Finite Group Databasebeta

← #20284346

Subgroups — #20284346

Order 6784.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#13133250C211yesmaximal, characteristic, complemented
2C2×M16×D53C211yesmaximal, characteristic, complemented
3#16811135C211yesmaximal, characteristic
4C2×#13133197C211yesmaximal, characteristic, complemented
5#16810882C212yesmaximal, complemented
6#16810870C212yesmaximal, complemented
7#16810865C212yesmaximal, complemented
8#16810853C212yesmaximal, complemented
9#16810631C211yesmaximal, characteristic, complemented
10#16810622C211yesmaximal, characteristic, complemented
11C53×#428C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15#13133250—24no—
16C2×Dic212C2212yescomplemented
17C2×#11719913C2212yescomplemented
18C2×#11719917C2211yescharacteristic
19C2×#11719912C2212yes—
20M16×D53—24no—
21C2×C8×D53C2212yes—
22#13133201—21no—
23#13133197—21no—
24#13133197—21no—
25#13133197C2212yes—
26#13133197C2211yescharacteristic
27#13133197C2212yes—
28#13133197C2211yescharacteristic
29#13133192C2211yescharacteristic
30#13133192C2211yescharacteristic
31#13133187—21no—
32C4×D212—21no—
33C4×D212—21no—
34C2×C4⋊Dic53—21no—
35C2×#11719921—21no—
36#13133140C2212yescomplemented
37#13133140C2212yescomplemented
38#13133140C2212yescomplemented
39#13133137C2212yescomplemented
40C2×M16×C53C2211yescharacteristic
41#117×C53C2211yescharacteristic
42C2×C4⋊C4×C53C2211yescharacteristic
43#107×C53C2212yescomplemented
44#106×C53C2212yescomplemented
45C23×D53D411yescharacteristic
46C22×C4×C53—21no—
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—21no—
49C22×Dic53—21no—
50C22×Dic53D411yescharacteristic
51#11719945—22no—
52#11719945—22no—
53C2×D212—21no—
54C2×D212—21no—
55C2×D212C2311yescharacteristic
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—21no—
59C2×C4×D53D411yescharacteristic
60C2×C4×D53D411yescharacteristic
61C2×C4×D53C2311yescharacteristic
62C2×C4×D53C2311yescharacteristic
63C2×Dic106—22no—
64C2×Dic106—22no—
65C2×Dic106—21no—
66C2×Dic106C2312yes—
67C2×Dic106C2311yescharacteristic
68M16×C53D414yescomplemented
69C2×C8×C53C2312yes—
70C4⋊C4×C53—21no—
71C4⋊C4×C53C2311yescharacteristic
72C4⋊C4×C53C2312yes—
73C4⋊C4×C53C2311yescharacteristic
74C22⋊C4×C53—21no—
75C42×C53—21no—
76#11719921—22no—
77#11719921—22no—
78#11719921—21no—
79#11719921—21no—
80#11719921—21no—
81#11719921—21no—
82C4⋊Dic53—22no—
83C4⋊Dic53—22no—
84C4⋊Dic53—21no—
85C4⋊Dic53—21no—
86#11719917—22no—
87C2×#88322C2312yes—
88Dic212—24no—
89#11719913—24no—
90#11719912—24no—
91C8×D53—24no—
92C23×C53C2×D411yescharacteristic
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—21no—
96C22×D53—41no—
97C22×D53C2×D411yescharacteristic
98C22×D53C2×D411yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—42no—
103C2×C4×C53C2×D412yes—
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53#4112yes—
106C2×C4×C53C2411yescharacteristic
107C22⋊D53—42no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—21no—
111C2×Dic53—42no—
112C2×Dic53C2×D411yescharacteristic
113C2×Dic53C2×D411yescharacteristic
114D212—21no—
115D212—21no—
116D212—42no—
117D212—42no—
118C4×D53—22no—
119C4×D53—22no—
120C4×D53—21no—
121C4×D53—21no—
122C4×D53—21no—
123C4×D53—21no—
124C4×D53—42no—
125Dic106—22no—
126Dic106—21no—
127Dic106—22no—
128Dic106—21no—
129Dic106—42no—
130C8×C53C2×D414yes—
131#88322—22no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53#12112yes—
135C22×C53C22×D411yescharacteristic
136C2×D53—21no—
137C2×D53—21no—
138C2×D53—21no—
139C2×D53—21no—
140C2×D53—21no—
141C2×D53—44no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—41no—
145C2×D53#13712yescomplemented
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—42no—
149C4×C53#12112yes—
150C4×C53C22×D411yescharacteristic
151C4×C53C2×#4111yescharacteristic
152Dic53—21no—
153Dic53—21no—
154Dic53—42no—
155Dic53—41no—
156#3056—531nomaximal, Sylow
157C2×C53—22no—
158C2×C53C2×#12111yescharacteristic
159C2×C53C2×#13711yescharacteristic
160C2×C53#57511yescharacteristic
161D53—22no—
162D53—41no—
163D53—81no—
164C2×#137—531no—
165C22×M16—531no—
166#525—531no—
167C2×C22⋊Q8—531no—
168#470—532no—
169#468—532no—
170#467—532no—
171#465—532no—
172#428D5311yescharacteristic, complemented
173#419—531no—
174#417—531no—
175C53#305611yescharacteristic, complemented, Sylow
176C2×#41—531no—
177C22×Q8—531no—
178C23×C4—531no—
179#137—1064no—
180C2×Q16—532no—
181C2×SD16—532no—
182C2×M16—531no—
183C2×M16—532no—
184C2×M16—1064no—
185C2×M16C2×D5311yescharacteristic
186C22×C8—532no—
187#123—1061no—
188C22⋊Q8—532no—
189C22⋊Q8—532no—
190C22⋊Q8—531no—
191C22⋊Q8—531no—
192C22⋊Q8—1061no—
193C22⋊Q8—1061no—
194#121—531no—
195#121—531no—
196#120—1061no—
197C4×D4—1061no—
198C4×D4—1061no—
199#117C2×D5311yescharacteristic
200C2×C4⋊C4—1061no—
201C2×C4⋊C4C2×D5311yescharacteristic
202C2×C22⋊C4—1061no—
203#107C2×D5312yescomplemented
204#106C2×D5312yescomplemented
205#103—532no—
206#103—532no—
207#103—532no—
208#102—532no—
209C24—531no—
210#41—1062no—
211#41—1062no—
212C2×Q8—532no—
213C2×Q8—531no—
214C2×Q8—1061no—
215C2×Q8—1062no—
216C2×Q8—1062no—
217C2×D4—531no—
218C2×D4—1061no—
219C2×D4—1061no—
220C2×D4—1061no—
221C22×C4—21no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1062no—
228C22×C4—1062no—
229C22×C4—1061no—
230C22×C4—1061no—
231C22×C4C22×D5311yescharacteristic
232Q16—1064no—
233SD16—1064no—
234M16—1062no—
235M16—1064no—
236M16D21214yescomplemented
237C2×C8—532no—
238C2×C8—1064no—
239C2×C8C22×D5312yes—
240C4⋊C4—21no—
241C4⋊C4—1062no—
242C4⋊C4—1062no—
243C4⋊C4—1061no—
244C4⋊C4—1061no—
245C4⋊C4C22×D5312yes—
246C4⋊C4C22×D5311yescharacteristic
247C4⋊C4C22×D5311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—1061no—
250C22⋊C4—1061no—
251C22⋊C4—1062no—
252C22⋊C4—1062no—
253C22⋊C4—1061no—
254C22⋊C4—1061no—
255C42—21no—
256C23—531no—
257C23—531no—
258C23—1062no—
259C23—1062no—
260C23—1061no—
261C23—2121no—
262C23D4×D5311yescharacteristic
263Q8—1062no—
264Q8—1061no—
265Q8—1062no—
266Q8—1061no—
267Q8—2122no—
268D4—1061no—
269D4—1061no—
270D4—2122no—
271D4—2122no—
272D4—2122no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—531no—
278C2×C4—531no—
279C2×C4—1062no—
280C2×C4—1062no—
281C2×C4—1062no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1062no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4C2×D21212yes—
291C2×C4D4×D5311yescharacteristic
292C2×C4#1171994912yes—
293C2×C4C23×D5311yescharacteristic
294C8—1062no—
295C8C2×D21214yes—
296C22—21no—
297C22—21no—
298C22—532no—
299C22—1061no—
300C22—1061no—
301C22—1061no—
302C22—1061no—
303C22—1061no—
304C22—2124no—
305C22—2121no—
306C22—2121no—
307C22—2121no—
308C22#1313319212yes—
309C22C2×D4×D5311yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—1061no—
314C4—1061no—
315C4—2121no—
316C4—2122no—
317C4#1313319212yes—
318C4C22×D21211yescharacteristic
319C4C2×#1171994911yescharacteristic
320C2—22no—
321C2—1062no—
322C2—4241no—
323C2—2121no—
324C2C2×#1313319211yescharacteristic
325C2#137×D5311yescharacteristic
326C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#428C2
socleC22×C53C22×D4
hypercenter#428D53
hypocenterC53#3056
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#13133250C2semidirect
2. C2×M16×D53C2semidirect
4. C2×#13133197C2semidirect
5. #16810882C2semidirect
6. #16810870C2semidirect
7. #16810865C2semidirect
8. #16810853C2semidirect
9. #16810631C2semidirect
10. #16810622C2semidirect
11. C53×#428C2semidirect
16. C2×Dic212C22semidirect
17. C2×#11719913C22semidirect
36. #13133140C22semidirect
37. #13133140C22semidirect
38. #13133140C22semidirect
39. #13133137C22semidirect
43. #107×C53C22semidirect
44. #106×C53C22semidirect
68. M16×C53D4semidirect
145. C2×D53#137semidirect
172. #428D53semidirect
175. C53#3056semidirect
203. #107C2×D53semidirect
204. #106C2×D53semidirect
236. M16D212semidirect