Group #21422912 — subgroups

Subgroups — #21422912

Order 7552.

160 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 71 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	Q₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#16837526	C₂	1	1	yes	maximal, characteristic
3	C₄×#11745672	C₂	1	1	yes	maximal, characteristic
4	#16837152	C₂	1	1	yes	maximal, characteristic
5	#16837134	C₂	1	1	yes	maximal, characteristic
6	#16837124	C₂	1	1	yes	maximal, characteristic
7	C₅₉×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745672	C₄	1	1	yes	characteristic
9	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
10	#14273660	C₄	1	1	yes	characteristic
11	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	Dic₅₉⋊C₈	—	2	1	no	—
14	#14273651	C₄	1	1	yes	characteristic
15	#14273650	C₄	1	1	yes	characteristic
16	C₄×C₅₉⋊C₈	—	2	1	no	—
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₅₉	—	2	1	no	—
20	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₉	—	2	1	no	—
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₄⋊C₄×C₅₉	—	2	1	no	—
28	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₉	—	2	1	no	—
30	C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	#11745672	C₈	1	4	yes	complemented
32	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
33	Dic₅₉⋊C₄	—	2	1	no	—
34	C₄×Dic₅₉	—	2	1	no	—
35	C₂×C₅₉⋊C₈	—	2	1	no	—
36	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	Q₈×C₅₉	—	2	1	no	—
41	Q₈×C₅₉	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₅₉	—	2	1	no	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂×Dic₅₉	—	2	1	no	—
48	C₂×Dic₅₉	—	2	1	no	—
49	Dic₁₁₈	—	2	1	no	—
50	Dic₁₁₈	C₂×C₈	1	2	yes	—
51	C₈×C₅₉	—	4	1	no	—
52	C₈×C₅₉	—	4	1	no	—
53	C₅₉⋊C₈	—	2	1	no	—
54	C₅₉⋊C₈	—	2	1	no	—
55	C₅₉⋊C₈	—	2	1	no	—
56	C₅₉⋊C₈	—	4	1	no	—
57	C₅₉⋊C₈	C₂×C₈	1	2	yes	—
58	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₅₉	—	2	2	no	—
60	C₄×C₅₉	—	2	1	no	—
61	C₄×C₅₉	—	4	1	no	—
62	C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₅₉	#131	1	1	yes	characteristic
64	C₄×C₅₉	#137	1	1	yes	characteristic
65	C₄×C₅₉	#137	1	1	yes	characteristic
66	Dic₅₉	—	2	2	no	—
67	Dic₅₉	—	4	1	no	—
68	#1579	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	C₈×D₄	1	1	yes	characteristic
70	C₂×C₅₉	#441	1	1	yes	characteristic
71	C₂×C₅₉	#444	1	1	yes	characteristic
72	C₈×Q₈	—	59	1	no	—
73	C₈×Q₈	—	59	1	no	—
74	C₄×Q₁₆	—	59	1	no	—
75	#334	—	59	1	no	—
76	Q₈⋊C₈	—	59	1	no	—
77	Q₈⋊C₈	D₅₉	1	1	yes	characteristic
78	#322	—	59	1	no	—
79	C₅₉	#1579	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	59	1	no	—
81	C₄×Q₈	—	59	1	no	—
82	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
83	#107	—	59	1	no	—
84	C₄⋊C₈	—	59	1	no	—
85	C₄⋊C₈	—	118	1	no	—
86	C₄⋊C₈	—	118	1	no	—
87	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
88	#103	—	59	1	no	—
89	#103	—	59	1	no	—
90	C₄×C₈	—	59	1	no	—
91	C₄×C₈	—	59	1	no	—
92	C₄×C₈	—	118	1	no	—
93	C₄×C₈	—	118	1	no	—
94	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
95	C₂×Q₈	—	59	1	no	—
96	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
97	Q₁₆	—	59	4	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	59	1	no	—
101	C₂×C₈	—	59	1	no	—
102	C₂×C₈	—	59	1	no	—
103	C₂×C₈	—	59	1	no	—
104	C₂×C₈	—	118	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	59	1	no	—
107	C₄⋊C₄	—	118	1	no	—
108	C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	118	1	no	—
111	C₄²	C₂²×D₅₉	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	59	2	no	—
114	Q₈	—	118	1	no	—
115	Q₈	C₈×D₅₉	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	118	1	no	—
119	C₂×C₄	—	118	1	no	—
120	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
121	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
122	C₂×C₄	#11745687	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	59	2	no	—
126	C₈	—	118	1	no	—
127	C₈	—	118	1	no	—
128	C₈	—	118	1	no	—
129	C₈	—	236	1	no	—
130	C₂²	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	118	2	no	—
135	C₄	—	236	1	no	—
136	C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
137	C₄	#14273735	1	1	yes	characteristic
138	C₄	#14273756	1	1	yes	characteristic
139	C₄	#14273759	1	1	yes	characteristic
140	C₂	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
141	C₂	#16837506	1	1	yes	characteristic
142	C₂	#16837521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic

#16837526

C₂

yes

maximal, characteristic

C₄×#11745672

C₂

yes

maximal, characteristic

#16837152

C₂

yes

maximal, characteristic

#16837134

C₂

yes

maximal, characteristic

#16837124

C₂

yes

maximal, characteristic

C₅₉×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₂×#11745672

C₄

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273660

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273651

C₄

yes

characteristic

#14273650

C₄

yes

characteristic

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₅₉

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×Dic₁₁₈

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745672

C₈

yes

complemented

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

C₂×C₈

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂×C₈

yes

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²×C₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1579

—

maximal, Sylow

C₂×C₅₉

C₈×D₄

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

D₅₉

yes

characteristic

#322

—

C₅₉

#1579

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

118

—

C₄×C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₄×D₅₉

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

118

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

118

—

108

C₄⋊C₄

C₄×D₅₉

yes

characteristic

109

C₄²

—

110

C₄²

—

118

—

111

C₄²

C₂²×D₅₉

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

118

—

115

Q₈

C₈×D₅₉

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

118

—

119

C₂×C₄

—

118

—

120

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

121

C₂×C₄

D₄×D₅₉

yes

characteristic

122

C₂×C₄

#11745687

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

118

—

127

C₈

—

118

—

128

C₈

—

118

—

129

C₈

—

236

—

130

C₂²

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

118

—

135

C₄

—

236

—

136

C₄

C₂×C₈×D₅₉

yes

characteristic

137

C₄

#14273735

yes

characteristic

138

C₄

#14273756

yes

characteristic

139

C₄

#14273759

yes

characteristic

140

C₂

(C₂²⋊D₅₉)⋊C₈

yes

characteristic

141

C₂

#16837506

yes

characteristic

142

C₂

#16837521

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₄×C₅₉	C₂²×C₈
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×Q₈⋊C₈	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	Q₈⋊C₈	D₅₉
hypocenter	C₅₉	#1579
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₄×C₅₉

C₂²×C₈

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×Q₈⋊C₈

C₂

socle

C₂²×C₅₉

C₄×D₄

hypercenter

Q₈⋊C₈

D₅₉

hypocenter

C₅₉

#1579

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
31. #11745672	C₈	semidirect
79. C₅₉	#1579	semidirect

H ≅ G/N

type

31. #11745672

C₈

semidirect

79. C₅₉

#1579

semidirect

Subgroups — #21422912

Conjugacy classes of subgroups

Complements