Group #21426329 — subgroups

Subgroups — #21426329

Order 7552.

138 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16837803	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837787	C₂	1	1	yes	maximal, characteristic
3	#16837206	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837204	C₂	1	1	yes	maximal, characteristic
5	#16837202	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745671	—	2	1	no	—
9	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
10	#14273718	—	2	1	no	—
11	#14273716	—	2	1	no	—
12	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14273661	C₂²	1	1	yes	characteristic
14	#14273658	—	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×D₂₃₆	—	2	1	no	—
21	Q₁₆×C₅₉	—	2	1	no	—
22	Q₁₆×C₅₉	—	2	1	no	—
23	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₉	—	2	1	no	—
25	#11745671	—	4	2	no	—
26	(C₂×D₅₉)⋊C₄	—	4	1	no	—
27	(C₂×D₅₉)⋊C₄	—	4	1	no	—
28	C₄⋊Dic₅₉	—	2	1	no	—
29	C₄⋊Dic₅₉	—	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	D₄₇₂	—	2	1	no	—
34	D₄₇₂	—	2	1	no	—
35	C₁₆×C₅₉	—	2	1	no	—
36	C₅₉⋊C₁₆	—	2	1	no	—
37	C₂²×D₅₉	—	4	1	no	—
38	Q₈×C₅₉	—	4	1	no	—
39	Q₈×C₅₉	—	4	1	no	—
40	C₂×C₄×C₅₉	—	4	1	no	—
41	C₂×C₄×C₅₉	—	4	1	no	—
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₅₉	—	4	1	no	—
44	C₂×Dic₅₉	D₈	1	1	yes	characteristic
45	C₂×Dic₅₉	D₈	1	1	yes	characteristic
46	D₂₃₆	—	4	1	no	—
47	D₂₃₆	—	4	1	no	—
48	C₈×C₅₉	#41	1	1	yes	characteristic
49	C₈×C₅₉	#41	1	1	yes	characteristic
50	C₅₉⋊C₈	SD₁₆	1	2	yes	—
51	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
52	C₂×D₅₉	—	8	1	no	—
53	C₂×D₅₉	—	8	1	no	—
54	C₂×D₅₉	—	8	1	no	—
55	C₄×C₅₉	—	8	1	no	—
56	C₄×C₅₉	—	8	1	no	—
57	C₄×C₅₉	#124	1	1	yes	characteristic
58	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
59	Dic₅₉	—	2	1	no	—
60	Dic₅₉	—	2	1	no	—
61	Dic₅₉	—	8	1	no	—
62	#2242	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	#486	1	1	yes	characteristic
64	C₂×C₅₉	#508	1	1	yes	characteristic
65	C₂×C₅₉	#508	1	1	yes	characteristic
66	D₅₉	—	16	1	no	—
67	#499	—	59	1	no	—
68	#495	—	59	1	no	—
69	#358	—	59	1	no	—
70	#358	D₅₉	1	1	yes	characteristic, complemented
71	#357	—	59	1	no	—
72	#357	—	59	1	no	—
73	C₄×C₁₆	—	59	1	no	—
74	C₅₉	#2242	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
76	C₂×SD₁₆	—	118	1	no	—
77	C₂×D₈	—	59	1	no	—
78	#125	—	118	1	no	—
79	#124	—	118	1	no	—
80	C₂×C₁₆	—	59	1	no	—
81	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
82	#107	—	59	1	no	—
83	#107	C₂×D₅₉	1	1	yes	characteristic
84	#106	—	118	1	no	—
85	C₄×C₈	—	59	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×D₄	—	118	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	2	1	no	—
90	SD₁₆	—	236	2	no	—
91	D₈	—	118	1	no	—
92	D₈	—	118	1	no	—
93	C₂×C₈	—	59	1	no	—
94	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	118	1	no	—
97	C₄⋊C₄	—	236	1	no	—
98	C₄⋊C₄	—	236	1	no	—
99	C₂²⋊C₄	—	236	1	no	—
100	C₂²⋊C₄	—	236	1	no	—
101	C₄²	—	59	1	no	—
102	C₁₆	—	2	1	no	—
103	C₁₆	—	118	1	no	—
104	C₂³	—	236	1	no	—
105	Q₈	—	4	1	no	—
106	Q₈	—	4	1	no	—
107	D₄	—	236	1	no	—
108	D₄	—	236	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	59	1	no	—
112	C₂×C₄	—	59	1	no	—
113	C₂×C₄	—	236	1	no	—
114	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
115	C₈	—	59	2	no	—
116	C₈	#11745685	1	1	yes	characteristic
117	C₈	#11745687	1	1	yes	characteristic
118	C₂²	—	472	1	no	—
119	C₂²	—	472	1	no	—
120	C₂²	—	472	1	no	—
121	C₂²	D₈×D₅₉	1	1	yes	characteristic
122	C₄	—	8	1	no	—
123	C₄	—	8	1	no	—
124	C₄	—	118	1	no	—
125	C₄	—	118	1	no	—
126	C₄	—	472	1	no	—
127	C₄	#14273713	1	1	yes	characteristic
128	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
129	C₂	—	944	1	no	—
130	C₂	#16837747	1	1	yes	characteristic
131	C₂	#16837833	1	1	yes	characteristic
132	C₂	#16837835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837803

C₂

yes

maximal, characteristic, complemented

#16837787

C₂

yes

maximal, characteristic

#16837206

C₂

yes

maximal, characteristic, complemented

#16837204

C₂

yes

maximal, characteristic

#16837202

C₂

yes

maximal, characteristic

C₁₆×Dic₅₉

C₂

yes

maximal, characteristic, complemented

C₅₉×#358

C₂

yes

maximal, characteristic, complemented

C₂×#11745671

—

C₂×D₄₇₂

C₂²

yes

characteristic

#14273718

—

#14273716

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273658

—

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×D₂₃₆

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745671

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

D₄₇₂

—

D₄₇₂

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

SD₁₆

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

#357

—

#357

—

C₄×C₁₆

—

C₅₉

#2242

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×SD₁₆

—

118

—

C₂×D₈

—

#125

—

118

—

#124

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#107

C₂×D₅₉

yes

characteristic

#106

—

118

—

C₄×C₈

—

C₂×Q₈

—

C₂×D₄

—

118

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

236

—

D₈

—

118

—

D₈

—

118

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

236

—

100

C₂²⋊C₄

—

236

—

101

C₄²

—

102

C₁₆

—

103

C₁₆

—

118

—

104

C₂³

—

236

—

105

Q₈

—

106

Q₈

—

107

D₄

—

236

—

108

D₄

—

236

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

236

—

114

C₂×C₄

D₄×D₅₉

yes

characteristic

115

C₈

—

116

C₈

#11745685

yes

characteristic

117

C₈

#11745687

yes

characteristic

118

C₂²

—

472

—

119

C₂²

—

472

—

120

C₂²

—

472

—

121

C₂²

D₈×D₅₉

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

118

—

125

C₄

—

118

—

126

C₄

—

472

—

127

C₄

#14273713

yes

characteristic

128

C₄

SD₁₆×D₅₉

yes

characteristic

129

C₂

—

944

—

130

C₂

#16837747

yes

characteristic

131

C₂

#16837833

yes

characteristic

132

C₂

#16837835

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	#2242
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
1. #16837803	C₂	semidirect
3. #16837206	C₂	semidirect
6. C₁₆×Dic₅₉	C₂	semidirect
7. C₅₉×#358	C₂	semidirect
70. #358	D₅₉	semidirect
74. C₅₉	#2242	semidirect

H ≅ G/N

type

1. #16837803

C₂

semidirect

3. #16837206

C₂

semidirect

6. C₁₆×Dic₅₉

C₂

semidirect

7. C₅₉×#358

C₂

semidirect

70. #358

D₅₉

semidirect

74. C₅₉

#2242

semidirect

Subgroups — #21426329

Conjugacy classes of subgroups

Complements