Group #24905369 — subgroups

Subgroups — #24905369

Order 10624.

200 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19127665	C₂	1	1	yes	maximal, characteristic
2	#19127665	C₂	1	1	yes	maximal, characteristic
3	#380×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13050574	—	2	1	no	—
5	C₂×#13050574	C₄	1	2	yes	complemented
6	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
9	C₂²⋊(C₈₃⋊C₈)	C₄	1	2	yes	complemented
10	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
12	C₂×C₂²⋊C₄×C₈₃	C₄	1	2	yes	—
13	#95×C₈₃	—	2	1	no	—
14	#13050574	—	4	2	no	—
15	#13050574	—	4	2	no	—
16	C₂×C₈₃⋊C₈	—	2	2	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂⁴×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₈₃	—	2	2	no	—
20	C₂²×C₄×C₈₃	—	4	2	no	—
21	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
27	C₄⋊C₄×C₈₃	—	4	2	no	—
28	C₂²⋊C₄×C₈₃	—	4	2	no	—
29	C₂³×C₈₃	—	2	1	no	—
30	C₂³×C₈₃	—	2	1	no	—
31	C₂³×C₈₃	—	4	1	no	—
32	C₂³×C₈₃	—	4	1	no	—
33	C₂³×C₈₃	C₄²	1	1	yes	characteristic
34	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	—	2	2	no	—
37	C₂×C₄×C₈₃	—	2	2	no	—
38	C₂×C₄×C₈₃	—	2	2	no	—
39	C₂×C₄×C₈₃	—	4	1	no	—
40	C₂×C₄×C₈₃	—	4	1	no	—
41	C₂×C₄×C₈₃	—	4	1	no	—
42	C₂×C₄×C₈₃	—	4	2	no	—
43	C₂×C₄×C₈₃	—	4	2	no	—
44	C₂×C₄×C₈₃	—	4	2	no	—
45	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
49	C₈₃⋊C₈	—	4	2	no	—
50	C₈₃⋊C₈	—	4	2	no	—
51	C₂²×C₈₃	—	2	1	no	—
52	C₂²×C₈₃	—	2	1	no	—
53	C₂²×C₈₃	—	2	1	no	—
54	C₂²×C₈₃	—	2	1	no	—
55	C₂²×C₈₃	—	2	1	no	—
56	C₂²×C₈₃	—	2	1	no	—
57	C₂²×C₈₃	—	2	1	no	—
58	C₂²×C₈₃	—	2	1	no	—
59	C₂²×C₈₃	—	4	2	no	—
60	C₂²×C₈₃	—	4	2	no	—
61	C₂²×C₈₃	#95	1	1	yes	characteristic
62	C₂²×C₈₃	#100	1	1	yes	characteristic
63	C₂²×C₈₃	#101	1	1	yes	characteristic
64	C₄×C₈₃	—	2	1	no	—
65	C₄×C₈₃	—	2	1	no	—
66	C₄×C₈₃	—	2	1	no	—
67	C₄×C₈₃	—	8	2	no	—
68	C₄×C₈₃	#99	1	2	yes	—
69	C₂×C₈₃	—	2	1	no	—
70	C₂×C₈₃	—	2	1	no	—
71	C₂×C₈₃	—	4	1	no	—
72	C₂×C₈₃	—	4	1	no	—
73	C₂×C₈₃	#342	1	1	yes	characteristic
74	C₂×C₈₃	#343	1	1	yes	characteristic
75	C₂×C₈₃	#344	1	1	yes	characteristic
76	#1305	—	83	1	no	maximal, Sylow
77	C₈₃	#1305	1	1	yes	characteristic, complemented, Sylow
78	#407	—	83	1	no	—
79	#407	—	83	1	no	—
80	#380	D₈₃	1	1	yes	characteristic
81	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
82	C₂×M₁₆	—	83	2	no	—
83	C₂×M₁₆	—	166	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₂²⋊C₄	Dic₈₃	1	2	yes	—
86	C₂²⋊C₈	—	83	2	no	—
87	C₂²⋊C₈	—	166	1	no	—
88	C₂²⋊C₈	—	166	1	no	—
89	C₂²⋊C₈	—	166	1	no	—
90	#95	—	2	1	no	—
91	C₂⁴	C₂×Dic₈₃	1	1	yes	characteristic
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
96	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
97	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
100	M₁₆	—	332	2	no	—
101	M₁₆	—	332	2	no	—
102	C₂×C₈	—	166	2	no	—
103	C₂×C₈	—	166	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₂²⋊C₄	—	4	2	no	—
106	C₂³	—	2	1	no	—
107	C₂³	—	2	1	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	C₄×Dic₈₃	1	1	yes	characteristic
111	C₂³	C₂²⋊Dic₈₃	1	1	yes	characteristic
112	C₂³	C₄⋊Dic₈₃	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
125	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
126	C₈	—	332	2	no	—
127	C₈	—	332	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₈₃⋊#95	1	1	yes	characteristic
139	C₂²	#14581709	1	1	yes	characteristic
140	C₂²	#14581711	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	8	2	no	—
145	C₄	#14581706	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	—	4	1	no	—
149	C₂	—	4	1	no	—
150	C₂	#19127467	1	1	yes	characteristic
151	C₂	#19127470	1	1	yes	characteristic
152	C₂	#19127471	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127665

C₂

yes

maximal, characteristic

#19127665

C₂

yes

maximal, characteristic

#380×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₂×#13050574

C₄

yes

complemented

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

C₄

yes

complemented

C₂³×C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₃

—

C₂×C₂²⋊C₄×C₈₃

C₄

yes

—

#95×C₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂⁴×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₄²

yes

characteristic

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#99

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#407

—

#407

—

#380

D₈₃

yes

characteristic

C₂³×C₄

C₂×D₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

166

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

Dic₈₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

166

—

C₂²⋊C₈

—

166

—

C₂²⋊C₈

—

166

—

#95

—

C₂⁴

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

100

M₁₆

—

332

—

101

M₁₆

—

332

—

102

C₂×C₈

—

166

—

103

C₂×C₈

—

166

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₄×Dic₈₃

yes

characteristic

111

C₂³

C₂²⋊Dic₈₃

yes

characteristic

112

C₂³

C₄⋊Dic₈₃

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

123

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

124

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

125

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

126

C₈

—

332

—

127

C₈

—

332

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₈₃⋊#95

yes

characteristic

139

C₂²

#14581709

yes

characteristic

140

C₂²

#14581711

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#14581706

yes

—

146

C₂

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

#19127467

yes

characteristic

151

C₂

#19127470

yes

characteristic

152

C₂

#19127471

yes

characteristic

	subgroup	G/H
center	C₂²	C₈₃⋊#95
commutator	C₂³×C₈₃	C₄²
Frattini	C₂³×C₄	C₂×D₈₃
Fitting	#380×C₈₃	C₂
socle	C₂²×C₈₃	#95
prime radical	C₂⁴×C₈₃	C₂×C₄
cosocle	C₂³×C₄×C₈₃	C₂²
hypercenter	#380	D₈₃
hypocenter	C₈₃	#1305
LCS	G ▷ C₂³×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂²

C₈₃⋊#95

commutator

C₂³×C₈₃

C₄²

Frattini

C₂³×C₄

C₂×D₈₃

Fitting

#380×C₈₃

C₂

socle

C₂²×C₈₃

#95

prime radical

C₂⁴×C₈₃

C₂×C₄

cosocle

C₂³×C₄×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
5. C₂×#13050574	C₄	semidirect
9. C₂²⋊(C₈₃⋊C₈)	C₄	semidirect
77. C₈₃	#1305	semidirect

H ≅ G/N

type

5. C₂×#13050574

C₄

semidirect

9. C₂²⋊(C₈₃⋊C₈)

C₄

semidirect

77. C₈₃

#1305

semidirect

Subgroups — #24905369

Conjugacy classes of subgroups

Complements