Group #21425538 — subgroups

Subgroups — #21425538

Order 7552.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₂×#14273661	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837404	C₂	1	2	yes	maximal
3	#16837368	C₂	1	1	yes	maximal, characteristic
4	#16837170	C₂	1	2	yes	maximal, complemented
5	C₅₉×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745685	C₂²	1	1	yes	characteristic
7	C₂×#11745664	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	#14273661	—	2	2	no	—
10	#14273652	—	2	2	no	—
11	#14273649	—	2	2	no	—
12	(C₂×D₅₉)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₅₉	—	2	1	no	—
17	C₂²×Dic₅₉	—	2	2	no	—
18	#11745685	—	2	2	no	—
19	#11745685	—	2	2	no	—
20	C₂×D₂₃₆	D₄	1	1	yes	characteristic
21	C₂×C₄×D₅₉	—	2	1	no	—
22	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
23	M₁₆×C₅₉	—	2	2	no	—
24	M₁₆×C₅₉	—	2	2	no	—
25	C₂×C₈×C₅₉	—	2	2	no	—
26	C₂×C₈×C₅₉	—	2	2	no	—
27	C₂×C₈×C₅₉	D₄	1	2	yes	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	#11745664	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	1	no	—
32	C₂³×C₅₉	#41	1	1	yes	characteristic
33	C₂²×D₅₉	—	2	1	no	—
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	#41	1	2	yes	—
37	C₂×C₄×C₅₉	#41	1	2	yes	—
38	C₂²⋊D₅₉	—	4	2	no	—
39	C₂×Dic₅₉	—	2	1	no	—
40	C₂×Dic₅₉	—	4	2	no	—
41	C₂×Dic₅₉	—	8	2	no	—
42	D₂₃₆	—	2	2	no	—
43	C₄×D₅₉	—	4	1	no	—
44	C₄×D₅₉	—	4	1	no	—
45	Dic₁₁₈	—	2	2	no	—
46	C₈×C₅₉	—	2	2	no	—
47	C₈×C₅₉	—	4	2	no	—
48	C₅₉⋊C₈	—	4	1	no	—
49	C₂²×C₅₉	—	2	1	no	—
50	C₂²×C₅₉	—	2	1	no	—
51	C₂²×C₅₉	#121	1	1	yes	characteristic
52	C₂²×C₅₉	#126	1	2	yes	—
53	C₂×D₅₉	—	4	1	no	—
54	C₂×D₅₉	—	4	1	no	—
55	C₂×D₅₉	—	4	1	no	—
56	C₄×C₅₉	#123	1	2	yes	—
57	C₄×C₅₉	#123	1	1	yes	characteristic
58	C₄×C₅₉	#124	1	1	yes	characteristic
59	Dic₅₉	—	4	1	no	—
60	Dic₅₉	—	8	2	no	—
61	#2078	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	2	no	—
63	C₂×C₅₉	#397	1	1	yes	characteristic
64	C₂×C₅₉	#472	1	1	yes	characteristic
65	C₂×C₅₉	#473	1	1	yes	characteristic
66	D₅₉	—	8	1	no	—
67	C₂×#107	—	59	1	no	—
68	#417	—	59	2	no	—
69	#408	—	59	1	no	—
70	#343	—	59	2	no	—
71	#341	D₅₉	1	1	yes	characteristic, complemented
72	C₅₉	#2078	1	1	yes	characteristic, complemented, Sylow
73	C₂×#41	—	59	1	no	—
74	C₂×M₁₆	—	59	1	no	—
75	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
76	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	59	2	no	—
78	#107	—	118	2	no	—
79	#103	—	118	2	no	—
80	#102	—	118	2	no	—
81	C₂²⋊C₈	—	118	2	no	—
82	#41	—	118	2	no	—
83	#41	—	118	2	no	—
84	C₂×Q₈	—	59	1	no	—
85	C₂×D₄	—	59	1	no	—
86	C₂×D₄	—	118	1	no	—
87	C₂²×C₄	—	118	1	no	—
88	C₂²×C₄	—	118	2	no	—
89	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	118	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	118	1	no	—
96	C₂×C₈	D₂₃₆	1	2	yes	—
97	C₄⋊C₄	—	118	2	no	—
98	C₄⋊C₄	—	118	2	no	—
99	C₂³	—	118	1	no	—
100	C₂³	#11745689	1	1	yes	characteristic
101	Q₈	—	118	2	no	—
102	D₄	—	118	2	no	—
103	D₄	—	236	2	no	—
104	C₂×C₄	—	118	1	no	—
105	C₂×C₄	—	472	2	no	—
106	C₂×C₄	—	236	2	no	—
107	C₂×C₄	—	236	1	no	—
108	C₂×C₄	—	236	1	no	—
109	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
110	C₂×C₄	#11745685	1	2	yes	—
111	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
112	C₂×C₄	#11745687	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₈	—	236	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	236	1	no	—
119	C₂²	—	236	1	no	—
120	C₂²	—	236	1	no	—
121	C₂²	#14273702	1	1	yes	characteristic
122	C₂²	#14273722	1	2	yes	—
123	C₄	—	472	2	no	—
124	C₄	—	236	1	no	—
125	C₄	#14273709	1	2	yes	—
126	C₄	#14273711	1	1	yes	characteristic
127	C₄	#14273715	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	472	1	no	—
130	C₂	#16837329	1	1	yes	characteristic
131	C₂	#16837673	1	1	yes	characteristic
132	C₂	#16837678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal, characteristic, complemented

#16837404

C₂

yes

maximal

#16837368

C₂

yes

maximal, characteristic

#16837170

C₂

yes

maximal, complemented

C₅₉×#341

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273661

—

#14273652

—

#14273649

—

(C₂×D₅₉)⋊C₈

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

C₂²×Dic₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

D₄

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

#126

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#123

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#2078

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#2078

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

118

—

#103

—

118

—

#102

—

118

—

C₂²⋊C₈

—

118

—

#41

—

118

—

#41

—

118

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

D₂₃₆

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₂³

—

118

—

100

C₂³

#11745689

yes

characteristic

101

Q₈

—

118

—

102

D₄

—

118

—

103

D₄

—

236

—

104

C₂×C₄

—

118

—

105

C₂×C₄

—

472

—

106

C₂×C₄

—

236

—

107

C₂×C₄

—

236

—

108

C₂×C₄

—

236

—

109

C₂×C₄

C₂×D₂₃₆

yes

characteristic

110

C₂×C₄

#11745685

yes

—

111

C₂×C₄

D₄×D₅₉

yes

characteristic

112

C₂×C₄

#11745687

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

236

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

236

—

119

C₂²

—

236

—

120

C₂²

—

236

—

121

C₂²

#14273702

yes

characteristic

122

C₂²

#14273722

yes

—

123

C₄

—

472

—

124

C₄

—

236

—

125

C₄

#14273709

yes

—

126

C₄

#14273711

yes

characteristic

127

C₄

#14273715

yes

characteristic

128

C₂

—

129

C₂

—

472

—

130

C₂

#16837329

yes

characteristic

131

C₂

#16837673

yes

characteristic

132

C₂

#16837678

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#341	C₂
socle	C₂²×C₅₉	#121
hypercenter	#341	D₅₉
hypocenter	C₅₉	#2078
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#14273702

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#341

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
1. C₂×#14273661	C₂	semidirect
4. #16837170	C₂	semidirect
5. C₅₉×#341	C₂	semidirect
71. #341	D₅₉	semidirect
72. C₅₉	#2078	semidirect

H ≅ G/N

type

1. C₂×#14273661

C₂

semidirect

4. #16837170

C₂

semidirect

5. C₅₉×#341

C₂

semidirect

71. #341

D₅₉

semidirect

72. C₅₉

#2078

semidirect

Subgroups — #21425538

Conjugacy classes of subgroups

Complements