Group #21426322 — subgroups

Subgroups — #21426322

Order 7552.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837806	C₂	1	1	yes	maximal, characteristic
2	#16837780	C₂	1	1	yes	maximal, characteristic
3	#16837207	C₂	1	1	yes	maximal, characteristic
4	#16837206	C₂	1	1	yes	maximal, characteristic
5	#16837205	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×#11745672	—	2	1	no	—
10	#14273726	—	2	1	no	—
11	#14273725	—	2	1	no	—
12	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14273661	C₂²	1	1	yes	characteristic
14	#14273660	—	2	1	no	—
15	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	#107×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₉	—	2	1	no	—
20	C₂×Dic₁₁₈	—	2	1	no	—
21	C₂×Dic₁₁₈	—	4	1	no	—
22	Q₁₆×C₅₉	—	2	1	no	—
23	Q₁₆×C₅₉	—	2	1	no	—
24	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₉	—	2	1	no	—
26	#11745672	—	4	2	no	—
27	C₄⋊Dic₅₉	—	2	1	no	—
28	Dic₅₉⋊C₄	—	4	1	no	—
29	Dic₅₉⋊C₄	—	4	1	no	—
30	Dic₅₉⋊C₄	—	4	1	no	—
31	C₄×Dic₅₉	D₄	1	1	yes	characteristic
32	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	Dic₂₃₆	—	2	1	no	—
34	Dic₂₃₆	—	2	1	no	—
35	C₁₆×C₅₉	—	2	1	no	—
36	C₅₉⋊C₁₆	—	2	1	no	—
37	Q₈×C₅₉	—	4	1	no	—
38	Q₈×C₅₉	—	4	1	no	—
39	C₂×C₄×C₅₉	—	4	1	no	—
40	C₂×C₄×C₅₉	—	4	1	no	—
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₅₉	—	4	1	no	—
43	C₂×Dic₅₉	—	4	1	no	—
44	C₂×Dic₅₉	D₈	1	1	yes	characteristic
45	C₂×Dic₅₉	D₈	1	1	yes	characteristic
46	Dic₁₁₈	—	4	1	no	—
47	Dic₁₁₈	—	4	1	no	—
48	Dic₁₁₈	—	8	2	no	—
49	C₈×C₅₉	#41	1	1	yes	characteristic
50	C₈×C₅₉	#41	1	1	yes	characteristic
51	C₅₉⋊C₈	SD₁₆	1	2	yes	—
52	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₄×C₅₉	—	8	1	no	—
54	C₄×C₅₉	—	8	1	no	—
55	C₄×C₅₉	#124	1	1	yes	characteristic
56	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
57	Dic₅₉	—	8	1	no	—
58	Dic₅₉	—	8	1	no	—
59	Dic₅₉	SD₃₂	1	2	yes	—
60	Dic₅₉	Q₃₂	1	2	yes	—
61	#2241	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	#486	1	1	yes	characteristic
63	C₂×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
64	C₂×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
65	#500	—	59	1	no	—
66	#494	—	59	1	no	—
67	#358	—	59	1	no	—
68	#358	—	59	1	no	—
69	#358	—	59	1	no	—
70	#358	D₅₉	1	1	yes	characteristic
71	C₄×C₁₆	—	59	1	no	—
72	C₅₉	#2241	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₁₆	—	59	1	no	—
74	C₂×Q₁₆	—	118	1	no	—
75	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
76	C₄⋊Q₈	—	118	1	no	—
77	C₄⋊Q₈	—	118	1	no	—
78	C₂×C₁₆	—	59	1	no	—
79	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
80	#107	—	59	1	no	—
81	#107	—	118	1	no	—
82	#107	C₂×D₅₉	1	1	yes	characteristic
83	C₄×C₈	—	59	1	no	—
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	118	1	no	—
86	C₂×Q₈	—	236	1	no	—
87	Q₁₆	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	118	1	no	—
90	Q₁₆	—	118	1	no	—
91	Q₁₆	—	236	2	no	—
92	C₂×C₈	—	59	1	no	—
93	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	118	1	no	—
96	C₄⋊C₄	—	236	1	no	—
97	C₄⋊C₄	—	236	1	no	—
98	C₄⋊C₄	—	236	1	no	—
99	C₄²	—	59	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	118	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	4	1	no	—
104	Q₈	—	472	2	no	—
105	Q₈	—	236	1	no	—
106	Q₈	—	236	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	59	1	no	—
110	C₂×C₄	—	59	1	no	—
111	C₂×C₄	—	236	1	no	—
112	C₂×C₄	—	236	1	no	—
113	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
114	C₈	—	59	2	no	—
115	C₈	#11745685	1	1	yes	characteristic
116	C₈	#11745687	1	1	yes	characteristic
117	C₂²	D₈×D₅₉	1	1	yes	characteristic
118	C₄	—	8	1	no	—
119	C₄	—	8	1	no	—
120	C₄	—	59	2	no	—
121	C₄	—	59	2	no	—
122	C₄	—	472	1	no	—
123	C₄	—	472	1	no	—
124	C₄	#14273713	1	1	yes	characteristic
125	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
126	C₂	#16837747	1	1	yes	characteristic
127	C₂	SD₃₂×D₅₉	1	1	yes	characteristic
128	C₂	Q₃₂×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837806

C₂

yes

maximal, characteristic

#16837780

C₂

yes

maximal, characteristic

#16837207

C₂

yes

maximal, characteristic

#16837206

C₂

yes

maximal, characteristic

#16837205

C₂

yes

maximal, characteristic

C₁₆×Dic₅₉

C₂

yes

maximal, characteristic

C₅₉×#358

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

—

—

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273660

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745672

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

SD₁₆

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

SD₃₂

yes

—

Dic₅₉

Q₃₂

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×SD₃₂

yes

characteristic

C₂×C₅₉

C₂×Q₃₂

yes

—

—

—

—

—

D₅₉

yes

characteristic

C₄×C₁₆

—

C₅₉

#2241

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

118

—

C₄⋊Q₈

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#107

—

118

—

#107

C₂×D₅₉

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

236

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

118

—

Q₁₆

—

118

—

Q₁₆

—

236

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

—

100

C₁₆

—

101

C₁₆

—

118

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

472

—

105

Q₈

—

236

—

106

Q₈

—

236

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

236

—

112

C₂×C₄

—

236

—

113

C₂×C₄

D₄×D₅₉

yes

characteristic

114

C₈

—

115

C₈

#11745685

yes

characteristic

116

C₈

#11745687

yes

characteristic

117

C₂²

D₈×D₅₉

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

472

—

123

C₄

—

472

—

124

C₄

#14273713

yes

characteristic

125

C₄

SD₁₆×D₅₉

yes

characteristic

126

C₂

#16837747

yes

characteristic

127

C₂

SD₃₂×D₅₉

yes

characteristic

128

C₂

Q₃₂×D₅₉

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#358	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#358	D₅₉
hypocenter	C₅₉	#2241
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#358

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
72. C₅₉	#2241	semidirect

H ≅ G/N

type

72. C₅₉

#2241

semidirect

Subgroups — #21426322

Conjugacy classes of subgroups

Complements