Finite Group Databasebeta

← #20284961

Subgroups — #20284961

Order 6784.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811233C211yesmaximal, characteristic, complemented
2#16811098C211yesmaximal, characteristic
3#16810965C211yesmaximal, characteristic, complemented
4#16810961C211yesmaximal, characteristic, complemented
5#16810925C211yesmaximal, characteristic, complemented
6#16810921C211yesmaximal, characteristic, complemented
7#16810826C211yesmaximal, characteristic, complemented
8#16810826C211yesmaximal, characteristic, complemented
9#16810774C211yesmaximal, characteristic
10#16810724C211yesmaximal, characteristic, complemented
11#16810720C211yesmaximal, characteristic, complemented
12#16810714C211yesmaximal, characteristic, complemented
13SD16×Dic53C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#450C211yesmaximal, characteristic
16C2×#11719923C2211yescharacteristic
17C2×#11719923C2211yescharacteristic
18#13133215C2211yescharacteristic, complemented
19#13133210—21no—
20#13133210—21no—
21#13133208—21no—
22#13133208C2211yescharacteristic, complemented
23#13133207—21no—
24#13133196C2211yescharacteristic
25Q8×Dic53C2211yescharacteristic, complemented
26#13133182C2211yescharacteristic, complemented
27D4×Dic53—21no—
28D4×Dic53—21no—
29D4×Dic53C2211yescharacteristic
30D4×Dic53C2211yescharacteristic
31D4×Dic53C2211yescharacteristic
32D4×Dic53C2211yescharacteristic
33#13133176C2211yescharacteristic
34#13133176C2211yescharacteristic
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133141C2211yescharacteristic, complemented
46#13133141C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C24×C53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C2×Dic106C2311yescharacteristic
72SD16×C53—22no—
73SD16×C53—22no—
74C2×C8×C53C2311yescharacteristic
75C2×C8×C53C2311yescharacteristic
76C4⋊C4×C53—21no—
77C4⋊C4×C53C2311yescharacteristic
78C22⋊C4×C53—21no—
79C22⋊Dic53—21no—
80C22⋊Dic53—21no—
81C22⋊Dic53—21no—
82C22⋊Dic53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53D411yescharacteristic
86C22⋊Dic53D411yescharacteristic
87#11719923—22no—
88#11719923—22no—
89C4⋊Dic53—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53D411yescharacteristic
92C4⋊Dic53D411yescharacteristic
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—24no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—81no—
112C23×C53—81no—
113C23×C53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116D4×C53—24no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53#4112yes—
120D4×C53#4112yes—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—81no—
132C2×Dic53—81no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139Dic106—21no—
140Dic106—21no—
141C8×C53—21no—
142C8×C53—21no—
143#88322—21no—
144#88322—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—84no—
155C22×C53C22×D411yescharacteristic
156C4×C53—21no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53C2×#4111yescharacteristic
160C4×C53C2×#4111yescharacteristic
161Dic53—21no—
162Dic53—21no—
163Dic53—21no—
164Dic53—21no—
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169#3123—531nomaximal, Sylow
170C2×C53—41no—
171C2×C53—42no—
172C2×C53—42no—
173C2×C53#52511yescharacteristic
174C2×C53#57711yescharacteristic
175C2×C53#57711yescharacteristic
176#541—531no—
177#518—531no—
178#485—531no—
179#485—531no—
180#478—531no—
181#478—531no—
182#461—531no—
183#461—531no—
184#450—531no—
185#450D5311yescharacteristic
186#440—531no—
187#440—531no—
188C4×SD16—531no—
189C4×SD16—531no—
190#433—531no—
191C53#312311yescharacteristic, complemented, Sylow
192C22×D4C2×D5311yescharacteristic
193C2×SD16—531no—
194C2×SD16—531no—
195C2×SD16C2×D5311yescharacteristic
196C2×SD16C2×D5311yescharacteristic
197C4⋊Q8—531no—
198#126—1061no—
199#126—1061no—
200#125—531no—
201#125—1061no—
202#125—1061no—
203C22⋊Q8—531no—
204C22⋊Q8C2×D5311yescharacteristic
205C4×Q8—531no—
206C4×Q8—531no—
207C4×D4—531no—
208C4×D4—531no—
209C4×D4—531no—
210C4×D4—531no—
211C4×D4—1061no—
212C4×D4—1061no—
213#117—531no—
214#117—531no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217#107—531no—
218#107—531no—
219#106—531no—
220#106—531no—
221C4⋊C8—531no—
222C4⋊C8—531no—
223#103—531no—
224#103—531no—
225#103—531no—
226#103—531no—
227#102—531no—
228#102—531no—
229#102C2×D5311yescharacteristic
230#102C2×D5311yescharacteristic
231C22⋊C8—531no—
232C22⋊C8C2×D5311yescharacteristic
233#97—531no—
234C4×C8—531no—
235C24—21no—
236C2×Q8—531no—
237C2×Q8C22×D5311yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D5311yescharacteristic
242C2×D4C22×D5311yescharacteristic
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4C22×D5311yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—1062no—
251SD16—1062no—
252C2×C8—531no—
253C2×C8—531no—
254C2×C8C22×D5311yescharacteristic
255C2×C8C22×D5311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—531no—
258C4⋊C4—531no—
259C4⋊C4—531no—
260C4⋊C4—531no—
261C4⋊C4—531no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4C22×D5311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—531no—
272C22⋊C4—531no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C42—531no—
280C42—531no—
281C42—1061no—
282C42—1061no—
283C42—1061no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D5311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—1061no—
293Q8—1061no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1171994712yes—
298D4#1171994712yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—531no—
304C2×C4—531no—
305C2×C4—531no—
306C2×C4—531no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—4241no—
312C2×C4—4241no—
313C2×C4—4241no—
314C2×C4—4241no—
315C2×C4D4×D5311yescharacteristic
316C2×C4C23×D5311yescharacteristic
317C8—21no—
318C8—21no—
319C8—1061no—
320C8—1061no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D5311yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—1061no—
336C4—1061no—
337C4—1061no—
338C4—1061no—
339C4—2121no—
340C4—2121no—
341C4—2121no—
342C4—2121no—
343C4C2×#1171994711yescharacteristic
344C4C2×#1171994711yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#1681112911yescharacteristic
349C2#1681142911yescharacteristic
350C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3123
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16811233C2semidirect
3. #16810965C2semidirect
4. #16810961C2semidirect
5. #16810925C2semidirect
6. #16810921C2semidirect
7. #16810826C2semidirect
8. #16810826C2semidirect
10. #16810724C2semidirect
11. #16810720C2semidirect
12. #16810714C2semidirect
13. SD16×Dic53C2semidirect
14. #16810681C2semidirect
18. #13133215C22semidirect
22. #13133208C22semidirect
25. Q8×Dic53C22semidirect
26. #13133182C22semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
45. #13133141C22semidirect
46. #13133141C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
191. C53#3123semidirect