Finite Group Databasebeta

← #20285000

Subgroups — #20285000

Order 6784.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811226C211yesmaximal, characteristic
2#16811101C211yesmaximal, characteristic, complemented
3#16810948C211yesmaximal, characteristic, complemented
4#16810944C211yesmaximal, characteristic, complemented
5#16810932C211yesmaximal, characteristic, complemented
6#16810928C211yesmaximal, characteristic, complemented
7#16810821C211yesmaximal, characteristic, complemented
8#16810821C211yesmaximal, characteristic, complemented
9#16810762C211yesmaximal, characteristic, complemented
10#16810726C211yesmaximal, characteristic, complemented
11#16810724C211yesmaximal, characteristic, complemented
12SD16×Dic53C211yesmaximal, characteristic, complemented
13#16810708C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#448C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18#13133213C2211yescharacteristic
19#13133208C2211yescharacteristic
20#13133201—21no—
21#13133199—21no—
22#13133191C2211yescharacteristic
23#13133189—21no—
24#13133188—21no—
25Q8×Dic53—21no—
26Q8×Dic53—21no—
27Q8×Dic53C2211yescharacteristic
28Q8×Dic53C2211yescharacteristic
29Q8×Dic53C2211yescharacteristic
30Q8×Dic53C2211yescharacteristic
31D4×Dic53C2211yescharacteristic
32#13133179C2211yescharacteristic
33#13133176—21no—
34#13133176—21no—
35#13133176C2211yescharacteristic
36#13133176C2211yescharacteristic
37#13133151C2211yescharacteristic
38#13133150C2211yescharacteristic
39#13133149C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C22×Q8×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C22×Dic53—21no—
72C2×D212C2311yescharacteristic
73C2×C4×D53—21no—
74SD16×C53—22no—
75SD16×C53—22no—
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53C2311yescharacteristic
79C22⋊C4×C53—21no—
80C22⋊Dic53—21no—
81C22⋊Dic53—21no—
82C22⋊Dic53D411yescharacteristic
83C22⋊Dic53D411yescharacteristic
84#11719924—22no—
85#11719924—22no—
86#11719921—21no—
87#11719921—21no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53D411yescharacteristic
94C4⋊Dic53D411yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99C4×Dic53—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53—21no—
111Q8×C53—24no—
112Q8×C53—21no—
113Q8×C53—21no—
114Q8×C53#4112yes—
115Q8×C53#4112yes—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—81no—
119C2×C4×C53—24no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—81no—
128C22⋊D53—81no—
129C22⋊D53—81no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—81no—
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—21no—
139D212—21no—
140C4×D53—81no—
141C8×C53—21no—
142C8×C53—21no—
143#88322—21no—
144#88322—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53C22×D411yescharacteristic
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53C2×#4111yescharacteristic
160C4×C53C2×#4111yescharacteristic
161Dic53—21no—
162Dic53—21no—
163Dic53—21no—
164Dic53—21no—
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168#3126—531nomaximal, Sylow
169C2×C53—41no—
170C2×C53—81no—
171C2×C53#52511yescharacteristic
172C2×C53#57711yescharacteristic
173C2×C53#57711yescharacteristic
174D53—81no—
175#540—531no—
176#519—531no—
177#482—531no—
178#482—531no—
179#479—531no—
180#479—531no—
181#460—531no—
182#460—531no—
183#448—531no—
184#448D5311yescharacteristic, complemented
185#440—531no—
186#440—531no—
187C4×SD16—531no—
188C4×SD16—531no—
189#433—531no—
190C53#312611yescharacteristic, complemented, Sylow
191C22×Q8C2×D5311yescharacteristic
192C2×SD16—531no—
193C2×SD16—531no—
194C2×SD16C2×D5311yescharacteristic
195C2×SD16C2×D5311yescharacteristic
196#127—531no—
197#125—531no—
198#123—1061no—
199#123—1061no—
200#121—531no—
201#121—1061no—
202#121—1061no—
203#121C2×D5311yescharacteristic
204C4×Q8—531no—
205C4×Q8—531no—
206C4×Q8—531no—
207C4×Q8—531no—
208C4×Q8—1061no—
209C4×Q8—1061no—
210C4×D4—531no—
211C4×D4—531no—
212#117—531no—
213#117—531no—
214#117—1061no—
215#117—1061no—
216#107—531no—
217#107—531no—
218#106—531no—
219#106—531no—
220C4⋊C8—531no—
221C4⋊C8—531no—
222#103—531no—
223#103—531no—
224#103C2×D5311yescharacteristic
225#103C2×D5311yescharacteristic
226#102—531no—
227#102—531no—
228#102—531no—
229#102—531no—
230C22⋊C8—531no—
231C22⋊C8C2×D5311yescharacteristic
232#97—531no—
233C4×C8—531no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D5311yescharacteristic
238C2×Q8C22×D5311yescharacteristic
239C2×D4—21no—
240C2×D4—531no—
241C2×D4—1061no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4C22×D5311yescharacteristic
245C22×C4—21no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4C22×D5311yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—1062no—
252SD16—1062no—
253C2×C8—531no—
254C2×C8—531no—
255C2×C8C22×D5311yescharacteristic
256C2×C8C22×D5311yescharacteristic
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⋊C4C22×D5311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—531no—
271C22⋊C4—531no—
272C22⋊C4—1061no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C42—531no—
277C42—531no—
278C42—1061no—
279C42—1061no—
280C42—1061no—
281C42—1061no—
282C42—1061no—
283C23—21no—
284C23—1061no—
285C23D4×D5311yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1171994712yes—
290Q8#1171994712yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—1061no—
295D4—1061no—
296D4—4241no—
297D4—4241no—
298D4—4241no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—531no—
306C2×C4—531no—
307C2×C4—531no—
308C2×C4—531no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—4241no—
313C2×C4—4241no—
314C2×C4D4×D5311yescharacteristic
315C2×C4C23×D5311yescharacteristic
316C8—21no—
317C8—21no—
318C8—1061no—
319C8—1061no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—2121no—
327C22—2121no—
328C22—2121no—
329C22C2×D4×D5311yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—1061no—
335C4—1061no—
336C4—1061no—
337C4—1061no—
338C4—2121no—
339C4—2121no—
340C4—2121no—
341C4C2×#1171994711yescharacteristic
342C4C2×#1171994711yescharacteristic
343C2—41no—
344C2—81no—
345C2—4241no—
346C2#1681112911yescharacteristic
347C2#1681142911yescharacteristic
348C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#448C2
socleC22×C53C22×D4
hypercenter#448D53
hypocenterC53#3126
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
2. #16811101C2semidirect
3. #16810948C2semidirect
4. #16810944C2semidirect
5. #16810932C2semidirect
6. #16810928C2semidirect
7. #16810821C2semidirect
8. #16810821C2semidirect
9. #16810762C2semidirect
10. #16810726C2semidirect
11. #16810724C2semidirect
12. SD16×Dic53C2semidirect
13. #16810708C2semidirect
14. #16810681C2semidirect
15. C53×#448C2semidirect
184. #448D53semidirect
190. C53#3126semidirect