Group #21422119 — subgroups

Subgroups — #21422119

Order 7552.

154 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 35 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	#16837338	C₂	1	1	yes	maximal, characteristic
2	#16837126	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#342	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic, complemented
5	C₂×Dic₅₉⋊C₄	—	2	1	no	—
6	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
7	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₅₉⋊#95	C₄	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
10	C₂×C₂²⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
11	C₂⁴×C₅₉	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₉	—	4	1	no	—
13	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
15	C₂²×Dic₅₉	—	2	1	no	—
16	C₂²×Dic₅₉	—	2	1	no	—
17	C₂²⋊C₄×C₅₉	—	2	2	no	—
18	C₂²⋊C₄×C₅₉	—	4	1	no	—
19	C₂²⋊C₄×C₅₉	—	4	1	no	—
20	C₂²⋊Dic₅₉	—	4	2	no	—
21	C₄⋊Dic₅₉	—	4	1	no	—
22	Dic₅₉⋊C₄	—	4	2	no	—
23	C₂×C₅₉⋊C₈	—	4	1	no	—
24	C₂³×C₅₉	—	2	1	no	—
25	C₂³×C₅₉	—	2	1	no	—
26	C₂³×C₅₉	—	4	1	no	—
27	C₂³×C₅₉	—	4	1	no	—
28	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂³×C₅₉	SD₁₆	1	1	yes	characteristic
30	C₂³×C₅₉	Q₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅₉	—	2	1	no	—
32	C₂×C₄×C₅₉	—	2	1	no	—
33	C₂×C₄×C₅₉	—	4	1	no	—
34	C₂×C₄×C₅₉	—	4	1	no	—
35	C₂×C₄×C₅₉	—	4	2	no	—
36	C₂×C₄×C₅₉	—	8	1	no	—
37	C₂×C₄×C₅₉	—	8	1	no	—
38	C₂×Dic₅₉	—	2	2	no	—
39	C₂×Dic₅₉	—	4	1	no	—
40	C₂×Dic₅₉	—	4	1	no	—
41	C₂×Dic₅₉	—	4	1	no	—
42	C₂×Dic₅₉	—	4	1	no	—
43	C₂×Dic₅₉	—	4	1	no	—
44	C₅₉⋊C₈	—	8	1	no	—
45	C₂²×C₅₉	—	2	1	no	—
46	C₂²×C₅₉	—	2	1	no	—
47	C₂²×C₅₉	—	2	1	no	—
48	C₂²×C₅₉	—	2	1	no	—
49	C₂²×C₅₉	—	4	1	no	—
50	C₂²×C₅₉	—	4	1	no	—
51	C₂²×C₅₉	—	4	1	no	—
52	C₂²×C₅₉	—	4	1	no	—
53	C₂²×C₅₉	—	8	1	no	—
54	C₂²×C₅₉	#99	1	1	yes	characteristic
55	C₂²×C₅₉	#103	1	1	yes	characteristic
56	C₂²×C₅₉	#104	1	1	yes	characteristic
57	C₄×C₅₉	—	4	1	no	—
58	C₄×C₅₉	—	4	1	no	—
59	C₄×C₅₉	—	8	2	no	—
60	Dic₅₉	—	4	2	no	—
61	Dic₅₉	—	8	1	no	—
62	#1352	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	—	2	1	no	—
64	C₂×C₅₉	—	2	1	no	—
65	C₂×C₅₉	—	4	1	no	—
66	C₂×C₅₉	—	4	1	no	—
67	C₂×C₅₉	#328	1	1	yes	characteristic
68	C₂×C₅₉	#352	1	1	yes	characteristic
69	C₂×C₅₉	#353	1	1	yes	characteristic
70	#399	—	59	1	no	—
71	#342	D₅₉	1	1	yes	characteristic
72	#323	—	59	1	no	—
73	C₅₉	#1352	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊C₄	—	59	1	no	—
75	C₂×C₄⋊C₄	—	118	1	no	—
76	C₂×C₂²⋊C₄	—	2	1	no	—
77	C₂×C₂²⋊C₄	—	118	1	no	—
78	C₂×C₂²⋊C₄	C₂×D₅₉	1	1	yes	characteristic
79	C₂²⋊C₈	—	118	1	no	—
80	#95	—	59	1	no	—
81	C₂⁴	C₂²⋊D₅₉	1	1	yes	characteristic
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	—	118	1	no	—
84	C₂²×C₄	—	118	1	no	—
85	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
86	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
87	C₂×C₈	—	236	1	no	—
88	C₄⋊C₄	—	236	1	no	—
89	C₄⋊C₄	—	236	2	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₂²⋊C₄	—	4	1	no	—
92	C₂²⋊C₄	—	4	1	no	—
93	C₂²⋊C₄	—	236	2	no	—
94	C₂³	—	2	1	no	—
95	C₂³	—	2	1	no	—
96	C₂³	—	4	1	no	—
97	C₂³	—	4	1	no	—
98	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
99	C₂³	#11745670	1	1	yes	characteristic
100	C₂³	#11745672	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	118	2	no	—
109	C₂×C₄	—	236	1	no	—
110	C₂×C₄	—	236	1	no	—
111	C₂×C₄	—	236	1	no	—
112	C₂×C₄	—	236	1	no	—
113	C₂×C₄	—	236	1	no	—
114	C₈	—	472	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	8	1	no	—
124	C₂²	#14273641	1	1	yes	characteristic
125	C₂²	#14273650	1	1	yes	characteristic
126	C₂²	#14273655	1	1	yes	characteristic
127	C₄	—	4	1	no	—
128	C₄	—	4	1	no	—
129	C₄	—	8	2	no	—
130	C₄	—	472	1	no	—
131	C₄	—	236	2	no	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	—	4	1	no	—
135	C₂	—	4	1	no	—
136	C₂	#16837138	1	1	yes	characteristic
137	C₂	#16837191	1	1	yes	characteristic
138	C₂	#16837194	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837338

C₂

yes

maximal, characteristic

#16837126

C₂

yes

maximal, characteristic

C₅₉×#342

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

C₄

yes

characteristic, complemented

C₂×Dic₅₉⋊C₄

—

C₂×C₂²⋊Dic₅₉

—

C₂²⋊(C₅₉⋊C₈)

—

C₅₉⋊#95

C₄

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₅₉

—

C₂×C₂²⋊C₄×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂³×C₅₉

SD₁₆

yes

characteristic

C₂³×C₅₉

Q₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

Dic₅₉

—

#1352

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

#323

—

C₅₉

#1352

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

118

—

C₂×C₂²⋊C₄

C₂×D₅₉

yes

characteristic

C₂²⋊C₈

—

118

—

#95

—

C₂⁴

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂×C₈

—

236

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

236

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

C₂³

#11745670

yes

characteristic

100

C₂³

#11745672

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

118

—

109

C₂×C₄

—

236

—

110

C₂×C₄

—

236

—

111

C₂×C₄

—

236

—

112

C₂×C₄

—

236

—

113

C₂×C₄

—

236

—

114

C₈

—

472

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#14273641

yes

characteristic

125

C₂²

#14273650

yes

characteristic

126

C₂²

#14273655

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

472

—

131

C₄

—

236

—

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

#16837138

yes

characteristic

137

C₂

#16837191

yes

characteristic

138

C₂

#16837194

yes

characteristic

	subgroup	G/H
center	C₂²	#14273641
commutator	C₂²×C₄×C₅₉	C₂×C₄
Frattini	C₂×C₂²⋊C₄	C₂×D₅₉
Fitting	C₅₉×#342	C₂
socle	C₂²×C₅₉	#99
hypercenter	#342	D₅₉
hypocenter	C₅₉	#1352
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #342

subgroup

G/H

center

C₂²

#14273641

commutator

C₂²×C₄×C₅₉

C₂×C₄

Frattini

C₂×C₂²⋊C₄

C₂×D₅₉

Fitting

C₅₉×#342

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #342

N	H ≅ G/N	type
4. C₂×C₄⋊Dic₅₉	C₄	semidirect
8. C₅₉⋊#95	C₄	semidirect
73. C₅₉	#1352	semidirect

H ≅ G/N

type

4. C₂×C₄⋊Dic₅₉

C₄

semidirect

8. C₅₉⋊#95

C₄

semidirect

73. C₅₉

#1352

semidirect

Subgroups — #21422119

Conjugacy classes of subgroups

Complements