Group #24907508 — subgroups

Subgroups — #24907508

Order 10624.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#14581800	C₂	1	1	yes	maximal, characteristic
2	#19127737	C₂	1	2	yes	maximal, complemented
3	#19127674	C₂	1	2	yes	maximal
4	#19127458	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,ℤ/4)×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13050590	C₂²	1	1	yes	characteristic
7	#14581800	—	2	2	no	—
8	#14581800	—	2	2	no	—
9	C₂×#13050573	C₄	1	2	yes	complemented
10	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₈₃	C₄	1	2	yes	complemented
12	#14581751	C₂²	1	2	yes	—
13	#14581725	—	2	2	no	—
14	#14581723	—	2	2	no	—
15	#14581706	—	2	4	no	—
16	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
17	#117×C₈₃	C₂²	1	2	yes	—
18	#13050590	—	2	2	no	—
19	#13050590	—	2	2	no	—
20	C₂×Dic₁₆₆	Q₈	1	1	yes	characteristic
21	C₂×D₃₃₂	Q₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₈₃	D₄	1	2	yes	—
23	C₂×C₄×D₈₃	C₂×C₄	1	2	yes	—
24	#13050573	—	2	4	no	—
25	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
28	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
29	C₈×D₈₃	—	2	2	no	—
30	C₈×D₈₃	—	2	2	no	—
31	C₄⋊Dic₈₃	—	2	2	no	—
32	C₂²⋊Dic₈₃	—	2	2	no	—
33	C₄×Dic₈₃	—	2	2	no	—
34	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
35	M₁₆×C₈₃	—	2	2	no	—
36	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₈₃	—	2	2	no	—
38	C₂²⋊C₄×C₈₃	—	2	2	no	—
39	C₄²×C₈₃	—	2	2	no	—
40	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₈₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₈₃	—	4	2	no	—
43	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈₃	#41	1	2	yes	—
47	C₂²⋊D₈₃	—	2	4	no	—
48	C₂×Dic₈₃	—	4	2	no	—
49	C₂×Dic₈₃	C₄⋊C₄	1	2	yes	—
50	D₃₃₂	—	2	2	no	—
51	C₄×D₈₃	—	2	2	no	—
52	C₄×D₈₃	—	2	2	no	—
53	Dic₁₆₆	—	2	2	no	—
54	C₈×C₈₃	—	2	2	no	—
55	C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₈₃	—	4	1	no	—
59	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₈₃	#121	1	2	yes	—
61	C₂×D₈₃	—	2	2	no	—
62	C₂×D₈₃	—	4	2	no	—
63	C₄×C₈₃	—	8	2	no	—
64	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₈₃	#117	1	1	yes	characteristic
66	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
67	Dic₈₃	—	2	2	no	—
68	Dic₈₃	—	8	2	no	—
69	C₂×C₈₃	—	2	2	no	—
70	C₂×C₈₃	—	2	1	no	—
71	C₂×C₈₃	#380	1	1	yes	characteristic
72	D₈₃	—	4	2	no	—
73	#1812	—	83	1	no	maximal, Sylow
74	C₈₃	#1812	1	1	yes	characteristic, complemented, Sylow
75	C₂×#131	—	83	1	no	—
76	#427	—	83	2	no	—
77	#410	—	83	2	no	—
78	H(3,ℤ/4)	—	83	1	no	—
79	H(3,ℤ/4)	D₈₃	1	1	yes	characteristic, complemented
80	C₂×#41	—	83	1	no	—
81	#131	—	166	2	no	—
82	#131	—	166	2	no	—
83	C₂×M₁₆	—	83	2	no	—
84	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
85	C₂²×C₈	—	83	1	no	—
86	C₂²×C₈	—	83	2	no	—
87	#117	—	83	2	no	—
88	#117	C₂×D₈₃	1	2	yes	—
89	#107	—	166	2	no	—
90	#106	—	166	2	no	—
91	#99	—	166	4	no	—
92	#41	—	166	2	no	—
93	#41	—	166	2	no	—
94	C₂×Q₈	—	83	1	no	—
95	C₂×D₄	—	83	2	no	—
96	C₂×D₄	—	83	1	no	—
97	C₂²×C₄	—	83	2	no	—
98	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	166	4	no	—
101	C₂×C₈	—	83	1	no	—
102	C₂×C₈	—	83	1	no	—
103	C₂×C₈	—	83	2	no	—
104	C₂×C₈	—	83	2	no	—
105	C₂×C₈	—	166	2	no	—
106	C₂×C₈	—	166	2	no	—
107	C₂×C₈	C₄×D₈₃	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	166	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	166	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	166	2	no	—
114	C₂³	—	83	2	no	—
115	C₂³	D₄×D₈₃	1	1	yes	characteristic
116	Q₈	—	166	2	no	—
117	D₄	—	166	4	no	—
118	D₄	—	166	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	332	2	no	—
121	C₂×C₄	—	83	2	no	—
122	C₂×C₄	—	166	2	no	—
123	C₂×C₄	—	166	2	no	—
124	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
125	C₂×C₄	D₄×D₈₃	1	2	yes	—
126	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
127	C₂×C₄	#13050590	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	83	1	no	—
130	C₈	—	83	1	no	—
131	C₈	—	83	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	332	2	no	—
134	C₂²	—	166	2	no	—
135	C₂²	C₄⋊C₄×D₈₃	1	1	yes	characteristic
136	C₂²	#14581765	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	664	2	no	—
139	C₄	—	166	2	no	—
140	C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
141	C₄	#14581749	1	1	yes	characteristic
142	C₄	#14581770	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	332	2	no	—
146	C₂	#19127560	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581800

C₂

yes

maximal, characteristic

#19127737

C₂

yes

maximal, complemented

#19127674

C₂

yes

maximal

#19127458

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050590

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₈₃

C₄

yes

complemented

#14581751

C₂²

yes

—

#14581725

—

#14581723

—

#14581706

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

—

#13050590

—

#13050590

—

C₂×Dic₁₆₆

Q₈

yes

characteristic

C₂×D₃₃₂

Q₈

yes

characteristic

C₂×C₂²⋊D₈₃

D₄

yes

—

C₂×C₄×D₈₃

C₂×C₄

yes

—

#13050573

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₈×D₈₃

—

C₈×D₈₃

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂²×D₈₃

C₄⋊C₄

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₄⋊C₄

yes

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₈₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₈₃⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

#121

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

#117

yes

characteristic

C₄×C₈₃

C₂²⋊Q₈

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#380

yes

characteristic

D₈₃

—

#1812

—

maximal, Sylow

C₈₃

#1812

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic, complemented

C₂×#41

—

#131

—

166

—

#131

—

166

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₈₃

yes

—

#107

—

166

—

#106

—

166

—

#99

—

166

—

#41

—

166

—

#41

—

166

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

100

M₁₆

—

166

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

166

—

106

C₂×C₈

—

166

—

107

C₂×C₈

C₄×D₈₃

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

166

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

166

—

112

C₄²

—

113

C₄²

—

166

—

114

C₂³

—

115

C₂³

D₄×D₈₃

yes

characteristic

116

Q₈

—

166

—

117

D₄

—

166

—

118

D₄

—

166

—

119

C₂×C₄

—

120

C₂×C₄

—

332

—

121

C₂×C₄

—

122

C₂×C₄

—

166

—

123

C₂×C₄

—

166

—

124

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

125

C₂×C₄

D₄×D₈₃

yes

—

126

C₂×C₄

Q₈×D₈₃

yes

characteristic

127

C₂×C₄

#13050590

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

332

—

134

C₂²

—

166

—

135

C₂²

C₄⋊C₄×D₈₃

yes

characteristic

136

C₂²

#14581765

yes

—

137

C₄

—

138

C₄

—

664

—

139

C₄

—

166

—

140

C₄

C₂²⋊C₄×D₈₃

yes

characteristic

141

C₄

#14581749

yes

characteristic

142

C₄

#14581770

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

332

—

146

C₂

#19127560

yes

characteristic

	subgroup	G/H
center	C₄	#14581749
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	H(3,ℤ/4)×C₈₃	C₂
socle	C₂×C₈₃	#380
prime radical	C₂×#13050590	C₂²
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	H(3,ℤ/4)	D₈₃
hypocenter	C₈₃	#1812
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#14581749

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

H(3,ℤ/4)×C₈₃

C₂

C₂×C₈₃

C₂²

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #19127737	C₂	semidirect
4. #19127458	C₂	semidirect
5. H(3,ℤ/4)×C₈₃	C₂	semidirect
9. C₂×#13050573	C₄	semidirect
11. C₂×C₈×D₈₃	C₄	semidirect
74. C₈₃	#1812	semidirect
79. H(3,ℤ/4)	D₈₃	semidirect

H ≅ G/N

type

2. #19127737

C₂

semidirect

4. #19127458

C₂

semidirect

5. H(3,ℤ/4)×C₈₃

C₂

semidirect

9. C₂×#13050573

C₄

semidirect

11. C₂×C₈×D₈₃

C₄

semidirect

74. C₈₃

#1812

semidirect

79. H(3,ℤ/4)

D₈₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14581800	2
#19127737	2
#19127674	2
#19127458	2
H(3,ℤ/4)×C₈₃	2
#1812	83

subgroup

index

H(3,ℤ/4)×C₈₃

Subgroups — #24907508

Conjugacy classes of subgroups

Complements

Maximal subgroups