Finite Group Databasebeta

← #24916699

Subgroups — #24916699

Order 10624.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×Q8×D83C211yesmaximal, characteristic, complemented
2C2×#14581825C211yesmaximal, characteristic, complemented
3C2×#14581823C211yesmaximal, characteristic, complemented
4#19128213C211yesmaximal, characteristic
5#19127858C214yesmaximal, complemented
6#19127841C214yesmaximal, complemented
7#19127709C211yesmaximal, characteristic, complemented
8#19127664C211yesmaximal, characteristic, complemented
9#419×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C22×Dic166—21no—
12C22×Dic166C2211yescharacteristic
13C2×Q8×D83—24no—
14C2×Q8×D83—22no—
15C2×Q8×D83—24no—
16C2×Q8×D83C2214yes—
17C22×C4×D83—21no—
18C22×C4×D83C2211yescharacteristic
19#14581825—24no—
20#14581823—24no—
21C2×Dic332C2212yescomplemented
22C2×#13050580C2212yescomplemented
23C2×#13050579C2212yescomplemented
24C2×#13050577C2212yescomplemented
25C2×#13050574C2211yescharacteristic
26#14581776—21no—
27#14581772C2212yes—
28#14581767C2212yes—
29#14581762—21no—
30C4×D332—22no—
31#14581715C2214yescomplemented
32(C2×D83)⋊C8C2214yes—
33C22×Q8×C83C2211yescharacteristic
34C2×M16×C83C2211yescharacteristic
35#117×C83C2211yescharacteristic
36#103×C83C2214yescomplemented
37C23×D83D411yescharacteristic
38#13050590—22no—
39#13050590—22no—
40C2×Dic166—28no—
41C2×Dic166—21no—
42C2×Dic166—22no—
43C2×Dic166—24no—
44C2×Dic166D414yescomplemented
45C2×Dic166C2311yescharacteristic
46C2×Dic166C2312yes—
47Q8×D83—216no—
48Q8×D83—28no—
49Q8×D83—28no—
50C2×D332—22no—
51C2×D332C2311yescharacteristic
52C2×C22⋊D83—21no—
53C22×Dic83—21no—
54C22×Dic83D411yescharacteristic
55C2×C4×D83—28no—
56C2×C4×D83—24no—
57C2×C4×D83—22no—
58C2×C4×D83—22no—
59C2×C4×D83—21no—
60C2×C4×D83D412yes—
61C2×C4×D83C2312yes—
62Dic332—24no—
63#13050580—24no—
64#13050579—24no—
65#13050577—24no—
66#13050574—22no—
67C2×C83⋊C8C2312yes—
68C4⋊Dic83—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—22no—
71C2×Q8×C83—22no—
72C2×Q8×C83—21no—
73C2×Q8×C83D414yescomplemented
74C2×Q8×C83C2312yes—
75C22×C4×C83—21no—
76C22×C4×C83C2311yescharacteristic
77M16×C83—22no—
78C2×C8×C83C2312yes—
79C4⋊C4×C83C2312yes—
80C22⋊C4×C83—21no—
81C42×C83—21no—
82C23×C83C2×D411yescharacteristic
83C22×D83—22no—
84C22×D83—24no—
85C22×D83—21no—
86C22×D83C2×D412yes—
87Q8×C83—24no—
88Q8×C83—22no—
89Q8×C83C2×D414yes—
90C2×C4×C83—24no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D412yes—
95C2×C4×C83C2×D412yes—
96C2×C4×C83C2411yescharacteristic
97C22⋊D83—42no—
98C2×Dic83—24no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83C2×D412yes—
103D332—21no—
104D332—21no—
105D332—44no—
106C4×D83—216no—
107C4×D83—24no—
108C4×D83—22no—
109C4×D83—22no—
110C4×D83—42no—
111Dic166—216no—
112Dic166—24no—
113Dic166—22no—
114Dic166—21no—
115Dic166—21no—
116Dic166C2×D414yes—
117C8×C83—22no—
118C83⋊C8—22no—
119C22×C83—22no—
120C22×C83#12012yes—
121C22×C83C22×D411yescharacteristic
122C2×D83—28no—
123C2×D83—22no—
124C2×D83—22no—
125C2×D83—41no—
126C2×D83—42no—
127C2×D83#13714yescomplemented
128C4×C83—24no—
129C4×C83—42no—
130C4×C83#12012yes—
131C4×C83C22×D411yescharacteristic
132C4×C83C22×D411yescharacteristic
133Dic83—22no—
134Dic83—24no—
135Dic83—41no—
136C2×C83—22no—
137C2×C83C2×#12011yescharacteristic
138C2×C83C2×#13712yes—
139D83—24no—
140D83—81no—
141#3004—831nomaximal, Sylow
142C83#300411yescharacteristic, complemented, Sylow
143C23×Q8—831no—
144C2×#137—831no—
145C2×#137—831no—
146#525—831no—
147#451—834no—
148#448—834no—
149#419—831no—
150#419D8311yescharacteristic, complemented
151#407—831no—
152C2×#41—831no—
153C22×Q8—834no—
154C22×Q8—831no—
155C22×Q8—1661no—
156C22×Q8—1662no—
157C22×Q8—1664no—
158C22×Q8—1664no—
159C22×Q8C2×D8311yescharacteristic
160C23×C4—831no—
161C23×C4—1661no—
162#137—1664no—
163#137—1664no—
164C2×Q16—832no—
165C2×Q16—832no—
166C2×SD16—832no—
167C2×SD16—832no—
168C2×M16—831no—
169C2×M16C2×D8311yescharacteristic
170#123—1661no—
171C22⋊Q8—832no—
172#121—832no—
173#120—1661no—
174C4×D4—1662no—
175#117C2×D8311yescharacteristic
176#103—834no—
177#103C2×D8314yescomplemented
178C22⋊C8—834no—
179C24—831no—
180#41—1662no—
181#41—1662no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—834no—
185C2×Q8—832no—
186C2×Q8—831no—
187C2×Q8—1664no—
188C2×Q8—1668no—
189C2×Q8—1668no—
190C2×Q8—16616no—
191C2×Q8—1668no—
192C2×Q8—1661no—
193C2×Q8—1662no—
194C2×Q8D33214yescomplemented
195C2×Q8C22×D8312yes—
196C2×D4—831no—
197C2×D4—1662no—
198C2×D4—1661no—
199C22×C4—21no—
200C22×C4—831no—
201C22×C4—832no—
202C22×C4—832no—
203C22×C4—1664no—
204C22×C4—1668no—
205C22×C4—1661no—
206C22×C4—1661no—
207C22×C4—1662no—
208C22×C4—1662no—
209C22×C4C22×D8311yescharacteristic
210Q16—1664no—
211Q16—1664no—
212SD16—1664no—
213SD16—1664no—
214M16—22no—
215M16—1662no—
216C2×C8—832no—
217C2×C8C22×D8312yes—
218C4⋊C4—1662no—
219C4⋊C4C22×D8312yes—
220C22⋊C4—21no—
221C22⋊C4—1662no—
222C22⋊C4—1662no—
223C42—21no—
224C23—832no—
225C23—1664no—
226C23—1661no—
227C23—1662no—
228C23D4×D8311yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—834no—
232Q8—16616no—
233Q8—1661no—
234Q8—1661no—
235Q8—1664no—
236Q8—1662no—
237Q8C2×D33214yes—
238D4—3324no—
239D4—3322no—
240D4—1661no—
241D4—1661no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—3322no—
246C2×C4—832no—
247C2×C4—16616no—
248C2×C4—1661no—
249C2×C4—1662no—
250C2×C4—1664no—
251C2×C4—1662no—
252C2×C4—1664no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4D4×D8311yescharacteristic
256C2×C4D4×D8312yes—
257C2×C4C2×D33212yes—
258C2×C4C23×D8311yescharacteristic
259C8—22no—
260C8—1662no—
261C22—22no—
262C22—3321no—
263C22—3322no—
264C22—834no—
265C22—1668no—
266C22—1662no—
267C22—1662no—
268C22#1458176212yes—
269C22C2×D4×D8311yescharacteristic
270C4—24no—
271C4—42no—
272C4—3321no—
273C4—1662no—
274C4—1664no—
275C4#1458176212yes—
276C4C2×D4×D8311yescharacteristic
277C4C22×D33211yescharacteristic
278C2—22no—
279C2—6641no—
280C2—1664no—
281C2C2×#1458176211yescharacteristic
282C2#137×D8312yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#419×C83C2
socleC22×C83C22×D4
prime radical#19128213C2
hypercenter#419D83
hypocenterC83#3004
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D83C2semidirect
2. C2×#14581825C2semidirect
3. C2×#14581823C2semidirect
5. #19127858C2semidirect
6. #19127841C2semidirect
7. #19127709C2semidirect
8. #19127664C2semidirect
9. #419×C83C2semidirect
21. C2×Dic332C22semidirect
22. C2×#13050580C22semidirect
23. C2×#13050579C22semidirect
24. C2×#13050577C22semidirect
31. #14581715C22semidirect
36. #103×C83C22semidirect
44. C2×Dic166D4semidirect
73. C2×Q8×C83D4semidirect
127. C2×D83#137semidirect
142. C83#3004semidirect
150. #419D83semidirect
177. #103C2×D83semidirect
194. C2×Q8D332semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D832
C2×#145818252
C2×#145818232
#191282132
#191278582
#191278412
#191277092
#191276642
#419×C832
#300483