Finite Group Databasebeta

← #24917424

Subgroups — #24917424

Order 10624.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128270C211yesmaximal, characteristic
2#19128181C211yesmaximal, characteristic, complemented
3#19128028C212yesmaximal, complemented
4#19128027C212yesmaximal, complemented
5#19128021C212yesmaximal, complemented
6#19128016C212yesmaximal, complemented
7#19127744C211yesmaximal, characteristic, complemented
8#19127705C211yesmaximal, characteristic, complemented
9C2×#14581714C211yesmaximal, characteristic, complemented
10#19127667C211yesmaximal, characteristic, complemented
11#427×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×D332C2211yescharacteristic
14C2×#13050574C2211yescharacteristic
15#14581789—21no—
16#14581780—21no—
17#14581767C2212yes—
18#14581767C2212yes—
19#14581762—22no—
20C4×Dic166—21no—
21C4×D332—21no—
22(C22⋊D83)⋊C4—22no—
23#14581750C2212yescomplemented
24C2×C4⋊Dic83C2211yescharacteristic
25C4⋊C4×D83C2212yescomplemented
26#14581725C2212yescomplemented
27#14581723C2212yescomplemented
28#14581714C2212yes—
29#14581714C2212yes—
30#14581712C2212yes—
31#14581712C2212yes—
32(C2×D83)⋊C8C2212yescomplemented
33(C2×D83)⋊C8C2212yescomplemented
34C22×C8×C83C2211yescharacteristic
35#117×C83C2211yescharacteristic
36#117×C83C2211yescharacteristic
37#107×C83C2211yescharacteristic, complemented
38#107×C83C2211yescharacteristic, complemented
39#106×C83C2211yescharacteristic, complemented
40#106×C83C2211yescharacteristic, complemented
41C23×D83—21no—
42#13050590—22no—
43#13050590—22no—
44C2×Dic166D411yescharacteristic
45C2×D332—22no—
46C2×D332—22no—
47C2×D332—21no—
48C2×D332—22no—
49C2×D332—22no—
50C2×D332D411yescharacteristic
51C2×D332C2312yes—
52C2×C22⋊D83D412yes—
53C22×Dic83—21no—
54C2×C4×D83—22no—
55C2×C4×D83C2312yes—
56#13050574—22no—
57C2×C83⋊C8C2312yes—
58C4⋊Dic83—21no—
59C4⋊Dic83C2312yes—
60Dic83⋊C4—22no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—22no—
64C4×Dic83—22no—
65C22×C4×C83C2311yescharacteristic
66C2×C8×C83—22no—
67C2×C8×C83C2311yescharacteristic
68C2×C8×C83C2311yescharacteristic
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2311yescharacteristic
71C4⋊C4×C83C2311yescharacteristic
72C4⋊C4×C83C2311yescharacteristic
73C22⋊C4×C83—21no—
74C22⋊C4×C83—21no—
75C42×C83—21no—
76C42×C83—21no—
77C23×C83C2×D411yescharacteristic
78C22×D83—22no—
79C22×D83—42no—
80C22×D83—81no—
81C22×D83—81no—
82C22×D83C2×D412yes—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83#4112yes—
89C2×C4×C83#4112yes—
90C2×C4×C83C2411yescharacteristic
91C22⋊D83—24no—
92C2×Dic83—22no—
93C2×Dic83—81no—
94C2×Dic83C2×D412yes—
95D332—22no—
96D332—22no—
97D332—22no—
98D332—42no—
99D332—82no—
100D332—82no—
101C4×D83—22no—
102C4×D83—22no—
103C4×D83—82no—
104Dic166—22no—
105C8×C83—21no—
106C8×C83—21no—
107C83⋊C8—22no—
108C22×C83—21no—
109C22×C83—21no—
110C22×C83#12312yes—
111C22×C83C22×D411yescharacteristic
112C2×D83—22no—
113C2×D83—22no—
114C2×D83—22no—
115C2×D83—42no—
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—84no—
119C4×C83—41no—
120C4×C83—41no—
121C4×C83—41no—
122C4×C83—41no—
123C4×C83#12312yes—
124C4×C83C2×#4111yescharacteristic
125C4×C83C2×#4111yescharacteristic
126Dic83—22no—
127Dic83—42no—
128C2×C83—22no—
129C2×C83C2×#12311yescharacteristic
130C2×C83#57611yescharacteristic
131C2×C83#57711yescharacteristic
132D83—42no—
133D83—82no—
134#3096—831nomaximal, Sylow
135C83#309611yescharacteristic, complemented, Sylow
136#535—831no—
137#520—831no—
138#482—832no—
139#482—832no—
140#481—832no—
141#480—832no—
142#428—831no—
143#427D8311yescharacteristic, complemented
144#418—831no—
145C2×#102—831no—
146#408—831no—
147C2×#41—831no—
148C22×D4—831no—
149C2×M16—831no—
150C22×C8C2×D8311yescharacteristic
151#127—1661no—
152#124—1661no—
153#121—832no—
154#121—832no—
155#120—1662no—
156C4×Q8—1661no—
157C4×D4—1661no—
158C4×D4—1662no—
159#117—832no—
160#117C2×D8311yescharacteristic
161#117C2×D8311yescharacteristic
162C2×C4⋊C4—831no—
163C2×C4⋊C4—832no—
164#107—832no—
165#107C2×D8311yescharacteristic, complemented
166#107C2×D8311yescharacteristic, complemented
167#106—832no—
168#106C2×D8311yescharacteristic, complemented
169#106C2×D8311yescharacteristic, complemented
170#102—832no—
171#102—832no—
172#102—832no—
173#102—832no—
174C22⋊C8—832no—
175C22⋊C8—832no—
176C24—1661no—
177#41—1662no—
178#41—1662no—
179C2×Q8—831no—
180C2×D4—832no—
181C2×D4—831no—
182C2×D4—832no—
183C2×D4—1662no—
184C2×D4—1662no—
185C2×D4—1661no—
186C2×D4—1662no—
187C2×D4—1662no—
188C22×C4—832no—
189C22×C4—1662no—
190C22×C4—1661no—
191C22×C4C22×D8311yescharacteristic
192M16—1662no—
193C2×C8—22no—
194C2×C8—832no—
195C2×C8C22×D8311yescharacteristic
196C2×C8C22×D8311yescharacteristic
197C4⋊C4—832no—
198C4⋊C4—1662no—
199C4⋊C4—1661no—
200C4⋊C4C22×D8311yescharacteristic
201C4⋊C4C22×D8311yescharacteristic
202C4⋊C4C22×D8311yescharacteristic
203C4⋊C4C22×D8311yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—1662no—
207C22⋊C4—1662no—
208C22⋊C4—1662no—
209C42—21no—
210C42—21no—
211C42—1662no—
212C23—3322no—
213C23—832no—
214C23—6641no—
215C23—6641no—
216C23—1662no—
217C23D4×D8311yescharacteristic
218Q8—1662no—
219D4—3322no—
220D4—6642no—
221D4—6642no—
222D4—1664no—
223D4—1662no—
224D4—1662no—
225D4—1662no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—832no—
231C2×C4—6642no—
232C2×C4—6641no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4D4×D8311yescharacteristic
237C2×C4#1305059012yes—
238C2×C4#1305059112yes—
239C2×C4C23×D8311yescharacteristic
240C8—21no—
241C8—21no—
242C8—1662no—
243C22—21no—
244C22—21no—
245C22—3322no—
246C22—3322no—
247C22—3322no—
248C22—6644no—
249C22—1662no—
250C22—1662no—
251C22—1662no—
252C22#1458177712yes—
253C22C2×D4×D8311yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—3322no—
259C4—1662no—
260C4#1458177712yes—
261C4C2×#1305059011yescharacteristic
262C4C2×#1305059111yescharacteristic
263C2—22no—
264C2—3322no—
265C2—6642no—
266C2C2×#1458177711yescharacteristic
267C2#1912850311yescharacteristic
268C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#427×C83C2
socleC22×C83C22×D4
prime radical#19128270C2
hypercenter#427D83
hypocenterC83#3096
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #19128181C2semidirect
3. #19128028C2semidirect
4. #19128027C2semidirect
5. #19128021C2semidirect
6. #19128016C2semidirect
7. #19127744C2semidirect
8. #19127705C2semidirect
9. C2×#14581714C2semidirect
10. #19127667C2semidirect
11. #427×C83C2semidirect
23. #14581750C22semidirect
25. C4⋊C4×D83C22semidirect
26. #14581725C22semidirect
27. #14581723C22semidirect
32. (C2×D83)⋊C8C22semidirect
33. (C2×D83)⋊C8C22semidirect
37. #107×C83C22semidirect
38. #107×C83C22semidirect
39. #106×C83C22semidirect
40. #106×C83C22semidirect
135. C83#3096semidirect
143. #427D83semidirect
165. #107C2×D83semidirect
166. #107C2×D83semidirect
168. #106C2×D83semidirect
169. #106C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282702
#191281812
#191280282
#191280272
#191280212
#191280162
#191277442
#191277052
C2×#145817142
#191276672
#427×C832
#309683