Finite Group Databasebeta

← #20286449

Subgroups — #20286449

Order 6784.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133238C211yesmaximal, characteristic, complemented
2C22×#11719925C211yesmaximal, characteristic, complemented
3#16811284C211yesmaximal, characteristic, complemented
4#16811279C211yesmaximal, characteristic, complemented
5#16811004C211yesmaximal, characteristic, complemented
6#16810863C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810848C211yesmaximal, characteristic, complemented
9#16810843C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810782C211yesmaximal, characteristic, complemented
12#16810764C211yesmaximal, characteristic, complemented
13#16810715C211yesmaximal, characteristic, complemented
14#16810686C211yesmaximal, characteristic, complemented
15C53×#448C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C22×Dic106C2211yescharacteristic
19#13133238—24no—
20C2×#11719925—24no—
21C2×#11719925C2211yescharacteristic
22C2×#11719925C2211yescharacteristic
23C2×#11719925C2211yescharacteristic
24C2×#11719925C2211yescharacteristic
25C2×#11719913C2211yescharacteristic
26C2×#11719913C2211yescharacteristic
27C2×#11719922C2211yescharacteristic
28C22×#88322C2211yescharacteristic
29C2×C8×D53C2211yescharacteristic
30#13133206—21no—
31#13133206C2211yescharacteristic
32#13133204C2211yescharacteristic
33#13133203—21no—
34#13133197C2211yescharacteristic
35#13133194—21no—
36#13133193—21no—
37#13133193C2211yescharacteristic
38#13133191—21no—
39#13133191C2211yescharacteristic
40#13133189—21no—
41Q8×Dic53C2211yescharacteristic
42#13133182C2211yescharacteristic
43D4×Dic53—21no—
44C4×C22⋊D53—21no—
45C4×C22⋊D53C2211yescharacteristic
46#13133150C2211yescharacteristic
47#13133146C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133142C2211yescharacteristic
50#13133138C2211yescharacteristic
51#13133137C2211yescharacteristic
52#13133129C2211yescharacteristic
53C8×Dic53C2211yescharacteristic
54C22×Q8×C53C2211yescharacteristic
55C2×SD16×C53C2211yescharacteristic
56C2×SD16×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59#103×C53C2211yescharacteristic
60C22⋊C8×C53C2211yescharacteristic
61C2×Q8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53C2311yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53C2311yescharacteristic
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75#11719949—22no—
76#11719949—41no—
77#11719947—22no—
78#11719947—41no—
79C2×D212—21no—
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53C2311yescharacteristic
83C2×Dic106—22no—
84C2×Dic106—21no—
85C2×Dic106—41no—
86C2×Dic106C2311yescharacteristic
87C2×Dic106C2311yescharacteristic
88SD16×C53—22no—
89SD16×C53—41no—
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53D411yescharacteristic
97#11719925—22no—
98#11719925—22no—
99#11719925—22no—
100#11719925D414yescomplemented
101#11719922—22no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921D411yescharacteristic
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53C2311yescharacteristic
111#11719919—21no—
112#11719919—21no—
113#11719919D411yescharacteristic
114C4×Dic53—21no—
115C4×Dic53—21no—
116C4×Dic53C2311yescharacteristic
117C2×#88322—22no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120#11719913—22no—
121#11719913—41no—
122C8×D53—22no—
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53C2×D411yescharacteristic
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—42no—
131Q8×C53C2×D412yes—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—42no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—42no—
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2411yescharacteristic
143C22⋊D53—22no—
144C22⋊D53—42no—
145C22⋊D53—42no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53C2×D411yescharacteristic
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154D212—21no—
155D212—21no—
156D212—42no—
157C4×D53—21no—
158C4×D53—21no—
159C4×D53—42no—
160Dic106—21no—
161Dic106—21no—
162Dic106—21no—
163Dic106—42no—
164Dic106C2×D412yes—
165C8×C53—21no—
166C8×C53—21no—
167#88322—21no—
168#88322C2×D412yes—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53#13512yescomplemented
175C22×C53C22×D411yescharacteristic
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C4×C53—21no—
183C4×C53—22no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53C22×D411yescharacteristic
187C4×C53H(5,𝔽2)11yescharacteristic
188Dic53—22no—
189Dic53—21no—
190Dic53—21no—
191Dic53—21no—
192Dic53—41no—
193Dic53—41no—
194#3285—531nomaximal, Sylow
195C2×C53—22no—
196C2×C53—81no—
197C2×C53D4211yescharacteristic
198C2×C53C2×#13511yescharacteristic
199C2×C53#57811yescharacteristic
200D53—41no—
201D53—81no—
202C2×#135—531no—
203C22×Q16—531no—
204#548—531no—
205#548—531no—
206#495—531no—
207#467—531no—
208#466—531no—
209#464—531no—
210#464—531no—
211#452—531no—
212#452—531no—
213#448—531no—
214#448D5311yescharacteristic, complemented
215C4×Q16—531no—
216C8×D4—531no—
217C53#328511yescharacteristic, complemented, Sylow
218C2×#41—531no—
219C2×#41—531no—
220C22×Q8—531no—
221C22×Q8C2×D5311yescharacteristic
222#135—1064no—
223C2×Q16—531no—
224C2×Q16—531no—
225C2×Q16—531no—
226C2×Q16—531no—
227C2×Q16—1064no—
228C2×SD16—531no—
229C2×SD16—531no—
230C2×SD16C2×D5311yescharacteristic
231C2×SD16C2×D5311yescharacteristic
232C2×D8—531no—
233C22×C8—531no—
234C22×C8—531no—
235#124—531no—
236#124—531no—
237#124—1061no—
238#124—1061no—
239C22⋊Q8—531no—
240C22⋊Q8—531no—
241C22⋊Q8—1061no—
242C22⋊Q8—1061no—
243#121—531no—
244#121—1061no—
245#121—1061no—
246#121C2×D5311yescharacteristic
247C4×Q8—531no—
248C4×Q8—531no—
249C4×D4—531no—
250C4×D4—1061no—
251C4×D4—1061no—
252#107—531no—
253C4⋊C8—531no—
254#103—531no—
255#103—531no—
256#103C2×D5311yescharacteristic
257#103C2×D5311yescharacteristic
258#102—531no—
259#102—531no—
260C22⋊C8—531no—
261C22⋊C8C2×D5311yescharacteristic
262C4×C8—531no—
263#41—1062no—
264#41—1062no—
265#41—2121no—
266#41—2121no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—531no—
271C2×Q8—531no—
272C2×Q8—1062no—
273C2×Q8—1061no—
274C2×Q8—2121no—
275C2×Q8C22×D5311yescharacteristic
276C2×Q8C22×D5311yescharacteristic
277C2×D4—21no—
278C2×D4—531no—
279C2×D4—531no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4C22×D5311yescharacteristic
284C22×C4—21no—
285C22×C4—531no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4C22×D5311yescharacteristic
290Q16—534no—
291Q16—1062no—
292Q16—1062no—
293Q16—1062no—
294SD16—22no—
295SD16—41no—
296SD16—1062no—
297SD16—2121no—
298D8—1062no—
299C2×C8—531no—
300C2×C8—531no—
301C2×C8—1062no—
302C2×C8—1062no—
303C2×C8C22×D5311yescharacteristic
304C2×C8C22×D5311yescharacteristic
305C4⋊C4—531no—
306C4⋊C4—531no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4C22×D5311yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—531no—
314C22⋊C4—531no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C42—531no—
323C42—1061no—
324C42—1061no—
325C23—21no—
326C23—531no—
327C23—1061no—
328C23D4×D5311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—532no—
334Q8—1061no—
335Q8—1061no—
336Q8—1061no—
337Q8—2122no—
338Q8D4×D5312yes—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—1062no—
343D4—1061no—
344D4—1061no—
345D4—2122no—
346D4—2122no—
347D4—2122no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—531no—
355C2×C4—531no—
356C2×C4—531no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4D4×D5311yescharacteristic
366C2×C4C23×D5311yescharacteristic
367C8—21no—
368C8—21no—
369C8—532no—
370C8—1061no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—1061no—
377C22—1061no—
378C22—1061no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22#1313323812yescomplemented
383C22C2×D4×D5311yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—1061no—
389C4—1061no—
390C4—1061no—
391C4—1062no—
392C4—2121no—
393C4—2121no—
394C4C2×D4×D5311yescharacteristic
395C4#1313326211yescharacteristic
396C2—22no—
397C2—81no—
398C2—4241no—
399C2—2121no—
400C2#1681125511yescharacteristic
401C2C2×#1313323811yescharacteristic
402C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#448C2
socleC22×C53C22×D4
hypercenter#448D53
hypocenterC53#3285
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#13133238C2semidirect
2. C22×#11719925C2semidirect
3. #16811284C2semidirect
4. #16811279C2semidirect
5. #16811004C2semidirect
6. #16810863C2semidirect
7. #16810857C2semidirect
8. #16810848C2semidirect
9. #16810843C2semidirect
10. #16810788C2semidirect
11. #16810782C2semidirect
12. #16810764C2semidirect
13. #16810715C2semidirect
14. #16810686C2semidirect
15. C53×#448C2semidirect
100. #11719925D4semidirect
174. C22×C53#135semidirect
214. #448D53semidirect
217. C53#3285semidirect
382. C22#13133238semidirect