Finite Group Databasebeta

← #24918513

Subgroups — #24918513

Order 10624.

420 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128306C211yesmaximal, characteristic, complemented
2#19128091C211yesmaximal, characteristic, complemented
3#19128077C211yesmaximal, characteristic, complemented
4#19128065C211yesmaximal, characteristic, complemented
5#19128055C211yesmaximal, characteristic, complemented
6#19127913C212yesmaximal, complemented
7#19127893C212yesmaximal, complemented
8#19127846C212yesmaximal, complemented
9#19127835C212yesmaximal, complemented
10#19127762C211yesmaximal, characteristic, complemented
11#535×C83C211yesmaximal, characteristic, complemented
12C2×#13050577C2212yescomplemented
13C2×#13050575C2212yes—
14C2×#13050575C2212yes—
15C2×#13050575C2212yescomplemented
16#14581789C2211yescharacteristic, complemented
17#14581784C2211yescharacteristic, complemented
18#14581777—22no—
19#14581766C2212yes—
20#14581765—22no—
21C4×D332C2212yescomplemented
22C4⋊(C83⋊C8)C2212yescomplemented
23#14581716C2212yescomplemented
24#14581713C2212yes—
25#14581713C2212yes—
26#14581713C2212yescomplemented
27C22⋊(C83⋊C8)C2212yes—
28#14581703C2211yescharacteristic, complemented
29C4×C83⋊C8C2211yescharacteristic, complemented
30C2×#41×C83C2211yescharacteristic
31C22×D4×C83C2211yescharacteristic
32#127×C83C2211yescharacteristic
33#127×C83C2211yescharacteristic
34#124×C83C2211yescharacteristic
35#124×C83C2211yescharacteristic
36#121×C83—22no—
37#120×C83—22no—
38#117×C83C2211yescharacteristic
39C2×D332—22no—
40C2×D332C2312yes—
41C2×C22⋊D83—22no—
42C2×C4×D83—22no—
43#13050577—42no—
44#13050575—42no—
45#13050575—42no—
46#13050575—42no—
47C2×C83⋊C8C2312yes—
48C2×C83⋊C8C2312yes—
49C4⋊Dic83C2312yes—
50Dic83⋊C4—22no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C24×C83—21no—
54#41×C83—42no—
55C2×Q8×C83C2311yescharacteristic
56C2×D4×C83—21no—
57C2×D4×C83—21no—
58C2×D4×C83—22no—
59C2×D4×C83—22no—
60C2×D4×C83—42no—
61C2×D4×C83C2311yescharacteristic
62C2×D4×C83C2311yescharacteristic
63C2×D4×C83C2311yescharacteristic
64C22×C4×C83—21no—
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83D412yes—
67C22⋊C4×C83—22no—
68C22⋊C4×C83—22no—
69C22⋊C4×C83D412yes—
70C42×C83C2311yescharacteristic
71C42×C83C2311yescharacteristic
72C23×C83—21no—
73C23×C83—21no—
74C23×C83—21no—
75C23×C83—42no—
76C23×C83—82no—
77C23×C83C2×D411yescharacteristic
78C22×D83—22no—
79Q8×C83—22no—
80D4×C83—22no—
81D4×C83—22no—
82D4×C83—22no—
83D4×C83—42no—
84D4×C83—42no—
85D4×C83—82no—
86D4×C83—82no—
87C2×C4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—42no—
90C2×C4×C83C2×D412yes—
91C2×C4×C83C2×D412yes—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2411yescharacteristic
94C22⋊D83—82no—
95C2×Dic83—22no—
96D332—22no—
97D332—22no—
98D332—82no—
99C4×D83—82no—
100C83⋊C8—22no—
101C83⋊C8—22no—
102C22×C83—21no—
103C22×C83—22no—
104C22×C83—41no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83—84no—
111C22×C83C22×D411yescharacteristic
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—42no—
115C4×C83—21no—
116C4×C83—22no—
117C4×C83—22no—
118C4×C83—41no—
119C4×C83H(5,𝔽2)12yes—
120Dic83—42no—
121C2×C83—41no—
122C2×C83—81no—
123C2×C83—81no—
124C2×C83—81no—
125C2×C83#53511yescharacteristic
126C2×C83#57612yes—
127D83—82no—
128#3220—831nomaximal, Sylow
129C83#322011yescharacteristic, complemented, Sylow
130#540—831no—
131#535D8311yescharacteristic, complemented
132#496—831no—
133#493—831no—
134#490—831no—
135#488—831no—
136#463—832no—
137#459—832no—
138#449—832no—
139#447—832no—
140#433—831no—
141C2×#41C2×D8311yescharacteristic
142C22×D4C2×D8311yescharacteristic
143C2×SD16—832no—
144C2×D8—832no—
145C2×D8—832no—
146C2×D8—832no—
147#127—831no—
148#127C2×D8311yescharacteristic
149#127C2×D8311yescharacteristic
150#125—831no—
151#124C2×D8311yescharacteristic
152#124C2×D8311yescharacteristic
153#123—1662no—
154#121—22no—
155#121—832no—
156#121—1662no—
157#120—22no—
158C4×D4—832no—
159#117C2×D8311yescharacteristic
160C4⋊C8—832no—
161#103—832no—
162#102—832no—
163#102—832no—
164#102—832no—
165C22⋊C8—832no—
166#97—831no—
167C4×C8—831no—
168C24—21no—
169#41—42no—
170C2×Q8C22×D8311yescharacteristic
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—42no—
176C2×D4—832no—
177C2×D4—1662no—
178C2×D4—1662no—
179C2×D4C22×D8311yescharacteristic
180C2×D4C22×D8311yescharacteristic
181C2×D4C22×D8311yescharacteristic
182C22×C4—21no—
183C22×C4—1662no—
184C22×C4C22×D8311yescharacteristic
185SD16—3322no—
186D8—3322no—
187D8—3322no—
188D8—3322no—
189C2×C8—832no—
190C2×C8—832no—
191C4⋊C4—832no—
192C4⋊C4—1662no—
193C4⋊C4C22⋊D8312yes—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—1662no—
197C22⋊C4—1662no—
198C22⋊C4C22⋊D8312yes—
199C42C22×D8311yescharacteristic
200C42C22×D8311yescharacteristic
201C23—21no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23—82no—
206C23—1662no—
207C23C2×C22⋊D8311yescharacteristic
208Q8—22no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—42no—
213D4—42no—
214D4—82no—
215D4—82no—
216D4—6642no—
217D4—6642no—
218D4—1662no—
219D4—1662no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—6642no—
224C2×C4—1662no—
225C2×C4C2×C22⋊D8312yes—
226C2×C4C2×C22⋊D8312yes—
227C2×C4C2×C22⋊D8311yescharacteristic
228C2×C4C23×D8311yescharacteristic
229C8—1662no—
230C8—1662no—
231C22—22no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—84no—
240C22—3322no—
241C22—3322no—
242C22—3322no—
243C22C22×C22⋊D8311yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—41no—
248C4—3322no—
249C4#1458183712yes—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—81no—
254C2—6642no—
255C2#1912826811yescharacteristic
256C2#1912850412yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#535×C83C2
socleC22×C83C22×D4
hypercenter#535D83
hypocenterC83#3220
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #19128306C2semidirect
2. #19128091C2semidirect
3. #19128077C2semidirect
4. #19128065C2semidirect
5. #19128055C2semidirect
6. #19127913C2semidirect
7. #19127893C2semidirect
8. #19127846C2semidirect
9. #19127835C2semidirect
10. #19127762C2semidirect
11. #535×C83C2semidirect
12. C2×#13050577C22semidirect
15. C2×#13050575C22semidirect
16. #14581789C22semidirect
17. #14581784C22semidirect
21. C4×D332C22semidirect
22. C4⋊(C83⋊C8)C22semidirect
23. #14581716C22semidirect
26. #14581713C22semidirect
28. #14581703C22semidirect
29. C4×C83⋊C8C22semidirect
129. C83#3220semidirect
131. #535D83semidirect

Maximal subgroups

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

subgroupindex
#191283062
#191280912
#191280772
#191280652
#191280552
#191279132
#191278932
#191278462
#191278352
#191277622
#535×C832
#322083