Group #24920607 — subgroups

Subgroups — #24920607

Order 10624.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#19128506	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128142	C₂	1	3	yes	maximal, complemented
3	#19128136	C₂	1	3	yes	maximal, complemented
4	#19128131	C₂	1	3	yes	maximal, complemented
5	C₂×#14581735	C₂	1	3	yes	maximal, complemented
6	#19128118	C₂	1	1	yes	maximal, characteristic, complemented
7	#578×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581838	—	2	1	no	—
9	#14581820	—	2	3	no	—
10	#14581814	C₂²	1	3	yes	—
11	C₂×D₆₆₄	C₂²	1	3	yes	—
12	#14581738	C₂²	1	3	yes	complemented
13	#14581736	C₂²	1	1	yes	characteristic, complemented
14	#14581735	C₂²	1	3	yes	complemented
15	#14581735	C₂²	1	6	yes	complemented
16	#14581733	C₂²	1	3	yes	complemented
17	#14581732	C₂²	1	3	yes	complemented
18	C₂×C₈₃⋊C₁₆	C₂²	1	3	yes	complemented
19	2₋¹⁺⁴×C₈₃	—	2	1	no	—
20	#137×C₈₃	—	2	3	no	—
21	#135×C₈₃	C₂²	1	3	yes	complemented
22	C₂×Q₁₆×C₈₃	C₂²	1	3	yes	complemented
23	#131×C₈₃	C₂²	1	1	yes	characteristic
24	#13050591	—	4	1	no	—
25	#13050590	—	2	3	no	—
26	C₂×D₃₃₂	—	2	3	no	—
27	D₄×D₈₃	—	4	3	no	—
28	Dic₃₃₂	C₂³	1	1	yes	characteristic
29	#13050579	—	2	3	no	—
30	D₆₆₄	—	2	3	no	—
31	D₆₆₄	C₂³	1	3	yes	—
32	C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
33	C₈₃⋊C₁₆	C₂³	1	3	yes	—
34	#41×C₈₃	—	2	3	no	—
35	#41×C₈₃	—	4	3	no	—
36	#41×C₈₃	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₈₃	—	2	3	no	—
38	C₂×Q₈×C₈₃	—	4	1	no	—
39	Q₁₆×C₈₃	—	2	3	no	—
40	Q₁₆×C₈₃	C₂³	1	3	yes	—
41	SD₁₆×C₈₃	—	2	3	no	—
42	D₈×C₈₃	C₂³	1	1	yes	characteristic
43	M₁₆×C₈₃	D₄	1	3	yes	—
44	C₂×C₈×C₈₃	C₂³	1	3	yes	—
45	C₂²×D₈₃	—	4	3	no	—
46	Q₈×C₈₃	—	2	3	no	—
47	Q₈×C₈₃	—	4	3	no	—
48	Q₈×C₈₃	D₈	1	1	yes	characteristic
49	D₄×C₈₃	—	2	1	no	—
50	D₄×C₈₃	—	4	3	no	—
51	D₄×C₈₃	D₈	1	3	yes	—
52	C₂×C₄×C₈₃	—	4	3	no	—
53	C₂×C₄×C₈₃	—	4	3	no	—
54	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
55	C₂²⋊D₈₃	—	4	3	no	—
56	D₃₃₂	—	2	3	no	—
57	D₃₃₂	—	4	3	no	—
58	C₄×D₈₃	—	4	3	no	—
59	Dic₁₆₆	—	2	1	no	—
60	C₈×C₈₃	C₂×D₄	1	3	yes	—
61	C₈×C₈₃	C₂⁴	1	1	yes	characteristic
62	C₂²×C₈₃	—	4	1	no	—
63	C₂²×C₈₃	C₂×D₈	1	3	yes	—
64	C₂×D₈₃	—	4	3	no	—
65	C₂×D₈₃	—	16	3	no	—
66	C₄×C₈₃	—	4	3	no	—
67	C₄×C₈₃	C₂×D₈	1	3	yes	—
68	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
69	Dic₈₃	—	4	1	no	—
70	C₂×C₈₃	—	2	3	no	—
71	C₂×C₈₃	—	8	1	no	—
72	C₂×C₈₃	C₂²×D₈	1	1	yes	characteristic
73	D₈₃	—	8	3	no	—
74	#3416	—	83	1	no	maximal, Sylow
75	C₈₃	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	D₈₃	1	1	yes	characteristic, complemented
77	#576	—	83	1	no	—
78	#510	—	83	3	no	—
79	#509	—	83	3	no	—
80	#508	—	83	3	no	—
81	C₂×SD₃₂	—	83	3	no	—
82	#504	—	83	1	no	—
83	2₋¹⁺⁴	—	2	1	no	—
84	H(5,𝔽₂)	—	166	1	no	—
85	#137	—	2	3	no	—
86	AGL(1,ℤ/8)	—	166	3	no	—
87	#135	—	83	3	no	—
88	#135	C₂×D₈₃	1	3	yes	complemented
89	C₂×Q₁₆	C₂×D₈₃	1	3	yes	complemented
90	C₂×D₈	—	83	3	no	—
91	#131	C₂×D₈₃	1	1	yes	characteristic
92	Q₃₂	—	83	3	no	—
93	SD₃₂	—	83	3	no	—
94	SD₃₂	—	83	6	no	—
95	SD₃₂	—	83	1	no	—
96	D₁₆	—	83	3	no	—
97	M₃₂	—	83	3	no	—
98	C₂×C₁₆	—	83	3	no	—
99	#41	—	2	3	no	—
100	#41	—	4	3	no	—
101	#41	—	332	1	no	—
102	#41	—	166	3	no	—
103	#41	C₂²⋊D₈₃	1	1	yes	characteristic
104	C₂×Q₈	—	2	3	no	—
105	C₂×Q₈	—	4	1	no	—
106	C₂×D₄	—	332	3	no	—
107	C₂×D₄	—	166	3	no	—
108	Q₁₆	—	2	3	no	—
109	Q₁₆	—	83	1	no	—
110	Q₁₆	C₂²×D₈₃	1	3	yes	—
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	166	3	no	—
113	D₈	—	83	3	no	—
114	D₈	—	166	3	no	—
115	D₈	C₂²×D₈₃	1	1	yes	characteristic
116	M₁₆	C₂²⋊D₈₃	1	3	yes	—
117	C₂×C₈	C₂²×D₈₃	1	3	yes	—
118	C₁₆	—	83	1	no	—
119	C₁₆	—	83	3	no	—
120	C₂³	—	332	3	no	—
121	Q₈	—	2	3	no	—
122	Q₈	—	4	3	no	—
123	Q₈	—	166	1	no	—
124	Q₈	#13050575	1	1	yes	characteristic
125	D₄	—	2	1	no	—
126	D₄	—	4	3	no	—
127	D₄	—	332	3	no	—
128	D₄	—	332	3	no	—
129	D₄	—	166	3	no	—
130	D₄	#13050575	1	3	yes	—
131	C₂×C₄	—	4	3	no	—
132	C₂×C₄	—	4	3	no	—
133	C₂×C₄	—	332	3	no	—
134	C₂×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
135	C₈	C₂×C₂²⋊D₈₃	1	3	yes	—
136	C₈	C₂³×D₈₃	1	1	yes	characteristic
137	C₂²	—	4	1	no	—
138	C₂²	—	1328	3	no	—
139	C₂²	—	332	3	no	—
140	C₂²	C₂×#13050575	1	3	yes	—
141	C₄	—	4	3	no	—
142	C₄	—	332	1	no	—
143	C₄	C₂×#13050575	1	3	yes	—
144	C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	8	1	no	—
147	C₂	—	664	3	no	—
148	C₂	C₂²×#13050575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128506

C₂

yes

maximal, characteristic, complemented

#19128142

C₂

yes

maximal, complemented

#19128136

C₂

yes

maximal, complemented

#19128131

C₂

yes

maximal, complemented

C₂×#14581735

C₂

yes

maximal, complemented

#19128118

C₂

yes

maximal, characteristic, complemented

#578×C₈₃

C₂

yes

maximal, characteristic, complemented

#14581838

—

#14581820

—

#14581814

C₂²

yes

—

C₂×D₆₆₄

C₂²

yes

—

#14581738

C₂²

yes

complemented

#14581736

C₂²

yes

characteristic, complemented

#14581735

C₂²

yes

complemented

#14581735

C₂²

yes

complemented

#14581733

C₂²

yes

complemented

#14581732

C₂²

yes

complemented

C₂×C₈₃⋊C₁₆

C₂²

yes

complemented

2₋¹⁺⁴×C₈₃

—

#137×C₈₃

—

#135×C₈₃

C₂²

yes

complemented

C₂×Q₁₆×C₈₃

C₂²

yes

complemented

#131×C₈₃

C₂²

yes

characteristic

#13050591

—

#13050590

—

C₂×D₃₃₂

—

D₄×D₈₃

—

Dic₃₃₂

C₂³

yes

characteristic

#13050579

—

D₆₆₄

—

D₆₆₄

C₂³

yes

—

C₈₃⋊C₁₆

C₂³

yes

characteristic

C₈₃⋊C₁₆

C₂³

yes

—

#41×C₈₃

—

#41×C₈₃

—

#41×C₈₃

D₄

yes

characteristic

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

C₂³

yes

—

SD₁₆×C₈₃

—

D₈×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

C₂³

yes

—

C₂²×D₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

D₈

yes

characteristic

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

D₈

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂²⋊D₈₃

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

C₂×D₄

yes

—

C₈×C₈₃

C₂⁴

yes

characteristic

C₂²×C₈₃

—

C₂²×C₈₃

C₂×D₈

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×D₈

yes

—

C₄×C₈₃

C₂²×D₄

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂²×D₈

yes

characteristic

D₈₃

—

#3416

—

maximal, Sylow

C₈₃

#3416

yes

characteristic, complemented, Sylow

#578

D₈₃

yes

characteristic, complemented

—

—

—

—

C₂×SD₃₂

—

#504

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

166

—

#137

—

AGL(1,ℤ/8)

—

166

—

#135

—

#135

C₂×D₈₃

yes

complemented

C₂×Q₁₆

C₂×D₈₃

yes

complemented

C₂×D₈

—

#131

C₂×D₈₃

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

#41

—

100

#41

—

101

#41

—

332

—

102

#41

—

166

—

103

#41

C₂²⋊D₈₃

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

332

—

107

C₂×D₄

—

166

—

108

Q₁₆

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₈₃

yes

—

111

SD₁₆

—

112

SD₁₆

—

166

—

113

D₈

—

114

D₈

—

166

—

115

D₈

C₂²×D₈₃

yes

characteristic

116

M₁₆

C₂²⋊D₈₃

yes

—

117

C₂×C₈

C₂²×D₈₃

yes

—

118

C₁₆

—

119

C₁₆

—

120

C₂³

—

332

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

166

—

124

Q₈

#13050575

yes

characteristic

125

D₄

—

126

D₄

—

127

D₄

—

332

—

128

D₄

—

332

—

129

D₄

—

166

—

130

D₄

#13050575

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

332

—

134

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

135

C₈

C₂×C₂²⋊D₈₃

yes

—

136

C₈

C₂³×D₈₃

yes

characteristic

137

C₂²

—

138

C₂²

—

1328

—

139

C₂²

—

332

—

140

C₂²

C₂×#13050575

yes

—

141

C₄

—

142

C₄

—

332

—

143

C₄

C₂×#13050575

yes

—

144

C₄

C₂²×C₂²⋊D₈₃

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

664

—

148

C₂

C₂²×#13050575

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#13050575
commutator, cosocle	C₈×C₈₃	C₂⁴
Frattini	C₈	C₂³×D₈₃
Fitting	#578×C₈₃	C₂
socle	C₂×C₈₃	C₂²×D₈
hypercenter	#578	D₈₃
hypocenter	C₈₃	#3416
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #578

subgroup

G/H

center

C₂

C₂²×#13050575

commutator, cosocle

C₈×C₈₃

C₂⁴

Frattini

C₈

C₂³×D₈₃

Fitting

#578×C₈₃

C₂

socle

C₂×C₈₃

C₂²×D₈

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #578

N	H ≅ G/N	type
1. #19128506	C₂	semidirect
2. #19128142	C₂	semidirect
3. #19128136	C₂	semidirect
4. #19128131	C₂	semidirect
5. C₂×#14581735	C₂	semidirect
6. #19128118	C₂	semidirect
7. #578×C₈₃	C₂	semidirect
12. #14581738	C₂²	semidirect
13. #14581736	C₂²	semidirect
14. #14581735	C₂²	semidirect
15. #14581735	C₂²	semidirect
16. #14581733	C₂²	semidirect
17. #14581732	C₂²	semidirect
18. C₂×C₈₃⋊C₁₆	C₂²	semidirect
21. #135×C₈₃	C₂²	semidirect
22. C₂×Q₁₆×C₈₃	C₂²	semidirect
75. C₈₃	#3416	semidirect
76. #578	D₈₃	semidirect
88. #135	C₂×D₈₃	semidirect
89. C₂×Q₁₆	C₂×D₈₃	semidirect

H ≅ G/N

type

1. #19128506

C₂

semidirect

2. #19128142

C₂

semidirect

3. #19128136

C₂

semidirect

4. #19128131

C₂

semidirect

5. C₂×#14581735

C₂

semidirect

6. #19128118

C₂

semidirect

7. #578×C₈₃

C₂

semidirect

12. #14581738

C₂²

semidirect

13. #14581736

C₂²

semidirect

14. #14581735

C₂²

semidirect

15. #14581735

C₂²

semidirect

16. #14581733

C₂²

semidirect

17. #14581732

C₂²

semidirect

18. C₂×C₈₃⋊C₁₆

C₂²

semidirect

21. #135×C₈₃

C₂²

semidirect

22. C₂×Q₁₆×C₈₃

C₂²

semidirect

75. C₈₃

#3416

semidirect

76. #578

D₈₃

semidirect

88. #135

C₂×D₈₃

semidirect

89. C₂×Q₁₆

C₂×D₈₃

semidirect

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

subgroup	index
#19128506	2
#19128142	2
#19128136	2
#19128131	2
C₂×#14581735	2
#19128118	2
#578×C₈₃	2
#3416	83

subgroup

index

#578×C₈₃

Subgroups — #24920607

Conjugacy classes of subgroups

Complements

Maximal subgroups