Finite Group Databasebeta

← #24917415

Subgroups — #24917415

Order 10624.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#19128184C211yesmaximal, characteristic, complemented
2#19128022C212yesmaximal, complemented
3#19128022C212yesmaximal, complemented
4#19128017C212yesmaximal, complemented
5#19128012C212yesmaximal, complemented
6#19127744C211yesmaximal, characteristic, complemented
7#19127737C211yesmaximal, characteristic, complemented
8#19127706C211yesmaximal, characteristic, complemented
9C2×#14581713C211yesmaximal, characteristic, complemented
10#19127668C211yesmaximal, characteristic, complemented
11#535×C83C211yesmaximal, characteristic
12C2×#13050574C2211yescharacteristic
13C22×C83⋊C8C2211yescharacteristic
14Q8×Dic83—21no—
15D4×Dic83—22no—
16D4×Dic83—21no—
17#14581751C2211yescharacteristic
18#14581751C2212yescomplemented
19C2×C4⋊Dic83C2211yescharacteristic
20C2×C4⋊Dic83C2212yescomplemented
21#14581725C2211yescharacteristic, complemented
22#14581725C2211yescharacteristic, complemented
23#14581725C2212yescomplemented
24#14581723C2211yescharacteristic, complemented
25#14581723C2211yescharacteristic, complemented
26#14581723C2212yescomplemented
27#14581713C2212yes—
28#14581713C2212yes—
29#14581713C2212yes—
30#14581713C2212yes—
31C22⋊(C83⋊C8)C2212yescomplemented
32C22⋊(C83⋊C8)C2212yescomplemented
33C2×#41×C83C2211yescharacteristic
34C22×D4×C83C2211yescharacteristic
35#127×C83—21no—
36#124×C83—21no—
37#121×C83C2212yes—
38#121×C83C2212yes—
39#120×C83—22no—
40#117×C83C2211yescharacteristic
41C22×Dic83—21no—
42C22×Dic83—22no—
43#13050574—22no—
44C2×C83⋊C8—22no—
45C2×C83⋊C8C2311yescharacteristic
46C2×C83⋊C8C2311yescharacteristic
47C2×C83⋊C8C2312yes—
48C4⋊Dic83—21no—
49C4⋊Dic83—22no—
50C4⋊Dic83C2312yes—
51C4⋊Dic83C2311yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C22⋊Dic83—21no—
54C22⋊Dic83—22no—
55C4×Dic83—21no—
56C4×Dic83—22no—
57C24×C83—21no—
58#41×C83—22no—
59#41×C83—22no—
60C2×Q8×C83D411yescharacteristic
61C2×D4×C83—22no—
62C2×D4×C83—22no—
63C2×D4×C83—21no—
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83D412yes—
67C2×D4×C83D411yescharacteristic
68C2×D4×C83C2312yes—
69C22×C4×C83C2312yes—
70C22×C4×C83C2311yescharacteristic
71C4⋊C4×C83C2311yescharacteristic
72C4⋊C4×C83C2311yescharacteristic
73C22⋊C4×C83—21no—
74C22⋊C4×C83—22no—
75C22⋊C4×C83—22no—
76C42×C83—21no—
77C23×C83—22no—
78C23×C83—42no—
79C23×C83—81no—
80C23×C83—81no—
81C23×C83C2×D412yes—
82C23×C83C2×D411yescharacteristic
83Q8×C83—22no—
84D4×C83—24no—
85D4×C83—22no—
86D4×C83—22no—
87D4×C83—22no—
88D4×C83—42no—
89D4×C83—82no—
90D4×C83—82no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83C2×D412yes—
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83#4112yes—
98C2×C4×C83#4112yes—
99C2×C4×C83C2411yescharacteristic
100C2×Dic83—21no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—81no—
104C2×Dic83—82no—
105C83⋊C8—21no—
106C83⋊C8—21no—
107C83⋊C8—22no—
108C22×C83—21no—
109C22×C83—21no—
110C22×C83—22no—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83—84no—
117C22×C83#12312yes—
118C22×C83C22×D411yescharacteristic
119C4×C83—22no—
120C4×C83—41no—
121C4×C83—41no—
122C4×C83#12312yes—
123C4×C83C2×#4111yescharacteristic
124C4×C83C2×#4111yescharacteristic
125Dic83—41no—
126Dic83—42no—
127Dic83—41no—
128C2×C83—22no—
129C2×C83—42no—
130C2×C83—82no—
131C2×C83C2×#12311yescharacteristic
132C2×C83#57611yescharacteristic
133C2×C83#57711yescharacteristic
134#3096—831nomaximal, Sylow
135C83#309611yescharacteristic, complemented, Sylow
136#535D8311yescharacteristic
137#520—831no—
138#482—832no—
139#482—832no—
140#481—832no—
141#480—832no—
142#428—831no—
143#427—831no—
144#418—831no—
145C2×#102—831no—
146#408—831no—
147C2×#41C2×D8311yescharacteristic
148C22×D4C2×D8311yescharacteristic
149C2×M16—831no—
150C22×C8—831no—
151#127—21no—
152#124—21no—
153#121C2×D8312yes—
154#121C2×D8312yes—
155#120—22no—
156C4×Q8—1661no—
157C4×D4—1662no—
158C4×D4—1661no—
159#117—831no—
160#117—832no—
161#117C2×D8311yescharacteristic
162C2×C4⋊C4—831no—
163C2×C4⋊C4—832no—
164#107—831no—
165#107—831no—
166#107—832no—
167#106—831no—
168#106—831no—
169#106—832no—
170#102—832no—
171#102—832no—
172#102—832no—
173#102—832no—
174C22⋊C8—832no—
175C22⋊C8—832no—
176C24—21no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D8311yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4C22⋊D8312yes—
186C2×D4C22⋊D8311yescharacteristic
187C2×D4C22×D8312yes—
188C22×C4—1662no—
189C22×C4—1661no—
190C22×C4C22×D8312yes—
191C22×C4C22×D8311yescharacteristic
192M16—1662no—
193C2×C8—831no—
194C2×C8—832no—
195C2×C8—831no—
196C2×C8—1662no—
197C4⋊C4—832no—
198C4⋊C4—831no—
199C4⋊C4—831no—
200C4⋊C4—1662no—
201C4⋊C4—1661no—
202C4⋊C4C22×D8311yescharacteristic
203C4⋊C4C22×D8311yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—1662no—
208C22⋊C4—1661no—
209C42—21no—
210C42—1662no—
211C42—1661no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23C2×C22⋊D8312yes—
217C23C2×C22⋊D8311yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4—22no—
223D4—42no—
224D4—82no—
225D4—82no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—6641no—
231C2×C4—6642no—
232C2×C4—1661no—
233C2×C4—1662no—
234C2×C4—1661no—
235C2×C4C2×C22⋊D8312yes—
236C2×C4C2×C22⋊D8311yescharacteristic
237C2×C4#1305058912yes—
238C2×C4#1305058912yes—
239C2×C4C23×D8311yescharacteristic
240C8—1661no—
241C8—1661no—
242C8—1662no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—84no—
252C22#1458177312yes—
253C22C22×C22⋊D8311yescharacteristic
254C4—22no—
255C4—41no—
256C4—41no—
257C4—3322no—
258C4—3321no—
259C4—3321no—
260C4#1458177312yes—
261C4C2×#1305058911yescharacteristic
262C4C2×#1305058911yescharacteristic
263C2—22no—
264C2—42no—
265C2—82no—
266C2C2×#1458177311yescharacteristic
267C2#1912850411yescharacteristic
268C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#535×C83C2
socleC22×C83C22×D4
hypercenter#535D83
hypocenterC83#3096
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #19128184C2semidirect
2. #19128022C2semidirect
3. #19128022C2semidirect
4. #19128017C2semidirect
5. #19128012C2semidirect
6. #19127744C2semidirect
7. #19127737C2semidirect
8. #19127706C2semidirect
9. C2×#14581713C2semidirect
10. #19127668C2semidirect
18. #14581751C22semidirect
20. C2×C4⋊Dic83C22semidirect
21. #14581725C22semidirect
22. #14581725C22semidirect
23. #14581725C22semidirect
24. #14581723C22semidirect
25. #14581723C22semidirect
26. #14581723C22semidirect
31. C22⋊(C83⋊C8)C22semidirect
32. C22⋊(C83⋊C8)C22semidirect
135. C83#3096semidirect

Maximal subgroups

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

subgroupindex
#191281842
#191280222
#191280172
#191280122
#191277442
#191277372
#191277062
C2×#145817132
#191276682
#535×C832
#309683