Group #21425032 — subgroups

Subgroups — #21425032

Order 7552.

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

normal subgroups only

This group has 292 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	C₂×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16837156	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₁₈	C₄	1	1	yes	characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₄×Dic₁₁₈	—	2	2	no	—
8	C₄×Dic₁₁₈	—	2	2	no	—
9	C₂×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
10	C₂×Dic₅₉⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
12	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
13	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
20	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₁₈	—	2	2	no	—
22	C₂×Dic₁₁₈	—	2	2	no	—
23	C₂×Dic₁₁₈	C₈	1	4	yes	complemented
24	C₂×C₈×C₅₉	—	2	2	no	—
25	C₂×C₈×C₅₉	—	2	2	no	—
26	C₂×C₈×C₅₉	Q₈	1	4	yes	complemented
27	C₄²×C₅₉	D₄	1	2	yes	—
28	C₄²×C₅₉	D₄	1	2	yes	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	Dic₅₉⋊C₄	—	2	4	no	—
31	C₄×Dic₅₉	—	2	2	no	—
32	C₄×Dic₅₉	—	2	2	no	—
33	C₂×C₅₉⋊C₈	—	2	2	no	—
34	C₂×C₅₉⋊C₈	—	2	4	no	—
35	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₉	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₅₉	M₁₆	1	2	yes	—
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₅₉	#41	1	2	yes	—
45	C₂×Dic₅₉	—	2	2	no	—
46	C₂×Dic₅₉	—	2	2	no	—
47	C₂×Dic₅₉	—	2	2	no	—
48	C₂×Dic₅₉	—	2	4	no	—
49	C₂×Dic₅₉	C₂×C₈	1	4	yes	—
50	Dic₁₁₈	—	2	8	no	—
51	C₈×C₅₉	—	2	4	no	—
52	C₅₉⋊C₈	—	4	2	no	—
53	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₅₉	C₄×Q₈	1	2	yes	—
55	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₅₉	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₅₉	#131	1	2	yes	—
58	C₄×C₅₉	—	2	2	no	—
59	C₄×C₅₉	C₂²⋊C₈	1	4	yes	—
60	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₅₉	#124	1	1	yes	characteristic
62	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₅₉	—	2	4	no	—
64	Dic₅₉	—	4	2	no	—
65	#1984	—	59	1	no	maximal, Sylow
66	C₂×C₅₉	#391	1	1	yes	characteristic
67	C₂×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₅₉	#408	1	1	yes	characteristic
69	C₂×C₅₉	C₈×Q₈	1	2	yes	—
70	C₂×C₅₉	#446	1	2	yes	—
71	C₂×C₄×Q₈	—	59	1	no	—
72	C₂×C₄⋊C₈	—	59	1	no	—
73	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
74	#336	—	59	4	no	—
75	C₅₉	#1984	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	59	1	no	—
77	C₂²×C₈	—	59	2	no	—
78	C₂²×C₈	C₂×D₅₉	1	2	yes	—
79	C₄×Q₈	—	118	2	no	—
80	C₄×Q₈	—	118	2	no	—
81	C₂×C₄⋊C₄	—	59	2	no	—
82	C₂×C₄⋊C₄	—	59	1	no	—
83	C₂×C₄²	—	59	2	no	—
84	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
85	C₄⋊C₈	—	118	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₂×Q₈	—	59	4	no	—
88	C₂×Q₈	—	118	2	no	—
89	C₂×Q₈	—	118	2	no	—
90	C₂²×C₄	—	59	2	no	—
91	C₂²×C₄	—	59	2	no	—
92	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	118	2	no	—
98	C₂×C₈	—	118	4	no	—
99	C₂×C₈	Dic₁₁₈	1	4	yes	complemented
100	C₄⋊C₄	—	118	4	no	—
101	C₄⋊C₄	—	118	2	no	—
102	C₄²	—	118	2	no	—
103	C₄²	—	118	2	no	—
104	C₄²	D₂₃₆	1	2	yes	—
105	C₄²	C₂²⋊D₅₉	1	2	yes	—
106	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
107	Q₈	—	118	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	59	4	no	—
110	C₂×C₄	—	118	2	no	—
111	C₂×C₄	—	118	2	no	—
112	C₂×C₄	—	118	2	no	—
113	C₂×C₄	—	118	4	no	—
114	C₂×C₄	C₈×D₅₉	1	2	yes	—
115	C₂×C₄	#11745659	1	2	yes	—
116	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
117	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
118	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
119	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
120	C₂×C₄	#11745685	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	236	2	no	—
124	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
125	C₂²	C₄×Dic₁₁₈	1	2	yes	—
126	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
127	C₂²	C₂×#11745659	1	1	yes	characteristic
128	C₂²	#14273733	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	118	4	no	—
131	C₄	—	236	2	no	—
132	C₄	(C₂×D₅₉)⋊C₈	1	4	yes	—
133	C₄	#14273706	1	2	yes	—
134	C₄	#14273715	1	1	yes	characteristic
135	C₄	#14273727	1	1	yes	characteristic
136	C₂	#16837308	1	1	yes	characteristic
137	C₂	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
138	C₂	#16837368	1	1	yes	characteristic
139	C₂	C₈×Dic₁₁₈	1	2	yes	—
140	C₂	#16837532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₁₈

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837156

C₂

yes

maximal

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

C₂×C₄⋊Dic₅₉

C₄

yes

characteristic

C₂×Dic₅₉⋊C₄

C₄

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₈

yes

complemented

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

Q₈

yes

complemented

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×C₈

yes

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂²×C₅₉

C₂×M₁₆

yes

characteristic

C₂²×C₅₉

#131

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊C₈

yes

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₅₉

#408

yes

characteristic

C₂×C₅₉

C₈×Q₈

yes

—

C₂×C₅₉

#446

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₅₉

yes

characteristic

#336

—

C₅₉

#1984

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

—

C₄×Q₈

—

118

—

C₄×Q₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

Dic₁₁₈

yes

complemented

100

C₄⋊C₄

—

118

—

101

C₄⋊C₄

—

118

—

102

C₄²

—

118

—

103

C₄²

—

118

—

104

C₄²

D₂₃₆

yes

—

105

C₄²

C₂²⋊D₅₉

yes

—

106

C₂³

C₂×C₄×D₅₉

yes

characteristic

107

Q₈

—

118

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

118

—

111

C₂×C₄

—

118

—

112

C₂×C₄

—

118

—

113

C₂×C₄

—

118

—

114

C₂×C₄

C₈×D₅₉

yes

—

115

C₂×C₄

#11745659

yes

—

116

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

117

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

118

C₂×C₄

C₂×Dic₁₁₈

yes

—

119

C₂×C₄

C₂×D₂₃₆

yes

characteristic

120

C₂×C₄

#11745685

yes

—

121

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

122

C₈

—

123

C₈

—

236

—

124

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

125

C₂²

C₄×Dic₁₁₈

yes

—

126

C₂²

C₂×C₈×D₅₉

yes

characteristic

127

C₂²

C₂×#11745659

yes

characteristic

128

C₂²

#14273733

yes

—

129

C₄

—

130

C₄

—

118

—

131

C₄

—

236

—

132

C₄

(C₂×D₅₉)⋊C₈

yes

—

133

C₄

#14273706

yes

—

134

C₄

#14273715

yes

characteristic

135

C₄

#14273727

yes

characteristic

136

C₂

#16837308

yes

characteristic

137

C₂

C₂×(C₂×D₅₉)⋊C₈

yes

characteristic

138

C₂

#16837368

yes

characteristic

139

C₂

C₈×Dic₁₁₈

yes

—

140

C₂

#16837532

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₈
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₄
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#1984
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₉

commutator

C₂²×C₅₉

C₂²×C₈

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂²×C₄

hypercenter

C₂×C₄×C₈

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
23. C₂×Dic₁₁₈	C₈	semidirect
26. C₂×C₈×C₅₉	Q₈	semidirect
75. C₅₉	#1984	semidirect
99. C₂×C₈	Dic₁₁₈	semidirect

H ≅ G/N

type

23. C₂×Dic₁₁₈

C₈

semidirect

26. C₂×C₈×C₅₉

Q₈

semidirect

75. C₅₉

#1984

semidirect

99. C₂×C₈

Dic₁₁₈

semidirect

Subgroups — #21425032

Conjugacy classes of subgroups

Complements