Finite Group Databasebeta

← #20286281

Subgroups — #20286281

Order 6784.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133247C211yesmaximal, characteristic, complemented
2C22×#11719925C211yesmaximal, characteristic, complemented
3D4×Dic106C211yesmaximal, characteristic, complemented
4#16811018C211yesmaximal, characteristic, complemented
5#16810879C211yesmaximal, characteristic, complemented
6#16810850C211yesmaximal, characteristic, complemented
7#16810842C211yesmaximal, characteristic, complemented
8#16810833C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810779C211yesmaximal, characteristic, complemented
11#16810778C211yesmaximal, characteristic, complemented
12#16810761C211yesmaximal, characteristic, complemented
13#16810728C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18#13133247—24no—
19C2×#11719925—24no—
20C2×#11719925C2211yescharacteristic
21C2×#11719925C2211yescharacteristic
22C2×#11719925C2211yescharacteristic
23C2×#11719925C2211yescharacteristic
24C2×#11719925C2211yescharacteristic
25C2×#11719925C2211yescharacteristic
26C2×#11719923C2211yescharacteristic
27C2×#11719923C2211yescharacteristic
28C2×#11719923C2211yescharacteristic
29C2×#11719917C2211yescharacteristic
30C22×#88322C2211yescharacteristic
31#13133217C2211yescharacteristic
32#13133216—21no—
33#13133196C2211yescharacteristic
34#13133196C2211yescharacteristic
35#13133194—21no—
36#13133194—21no—
37C4×Dic106C2211yescharacteristic
38D4×Dic53—21no—
39#13133148C2211yescharacteristic
40C4⋊#88322C2211yescharacteristic
41#13133141C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133140C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133138C2211yescharacteristic
47C22⋊#88322C2211yescharacteristic
48C22⋊#88322C2211yescharacteristic
49#13133128C2211yescharacteristic
50C2×#41×C53C2211yescharacteristic
51C22×Q8×C53C2211yescharacteristic
52#124×C53—21no—
53#124×C53C2211yescharacteristic
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53C2211yescharacteristic
56#121×C53—21no—
57#121×C53C2211yescharacteristic
58C4×Q8×C53C2211yescharacteristic
59C4×D4×C53—21no—
60C4×D4×C53C2211yescharacteristic
61#41×C53—22no—
62#41×C53—41no—
63C2×Q8×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—41no—
66C2×Q8×C53C2311yescharacteristic
67C2×Q8×C53C2311yescharacteristic
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53D411yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C2×Dic106—22no—
79C2×Dic106—22no—
80C2×Dic106—21no—
81C2×Dic106—21no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×Dic106C2311yescharacteristic
85C2×Dic106C2311yescharacteristic
86C2×Dic106C2311yescharacteristic
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53D411yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C22⋊C4×C53D411yescharacteristic
96C22⋊C4×C53D411yescharacteristic
97C42×C53—21no—
98C42×C53C2311yescharacteristic
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101#11719925—22no—
102#11719925—22no—
103#11719925—22no—
104#11719925—22no—
105#11719925—41no—
106#11719925D414yescomplemented
107#11719923—22no—
108#11719923—22no—
109#11719923—41no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53C2311yescharacteristic
113#11719919—21no—
114#11719919—21no—
115#11719919—21no—
116C4×Dic53—21no—
117#11719917—22no—
118C2×#88322—22no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122C2×#88322C2311yescharacteristic
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C23×C53C2×D411yescharacteristic
126Q8×C53—21no—
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—42no—
130Q8×C53C2×D412yes—
131D4×C53—22no—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—42no—
135D4×C53—42no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53C2×D411yescharacteristic
146C2×C4×C53C2×D411yescharacteristic
147C2×C4×C53C2×D411yescharacteristic
148C2×C4×C53C2×D411yescharacteristic
149C2×C4×C53C2411yescharacteristic
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156Dic106—21no—
157Dic106—21no—
158Dic106—21no—
159Dic106—21no—
160Dic106—21no—
161Dic106—42no—
162Dic106—42no—
163Dic106C2×D412yes—
164#88322—21no—
165#88322—21no—
166#88322—21no—
167#88322C2×D412yes—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53#13712yescomplemented
177C22×C53C22×D411yescharacteristic
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—22no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53C22×D411yescharacteristic
186C4×C53H(5,𝔽2)11yescharacteristic
187Dic53—22no—
188Dic53—41no—
189Dic53—41no—
190Dic53—41no—
191#3270—531nomaximal, Sylow
192C2×C53—22no—
193C2×C53—41no—
194C2×C53—81no—
195C2×C53D4211yescharacteristic
196C2×C53C2×#13711yescharacteristic
197C2×C53#57811yescharacteristic
198C2×#137—531no—
199C22×Q16—531no—
200D4×Q8—531no—
201#548D5311yescharacteristic
202#497—531no—
203#470—531no—
204#465—531no—
205#464—531no—
206#462—531no—
207#452—531no—
208#451—531no—
209#451—531no—
210#448—531no—
211#441—531no—
212#435—531no—
213C53#327011yescharacteristic, complemented, Sylow
214C2×#41C2×D5311yescharacteristic
215C22×Q8—531no—
216C22×Q8—531no—
217C22×Q8C2×D5311yescharacteristic
218#137—1064no—
219C2×Q16—531no—
220C2×Q16—531no—
221C2×Q16—531no—
222C2×Q16—531no—
223C2×Q16—531no—
224C2×Q16—531no—
225C2×Q16—1064no—
226C2×SD16—531no—
227C2×SD16—531no—
228C2×SD16—531no—
229C2×M16—531no—
230C22×C8—531no—
231C4⋊Q8—531no—
232C4⋊Q8—1061no—
233#124—21no—
234#124C2×D5311yescharacteristic
235C22⋊Q8—21no—
236C22⋊Q8—531no—
237C22⋊Q8—531no—
238C22⋊Q8—1061no—
239C22⋊Q8—1061no—
240C22⋊Q8C2×D5311yescharacteristic
241#121—21no—
242#121C2×D5311yescharacteristic
243C4×Q8—531no—
244C4×Q8C2×D5311yescharacteristic
245C4×D4—21no—
246C4×D4—1061no—
247C4×D4C2×D5311yescharacteristic
248#106—531no—
249C4⋊C8—531no—
250#103—531no—
251#103—531no—
252#103—531no—
253#103—531no—
254#103—531no—
255#102—531no—
256C22⋊C8—531no—
257C22⋊C8—531no—
258#97—531no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—531no—
265C2×Q8—531no—
266C2×Q8—531no—
267C2×Q8—1062no—
268C2×Q8—1062no—
269C2×Q8—1061no—
270C2×Q8—1061no—
271C2×Q8—2121no—
272C2×Q8—2121no—
273C2×Q8C22×D5311yescharacteristic
274C2×Q8C22×D5311yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C22⋊D5311yescharacteristic
278C2×D4C22×D5311yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4C22×D5311yescharacteristic
284C22×C4C22×D5311yescharacteristic
285Q16—534no—
286Q16—1062no—
287Q16—1062no—
288Q16—1062no—
289Q16—1062no—
290Q16—2121no—
291SD16—1062no—
292SD16—1062no—
293SD16—2121no—
294M16—1062no—
295C2×C8—531no—
296C2×C8—531no—
297C2×C8—531no—
298C2×C8—531no—
299C2×C8—1062no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—531no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C4⋊C4C22⋊D5311yescharacteristic
309C4⋊C4C22×D5311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4C22⋊D5311yescharacteristic
317C22⋊C4C22⋊D5311yescharacteristic
318C42—21no—
319C42—1061no—
320C42C22×D5311yescharacteristic
321C23—21no—
322C23C2×C22⋊D5311yescharacteristic
323C23C2×C22⋊D5311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—532no—
329Q8—1061no—
330Q8—1061no—
331Q8—1061no—
332Q8—1061no—
333Q8—1061no—
334Q8—2122no—
335Q8—2122no—
336Q8D4×D5312yes—
337D4—22no—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—42no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4C2×C22⋊D5311yescharacteristic
358C2×C4C2×C22⋊D5311yescharacteristic
359C2×C4C2×C22⋊D5311yescharacteristic
360C2×C4C2×C22⋊D5311yescharacteristic
361C2×C4C23×D5311yescharacteristic
362C8—532no—
363C8—1061no—
364C8—1061no—
365C8—1061no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22#1313324712yescomplemented
375C22C22×C22⋊D5311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—1062no—
384C4—2121no—
385C4—2121no—
386C4—2121no—
387C4C2×D4×D5311yescharacteristic
388C4#1313326211yescharacteristic
389C2—22no—
390C2—41no—
391C2—81no—
392C2D4×C22⋊D5311yescharacteristic
393C2C2×#1313324711yescharacteristic
394C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#548C2
socleC22×C53C22×D4
hypercenter#548D53
hypocenterC53#3270
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#13133247C2semidirect
2. C22×#11719925C2semidirect
3. D4×Dic106C2semidirect
4. #16811018C2semidirect
5. #16810879C2semidirect
6. #16810850C2semidirect
7. #16810842C2semidirect
8. #16810833C2semidirect
9. #16810785C2semidirect
10. #16810779C2semidirect
11. #16810778C2semidirect
12. #16810761C2semidirect
13. #16810728C2semidirect
14. #16810695C2semidirect
106. #11719925D4semidirect
176. C22×C53#137semidirect
213. C53#3270semidirect
374. C22#13133247semidirect