Group #21425541 — subgroups

Subgroups — #21425541

Order 7552.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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×#14273659	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837403	C₂	1	2	yes	maximal
3	#16837170	C₂	1	2	yes	maximal, complemented
4	#16837167	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#408	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	#14273659	—	2	2	no	—
9	#14273651	—	2	2	no	—
10	#14273648	—	2	2	no	—
11	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₅₉	—	2	2	no	—
15	#41×C₅₉	—	2	2	no	—
16	#41×C₅₉	—	2	2	no	—
17	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×D₄×C₅₉	—	2	1	no	—
19	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	—	2	1	no	—
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₉	—	2	2	no	—
23	M₁₆×C₅₉	—	2	2	no	—
24	C₂×C₈×C₅₉	—	2	2	no	—
25	C₂×C₈×C₅₉	—	2	1	no	—
26	C₂×C₈×C₅₉	D₄	1	2	yes	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	#11745664	—	2	2	no	—
30	#11745664	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂³×C₅₉	—	2	1	no	—
33	C₂³×C₅₉	#41	1	1	yes	characteristic
34	Q₈×C₅₉	—	2	2	no	—
35	D₄×C₅₉	—	2	2	no	—
36	D₄×C₅₉	—	4	2	no	—
37	C₂×C₄×C₅₉	—	2	1	no	—
38	C₂×C₄×C₅₉	—	4	1	no	—
39	C₂×C₄×C₅₉	—	4	1	no	—
40	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	#41	1	2	yes	—
43	C₂×C₄×C₅₉	#41	1	2	yes	—
44	C₂×Dic₅₉	—	4	2	no	—
45	C₂×Dic₅₉	—	8	2	no	—
46	C₈×C₅₉	—	2	2	no	—
47	C₈×C₅₉	—	4	1	no	—
48	C₅₉⋊C₈	—	4	2	no	—
49	C₂²×C₅₉	—	2	2	no	—
50	C₂²×C₅₉	—	4	1	no	—
51	C₂²×C₅₉	—	4	2	no	—
52	C₂²×C₅₉	#121	1	1	yes	characteristic
53	C₂²×C₅₉	#126	1	2	yes	—
54	C₄×C₅₉	—	4	1	no	—
55	C₄×C₅₉	#123	1	1	yes	characteristic
56	C₄×C₅₉	#123	1	2	yes	—
57	C₄×C₅₉	#124	1	1	yes	characteristic
58	Dic₅₉	—	8	2	no	—
59	#2079	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	—	2	2	no	—
61	C₂×C₅₉	—	8	1	no	—
62	C₂×C₅₉	#397	1	1	yes	characteristic
63	C₂×C₅₉	#471	1	2	yes	—
64	C₂×#106	—	59	1	no	—
65	#417	—	59	2	no	—
66	#408	D₅₉	1	1	yes	characteristic
67	#343	—	59	2	no	—
68	#341	—	59	1	no	—
69	C₅₉	#2079	1	1	yes	characteristic, complemented, Sylow
70	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
71	C₂×M₁₆	—	59	2	no	—
72	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	59	2	no	—
75	#106	—	118	2	no	—
76	#103	—	118	2	no	—
77	#102	—	118	2	no	—
78	C₂²⋊C₈	—	2	2	no	—
79	#41	—	2	2	no	—
80	#41	—	2	2	no	—
81	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	D₂₃₆	1	1	yes	characteristic
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	118	2	no	—
86	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	118	2	no	—
89	M₁₆	—	118	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	118	2	no	—
93	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
94	C₄⋊C₄	—	118	2	no	—
95	C₄⋊C₄	—	118	2	no	—
96	C₂³	—	2	1	no	—
97	C₂³	#11745685	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	4	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	472	2	no	—
105	C₂×C₄	—	236	2	no	—
106	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
107	C₂×C₄	#11745687	1	2	yes	—
108	C₂×C₄	#11745687	1	2	yes	—
109	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	4	1	no	—
112	C₈	—	236	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	4	2	no	—
115	C₂²	—	4	1	no	—
116	C₂²	#14273701	1	1	yes	characteristic
117	C₂²	#14273720	1	2	yes	—
118	C₄	—	4	1	no	—
119	C₄	—	472	2	no	—
120	C₄	#14273708	1	1	yes	characteristic
121	C₄	#14273711	1	2	yes	—
122	C₄	#14273714	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	8	1	no	—
125	C₂	#16837327	1	1	yes	characteristic
126	C₂	#16837665	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273659

C₂

yes

maximal, characteristic, complemented

#16837403

C₂

yes

maximal

#16837170

C₂

yes

maximal, complemented

#16837167

C₂

yes

maximal, characteristic, complemented

C₅₉×#408

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273659

—

#14273651

—

#14273648

—

C₂×#41×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²⋊C₈×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

#41

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

#126

yes

—

C₄×C₅₉

—

C₄×C₅₉

#123

yes

characteristic

C₄×C₅₉

#123

yes

—

C₄×C₅₉

#124

yes

characteristic

Dic₅₉

—

#2079

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#397

yes

characteristic

C₂×C₅₉

#471

yes

—

C₂×#106

—

#417

—

#408

D₅₉

yes

characteristic

#343

—

#341

—

C₅₉

#2079

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

#106

—

118

—

#103

—

118

—

#102

—

118

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

D₂₃₆

yes

characteristic

C₂×D₄

—

C₂×D₄

D₂₃₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₂³

—

C₂³

#11745685

yes

characteristic

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

472

—

105

C₂×C₄

—

236

—

106

C₂×C₄

C₂×D₂₃₆

yes

characteristic

107

C₂×C₄

#11745687

yes

—

108

C₂×C₄

#11745687

yes

—

109

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

236

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#14273701

yes

characteristic

117

C₂²

#14273720

yes

—

118

C₄

—

119

C₄

—

472

—

120

C₄

#14273708

yes

characteristic

121

C₄

#14273711

yes

—

122

C₄

#14273714

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#16837327

yes

characteristic

126

C₂

#16837665

yes

—

	subgroup	G/H
center	C₂²	#14273701
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#408	C₂
socle	C₂²×C₅₉	#121
hypercenter	#408	D₅₉
hypocenter	C₅₉	#2079
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#14273701

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#408

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. C₂×#14273659	C₂	semidirect
3. #16837170	C₂	semidirect
4. #16837167	C₂	semidirect
69. C₅₉	#2079	semidirect

H ≅ G/N

type

1. C₂×#14273659

C₂

semidirect

3. #16837170

C₂

semidirect

4. #16837167

C₂

semidirect

69. C₅₉

#2079

semidirect

Subgroups — #21425541

Conjugacy classes of subgroups

Complements