Finite Group Databasebeta

← #20286221

Subgroups — #20286221

Order 6784.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133247C211yesmaximal, characteristic, complemented
2C22×#11719923C211yesmaximal, characteristic, complemented
3D4×Dic106C211yesmaximal, characteristic, complemented
4#16811018C211yesmaximal, characteristic, complemented
5#16810867C211yesmaximal, characteristic, complemented
6#16810862C211yesmaximal, characteristic, complemented
7#16810842C211yesmaximal, characteristic, complemented
8#16810828C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810778C211yesmaximal, characteristic, complemented
11#16810774C211yesmaximal, characteristic, complemented
12#16810761C211yesmaximal, characteristic, complemented
13#16810723C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15C53×#546C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C22×Dic106C2211yescharacteristic, complemented
18#13133247—24no—
19C2×#11719925C2211yescharacteristic, complemented
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719925C2211yescharacteristic, complemented
22C2×#11719923—24no—
23C2×#11719923C2211yescharacteristic
24C2×#11719923C2211yescharacteristic
25C2×#11719923C2211yescharacteristic
26C2×#11719923C2211yescharacteristic
27C2×#11719923C2211yescharacteristic
28C2×#11719923C2211yescharacteristic
29C2×#11719917C2211yescharacteristic, complemented
30C22×#88322C2211yescharacteristic
31#13133217C2211yescharacteristic, complemented
32#13133216—21no—
33#13133196C2211yescharacteristic
34#13133196C2211yescharacteristic, complemented
35#13133194—21no—
36#13133194—21no—
37C4×Dic106C2211yescharacteristic, complemented
38D4×Dic53—21no—
39#13133150C2211yescharacteristic, complemented
40C4⋊#88322C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133140C2211yescharacteristic, complemented
43#13133140C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47C22⋊#88322C2211yescharacteristic
48C22⋊#88322C2211yescharacteristic, complemented
49#13133128C2211yescharacteristic, complemented
50C2×#41×C53C2211yescharacteristic
51C22×D4×C53C2211yescharacteristic
52#124×C53C2211yescharacteristic
53#123×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53—21no—
56#121×C53C2211yescharacteristic
57#120×C53—21no—
58C4×D4×C53C2211yescharacteristic
59C4×D4×C53C2211yescharacteristic
60C2×C22⋊C4×C53—21no—
61C24×C53—21no—
62#41×C53—22no—
63#41×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—41no—
70C2×D4×C53D411yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C2×D4×C53C2311yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C2×Dic106—22no—
80C2×Dic106—22no—
81C2×Dic106—21no—
82C2×Dic106—21no—
83C2×Dic106—41no—
84C2×Dic106—41no—
85C2×Dic106C2311yescharacteristic
86C2×Dic106C2311yescharacteristic
87C2×Dic106C2311yescharacteristic
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×C53—21no—
96C22⋊C4×C53D411yescharacteristic
97C22⋊C4×C53D411yescharacteristic
98C42×C53C2311yescharacteristic
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101#11719925—22no—
102#11719925—22no—
103#11719925—41no—
104#11719923—22no—
105#11719923—22no—
106#11719923—22no—
107#11719923—22no—
108#11719923—41no—
109#11719923D414yescomplemented
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×C53—21no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—81no—
128C23×C53C2×D411yescharacteristic
129C23×C53C2×D411yescharacteristic
130Q8×C53—21no—
131Q8×C53—21no—
132D4×C53—22no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—81no—
139D4×C53C2×D412yes—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—42no—
147C2×C4×C53—81no—
148C2×C4×C53C2×D411yescharacteristic
149C2×C4×C53C2×D411yescharacteristic
150C2×C4×C53C2×D411yescharacteristic
151C2×C4×C53C2×D411yescharacteristic
152C2×C4×C53C2411yescharacteristic
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159Dic106—21no—
160Dic106—21no—
161Dic106—21no—
162Dic106—21no—
163Dic106—21no—
164Dic106—42no—
165Dic106—42no—
166Dic106C2×D412yes—
167#88322—21no—
168#88322—21no—
169#88322—21no—
170#88322C2×D412yes—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—22no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—84no—
183C22×C53#13712yescomplemented
184C22×C53C22×D411yescharacteristic
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53C22×D411yescharacteristic
192C4×C53H(5,𝔽2)11yescharacteristic
193Dic53—22no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#3266—531nomaximal, Sylow
198C2×C53—22no—
199C2×C53—41no—
200C2×C53—42no—
201C2×C53—81no—
202C2×C53D4211yescharacteristic
203C2×C53C2×#13711yescharacteristic
204C2×C53#57711yescharacteristic
205C2×#137—531no—
206C22×SD16—531no—
207D4×Q8—531no—
208#546D5311yescharacteristic
209#497—531no—
210#468—531no—
211#467—531no—
212#464—531no—
213#461—531no—
214#452—531no—
215#451—531no—
216#450—531no—
217#448—531no—
218#440—531no—
219#435—531no—
220C53#326611yescharacteristic, complemented, Sylow
221C2×#41C2×D5311yescharacteristic
222C22×Q8—531no—
223C22×Q8—531no—
224C22×D4C2×D5311yescharacteristic
225#137—1064no—
226C2×Q16—531no—
227C2×Q16—531no—
228C2×Q16—531no—
229C2×SD16—531no—
230C2×SD16—531no—
231C2×SD16—531no—
232C2×SD16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—1064no—
236C2×M16—531no—
237C22×C8—531no—
238C4⋊Q8—531no—
239C4⋊Q8—1061no—
240#124C2×D5311yescharacteristic
241#123—21no—
242C22⋊Q8—531no—
243C22⋊Q8—531no—
244C22⋊Q8—1061no—
245C22⋊Q8—1061no—
246C22⋊Q8C2×D5311yescharacteristic
247#121—21no—
248#121C2×D5311yescharacteristic
249#120—21no—
250C4×Q8—531no—
251C4×D4—1061no—
252C4×D4C2×D5311yescharacteristic
253C4×D4C2×D5311yescharacteristic
254C2×C22⋊C4—21no—
255#107—531no—
256C4⋊C8—531no—
257#103—531no—
258#103—531no—
259#103—531no—
260#103—531no—
261#102—531no—
262#102—531no—
263C22⋊C8—531no—
264C22⋊C8—531no—
265#97—531no—
266C24—21no—
267#41—22no—
268#41—41no—
269C2×Q8—531no—
270C2×Q8—531no—
271C2×Q8—531no—
272C2×Q8—1062no—
273C2×Q8—1062no—
274C2×Q8—1061no—
275C2×Q8—1061no—
276C2×Q8—2121no—
277C2×Q8—2121no—
278C2×Q8C22×D5311yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D5311yescharacteristic
285C2×D4C22×D5311yescharacteristic
286C2×D4C22×D5311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4C22×D5311yescharacteristic
292C22×C4C22×D5311yescharacteristic
293Q16—1062no—
294Q16—1062no—
295Q16—2121no—
296SD16—534no—
297SD16—1062no—
298SD16—1062no—
299SD16—1062no—
300SD16—1062no—
301SD16—2121no—
302M16—1062no—
303C2×C8—531no—
304C2×C8—531no—
305C2×C8—531no—
306C2×C8—531no—
307C2×C8—1062no—
308C4⋊C4—21no—
309C4⋊C4—531no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4C22⋊D5311yescharacteristic
316C4⋊C4C22×D5311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4C22⋊D5311yescharacteristic
325C22⋊C4C22⋊D5311yescharacteristic
326C42—1061no—
327C42C22×D5311yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D5311yescharacteristic
334C23C2×C22⋊D5311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—532no—
338Q8—1061no—
339Q8—1061no—
340Q8—1061no—
341Q8—1061no—
342Q8—1061no—
343Q8—2122no—
344Q8—2122no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D5312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—2122no—
366C2×C4—2122no—
367C2×C4C2×C22⋊D5311yescharacteristic
368C2×C4C2×C22⋊D5311yescharacteristic
369C2×C4C2×C22⋊D5311yescharacteristic
370C2×C4C2×C22⋊D5311yescharacteristic
371C2×C4C23×D5311yescharacteristic
372C8—532no—
373C8—1061no—
374C8—1061no—
375C8—1061no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#1313324712yescomplemented
389C22C22×C22⋊D5311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—1062no—
397C4—2121no—
398C4—2121no—
399C4—2121no—
400C4C2×D4×D5311yescharacteristic
401C4#1313326211yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D5311yescharacteristic
407C2C2×#1313324711yescharacteristic
408C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#546C2
socleC22×C53C22×D4
hypercenter#546D53
hypocenterC53#3266
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#13133247C2semidirect
2. C22×#11719923C2semidirect
3. D4×Dic106C2semidirect
4. #16811018C2semidirect
5. #16810867C2semidirect
6. #16810862C2semidirect
7. #16810842C2semidirect
8. #16810828C2semidirect
9. #16810785C2semidirect
10. #16810778C2semidirect
11. #16810774C2semidirect
12. #16810761C2semidirect
13. #16810723C2semidirect
14. #16810695C2semidirect
17. C22×Dic106C22semidirect
19. C2×#11719925C22semidirect
20. C2×#11719925C22semidirect
21. C2×#11719925C22semidirect
29. C2×#11719917C22semidirect
31. #13133217C22semidirect
34. #13133196C22semidirect
37. C4×Dic106C22semidirect
39. #13133150C22semidirect
40. C4⋊#88322C22semidirect
41. #13133141C22semidirect
42. #13133140C22semidirect
43. #13133140C22semidirect
44. #13133140C22semidirect
48. C22⋊#88322C22semidirect
49. #13133128C22semidirect
109. #11719923D4semidirect
183. C22×C53#137semidirect
220. C53#3266semidirect
388. C22#13133247semidirect