Group #21424137 — subgroups

Subgroups — #21424137

Order 7552.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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×#11745659	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273662	C₂	1	2	yes	maximal, complemented
3	#16837365	C₂	1	2	yes	maximal
4	#16837167	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₉	C₂²	1	1	yes	characteristic
7	C₂×#11745664	C₂²	1	2	yes	—
8	C₂×#11745659	—	2	4	no	—
9	C₂×#11745659	C₄	1	4	yes	—
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	#14273662	—	2	4	no	—
12	(C₂×D₅₉)⋊C₈	—	2	4	no	—
13	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂³×D₅₉	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₉	Q₈	1	1	yes	characteristic
18	C₂×C₄×D₅₉	—	2	2	no	—
19	C₂×C₄×D₅₉	—	2	2	no	—
20	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₅₉	D₄	1	2	yes	—
22	M₁₆×C₅₉	—	2	2	no	—
23	M₁₆×C₅₉	—	2	2	no	—
24	C₂×C₈×C₅₉	—	2	2	no	—
25	C₂×C₈×C₅₉	—	2	2	no	—
26	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
27	#11745664	—	2	2	no	—
28	#11745664	—	2	2	no	—
29	C₂×C₅₉⋊C₈	—	2	2	no	—
30	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
32	#11745659	—	2	8	no	—
33	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₂²×D₅₉	—	2	2	no	—
35	C₂²×D₅₉	—	4	2	no	—
36	C₂²×D₅₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	#41	1	2	yes	—
41	C₂×Dic₅₉	—	2	2	no	—
42	C₂×Dic₅₉	C₄⋊C₄	1	2	yes	—
43	C₄×D₅₉	—	2	2	no	—
44	C₄×D₅₉	—	2	2	no	—
45	C₄×D₅₉	—	2	4	no	—
46	C₈×C₅₉	—	2	2	no	—
47	C₈×C₅₉	—	4	2	no	—
48	C₅₉⋊C₈	—	4	2	no	—
49	C₅₉⋊C₈	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₅₉	—	2	2	no	—
51	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₅₉	—	2	2	no	—
54	C₂×D₅₉	—	2	4	no	—
55	C₂×D₅₉	—	4	4	no	—
56	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₅₉	#117	1	1	yes	characteristic
58	C₄×C₅₉	#121	1	2	yes	—
59	Dic₅₉	—	2	2	no	—
60	#1811	—	59	1	no	maximal, Sylow
61	C₂×C₅₉	—	2	2	no	—
62	C₂×C₅₉	#380	1	1	yes	characteristic
63	C₂×C₅₉	#430	1	2	yes	—
64	D₅₉	—	4	2	no	—
65	C₂²×M₁₆	—	59	1	no	—
66	C₂×#108	—	59	2	no	—
67	#407	—	59	2	no	—
68	#341	—	59	1	no	—
69	#341	D₅₉	1	1	yes	characteristic, complemented
70	C₅₉	#1811	1	1	yes	characteristic, complemented, Sylow
71	C₂³×C₄	—	59	1	no	—
72	C₂×M₁₆	—	59	4	no	—
73	C₂×M₁₆	—	59	2	no	—
74	C₂×M₁₆	—	118	4	no	—
75	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
76	C₂²×C₈	—	59	1	no	—
77	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
78	#108	—	118	4	no	—
79	C₂²⋊C₈	—	118	4	no	—
80	C₂⁴	—	59	1	no	—
81	C₂²×C₄	—	59	1	no	—
82	C₂²×C₄	—	59	2	no	—
83	C₂²×C₄	—	59	2	no	—
84	C₂²×C₄	—	118	2	no	—
85	C₂²×C₄	—	118	2	no	—
86	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	118	8	no	—
90	M₁₆	—	118	2	no	—
91	M₁₆	—	118	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	59	2	no	—
95	C₂×C₈	—	59	4	no	—
96	C₂×C₈	—	118	2	no	—
97	C₂×C₈	C₄×D₅₉	1	2	yes	—
98	C₂³	—	59	2	no	—
99	C₂³	—	118	2	no	—
100	C₂³	—	236	2	no	—
101	C₂³	Q₈×D₅₉	1	1	yes	characteristic
102	C₂×C₄	—	59	2	no	—
103	C₂×C₄	—	118	2	no	—
104	C₂×C₄	—	118	2	no	—
105	C₂×C₄	—	118	2	no	—
106	C₂×C₄	—	118	4	no	—
107	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
108	C₂×C₄	#11745685	1	2	yes	—
109	C₂×C₄	D₄×D₅₉	1	2	yes	—
110	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	59	4	no	—
114	C₈	—	236	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	118	4	no	—
117	C₂²	—	118	2	no	—
118	C₂²	—	236	4	no	—
119	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
120	C₂²	#14273705	1	2	yes	—
121	C₄	—	118	2	no	—
122	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
123	C₄	#14273684	1	1	yes	characteristic
124	C₄	#14273700	1	2	yes	—
125	C₂	—	2	2	no	—
126	C₂	—	236	2	no	—
127	C₂	#16837261	1	1	yes	characteristic
128	C₂	#16837449	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745659

C₂

yes

maximal, characteristic

C₂×#14273662

C₂

yes

maximal, complemented

#16837365

C₂

yes

maximal

#16837167

C₂

yes

maximal, characteristic, complemented

C₅₉×#341

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₉

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

—

C₂×#11745659

—

C₂×#11745659

C₄

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273662

—

(C₂×D₅₉)⋊C₈

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂³×D₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

Q₈

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₂×C₄

yes

—

C₂×C₄×D₅₉

D₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

#11745659

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

#1811

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#380

yes

characteristic

C₂×C₅₉

#430

yes

—

D₅₉

—

C₂²×M₁₆

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#1811

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#108

—

118

—

C₂²⋊C₈

—

118

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

—

C₂³

—

C₂³

—

118

—

100

C₂³

—

236

—

101

C₂³

Q₈×D₅₉

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

118

—

104

C₂×C₄

—

118

—

105

C₂×C₄

—

118

—

106

C₂×C₄

—

118

—

107

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

108

C₂×C₄

#11745685

yes

—

109

C₂×C₄

D₄×D₅₉

yes

—

110

C₂×C₄

D₄×D₅₉

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

236

—

115

C₂²

—

116

C₂²

—

118

—

117

C₂²

—

118

—

118

C₂²

—

236

—

119

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

120

C₂²

#14273705

yes

—

121

C₄

—

118

—

122

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

123

C₄

#14273684

yes

characteristic

124

C₄

#14273700

yes

—

125

C₂

—

126

C₂

—

236

—

127

C₂

#16837261

yes

characteristic

128

C₂

#16837449

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#341	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#341	D₅₉
hypocenter	C₅₉	#1811
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#341

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#14273662	C₂	semidirect
4. #16837167	C₂	semidirect
5. C₅₉×#341	C₂	semidirect
69. #341	D₅₉	semidirect
70. C₅₉	#1811	semidirect

H ≅ G/N

type

2. C₂×#14273662

C₂

semidirect

4. #16837167

C₂

semidirect

5. C₅₉×#341

C₂

semidirect

69. #341

D₅₉

semidirect

70. C₅₉

#1811

semidirect

Subgroups — #21424137

Conjugacy classes of subgroups

Complements