Finite Group Databasebeta

← #20286790

Subgroups — #20286790

Order 6784.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811269C211yesmaximal, characteristic
2#16810985C211yesmaximal, characteristic, complemented
3#16810958C211yesmaximal, characteristic, complemented
4#16810936C211yesmaximal, characteristic, complemented
5#16810900C211yesmaximal, characteristic, complemented
6#16810871C211yesmaximal, characteristic, complemented
7#16810862C211yesmaximal, characteristic, complemented
8#16810845C211yesmaximal, characteristic, complemented
9#16810783C211yesmaximal, characteristic, complemented
10#16810773C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810622C211yesmaximal, characteristic, complemented
14C2×#13133137C211yesmaximal, characteristic, complemented
15C53×#554C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×#11719925C2211yescharacteristic, complemented
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719917C2211yescharacteristic, complemented
22C22×#88322C2211yescharacteristic
23#13133205C2211yescharacteristic
24#13133202—21no—
25#13133196C2211yescharacteristic
26#13133191C2211yescharacteristic
27#13133190—21no—
28#13133185—21no—
29C4×D212C2211yescharacteristic
30C22⋊C4×D53—21no—
31#13133150C2211yescharacteristic, complemented
32#13133150C2211yescharacteristic, complemented
33#13133150C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133141C2211yescharacteristic, complemented
36#13133140C2211yescharacteristic, complemented
37#13133140C2211yescharacteristic, complemented
38#13133140C2211yescharacteristic, complemented
39#13133137C2211yescharacteristic
40#13133137C2211yescharacteristic
41#13133137C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic, complemented
47#13133128C2211yescharacteristic, complemented
48C4⋊Q8×C53C2211yescharacteristic, complemented
49#125×C53—21no—
50C22⋊Q8×C53—21no—
51C22⋊Q8×C53—21no—
52C22⋊Q8×C53C2211yescharacteristic
53C22⋊Q8×C53C2211yescharacteristic, complemented
54C4×Q8×C53C2211yescharacteristic, complemented
55C4×D4×C53—21no—
56C4×D4×C53C2211yescharacteristic
57C2×C4⋊C4×C53C2211yescharacteristic
58C2×C4⋊C4×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53—21no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53D411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69#11719945—22no—
70#11719945—41no—
71C2×D212—22no—
72C2×D212—21no—
73C2×D212—41no—
74C2×D212C2311yescharacteristic
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×Dic106C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53D411yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53D411yescharacteristic
92C22⋊C4×C53D411yescharacteristic
93C42×C53—21no—
94C42×C53C2311yescharacteristic
95C22⋊Dic53—21no—
96#11719925—41no—
97#11719924—22no—
98#11719924—41no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53C2311yescharacteristic
104#11719919—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—81no—
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—81no—
121D4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2411yescharacteristic
137C22⋊D53—42no—
138C22⋊D53—81no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—42no—
145D212#4112yes—
146C4×D53—42no—
147C4×D53—81no—
148Dic106—21no—
149Dic106—21no—
150#88322—21no—
151#88322—21no—
152#88322—21no—
153#88322—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#13712yes—
160C22×C53C22×D411yescharacteristic
161C2×D53—22no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—84no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53C2×#4111yescharacteristic
177C4×C53H(5,𝔽2)11yescharacteristic
178Dic53—41no—
179Dic53—41no—
180#3314—531nomaximal, Sylow
181C2×C53—22no—
182C2×C53—41no—
183C2×C53#54611yescharacteristic
184C2×C53C2×#13711yescharacteristic
185C2×C53#57611yescharacteristic
186D53—42no—
187D53—81no—
188#554D5311yescharacteristic, complemented
189#546—531no—
190#489—531no—
191#484—531no—
192#480—531no—
193#474—531no—
194#468—531no—
195#467—531no—
196#464—531no—
197#452—531no—
198#450—531no—
199#440—531no—
200#435—531no—
201#417—531no—
202C2×#102—531no—
203C53#331411yescharacteristic, complemented, Sylow
204C2×#41—531no—
205C22×D4—531no—
206C2×Q16—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×M16—531no—
210C22×C8—531no—
211C4⋊Q8C2×D5311yescharacteristic, complemented
212#125—21no—
213#124—531no—
214#123—1061no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—531no—
218C22⋊Q8C2×D5311yescharacteristic
219C22⋊Q8C2×D5311yescharacteristic, complemented
220#121—531no—
221#121—1061no—
222#120—1061no—
223C4×Q8C2×D5311yescharacteristic, complemented
224C4×D4—21no—
225C4×D4—531no—
226C4×D4C2×D5311yescharacteristic
227C2×C4⋊C4C2×D5311yescharacteristic
228C2×C4⋊C4C2×D5311yescharacteristic, complemented
229C2×C22⋊C4—1061no—
230#107—531no—
231#107—531no—
232#107—531no—
233C4⋊C8—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244C22⋊C8—531no—
245C22⋊C8—531no—
246#97—531no—
247C24—1061no—
248#41—1062no—
249#41—2121no—
250C2×Q8—21no—
251C2×Q8—531no—
252C2×Q8C22×D5311yescharacteristic
253C2×D4—531no—
254C2×D4—531no—
255C2×D4—1062no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4—2121no—
260C2×D4C22⋊D5311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4C22×D5311yescharacteristic
266C22×C4C22×D5311yescharacteristic
267Q16—2121no—
268SD16—1062no—
269SD16—2121no—
270M16—2121no—
271C2×C8—531no—
272C2×C8—531no—
273C2×C8—531no—
274C2×C8—531no—
275C2×C8—2121no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—531no—
283C4⋊C4—1061no—
284C4⋊C4C22⋊D5311yescharacteristic
285C4⋊C4C22×D5311yescharacteristic
286C4⋊C4C22×D5311yescharacteristic
287C4⋊C4C22×D5311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4C22⋊D5311yescharacteristic
296C22⋊C4C22⋊D5311yescharacteristic
297C42—21no—
298C42C22×D5311yescharacteristic
299C23—1061no—
300C23—1061no—
301C23—4241no—
302C23—2122no—
303C23—2122no—
304C23C2×C22⋊D5311yescharacteristic
305C23C2×C22⋊D5311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—81no—
309Q8—1061no—
310Q8—1061no—
311D4—22no—
312D4—532no—
313D4—1061no—
314D4—1061no—
315D4—1061no—
316D4—4241no—
317D4—2122no—
318D4—2122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—4241no—
332C2×C4—2122no—
333C2×C4C2×C22⋊D5311yescharacteristic
334C2×C4C2×C22⋊D5311yescharacteristic
335C2×C4C2×C22⋊D5311yescharacteristic
336C2×C4C2×C22⋊D5311yescharacteristic
337C2×C4C23×D5311yescharacteristic
338C8—1061no—
339C8—1061no—
340C8—1061no—
341C8—1061no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—1062no—
348C22—4244no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22—2121no—
354C22#1313324812yes—
355C22C22×C22⋊D5311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—2121no—
365C4—2121no—
366C4#41×D5311yescharacteristic
367C4#1313326311yescharacteristic
368C2—22no—
369C2—41no—
370C2—4241no—
371C2—2122no—
372C2#1681126111yescharacteristic
373C2C2×#1313324811yescharacteristic
374C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#554C2
socleC22×C53C22×D4
hypercenter#554D53
hypocenterC53#3314
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #16810985C2semidirect
3. #16810958C2semidirect
4. #16810936C2semidirect
5. #16810900C2semidirect
6. #16810871C2semidirect
7. #16810862C2semidirect
8. #16810845C2semidirect
9. #16810783C2semidirect
10. #16810773C2semidirect
11. #16810721C2semidirect
12. #16810695C2semidirect
13. #16810622C2semidirect
14. C2×#13133137C2semidirect
15. C53×#554C2semidirect
18. C2×#11719925C22semidirect
21. C2×#11719917C22semidirect
31. #13133150C22semidirect
32. #13133150C22semidirect
33. #13133150C22semidirect
34. C4⋊#88322C22semidirect
35. #13133141C22semidirect
36. #13133140C22semidirect
37. #13133140C22semidirect
38. #13133140C22semidirect
46. C22⋊#88322C22semidirect
47. #13133128C22semidirect
48. C4⋊Q8×C53C22semidirect
53. C22⋊Q8×C53C22semidirect
54. C4×Q8×C53C22semidirect
58. C2×C4⋊C4×C53C22semidirect
188. #554D53semidirect
203. C53#3314semidirect
211. C4⋊Q8C2×D53semidirect
219. C22⋊Q8C2×D53semidirect
223. C4×Q8C2×D53semidirect
228. C2×C4⋊C4C2×D53semidirect