Group #21432368 — subgroups

Subgroups — #21432368

Order 7552.

356 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16837344	C₂	1	4	yes	maximal
2	#16837341	C₂	1	2	yes	maximal
3	#16837323	C₂	1	4	yes	maximal
4	#16837322	C₂	1	2	yes	maximal
5	#16837268	C₂	1	2	yes	maximal
6	C₅₉×#382	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
11	C₂×Dic₅₉⋊C₄	C₂²	1	4	yes	—
12	C₅₉⋊#95	C₂²	1	4	yes	—
13	C₅₉⋊#95	C₂²	1	4	yes	—
14	C₅₉⋊#95	C₂²	1	2	yes	—
15	C₅₉⋊#95	C₂²	1	4	yes	—
16	C₅₉⋊#95	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
20	#95×C₅₉	C₂²	1	4	yes	—
21	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
22	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
23	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₉	C₂³	1	4	yes	—
26	C₂²×Dic₅₉	C₂³	1	4	yes	—
27	C₂×Dic₁₁₈	—	4	4	no	—
28	C₂×Dic₁₁₈	—	4	2	no	—
29	C₄⋊C₄×C₅₉	—	4	1	no	—
30	C₄⋊C₄×C₅₉	—	4	1	no	—
31	C₄²×C₅₉	Q₈	1	4	yes	complemented
32	C₄⋊Dic₅₉	—	4	2	no	—
33	C₄⋊Dic₅₉	—	4	2	no	—
34	C₄⋊Dic₅₉	—	4	2	no	—
35	Dic₅₉⋊C₄	—	4	4	no	—
36	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	—	2	1	no	—
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	2	1	no	—
41	C₂×C₄×C₅₉	—	2	4	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	2	4	no	—
45	C₂×C₄×C₅₉	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
47	C₂×Dic₅₉	—	2	4	no	—
48	C₂×Dic₅₉	—	2	4	no	—
49	C₂×Dic₅₉	—	2	4	no	—
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	—	2	4	no	—
52	C₂×Dic₅₉	—	2	4	no	—
53	Dic₁₁₈	—	4	8	no	—
54	C₂²×C₅₉	C₂²×Q₈	1	1	yes	characteristic
55	C₂²×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
56	C₂²×C₅₉	H(5,𝔽₂)	1	2	yes	—
57	C₂²×C₅₉	2₋¹⁺⁴	1	2	yes	—
58	C₂²×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
59	C₄×C₅₉	—	2	4	no	—
60	C₄×C₅₉	—	2	2	no	—
61	C₄×C₅₉	—	4	2	no	—
62	C₄×C₅₉	—	4	2	no	—
63	Dic₅₉	—	4	4	no	—
64	Dic₅₉	—	4	4	no	—
65	#2844	—	59	1	no	maximal, Sylow
66	C₂×C₅₉	#544	1	2	yes	—
67	C₂×C₅₉	#544	1	1	yes	characteristic
68	C₂×C₅₉	H(3,𝔽₄)	1	1	yes	characteristic
69	C₂×C₅₉	#563	1	1	yes	characteristic
70	C₂×C₅₉	#563	1	2	yes	—
71	#400	—	59	2	no	—
72	#400	—	59	4	no	—
73	#395	—	59	2	no	—
74	#395	—	59	4	no	—
75	#382	—	59	2	no	—
76	#382	D₅₉	1	1	yes	characteristic
77	C₅₉	#2844	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	—	59	2	no	—
79	C₂×C₄⋊C₄	—	59	2	no	—
80	C₂×C₄⋊C₄	—	59	2	no	—
81	C₂×C₄⋊C₄	—	59	4	no	—
82	C₂×C₄⋊C₄	—	59	2	no	—
83	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
85	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
86	#95	—	59	4	no	—
87	#95	—	59	4	no	—
88	#95	—	59	2	no	—
89	#95	—	59	4	no	—
90	#95	—	59	2	no	—
91	#95	C₂×D₅₉	1	4	yes	—
92	C₂×Q₈	—	236	2	no	—
93	C₂×Q₈	—	236	4	no	—
94	C₂²×C₄	—	59	4	no	—
95	C₂²×C₄	—	59	4	no	—
96	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
97	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
98	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	236	4	no	—
103	C₄⋊C₄	—	236	2	no	—
104	C₄⋊C₄	—	236	2	no	—
105	C₄⋊C₄	—	236	2	no	—
106	C₄²	Dic₁₁₈	1	4	yes	complemented
107	C₂³	C₂³×D₅₉	1	1	yes	characteristic
108	Q₈	—	236	8	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	118	4	no	—
118	C₂×C₄	—	118	4	no	—
119	C₂×C₄	—	118	4	no	—
120	C₂×C₄	—	118	4	no	—
121	C₂×C₄	—	118	4	no	—
122	C₂×C₄	—	118	4	no	—
123	C₂×C₄	C₂×Dic₁₁₈	1	4	yes	—
124	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
125	C₂²	C₂²×Dic₁₁₈	1	1	yes	characteristic
126	C₂²	#14273772	1	1	yes	characteristic
127	C₂²	#14273773	1	2	yes	—
128	C₂²	#14273774	1	2	yes	—
129	C₂²	#14273775	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	236	4	no	—
135	C₄	—	236	4	no	—
136	C₂	#16838030	1	2	yes	—
137	C₂	#16838033	1	1	yes	characteristic
138	C₂	#16838144	1	1	yes	characteristic
139	C₂	#16838152	1	1	yes	characteristic
140	C₂	#16838156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837344

C₂

yes

maximal

#16837341

C₂

yes

maximal

#16837323

C₂

yes

maximal

#16837322

C₂

yes

maximal

#16837268

C₂

yes

maximal

C₅₉×#382

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×Dic₅₉⋊C₄

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

Q₈

yes

complemented

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₂³×C₅₉

C₂⁴

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₂²×C₅₉

C₂²×Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

2₋¹⁺⁴

yes

—

C₂²×C₅₉

2₋¹⁺⁴

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

Dic₅₉

—

#2844

—

maximal, Sylow

C₂×C₅₉

#544

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

—

—

—

D₅₉

yes

characteristic

C₅₉

#2844

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄²

C₂×D₅₉

yes

—

—

—

—

—

C₂×D₅₉

yes

—

C₂×Q₈

—

236

—

C₂×Q₈

—

236

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

236

—

103

C₄⋊C₄

—

236

—

104

C₄⋊C₄

—

236

—

105

C₄⋊C₄

—

236

—

106

C₄²

Dic₁₁₈

yes

complemented

107

C₂³

C₂³×D₅₉

yes

characteristic

108

Q₈

—

236

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

—

118

C₂×C₄

—

118

—

119

C₂×C₄

—

118

—

120

C₂×C₄

—

118

—

121

C₂×C₄

—

118

—

122

C₂×C₄

—

118

—

123

C₂×C₄

C₂×Dic₁₁₈

yes

—

124

C₂×C₄

C₂×Dic₁₁₈

yes

—

125

C₂²

C₂²×Dic₁₁₈

yes

characteristic

126

C₂²

#14273772

yes

characteristic

127

C₂²

#14273773

yes

—

128

C₂²

#14273774

yes

—

129

C₂²

#14273775

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

236

—

135

C₄

—

236

—

136

C₂

#16838030

yes

—

137

C₂

#16838033

yes

characteristic

138

C₂

#16838144

yes

characteristic

139

C₂

#16838152

yes

characteristic

140

C₂

#16838156

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator, socle	C₂³×C₅₉	C₂⁴
Fitting	C₅₉×#382	C₂
hypercenter	#382	D₅₉
hypocenter	C₅₉	#2844
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator, socle

C₂³×C₅₉

C₂⁴

Fitting

C₅₉×#382

C₂

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
31. C₄²×C₅₉	Q₈	semidirect
77. C₅₉	#2844	semidirect
106. C₄²	Dic₁₁₈	semidirect

H ≅ G/N

type

31. C₄²×C₅₉

Q₈

semidirect

77. C₅₉

#2844

semidirect

106. C₄²

Dic₁₁₈

semidirect

Subgroups — #21432368

Conjugacy classes of subgroups

Complements