Group #24908770 — subgroups

Subgroups — #24908770

Order 10624.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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	#19128501	C₂	1	1	yes	maximal, characteristic
2	#19127729	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127695	C₂	1	2	yes	maximal, complemented
4	#19127674	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13050590	C₂²	1	2	yes	—
7	#14581825	—	2	2	no	—
8	#14581820	—	2	2	no	—
9	#14581814	—	2	2	no	—
10	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	#14581751	C₂²	1	1	yes	characteristic
12	Dic₈₃⋊C₈	—	2	2	no	—
13	#14581715	—	2	2	no	—
14	#14581712	—	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	#13050590	—	2	2	no	—
21	#13050590	—	2	2	no	—
22	C₂×Dic₁₆₆	D₄	1	2	yes	—
23	C₂×D₃₃₂	D₄	1	2	yes	—
24	C₂×C₂²⋊D₈₃	—	2	2	no	—
25	C₂×C₄×D₈₃	—	2	2	no	—
26	Dic₃₃₂	—	2	2	no	—
27	#13050579	—	2	4	no	—
28	D₆₆₄	—	2	2	no	—
29	C₂×C₈₃⋊C₈	—	2	2	no	—
30	C₂×C₈₃⋊C₈	—	2	1	no	—
31	C₄⋊Dic₈₃	—	2	1	no	—
32	C₂²⋊Dic₈₃	D₄	1	2	yes	—
33	C₄×Dic₈₃	—	2	1	no	—
34	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
35	M₁₆×C₈₃	—	2	2	no	—
36	C₂×C₈×C₈₃	D₄	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	—	2	2	no	—
42	C₂×C₄×C₈₃	—	4	2	no	—
43	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	#41	1	2	yes	—
46	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
47	C₂²⋊D₈₃	—	4	2	no	—
48	C₂²⋊D₈₃	—	4	2	no	—
49	C₂×Dic₈₃	—	2	2	no	—
50	C₂×Dic₈₃	—	2	2	no	—
51	D₃₃₂	—	2	2	no	—
52	D₃₃₂	—	2	2	no	—
53	C₄×D₈₃	—	4	2	no	—
54	C₄×D₈₃	—	4	2	no	—
55	Dic₁₆₆	—	2	2	no	—
56	Dic₁₆₆	—	2	2	no	—
57	C₈×C₈₃	—	2	2	no	—
58	C₈₃⋊C₈	—	4	1	no	—
59	C₂²×C₈₃	—	4	1	no	—
60	C₂²×C₈₃	#121	1	2	yes	—
61	C₂²×C₈₃	#121	1	1	yes	characteristic
62	C₂×D₈₃	—	4	2	no	—
63	C₂×D₈₃	—	8	2	no	—
64	C₄×C₈₃	—	8	2	no	—
65	C₄×C₈₃	#120	1	1	yes	characteristic
66	C₄×C₈₃	#123	1	2	yes	—
67	C₄×C₈₃	#124	1	1	yes	characteristic
68	Dic₈₃	—	4	2	no	—
69	Dic₈₃	—	4	2	no	—
70	C₂×C₈₃	—	2	1	no	—
71	C₂×C₈₃	—	2	2	no	—
72	C₂×C₈₃	#394	1	1	yes	characteristic
73	D₈₃	—	8	2	no	—
74	#2053	—	83	1	no	maximal, Sylow
75	C₈₃	#2053	1	1	yes	characteristic, complemented, Sylow
76	#575	—	83	1	no	—
77	#424	—	83	1	no	—
78	#416	—	83	2	no	—
79	#410	—	83	2	no	—
80	H(3,ℤ/4)	D₈₃	1	1	yes	characteristic, complemented
81	C₂×#41	—	83	2	no	—
82	#137	—	166	2	no	—
83	AGL(1,ℤ/8)	—	166	2	no	—
84	#135	—	166	2	no	—
85	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
86	C₂²×C₈	—	83	1	no	—
87	#117	—	83	1	no	—
88	#117	C₂×D₈₃	1	2	yes	—
89	C₄⋊C₈	—	166	2	no	—
90	#103	—	166	2	no	—
91	#102	—	166	2	no	—
92	#99	—	166	4	no	—
93	#41	—	166	2	no	—
94	#41	—	166	2	no	—
95	#41	—	166	2	no	—
96	#41	—	166	2	no	—
97	C₂×Q₈	—	83	2	no	—
98	C₂×D₄	—	83	2	no	—
99	C₂×D₄	—	166	2	no	—
100	C₂²×C₄	—	166	2	no	—
101	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
102	Q₁₆	—	166	2	no	—
103	SD₁₆	—	166	4	no	—
104	D₈	—	166	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	166	2	no	—
107	C₂×C₈	—	166	1	no	—
108	C₂×C₈	D₃₃₂	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	166	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	83	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	166	1	no	—
115	C₂³	—	166	2	no	—
116	C₂³	D₄×D₈₃	1	1	yes	characteristic
117	Q₈	—	166	2	no	—
118	Q₈	—	166	2	no	—
119	D₄	—	332	2	no	—
120	D₄	—	332	2	no	—
121	D₄	—	166	2	no	—
122	D₄	—	166	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	332	2	no	—
125	C₂×C₄	—	332	2	no	—
126	C₂×C₄	—	166	2	no	—
127	C₂×C₄	—	166	2	no	—
128	C₂×C₄	D₄×D₈₃	1	2	yes	—
129	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
130	C₂×C₄	#13050590	1	2	yes	—
131	C₂×C₄	#13050591	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	332	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	332	2	no	—
136	C₂²	—	664	2	no	—
137	C₂²	#14581765	1	2	yes	—
138	C₂²	#14581767	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	332	2	no	—
141	C₄	—	332	2	no	—
142	C₄	#14581762	1	1	yes	characteristic
143	C₄	#14581777	1	2	yes	—
144	C₄	#14581780	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	664	2	no	—
148	C₂	#19127619	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128501

C₂

yes

maximal, characteristic

#19127729

C₂

yes

maximal, characteristic, complemented

#19127695

C₂

yes

maximal, complemented

#19127674

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050590

C₂²

yes

—

#14581825

—

#14581820

—

#14581814

—

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581751

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

#14581715

—

#14581712

—

#14581706

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

—

—

—

—

—

C₂×Dic₁₆₆

D₄

yes

—

C₂×D₃₃₂

D₄

yes

—

C₂×C₂²⋊D₈₃

—

C₂×C₄×D₈₃

—

Dic₃₃₂

—

#13050579

—

D₆₆₄

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

D₄

yes

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂²×D₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

characteristic

C₂²⋊D₈₃

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

—

C₂²×C₈₃

#121

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#120

yes

characteristic

C₄×C₈₃

#123

yes

—

C₄×C₈₃

#124

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#394

yes

characteristic

D₈₃

—

#2053

—

maximal, Sylow

C₈₃

#2053

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic, complemented

C₂×#41

—

#137

—

166

—

AGL(1,ℤ/8)

—

166

—

#135

—

166

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×D₈₃

yes

—

C₄⋊C₈

—

166

—

#103

—

166

—

#102

—

166

—

#99

—

166

—

#41

—

166

—

#41

—

166

—

#41

—

166

—

#41

—

166

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

166

—

100

C₂²×C₄

—

166

—

101

C₂²×C₄

C₂²×D₈₃

yes

characteristic

102

Q₁₆

—

166

—

103

SD₁₆

—

166

—

104

D₈

—

166

—

105

M₁₆

—

106

C₂×C₈

—

166

—

107

C₂×C₈

—

166

—

108

C₂×C₈

D₃₃₂

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

166

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

166

—

115

C₂³

—

166

—

116

C₂³

D₄×D₈₃

yes

characteristic

117

Q₈

—

166

—

118

Q₈

—

166

—

119

D₄

—

332

—

120

D₄

—

332

—

121

D₄

—

166

—

122

D₄

—

166

—

123

C₂×C₄

—

124

C₂×C₄

—

332

—

125

C₂×C₄

—

332

—

126

C₂×C₄

—

166

—

127

C₂×C₄

—

166

—

128

C₂×C₄

D₄×D₈₃

yes

—

129

C₂×C₄

C₂×D₃₃₂

yes

characteristic

130

C₂×C₄

#13050590

yes

—

131

C₂×C₄

#13050591

yes

characteristic

132

C₈

—

133

C₈

—

332

—

134

C₂²

—

135

C₂²

—

332

—

136

C₂²

—

664

—

137

C₂²

#14581765

yes

—

138

C₂²

#14581767

yes

characteristic

139

C₄

—

140

C₄

—

332

—

141

C₄

—

332

—

142

C₄

#14581762

yes

characteristic

143

C₄

#14581777

yes

—

144

C₄

#14581780

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

664

—

148

C₂

#19127619

yes

characteristic

	subgroup	G/H
center	C₄	#14581780
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	H(3,ℤ/4)×C₈₃	C₂
socle	C₂×C₈₃	#394
prime radical	#19128501	C₂
hypercenter	H(3,ℤ/4)	D₈₃
hypocenter	C₈₃	#2053
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#14581780

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

H(3,ℤ/4)×C₈₃

C₂

C₂×C₈₃

C₂

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #19127729	C₂	semidirect
3. #19127695	C₂	semidirect
4. #19127674	C₂	semidirect
5. H(3,ℤ/4)×C₈₃	C₂	semidirect
75. C₈₃	#2053	semidirect
80. H(3,ℤ/4)	D₈₃	semidirect

H ≅ G/N

type

2. #19127729

C₂

semidirect

3. #19127695

C₂

semidirect

4. #19127674

C₂

semidirect

5. H(3,ℤ/4)×C₈₃

C₂

semidirect

75. C₈₃

#2053

semidirect

80. 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
#19128501	2
#19127729	2
#19127695	2
#19127674	2
H(3,ℤ/4)×C₈₃	2
#2053	83

subgroup

index

H(3,ℤ/4)×C₈₃

Subgroups — #24908770

Conjugacy classes of subgroups

Complements

Maximal subgroups