Finite Group Databasebeta

← #20286355

Subgroups — #20286355

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×#13133246C211yesmaximal, characteristic, complemented
2Q8×C22⋊D53C211yesmaximal, characteristic, complemented
3#16811272C211yesmaximal, characteristic
4#16811001C211yesmaximal, characteristic, complemented
5#16810881C211yesmaximal, characteristic, complemented
6#16810878C211yesmaximal, characteristic, complemented
7#16810831C211yesmaximal, characteristic, complemented
8#16810826C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810782C211yesmaximal, characteristic, complemented
11#16810779C211yesmaximal, characteristic, complemented
12#16810777C211yesmaximal, characteristic, complemented
13SD16×Dic53C211yesmaximal, characteristic, complemented
14#16810699C211yesmaximal, characteristic, complemented
15C2×#137×C53C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18#13133246—24no—
19C2×Dic212C2211yescharacteristic
20C2×#11719925C2211yescharacteristic
21C2×#11719925C2211yescharacteristic
22C2×#11719925C2211yescharacteristic
23C2×#11719923C2211yescharacteristic
24C2×#11719912C2211yescharacteristic
25#13133215—21no—
26#13133215C2211yescharacteristic
27#13133215C2211yescharacteristic
28#13133200—21no—
29#13133198—21no—
30#13133196—21no—
31#13133196C2211yescharacteristic
32#13133196C2211yescharacteristic
33#13133193—21no—
34#13133193C2211yescharacteristic
35#13133193C2211yescharacteristic
36#13133189—21no—
37Q8×Dic53C2211yescharacteristic
38D4×Dic53C2211yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53C2211yescharacteristic
41C2×#11719919—21no—
42#13133149C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133141C2211yescharacteristic
49#13133138C2211yescharacteristic
50C22⋊#88322C2211yescharacteristic
51#13133129C2211yescharacteristic
52C8×Dic53C2211yescharacteristic
53C2×#41×C53C2211yescharacteristic
54C22×Q8×C53C2211yescharacteristic
55#137×C53—24no—
56C2×Q16×C53C2211yescharacteristic
57C2×Q16×C53C2211yescharacteristic
58C2×SD16×C53C2211yescharacteristic
59C2×SD16×C53C2211yescharacteristic
60C2×M16×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×Q8×C53C2311yescharacteristic
69C2×D4×C53—21no—
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—21no—
75C2×C22⋊D53D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78Q8×D53—22no—
79Q8×D53—41no—
80#11719947—22no—
81#11719947—41no—
82C2×C4×D53—21no—
83C2×C4×D53C2311yescharacteristic
84C2×Dic106—21no—
85C2×Dic106C2311yescharacteristic
86C2×Dic106C2311yescharacteristic
87Q16×C53—22no—
88Q16×C53—22no—
89SD16×C53—22no—
90SD16×C53D414yescomplemented
91M16×C53—22no—
92C2×C8×C53C2311yescharacteristic
93C2×C8×C53C2311yescharacteristic
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53D411yescharacteristic
98#11719925—22no—
99#11719925—41no—
100#11719925—41no—
101#11719923—22no—
102#11719921—21no—
103#11719921—21no—
104#11719921D411yescharacteristic
105C4⋊Dic53—21no—
106C4⋊Dic53C2311yescharacteristic
107C4⋊Dic53C2311yescharacteristic
108#11719919—21no—
109#11719919—21no—
110#11719919—21no—
111#11719919—21no—
112#11719919—21no—
113#11719919—21no—
114#11719919—21no—
115#11719919—21no—
116#11719919D411yescharacteristic
117C4×Dic53—21no—
118C4×Dic53C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121Dic212—22no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C22×D53C2×D411yescharacteristic
126Q8×C53—21no—
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—21no—
131Q8×C53—42no—
132Q8×C53C2×D412yes—
133D4×C53—21no—
134D4×C53—42no—
135D4×C53C2×D412yes—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2411yescharacteristic
145C22⋊D53—22no—
146C22⋊D53—42no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53C2×D411yescharacteristic
154C2×Dic53C2×D411yescharacteristic
155C2×Dic53C2×D411yescharacteristic
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—42no—
159Dic106—21no—
160Dic106—21no—
161Dic106—21no—
162Dic106—21no—
163Dic106—42no—
164C8×C53—21no—
165C8×C53C2×D412yes—
166#88322—21no—
167#88322—21no—
168C22×C53—22no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53C22×D411yescharacteristic
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C4×C53—22no—
179C4×C53—21no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53C22×D411yescharacteristic
183C4×C53H(5,𝔽2)11yescharacteristic
184Dic53—21no—
185Dic53—21no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190Dic53#13712yescomplemented
191#3276—531nomaximal, Sylow
192C2×C53—41no—
193C2×C53—42no—
194C2×C53D4211yescharacteristic
195C2×C53C2×#13711yescharacteristic
196C2×C53#57811yescharacteristic
197D53—41no—
198C2×#137—531no—
199C2×#137D5311yescharacteristic, complemented
200D4×Q8—531no—
201#547—531no—
202#494—531no—
203#470—531no—
204#470—531no—
205#462—531no—
206#461—531no—
207#452—531no—
208#452—531no—
209#451—531no—
210#451—531no—
211C4×SD16—531no—
212#436—531no—
213C53#327611yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C2×#41C2×D5311yescharacteristic
216C22×Q8—531no—
217C22×Q8C2×D5311yescharacteristic
218#137—24no—
219#137—1064no—
220C2×Q16—531no—
221C2×Q16—531no—
222C2×Q16—531no—
223C2×Q16—531no—
224C2×Q16C2×D5311yescharacteristic
225C2×Q16C2×D5311yescharacteristic
226C2×SD16—531no—
227C2×SD16C2×D5311yescharacteristic
228C2×SD16C2×D5311yescharacteristic
229C2×M16—531no—
230C2×M16C2×D5311yescharacteristic
231C4⋊Q8—531no—
232C4⋊Q8—531no—
233C4⋊Q8—1061no—
234#123—1061no—
235#123—1061no—
236C22⋊Q8—531no—
237C22⋊Q8—531no—
238C22⋊Q8—531no—
239C22⋊Q8—531no—
240C22⋊Q8—1061no—
241C22⋊Q8—1061no—
242#121—1061no—
243C4×Q8—531no—
244C4×D4—531no—
245C4×D4—531no—
246C4×D4—1061no—
247C2×C4⋊C4—1061no—
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—
258C4×C8—531no—
259#41—22no—
260#41—41no—
261#41—1062no—
262#41—2121no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—531no—
267C2×Q8—531no—
268C2×Q8—1061no—
269C2×Q8—1062no—
270C2×Q8—2121no—
271C2×Q8C22×D5311yescharacteristic
272C2×Q8C22×D5311yescharacteristic
273C2×Q8C22×D5311yescharacteristic
274C2×D4—21no—
275C2×D4—531no—
276C2×D4—1061no—
277C2×D4C22×D5311yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—531no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4C22×D5311yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—1062no—
288Q16—1062no—
289Q16—2121no—
290Q16—2121no—
291SD16—22no—
292SD16—1062no—
293SD16C22⋊D5314yescomplemented
294M16—22no—
295M16—1062no—
296C2×C8—531no—
297C2×C8—531no—
298C2×C8C22×D5311yescharacteristic
299C2×C8C22×D5311yescharacteristic
300C4⋊C4—531no—
301C4⋊C4—531no—
302C4⋊C4—531no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C22⋊C4—531no—
313C22⋊C4—531no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C42—531no—
320C42—1061no—
321C23—21no—
322C23—531no—
323C23D4×D5311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—1061no—
331Q8—1061no—
332Q8—1061no—
333Q8—1061no—
334Q8—2122no—
335Q8C2×C22⋊D5312yes—
336D4—21no—
337D4—42no—
338D4—1062no—
339D4—2122no—
340D4C2×C22⋊D5312yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—531no—
349C2×C4—531no—
350C2×C4—531no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4D4×D5311yescharacteristic
361C2×C4C23×D5311yescharacteristic
362C8—21no—
363C8—1061no—
364C8—1061no—
365C8C2×C22⋊D5312yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—1061no—
373C22—1061no—
374C22—1061no—
375C22C2×D4×D5311yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—532no—
381C4—1061no—
382C4—1061no—
383C4—2121no—
384C4—2121no—
385C4—2121no—
386C4—2121no—
387C4C22×C22⋊D5311yescharacteristic
388C4#1313326211yescharacteristic
389C2—41no—
390C2—42no—
391C2—2121no—
392C2D4×C22⋊D5311yescharacteristic
393C2#137×D5311yescharacteristic
394C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#137×C53C2
socleC22×C53C22×D4
hypercenterC2×#137D53
hypocenterC53#3276
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
2. Q8×C22⋊D53C2semidirect
4. #16811001C2semidirect
5. #16810881C2semidirect
6. #16810878C2semidirect
7. #16810831C2semidirect
8. #16810826C2semidirect
9. #16810785C2semidirect
10. #16810782C2semidirect
11. #16810779C2semidirect
12. #16810777C2semidirect
13. SD16×Dic53C2semidirect
14. #16810699C2semidirect
15. C2×#137×C53C2semidirect
90. SD16×C53D4semidirect
190. Dic53#137semidirect
199. C2×#137D53semidirect
213. C53#3276semidirect
293. SD16C22⋊D53semidirect