Finite Group Databasebeta

← #24922244

Subgroups — #24922244

Order 10624.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128537C211yesmaximal, characteristic, complemented
2C2×#14581838C211yesmaximal, characteristic, complemented
3#19128513C218yesmaximal, complemented
4#19128506C218yesmaximal, complemented
5#19128501C216yesmaximal, complemented
6C2×#14581820C216yesmaximal, complemented
7#568×C83C211yesmaximal, characteristic, complemented
8#14581840—21no—
9#14581839C2214yescomplemented
10#14581838—24no—
11#14581838C2218yescomplemented
12#14581838C2214yescomplemented
13#14581837—23no—
14C2×#13050591—21no—
15C2×#13050590C2213yescomplemented
16C2×#13050590C2216yescomplemented
17#41×D83—24no—
18C22×D332C2213yescomplemented
19C2×D4×D83—23no—
20#14581825C2214yescomplemented
21#14581825C22112yescomplemented
22#14581820C22112yescomplemented
23#14581820C22124yescomplemented
24#14581820C22112yescomplemented
25#14581814C22124yescomplemented
26C2×#13050579C22112yescomplemented
27C2×D664C22112yescomplemented
28C2×#41×C83C2211yescharacteristic
29#131×C83C2218yescomplemented
30C2×M16×C83C2216yescomplemented
31C23×D83—23no—
32#13050591—24no—
33#13050591—28no—
34#13050590—26no—
35#13050590—26no—
36#13050590—26no—
37#13050590C23112yes—
38#13050590C2314yes—
39#13050590C23112yescomplemented
40#13050589—212no—
41C2×Dic166C2311yescharacteristic
42C2×Dic166C2316yescomplemented
43Q8×D83—24no—
44C2×D332—23no—
45C2×D332—26no—
46C2×D332C23112yes—
47C2×D332C2313yes—
48C2×D332C2316yescomplemented
49C2×C22⋊D83—23no—
50C2×C22⋊D83—26no—
51D4×D83—224no—
52D4×D83—212no—
53C2×C4×D83—23no—
54C2×C4×D83—26no—
55Dic332C2318yescomplemented
56#13050579C23124yescomplemented
57#13050579C2318yescomplemented
58D664C23124yescomplemented
59#41×C83D414yes—
60#41×C83C2314yes—
61C2×Q8×C83D411yescharacteristic
62C2×D4×C83D413yes—
63C22×C4×C83C2313yes—
64M16×C83C2314yes—
65M16×C83C23112yescomplemented
66C2×C8×C83C23112yescomplemented
67C23×C83C2×D413yes—
68C22×D83—23no—
69C22×D83—26no—
70C22×D83—212no—
71C22×D83—86no—
72Q8×C83C2×D414yes—
73D4×C83C2×D4112yes—
74C2×C4×C83C2×D413yes—
75C2×C4×C83C2×D416yes—
76C2×C4×C83C2416yes—
77C2×C4×C83C2411yescharacteristic
78C22⋊D83—212no—
79C22⋊D83—212no—
80C22⋊D83—24no—
81C2×Dic83—21no—
82C2×Dic83—26no—
83D332—212no—
84D332—26no—
85D332—23no—
86D332C2414yes—
87D332C2416yes—
88C4×D83—24no—
89C4×D83—212no—
90C4×D83—212no—
91Dic166—26no—
92Dic166—21no—
93Dic166C2414yes—
94Dic166C2412yes—
95C8×C83C2418yes—
96C22×C83—43no—
97C22×C83C22×D411yescharacteristic
98C22×C83C22×D416yes—
99C2×D83—24no—
100C2×D83—26no—
101C2×D83—46no—
102C2×D83—86no—
103C2×D83—812no—
104C4×C83C22×D411yescharacteristic
105C4×C83C22×D416yes—
106C4×C83C2511yescharacteristic
107Dic83—24no—
108Dic83—22no—
109C2×C83—21no—
110C2×C83—26no—
111C2×C83C23×D411yescharacteristic
112D83—46no—
113D83—44no—
114#3585—831nomaximal, Sylow
115C83#358511yescharacteristic, complemented, Sylow
116#585—831no—
117C2×H(5,𝔽2)—831no—
118#577—838no—
119#576—838no—
120#575—836no—
121C2×AGL(1,ℤ/8)—836no—
122#568D8311yescharacteristic, complemented
1232−1+4—834no—
1242−1+4—1661no—
125H(5,𝔽2)—834no—
126H(5,𝔽2)—838no—
127H(5,𝔽2)—1664no—
128H(5,𝔽2)—1663no—
129C2×#41—836no—
130C2×#41—833no—
131C2×#41—1664no—
132C2×#41—1661no—
133C2×#41C2×D8311yescharacteristic
134C22×D4—833no—
135C22×D4—1663no—
136#137—834no—
137#137—8312no—
138AGL(1,ℤ/8)—8312no—
139AGL(1,ℤ/8)—8324no—
140AGL(1,ℤ/8)—8312no—
141#135—8324no—
142C2×SD16—8312no—
143C2×D8—8312no—
144#131C2×D8318yescomplemented
145C2×M16C2×D8316yescomplemented
146C24—1663no—
147#41—8312no—
148#41—8312no—
149#41—834no—
150#41—1668no—
151#41—1664no—
152#41—16612no—
153#41—1666no—
154#41—1666no—
155#41—1666no—
156#41D33214yes—
157#41C22×D8314yes—
158C2×Q8—836no—
159C2×Q8—831no—
160C2×Q8—1664no—
161C2×Q8D33211yescharacteristic
162C2×D4—836no—
163C2×D4—8312no—
164C2×D4—833no—
165C2×D4—1663no—
166C2×D4—1666no—
167C2×D4—16624no—
168C2×D4—16612no—
169C2×D4—1663no—
170C2×D4—1666no—
171C2×D4D33213yes—
172C22×C4—1663no—
173C22×C4—1666no—
174C22×C4C22×D8313yes—
175Q16—838no—
176SD16—8324no—
177SD16—838no—
178D8—8324no—
179M16C22×D8314yes—
180M16C22×D83112yescomplemented
181C2×C8C22×D83112yescomplemented
182C23—6646no—
183C23—1663no—
184C23—1666no—
185C23—16612no—
186C23C2×D33213yes—
187Q8—834no—
188Q8—832no—
189Q8—1666no—
190Q8—1661no—
191Q8C2×D33214yes—
192D4—834no—
193D4—836no—
194D4—16612no—
195D4—1664no—
196D4—16612no—
197D4—1663no—
198D4—16612no—
199D4—1666no—
200D4C2×D332112yes—
201C2×C4—1661no—
202C2×C4—1666no—
203C2×C4—1664no—
204C2×C4—16612no—
205C2×C4—16612no—
206C2×C4C2×D33213yes—
207C2×C4C2×D33216yes—
208C2×C4C23×D8316yes—
209C2×C4C23×D8311yescharacteristic
210C8C23×D8318yes—
211C22—43no—
212C22—3326no—
213C22—66412no—
214C22—6646no—
215C22—1664no—
216C22—1666no—
217C22C22×D33211yescharacteristic
218C22C22×D33216yes—
219C4—1662no—
220C4—1664no—
221C4C22×D33211yescharacteristic
222C4C22×D33216yes—
223C4C24×D8311yescharacteristic
224C2—21no—
225C2—26no—
226C2—3326no—
227C2—3324no—
228C2C23×D33211yescharacteristic
subgroupG/H
centerC2C23×D332
commutator, cosocleC4×C83C25
FrattiniC4C24×D83
Fitting#568×C83C2
socleC2×C83C23×D4
hypercenter#568D83
hypocenterC83#3585
LCSG ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #19128537C2semidirect
2. C2×#14581838C2semidirect
3. #19128513C2semidirect
4. #19128506C2semidirect
5. #19128501C2semidirect
6. C2×#14581820C2semidirect
7. #568×C83C2semidirect
9. #14581839C22semidirect
11. #14581838C22semidirect
12. #14581838C22semidirect
15. C2×#13050590C22semidirect
16. C2×#13050590C22semidirect
18. C22×D332C22semidirect
20. #14581825C22semidirect
21. #14581825C22semidirect
22. #14581820C22semidirect
23. #14581820C22semidirect
24. #14581820C22semidirect
25. #14581814C22semidirect
26. C2×#13050579C22semidirect
27. C2×D664C22semidirect
29. #131×C83C22semidirect
30. C2×M16×C83C22semidirect
39. #13050590C23semidirect
42. C2×Dic166C23semidirect
48. C2×D332C23semidirect
55. Dic332C23semidirect
56. #13050579C23semidirect
57. #13050579C23semidirect
58. D664C23semidirect
65. M16×C83C23semidirect
66. C2×C8×C83C23semidirect
115. C83#3585semidirect
122. #568D83semidirect
144. #131C2×D83semidirect
145. C2×M16C2×D83semidirect
180. M16C22×D83semidirect
181. C2×C8C22×D83semidirect

Maximal subgroups

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

subgroupindex
#191285372
C2×#145818382
#191285132
#191285062
#191285012
C2×#145818202
#568×C832
#358583