Group #21426300 — subgroups

Subgroups — #21426300

Order 7552.

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

normal subgroups only

This group has 140 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	#16837644	C₂	1	1	yes	maximal, characteristic
2	#107×D₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837227	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837207	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837205	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
10	#14273707	—	2	1	no	—
11	C₄⋊C₄×D₅₉	—	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	#14273650	—	2	1	no	—
16	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	#107×C₅₉	C₂²	1	1	yes	characteristic
18	#107×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₅₉	—	4	1	no	—
20	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
21	C₂×Dic₁₁₈	—	2	1	no	—
22	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₅₉	—	2	1	no	—
24	C₄⋊C₄×C₅₉	—	2	1	no	—
25	(C₂×D₅₉)⋊C₄	—	4	1	no	—
26	C₄⋊Dic₅₉	—	2	1	no	—
27	C₄⋊Dic₅₉	—	4	1	no	—
28	Dic₅₉⋊C₄	—	4	1	no	—
29	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
30	Dic₂₃₆	—	2	1	no	—
31	Dic₂₃₆	—	2	1	no	—
32	C₈×D₅₉	—	2	1	no	—
33	C₈×D₅₉	—	2	1	no	—
34	C₁₆×C₅₉	—	2	1	no	—
35	C₅₉⋊C₁₆	—	2	1	no	—
36	C₂²×D₅₉	D₈	1	1	yes	characteristic
37	C₂×C₄×C₅₉	—	4	1	no	—
38	C₂×C₄×C₅₉	—	4	1	no	—
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₉	—	4	1	no	—
41	C₂×Dic₅₉	—	4	1	no	—
42	C₂×Dic₅₉	D₈	1	1	yes	characteristic
43	C₄×D₅₉	—	2	1	no	—
44	C₄×D₅₉	—	2	1	no	—
45	C₄×D₅₉	—	8	2	no	—
46	Dic₁₁₈	—	4	1	no	—
47	Dic₁₁₈	—	4	1	no	—
48	C₈×C₅₉	#41	1	1	yes	characteristic
49	C₈×C₅₉	#41	1	1	yes	characteristic
50	C₅₉⋊C₈	—	2	1	no	—
51	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
52	C₂×D₅₉	—	2	1	no	—
53	C₂×D₅₉	—	2	1	no	—
54	C₂×D₅₉	Q₃₂	1	2	yes	—
55	C₄×C₅₉	—	8	1	no	—
56	C₄×C₅₉	—	8	1	no	—
57	C₄×C₅₉	#123	1	1	yes	characteristic
58	C₄×C₅₉	#137	1	1	yes	characteristic
59	Dic₅₉	—	2	1	no	—
60	Dic₅₉	—	8	1	no	—
61	Dic₅₉	—	8	1	no	—
62	#2236	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	#480	1	1	yes	characteristic
64	C₂×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
65	C₂×C₅₉	#509	1	1	yes	characteristic
66	D₅₉	—	2	2	no	—
67	#467	—	59	1	no	—
68	C₂×#107	—	59	1	no	—
69	#366	—	59	1	no	—
70	#366	D₅₉	1	1	yes	characteristic, complemented
71	#358	—	59	1	no	—
72	#358	—	59	1	no	—
73	C₂²⋊C₁₆	—	59	1	no	—
74	C₅₉	#2236	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	59	1	no	—
76	C₂²×C₈	—	59	1	no	—
77	C₂²⋊Q₈	—	118	1	no	—
78	C₂×C₄⋊C₄	—	118	1	no	—
79	C₂×C₁₆	—	59	1	no	—
80	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
81	#107	—	59	1	no	—
82	#107	—	118	1	no	—
83	#107	C₂×D₅₉	1	1	yes	characteristic
84	#107	C₂×D₅₉	1	1	yes	characteristic
85	#103	—	118	1	no	—
86	C₂×Q₈	—	118	1	no	—
87	C₂²×C₄	—	59	1	no	—
88	C₂²×C₄	—	236	1	no	—
89	Q₁₆	—	118	1	no	—
90	Q₁₆	—	118	1	no	—
91	C₂×C₈	—	59	1	no	—
92	C₂×C₈	—	118	1	no	—
93	C₂×C₈	—	118	1	no	—
94	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	118	1	no	—
98	C₄⋊C₄	—	236	1	no	—
99	C₄⋊C₄	—	236	1	no	—
100	C₂²⋊C₄	—	236	1	no	—
101	C₁₆	—	2	1	no	—
102	C₁₆	—	118	1	no	—
103	C₂³	—	59	1	no	—
104	Q₈	—	236	1	no	—
105	Q₈	—	236	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	59	1	no	—
109	C₂×C₄	—	118	1	no	—
110	C₂×C₄	—	118	1	no	—
111	C₂×C₄	—	472	2	no	—
112	C₂×C₄	—	236	1	no	—
113	C₂×C₄	—	236	1	no	—
114	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
115	C₈	—	118	1	no	—
116	C₈	#11745685	1	1	yes	characteristic
117	C₈	#11745689	1	1	yes	characteristic
118	C₂²	—	59	2	no	—
119	C₂²	—	118	1	no	—
120	C₂²	—	118	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	—	472	1	no	—
126	C₄	—	472	1	no	—
127	C₄	#14273712	1	1	yes	characteristic
128	C₄	#14273759	1	1	yes	characteristic
129	C₂	—	118	2	no	—
130	C₂	#16837717	1	1	yes	characteristic
131	C₂	Q₃₂×D₅₉	1	1	yes	characteristic
132	C₂	#16837836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837644

C₂

yes

maximal, characteristic

#107×D₅₉

C₂

yes

maximal, characteristic

#16837227

C₂

yes

maximal, characteristic, complemented

#16837207

C₂

yes

maximal, characteristic, complemented

#16837205

C₂

yes

maximal, characteristic, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal, characteristic

C₅₉×#366

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273707

—

C₄⋊C₄×D₅₉

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273660

—

#14273650

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

Q₃₂

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×Q₃₂

yes

characteristic

C₂×C₅₉

#509

yes

characteristic

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

#358

—

#358

—

C₂²⋊C₁₆

—

C₅₉

#2236

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

—

C₂²⋊Q₈

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#107

—

118

—

#107

C₂×D₅₉

yes

characteristic

#107

C₂×D₅₉

yes

characteristic

#103

—

118

—

C₂×Q₈

—

118

—

C₂²×C₄

—

C₂²×C₄

—

236

—

Q₁₆

—

118

—

Q₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

100

C₂²⋊C₄

—

236

—

101

C₁₆

—

102

C₁₆

—

118

—

103

C₂³

—

104

Q₈

—

236

—

105

Q₈

—

236

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

118

—

110

C₂×C₄

—

118

—

111

C₂×C₄

—

472

—

112

C₂×C₄

—

236

—

113

C₂×C₄

—

236

—

114

C₂×C₄

D₄×D₅₉

yes

characteristic

115

C₈

—

118

—

116

C₈

#11745685

yes

characteristic

117

C₈

#11745689

yes

characteristic

118

C₂²

—

119

C₂²

—

118

—

120

C₂²

—

118

—

121

C₂²

D₈×D₅₉

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

118

—

125

C₄

—

472

—

126

C₄

—

472

—

127

C₄

#14273712

yes

characteristic

128

C₄

#14273759

yes

characteristic

129

C₂

—

118

—

130

C₂

#16837717

yes

characteristic

131

C₂

Q₃₂×D₅₉

yes

characteristic

132

C₂

#16837836

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#366	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#366	D₅₉
hypocenter	C₅₉	#2236
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#366

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
3. #16837227	C₂	semidirect
4. #16837207	C₂	semidirect
5. #16837205	C₂	semidirect
7. C₅₉×#366	C₂	semidirect
70. #366	D₅₉	semidirect
74. C₅₉	#2236	semidirect

H ≅ G/N

type

3. #16837227

C₂

semidirect

4. #16837207

C₂

semidirect

5. #16837205

C₂

semidirect

7. C₅₉×#366

C₂

semidirect

70. #366

D₅₉

semidirect

74. C₅₉

#2236

semidirect

Subgroups — #21426300

Conjugacy classes of subgroups

Complements