Group #21422397 — subgroups

Subgroups — #21422397

Order 7552.

178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16837453	C₂	1	1	yes	maximal, characteristic
2	#16837453	C₂	1	1	yes	maximal, characteristic
3	#16837453	C₂	1	1	yes	maximal, characteristic
4	#16837124	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837123	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×D₅₉	C₂²	1	1	yes	characteristic
9	Dic₅₉⋊C₈	C₄	1	2	yes	—
10	Dic₅₉⋊C₈	C₄	1	2	yes	—
11	Dic₅₉⋊C₈	C₄	1	2	yes	—
12	(C₂×D₅₉)⋊C₈	C₄	1	2	yes	—
13	(C₂×D₅₉)⋊C₈	C₄	1	2	yes	—
14	(C₂×D₅₉)⋊C₈	C₄	1	2	yes	—
15	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
27	C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₂²×D₅₉	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₅₉	C₄²	1	1	yes	characteristic
39	C₂×Dic₅₉	C₄²	1	1	yes	characteristic
40	C₂×Dic₅₉	C₄²	1	1	yes	characteristic
41	C₄×D₅₉	—	2	2	no	—
42	C₄×D₅₉	—	2	2	no	—
43	C₄×D₅₉	—	2	2	no	—
44	C₈×C₅₉	—	2	2	no	—
45	C₈×C₅₉	—	2	2	no	—
46	C₈×C₅₉	—	2	2	no	—
47	C₅₉⋊C₈	—	2	2	no	—
48	C₅₉⋊C₈	—	2	2	no	—
49	C₅₉⋊C₈	—	2	2	no	—
50	C₂²×C₅₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×D₅₉	—	2	1	no	—
52	C₂×D₅₉	—	2	1	no	—
53	C₂×D₅₉	—	2	1	no	—
54	C₄×C₅₉	#131	1	2	yes	—
55	C₄×C₅₉	#131	1	2	yes	—
56	C₄×C₅₉	#131	1	2	yes	—
57	Dic₅₉	—	2	1	no	—
58	Dic₅₉	—	2	1	no	—
59	Dic₅₉	—	2	1	no	—
60	#1454	—	59	1	no	maximal, Sylow
61	C₂×C₅₉	#405	1	1	yes	characteristic
62	C₂×C₅₉	#405	1	1	yes	characteristic
63	C₂×C₅₉	#405	1	1	yes	characteristic
64	D₅₉	—	4	1	no	—
65	#431	—	59	1	no	—
66	#431	—	59	1	no	—
67	#431	—	59	1	no	—
68	#322	—	59	1	no	—
69	#322	—	59	1	no	—
70	#322	D₅₉	1	1	yes	characteristic, complemented
71	C₈²	—	59	1	no	—
72	C₅₉	#1454	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄²	—	59	1	no	—
74	C₄⋊C₈	—	59	2	no	—
75	C₄⋊C₈	—	59	2	no	—
76	C₄⋊C₈	—	59	2	no	—
77	C₂²⋊C₈	—	59	2	no	—
78	C₂²⋊C₈	—	59	2	no	—
79	C₂²⋊C₈	—	59	2	no	—
80	C₄×C₈	—	59	1	no	—
81	C₄×C₈	—	59	1	no	—
82	C₄×C₈	—	59	1	no	—
83	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
84	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
85	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
86	C₂²×C₄	—	59	1	no	—
87	C₂²×C₄	—	59	1	no	—
88	C₂²×C₄	—	59	1	no	—
89	C₂×C₈	—	59	2	no	—
90	C₂×C₈	—	59	2	no	—
91	C₂×C₈	—	59	2	no	—
92	C₂×C₈	C₄×D₅₉	1	2	yes	—
93	C₂×C₈	C₄×D₅₉	1	2	yes	—
94	C₂×C₈	C₄×D₅₉	1	2	yes	—
95	C₄²	—	59	1	no	—
96	C₄²	—	59	1	no	—
97	C₄²	—	59	1	no	—
98	C₄²	C₂²×D₅₉	1	1	yes	characteristic
99	C₂³	—	59	1	no	—
100	C₂×C₄	—	59	1	no	—
101	C₂×C₄	—	59	1	no	—
102	C₂×C₄	—	59	1	no	—
103	C₂×C₄	—	118	2	no	—
104	C₂×C₄	—	118	2	no	—
105	C₂×C₄	—	118	2	no	—
106	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	118	2	no	—
113	C₈	—	118	2	no	—
114	C₈	—	118	2	no	—
115	C₂²	—	118	1	no	—
116	C₂²	—	118	1	no	—
117	C₂²	—	118	1	no	—
118	C₂²	C₄²×D₅₉	1	1	yes	characteristic
119	C₄	—	118	1	no	—
120	C₄	—	118	1	no	—
121	C₄	—	118	1	no	—
122	C₄	#14273733	1	2	yes	—
123	C₄	#14273735	1	2	yes	—
124	C₄	#14273735	1	2	yes	—
125	C₂	—	236	1	no	—
126	C₂	#16837356	1	1	yes	characteristic
127	C₂	#16837359	1	1	yes	characteristic
128	C₂	#16837359	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837453

C₂

yes

maximal, characteristic

#16837453

C₂

yes

maximal, characteristic

#16837453

C₂

yes

maximal, characteristic

#16837124

C₂

yes

maximal, characteristic, complemented

#16837123

C₂

yes

maximal, characteristic, complemented

C₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₅₉×#322

C₂

yes

maximal, characteristic, complemented

C₄²×D₅₉

C₂²

yes

characteristic

Dic₅₉⋊C₈

C₄

yes

—

Dic₅₉⋊C₈

C₄

yes

—

Dic₅₉⋊C₈

C₄

yes

—

(C₂×D₅₉)⋊C₈

C₄

yes

—

(C₂×D₅₉)⋊C₈

C₄

yes

—

(C₂×D₅₉)⋊C₈

C₄

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂²×D₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×Dic₅₉

C₄²

yes

characteristic

C₂×Dic₅₉

C₄²

yes

characteristic

C₂×Dic₅₉

C₄²

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄²

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

#131

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₈²

—

C₅₉

#1454

yes

characteristic, complemented, Sylow

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

C₄×D₅₉

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

118

—

104

C₂×C₄

—

118

—

105

C₂×C₄

—

118

—

106

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

107

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

108

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

118

—

113

C₈

—

118

—

114

C₈

—

118

—

115

C₂²

—

118

—

116

C₂²

—

118

—

117

C₂²

—

118

—

118

C₂²

C₄²×D₅₉

yes

characteristic

119

C₄

—

118

—

120

C₄

—

118

—

121

C₄

—

118

—

122

C₄

#14273733

yes

—

123

C₄

#14273735

yes

—

124

C₄

#14273735

yes

—

125

C₂

—

236

—

126

C₂

#16837356

yes

characteristic

127

C₂

#16837359

yes

characteristic

128

C₂

#16837359

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂×C₄²
Fitting	C₅₉×#322	C₂
hypercenter	#322	D₅₉
hypocenter	C₅₉	#1454
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂×C₄²

Fitting

C₅₉×#322

C₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #16837124	C₂	semidirect
5. #16837123	C₂	semidirect
6. C₈×C₅₉⋊C₈	C₂	semidirect
7. C₅₉×#322	C₂	semidirect
70. #322	D₅₉	semidirect
72. C₅₉	#1454	semidirect

H ≅ G/N

type

4. #16837124

C₂

semidirect

5. #16837123

C₂

semidirect

6. C₈×C₅₉⋊C₈

C₂

semidirect

7. C₅₉×#322

C₂

semidirect

70. #322

D₅₉

semidirect

72. C₅₉

#1454

semidirect

Subgroups — #21422397

Conjugacy classes of subgroups

Complements