Group #24906335 — subgroups

Subgroups — #24906335

Order 10624.

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

normal subgroups only

This group has 152 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	#19127832	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#19127452	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#19127423	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
9	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
10	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	#14581703	—	2	1	no	—
13	C₄×C₈₃⋊C₈	—	2	1	no	—
14	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₈₃	C₄	1	1	yes	characteristic
17	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
19	#107×C₈₃	C₄	1	1	yes	characteristic
20	#103×C₈₃	C₄	1	1	yes	characteristic
21	#103×C₈₃	C₄	1	1	yes	characteristic
22	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×C₈₃⋊C₈	—	2	1	no	—
24	C₂×C₈₃⋊C₈	—	2	1	no	—
25	C₂×C₈₃⋊C₈	—	2	1	no	—
26	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
28	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	Q₁₆×C₈₃	—	2	2	no	—
31	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₈₃	—	2	1	no	—
34	C₄⋊C₄×C₈₃	—	2	1	no	—
35	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₈₃	—	2	1	no	—
38	C₄²×C₈₃	—	2	1	no	—
39	C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	Q₈×C₈₃	—	2	1	no	—
41	Q₈×C₈₃	—	2	1	no	—
42	Q₈×C₈₃	—	2	1	no	—
43	Q₈×C₈₃	M₁₆	1	2	yes	—
44	C₂×C₄×C₈₃	—	2	1	no	—
45	C₂×C₄×C₈₃	—	2	1	no	—
46	C₂×C₄×C₈₃	—	2	1	no	—
47	C₂×C₄×C₈₃	—	2	1	no	—
48	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
51	C₈×C₈₃	—	2	1	no	—
52	C₈×C₈₃	—	2	1	no	—
53	C₈₃⋊C₈	—	2	2	no	—
54	C₈₃⋊C₈	—	2	1	no	—
55	C₈₃⋊C₈	—	2	1	no	—
56	C₈₃⋊C₈	—	4	1	no	—
57	C₈₃⋊C₈	—	4	1	no	—
58	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₈₃	—	2	2	no	—
60	C₄×C₈₃	—	2	1	no	—
61	C₄×C₈₃	—	4	1	no	—
62	C₄×C₈₃	—	4	1	no	—
63	C₄×C₈₃	—	4	1	no	—
64	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
65	C₄×C₈₃	#131	1	1	yes	characteristic
66	C₄×C₈₃	#137	1	1	yes	characteristic
67	C₄×C₈₃	#137	1	1	yes	characteristic
68	C₂×C₈₃	#435	1	1	yes	characteristic
69	C₂×C₈₃	#441	1	1	yes	characteristic
70	C₂×C₈₃	#443	1	1	yes	characteristic
71	#1587	—	83	1	no	maximal, Sylow
72	C₈₃	#1587	1	1	yes	characteristic, complemented, Sylow
73	#446	—	83	1	no	—
74	C₈×Q₈	—	83	1	no	—
75	C₄×Q₁₆	D₈₃	1	1	yes	characteristic
76	#334	—	83	1	no	—
77	Q₈⋊C₈	—	83	1	no	—
78	Q₈⋊C₈	—	83	1	no	—
79	#322	—	83	1	no	—
80	C₂×Q₁₆	Dic₈₃	1	1	yes	characteristic
81	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
83	#107	Dic₈₃	1	1	yes	characteristic
84	C₄⋊C₈	—	83	1	no	—
85	C₄⋊C₈	—	83	1	no	—
86	C₄⋊C₈	—	166	1	no	—
87	C₄⋊C₈	—	166	1	no	—
88	#103	Dic₈₃	1	1	yes	characteristic
89	#103	Dic₈₃	1	1	yes	characteristic
90	#97	—	166	1	no	—
91	C₄×C₈	—	83	1	no	—
92	C₄×C₈	—	83	1	no	—
93	C₄×C₈	—	166	1	no	—
94	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
95	C₂×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
96	C₂×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
97	Q₁₆	—	2	2	no	—
98	C₂×C₈	—	83	1	no	—
99	C₂×C₈	—	83	1	no	—
100	C₂×C₈	—	166	1	no	—
101	C₂×C₈	—	166	1	no	—
102	C₂×C₈	—	166	1	no	—
103	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	C₂×Dic₈₃	1	1	yes	characteristic
108	C₄⋊C₄	C₂×Dic₈₃	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	C₂²×D₈₃	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	#13050574	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
121	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
122	C₂×C₄	#13050589	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	332	1	no	—
126	C₈	—	332	1	no	—
127	C₈	—	166	2	no	—
128	C₈	—	166	1	no	—
129	C₈	—	166	1	no	—
130	C₂²	D₄×Dic₈₃	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	4	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	4	1	no	—
136	C₄	C₂×#13050574	1	1	yes	characteristic
137	C₄	#14581799	1	1	yes	characteristic
138	C₄	#14581824	1	1	yes	characteristic
139	C₄	#14581824	1	1	yes	characteristic
140	C₂	#19127773	1	1	yes	characteristic
141	C₂	#19127808	1	1	yes	characteristic
142	C₂	#19127818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127832

C₂

yes

maximal, characteristic

Q₈×C₈₃⋊C₈

C₂

yes

maximal, characteristic

#19127452

C₂

yes

maximal, characteristic

Q₈⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

Q₈⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

#19127423

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₈₃

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

#14581703

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₄

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

characteristic

#107×C₈₃

C₄

yes

characteristic

#103×C₈₃

C₄

yes

characteristic

#103×C₈₃

C₄

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×Q₈×C₈₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₈₃

C₂×C₄

yes

characteristic

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#446

—

C₈×Q₈

—

C₄×Q₁₆

D₈₃

yes

characteristic

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₂×Q₁₆

Dic₈₃

yes

characteristic

C₄×Q₈

C₂×D₈₃

yes

characteristic

C₄×Q₈

C₂×D₈₃

yes

characteristic

#107

Dic₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

—

166

—

#103

Dic₈₃

yes

characteristic

#103

Dic₈₃

yes

characteristic

#97

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

C₂×Dic₈₃

yes

characteristic

C₂×Q₈

C₂×Dic₈₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

166

—

101

C₂×C₈

—

166

—

102

C₂×C₈

—

166

—

103

C₂×C₈

C₂×Dic₈₃

yes

characteristic

104

C₂×C₈

C₂×Dic₈₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×Dic₈₃

yes

characteristic

108

C₄⋊C₄

C₂×Dic₈₃

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₈₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

#13050574

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

121

C₂×C₄

D₄×D₈₃

yes

characteristic

122

C₂×C₄

#13050589

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

332

—

126

C₈

—

332

—

127

C₈

—

166

—

128

C₈

—

166

—

129

C₈

—

166

—

130

C₂²

D₄×Dic₈₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×#13050574

yes

characteristic

137

C₄

#14581799

yes

characteristic

138

C₄

#14581824

yes

characteristic

139

C₄

#14581824

yes

characteristic

140

C₂

#19127773

yes

characteristic

141

C₂

#19127808

yes

characteristic

142

C₂

#19127818

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	C₄×Q₁₆×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₄×Q₁₆	D₈₃
hypocenter	C₈₃	#1587
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

C₄×Q₁₆×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₄×D₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₄×Q₁₆

D₈₃

hypocenter

C₈₃

#1587

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
72. C₈₃	#1587	semidirect

H ≅ G/N

type

72. C₈₃

#1587

semidirect

Subgroups — #24906335

Conjugacy classes of subgroups

Complements