Group #24917995 — subgroups

Subgroups — #24917995

Order 10624.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#19128329	C₂	1	2	yes	maximal
2	#19128109	C₂	1	4	yes	maximal, complemented
3	#19128101	C₂	1	2	yes	maximal, complemented
4	#19128098	C₂	1	2	yes	maximal, complemented
5	#19127744	C₂	1	2	yes	maximal
6	C₂×#14581724	C₂	1	1	yes	maximal, characteristic
7	#19127656	C₂	1	1	yes	maximal, characteristic
8	#427×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13050574	C₂²	1	1	yes	characteristic
10	#14581792	—	2	2	no	—
11	#14581791	C₂²	1	4	yes	—
12	#14581784	—	2	2	no	—
13	#14581783	C₂²	1	4	yes	—
14	#14581769	—	2	4	no	—
15	#14581751	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
17	#14581725	C₂²	1	4	yes	—
18	#14581724	C₂²	1	2	yes	—
19	#14581724	C₂²	1	2	yes	—
20	#14581723	C₂²	1	4	yes	—
21	#14581702	C₂²	1	2	yes	—
22	C₈×Dic₈₃	C₂²	1	2	yes	—
23	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	#117×C₈₃	C₂²	1	2	yes	—
25	#107×C₈₃	C₂²	1	2	yes	—
26	#106×C₈₃	C₂²	1	1	yes	characteristic
27	#106×C₈₃	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₆₆	—	2	4	no	—
29	C₂²×Dic₈₃	—	2	2	no	—
30	#13050574	Q₈	1	4	yes	complemented
31	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₈₃	—	2	2	no	—
33	C₄⋊Dic₈₃	—	2	4	no	—
34	C₄⋊Dic₈₃	D₄	1	2	yes	—
35	C₄⋊Dic₈₃	C₂³	1	4	yes	—
36	Dic₈₃⋊C₄	—	2	4	no	—
37	Dic₈₃⋊C₄	—	2	4	no	—
38	C₂²⋊Dic₈₃	D₄	1	2	yes	—
39	C₄×Dic₈₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₈₃	Q₈	1	4	yes	complemented
42	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
45	C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₈₃	—	2	2	no	—
47	C₄²×C₈₃	—	2	2	no	—
48	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	—	2	2	no	—
50	C₂×C₄×C₈₃	—	2	2	no	—
51	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₈₃	—	2	4	no	—
56	C₂×Dic₈₃	—	8	2	no	—
57	C₂×Dic₈₃	C₂×D₄	1	4	yes	—
58	Dic₁₆₆	—	8	4	no	—
59	C₈×C₈₃	C₂×Q₈	1	2	yes	—
60	C₈×C₈₃	C₂×Q₈	1	2	yes	—
61	C₈₃⋊C₈	C₂×Q₈	1	4	yes	—
62	C₂²×C₈₃	—	2	2	no	—
63	C₂²×C₈₃	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₈₃	—	4	2	no	—
66	C₄×C₈₃	—	4	2	no	—
67	C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
68	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
70	Dic₈₃	—	2	4	no	—
71	Dic₈₃	—	4	4	no	—
72	C₂×C₈₃	—	2	2	no	—
73	C₂×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₈₃	#577	1	2	yes	—
75	#3163	—	83	1	no	maximal, Sylow
76	C₈₃	#3163	1	1	yes	characteristic, complemented, Sylow
77	#544	—	83	2	no	—
78	#501	—	83	4	no	—
79	#499	—	83	2	no	—
80	#498	—	83	2	no	—
81	#428	—	83	2	no	—
82	#427	D₈₃	1	1	yes	characteristic
83	C₂×#106	—	83	1	no	—
84	#405	—	83	1	no	—
85	C₂×M₁₆	—	83	1	no	—
86	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
87	C₄⋊Q₈	—	83	4	no	—
88	C₄⋊Q₈	—	166	2	no	—
89	#125	—	83	4	no	—
90	#125	—	166	2	no	—
91	C₂²⋊Q₈	—	166	4	no	—
92	#117	—	83	1	no	—
93	#117	C₂×D₈₃	1	2	yes	—
94	C₂×C₄⋊C₄	—	83	2	no	—
95	#107	—	83	4	no	—
96	#107	C₂×D₈₃	1	2	yes	—
97	#106	—	83	2	no	—
98	#106	—	83	2	no	—
99	#106	—	83	4	no	—
100	#106	C₂×D₈₃	1	1	yes	characteristic
101	#106	C₂×D₈₃	1	1	yes	characteristic
102	#97	—	83	2	no	—
103	C₄×C₈	—	83	2	no	—
104	C₂×Q₈	—	166	4	no	—
105	C₂²×C₄	—	166	2	no	—
106	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
107	M₁₆	—	83	4	no	—
108	C₂×C₈	—	83	2	no	—
109	C₂×C₈	Dic₁₆₆	1	4	yes	complemented
110	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
112	C₄⋊C₄	—	83	2	no	—
113	C₄⋊C₄	—	83	4	no	—
114	C₄⋊C₄	—	166	4	no	—
115	C₄⋊C₄	—	166	4	no	—
116	C₄⋊C₄	—	166	4	no	—
117	C₄⋊C₄	—	166	2	no	—
118	C₄⋊C₄	C₂²×D₈₃	1	2	yes	—
119	C₄⋊C₄	C₂²×D₈₃	1	2	yes	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	83	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	83	2	no	—
124	C₂³	D₄×D₈₃	1	1	yes	characteristic
125	Q₈	—	664	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	83	4	no	—
129	C₂×C₄	—	664	2	no	—
130	C₂×C₄	—	166	4	no	—
131	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
132	C₂×C₄	Q₈×D₈₃	1	2	yes	—
133	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
134	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
135	C₈	—	83	4	no	—
136	C₈	C₂×Dic₁₆₆	1	2	yes	—
137	C₈	C₂×Dic₁₆₆	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#14581791	1	2	yes	—
140	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	332	4	no	—
144	C₄	—	166	4	no	—
145	C₄	#14581791	1	2	yes	—
146	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
147	C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#14581791	1	1	yes	characteristic
150	C₂	#19128508	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128329

C₂

yes

maximal

#19128109

C₂

yes

maximal, complemented

#19128101

C₂

yes

maximal, complemented

#19128098

C₂

yes

maximal, complemented

#19127744

C₂

yes

maximal

C₂×#14581724

C₂

yes

maximal, characteristic

#19127656

C₂

yes

maximal, characteristic

#427×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

#14581792

—

#14581791

C₂²

yes

—

#14581784

—

#14581783

C₂²

yes

—

#14581769

—

#14581751

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

#14581725

C₂²

yes

—

#14581724

C₂²

yes

—

#14581724

C₂²

yes

—

#14581723

C₂²

yes

—

#14581702

C₂²

yes

—

C₈×Dic₈₃

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

—

#107×C₈₃

C₂²

yes

—

#106×C₈₃

C₂²

yes

characteristic

#106×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

#13050574

Q₈

yes

complemented

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

D₄

yes

—

C₄×Dic₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

Q₈

yes

complemented

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

Dic₁₆₆

—

C₈×C₈₃

C₂×Q₈

yes

—

C₈×C₈₃

C₂×Q₈

yes

—

C₈₃⋊C₈

C₂×Q₈

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄⋊Q₈

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄⋊Q₈

yes

—

C₄×C₈₃

C₂²×Q₈

yes

characteristic

C₄×C₈₃

C₂²×Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₈₃

#577

yes

—

#3163

—

maximal, Sylow

C₈₃

#3163

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₃

yes

characteristic

C₂×#106

—

#405

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

#125

—

#125

—

166

—

C₂²⋊Q₈

—

166

—

#117

—

#117

C₂×D₈₃

yes

—

C₂×C₄⋊C₄

—

#107

—

#107

C₂×D₈₃

yes

—

#106

—

#106

—

#106

—

100

#106

C₂×D₈₃

yes

characteristic

101

#106

C₂×D₈₃

yes

characteristic

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

166

—

105

C₂²×C₄

—

166

—

106

C₂²×C₄

C₂²×D₈₃

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

Dic₁₆₆

yes

complemented

110

C₂×C₈

C₂²×D₈₃

yes

characteristic

111

C₂×C₈

C₂²×D₈₃

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

166

—

115

C₄⋊C₄

—

166

—

116

C₄⋊C₄

—

166

—

117

C₄⋊C₄

—

166

—

118

C₄⋊C₄

C₂²×D₈₃

yes

—

119

C₄⋊C₄

C₂²×D₈₃

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₂³

D₄×D₈₃

yes

characteristic

125

Q₈

—

664

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

664

—

130

C₂×C₄

—

166

—

131

C₂×C₄

D₄×D₈₃

yes

characteristic

132

C₂×C₄

Q₈×D₈₃

yes

—

133

C₂×C₄

C₂×Dic₁₆₆

yes

—

134

C₂×C₄

C₂³×D₈₃

yes

characteristic

135

C₈

—

136

C₈

C₂×Dic₁₆₆

yes

—

137

C₈

C₂×Dic₁₆₆

yes

—

138

C₂²

—

139

C₂²

#14581791

yes

—

140

C₂²

C₂×D₄×D₈₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

332

—

144

C₄

—

166

—

145

C₄

#14581791

yes

—

146

C₄

C₂×Q₈×D₈₃

yes

characteristic

147

C₄

C₂²×Dic₁₆₆

yes

characteristic

148

C₂

—

149

C₂

C₂×#14581791

yes

characteristic

150

C₂

#19128508

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#427×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#427	D₈₃
hypocenter	C₈₃	#3163
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#427×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #19128109	C₂	semidirect
3. #19128101	C₂	semidirect
4. #19128098	C₂	semidirect
30. #13050574	Q₈	semidirect
41. C₂×C₈×C₈₃	Q₈	semidirect
76. C₈₃	#3163	semidirect
109. C₂×C₈	Dic₁₆₆	semidirect

H ≅ G/N

type

2. #19128109

C₂

semidirect

3. #19128101

C₂

semidirect

4. #19128098

C₂

semidirect

30. #13050574

Q₈

semidirect

41. C₂×C₈×C₈₃

Q₈

semidirect

76. C₈₃

#3163

semidirect

109. C₂×C₈

Dic₁₆₆

semidirect

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

subgroup	index
#19128329	2
#19128109	2
#19128101	2
#19128098	2
#19127744	2
C₂×#14581724	2
#19127656	2
#427×C₈₃	2
#3163	83

subgroup

index

#427×C₈₃

Subgroups — #24917995

Conjugacy classes of subgroups

Complements

Maximal subgroups