Finite Group Databasebeta

← #20284401

Subgroups — #20284401

Order 6784.

472 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133246C211yesmaximal, characteristic, complemented
2C2×#13133225C211yesmaximal, characteristic, complemented
3#16811202C211yesmaximal, characteristic
4#16811196C211yesmaximal, characteristic, complemented
5#16810881C211yesmaximal, characteristic, complemented
6#16810880C211yesmaximal, characteristic, complemented
7#16810868C211yesmaximal, characteristic, complemented
8#16810865C211yesmaximal, characteristic, complemented
9#16810862C211yesmaximal, characteristic, complemented
10#16810850C211yesmaximal, characteristic, complemented
11#16810849C211yesmaximal, characteristic, complemented
12#16810664C211yesmaximal, characteristic, complemented
13#16810616C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19#13133246—24no—
20C2×Dic212C2211yescharacteristic
21C2×#11719925C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23#13133225—24no—
24C2×#11719912C2211yescharacteristic
25C2×#11719912C2211yescharacteristic
26C22×#88322C2211yescharacteristic
27C2×C8×D53C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic
29#13133216—21no—
30#13133203—21no—
31#13133202—21no—
32#13133199—21no—
33#13133197C2211yescharacteristic
34#13133197C2211yescharacteristic
35#13133196C2211yescharacteristic
36#13133196C2211yescharacteristic
37#13133194—21no—
38#13133193C2211yescharacteristic
39#13133190—21no—
40#13133189—21no—
41#13133188—21no—
42#13133188C2211yescharacteristic
43#13133175C2211yescharacteristic
44C4⋊C4×D53C2211yescharacteristic
45#13133150C2211yescharacteristic
46#13133149C2211yescharacteristic
47#13133148C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133142C2211yescharacteristic
50#13133141C2211yescharacteristic
51#13133140C2211yescharacteristic
52#13133140C2211yescharacteristic
53#13133138C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C2×M16×C53C2211yescharacteristic
56C22⋊Q8×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58#107×C53C2211yescharacteristic
59#103×C53C2211yescharacteristic
60#102×C53C2211yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53D411yescharacteristic
68C2×C22⋊D53D411yescharacteristic
69C22×Dic53—21no—
70Q8×D53—22no—
71Q8×D53—22no—
72#11719947—22no—
73#11719947—22no—
74#11719945—22no—
75#11719945—22no—
76C2×D212D411yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—21no—
82C2×Dic106D411yescharacteristic
83C2×Dic106C2311yescharacteristic
84C2×Dic106C2311yescharacteristic
85SD16×C53—22no—
86M16×C53—22no—
87C2×C8×C53C2311yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C4⋊C4×C53—21no—
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719925—22no—
98#11719923—22no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108#11719919—21no—
109#11719919—21no—
110#11719919—21no—
111C4×Dic53—21no—
112C2×#88322—22no—
113C2×#88322C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115Dic212—22no—
116#11719912—22no—
117#11719912D414yescomplemented
118C8×D53—22no—
119C8×D53—22no—
120C23×C53—21no—
121C23×C53C2×D411yescharacteristic
122C22×D53C2×D411yescharacteristic
123C22×D53C2×D411yescharacteristic
124Q8×C53—21no—
125Q8×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—81no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2411yescharacteristic
136C22⋊D53—22no—
137C22⋊D53—22no—
138C22⋊D53—42no—
139C22⋊D53—81no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—42no—
145C2×Dic53C2×D411yescharacteristic
146C2×Dic53C2×D411yescharacteristic
147D212—22no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—42no—
151C4×D53—81no—
152C4×D53C2×D412yes—
153C4×D53#4112yes—
154Dic106—22no—
155Dic106—21no—
156Dic106—21no—
157Dic106—21no—
158Dic106—21no—
159Dic106—42no—
160C8×C53—21no—
161C8×C53C2×D412yes—
162#88322—21no—
163#88322C2×D412yes—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53C22×D411yescharacteristic
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—21no—
176C2×D53#12112yes—
177C4×C53—21no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53C22×D411yescharacteristic
182C4×C53C2×#4111yescharacteristic
183Dic53—21no—
184Dic53—41no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188Dic53#12112yes—
189#3063—531nomaximal, Sylow
190C2×C53—41no—
191C2×C53—81no—
192C2×C53C2×#12111yescharacteristic
193C2×C53#57711yescharacteristic
194C2×C53#57811yescharacteristic
195D53—22no—
196D53—41no—
197C2×#137—531no—
198C2×#131—531no—
199#537—531no—
200#536—531no—
201#470—531no—
202#470—531no—
203#468—531no—
204#468D5311yescharacteristic, complemented
205#467—531no—
206#467—531no—
207#465—531no—
208#465—531no—
209#428—531no—
210#416—531no—
211C2×#103—531no—
212C53#306311yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C2×#41—531no—
215C22×Q8—531no—
216#137—1064no—
217C2×Q16—531no—
218C2×Q16—531no—
219C2×SD16—531no—
220C2×SD16C2×D5311yescharacteristic
221#131—1064no—
222C2×M16—531no—
223C2×M16—531no—
224C2×M16C2×D5311yescharacteristic
225C22×C8—531no—
226C22×C8—531no—
227C22×C8—531no—
228C4⋊Q8—1061no—
229#124—1061no—
230#123—1061no—
231#123—1061no—
232C22⋊Q8—531no—
233C22⋊Q8—531no—
234C22⋊Q8—531no—
235C22⋊Q8—531no—
236C22⋊Q8—531no—
237C22⋊Q8—1061no—
238C22⋊Q8C2×D5311yescharacteristic
239#121—531no—
240#121—1061no—
241#121—1061no—
242#121—1061no—
243#121C2×D5311yescharacteristic
244#117—531no—
245C2×C4⋊C4—531no—
246#107—531no—
247#107C2×D5311yescharacteristic
248#106—531no—
249#106—531no—
250#103—531no—
251#103—531no—
252#103—531no—
253#103—531no—
254#103—531no—
255#103C2×D5311yescharacteristic
256#102—531no—
257#102C2×D5311yescharacteristic
258#41—1062no—
259#41—1062no—
260#41—1062no—
261#41—1062no—
262C2×Q8—531no—
263C2×Q8—531no—
264C2×Q8—531no—
265C2×Q8—1061no—
266C2×Q8—1062no—
267C2×Q8—1062no—
268C2×Q8C22×D5311yescharacteristic
269C2×D4—21no—
270C2×D4—531no—
271C2×D4—531no—
272C2×D4—531no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4C22×D5311yescharacteristic
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4C22×D5311yescharacteristic
282Q16—1062no—
283Q16—1062no—
284SD16—22no—
285SD16—1062no—
286M16—22no—
287M16—534no—
288M16—1062no—
289C2×C8—531no—
290C2×C8—531no—
291C2×C8—1062no—
292C2×C8—1062no—
293C2×C8—1062no—
294C2×C8C22×D5311yescharacteristic
295C2×C8C22×D5311yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—531no—
298C4⋊C4—531no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4C22×D5311yescharacteristic
306C4⋊C4C22×D5311yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C42—1061no—
317C23—21no—
318C23—531no—
319C23—531no—
320C23D4×D5311yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—1062no—
324Q8—1061no—
325Q8—1061no—
326Q8—1061no—
327Q8—1061no—
328Q8—2122no—
329D4—21no—
330D4—21no—
331D4—81no—
332D4—1062no—
333D4—1062no—
334D4—1062no—
335D4—4241no—
336D4—2122no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—531no—
343C2×C4—532no—
344C2×C4—532no—
345C2×C4—531no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—4241no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4D4×D5311yescharacteristic
356C2×C4C23×D5311yescharacteristic
357C8—21no—
358C8—532no—
359C8—1061no—
360C8D4×D5312yes—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—532no—
368C22—1061no—
369C22—1061no—
370C22—1061no—
371C22—1061no—
372C22—1061no—
373C22C2×D4×D5311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—532no—
379C4—1061no—
380C4—2121no—
381C4—2121no—
382C4—2121no—
383C4—2121no—
384C4C2×D4×D5311yescharacteristic
385C4#41×D5311yescharacteristic
386C2—41no—
387C2—81no—
388C2—1062no—
389C2—2121no—
390C2#121×D5311yescharacteristic
391C2#1681143311yescharacteristic
392C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3063
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
2. C2×#13133225C2semidirect
4. #16811196C2semidirect
5. #16810881C2semidirect
6. #16810880C2semidirect
7. #16810868C2semidirect
8. #16810865C2semidirect
9. #16810862C2semidirect
10. #16810850C2semidirect
11. #16810849C2semidirect
12. #16810664C2semidirect
13. #16810616C2semidirect
14. #103×D53C2semidirect
15. C53×#468C2semidirect
117. #11719912D4semidirect
204. #468D53semidirect
212. C53#3063semidirect