	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	236	4	no	—
124	C₄	—	236	4	no	—
125	C₄	#14273760	1	4	yes	—
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	C₂×#14273760	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal

#16837688

C₂

yes

maximal, complemented

#16837660

C₂

yes

maximal

#16837613

C₂

yes

maximal, complemented

C₅₉×#423

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

—

#14273727

C₂²

yes

—

#14273727

C₂²

yes

—

#14273719

—

#14273706

—

#14273706

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273686

—

#14273659

C₂²

yes

—

#14273652

C₂²

yes

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#137

yes

—

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#3255

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#537

yes

characteristic

C₂×C₅₉

C₂×#137

yes

—

—

—

—

—

D₅₉

yes

characteristic

C₅₉

#3255

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

118

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

#117

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#106

—

C₄⋊C₈

C₂×D₅₉

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

Q₁₆

—

236

—

M₁₆

—

C₂×C₈

C₂²×D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

100

C₄²

—

118

—

101

C₄²

D₂₃₆

yes

characteristic

102

C₄²

D₂₃₆

yes

characteristic

103

C₄²

C₂²×D₅₉

yes

—

104

C₂³

C₂×D₂₃₆

yes

characteristic

105

Q₈

—

118

—

106

Q₈

—

118

—

107

Q₈

—

236

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

118

—

112

C₂×C₄

—

118

—

113

C₂×C₄

C₂×D₂₃₆

yes

characteristic

114

C₂×C₄

C₂×D₂₃₆

yes

—

115

C₂×C₄

C₂×D₂₃₆

yes

—

116

C₂×C₄

C₂³×D₅₉

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₂₃₆

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

236

—

124

C₄

—

236

—

125

C₄

#14273760

yes

—

126

C₄

#14273772

yes

characteristic

127

C₄

#14273775

yes

characteristic

128

C₂

—

129

C₂

#16837983

yes

characteristic

130

C₂

C₂×#14273760

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#423	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#423	D₅₉
hypocenter	C₅₉	#3255
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#423

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #16837688	C₂	semidirect
4. #16837613	C₂	semidirect
71. C₅₉	#3255	semidirect

H ≅ G/N

type

2. #16837688

C₂

semidirect

4. #16837613

C₂

semidirect

71. C₅₉

#3255

semidirect

Subgroups — #21435490

Conjugacy classes of subgroups

Complements