Finite Group Databasebeta

← #20286782

Subgroups — #20286782

Order 6784.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811341C211yesmaximal, characteristic, complemented
2#16811284C211yesmaximal, characteristic
3#16810984C211yesmaximal, characteristic, complemented
4#16810963C211yesmaximal, characteristic, complemented
5#16810947C211yesmaximal, characteristic, complemented
6#16810916C211yesmaximal, characteristic, complemented
7#16810869C211yesmaximal, characteristic, complemented
8#16810863C211yesmaximal, characteristic, complemented
9#16810848C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810779C211yesmaximal, characteristic, complemented
12#16810727C211yesmaximal, characteristic, complemented
13#16810692C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#419C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133211—21no—
23#13133210—21no—
24#13133206—21no—
25#13133206C2211yescharacteristic
26#13133203—21no—
27#13133197C2211yescharacteristic
28#13133196C2211yescharacteristic
29#13133193—21no—
30#13133193C2211yescharacteristic
31#13133191—21no—
32#13133191C2211yescharacteristic
33Q8×Dic53C2211yescharacteristic
34C4×Dic106—21no—
35#13133182C2211yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53C2211yescharacteristic
38#13133175C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133148C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic
52C22×Q8×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54#117×C53C2211yescharacteristic
55#103×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53D411yescharacteristic
68#11719949—22no—
69#11719949—41no—
70C2×D212—21no—
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106C2311yescharacteristic
76M16×C53—41no—
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C22⋊C4×C53—21no—
82C42×C53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53D411yescharacteristic
85#11719925—22no—
86#11719925—41no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921D411yescharacteristic
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719913—41no—
110C8×D53—41no—
111C23×C53C2×D411yescharacteristic
112C22×D53—21no—
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—42no—
118Q8×C53#4112yes—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—42no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—41no—
129C22⋊D53—42no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136D212—21no—
137D212—21no—
138D212—42no—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—42no—
142Dic106—21no—
143Dic106—21no—
144Dic106—42no—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53C22×D411yescharacteristic
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—22no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53C2×#4111yescharacteristic
165C4×C53H(5,𝔽2)11yescharacteristic
166Dic53—21no—
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#3313—531nomaximal, Sylow
173C2×C53—41no—
174C2×C53#54611yescharacteristic
175C2×C53#57511yescharacteristic
176C2×C53#57811yescharacteristic
177D53—41no—
178D53—81no—
179#556—531no—
180#548—531no—
181#489—531no—
182#485—531no—
183#482—531no—
184#477—531no—
185#468—531no—
186#467—531no—
187#464—531no—
188#452—531no—
189#451—531no—
190#441—531no—
191#435—531no—
192#419D5311yescharacteristic, complemented
193#416—531no—
194C53#331311yescharacteristic, complemented, Sylow
195C2×#41—531no—
196C22×Q8C2×D5311yescharacteristic
197C2×Q16—531no—
198C2×Q16—531no—
199C2×SD16—531no—
200C2×M16C2×D5311yescharacteristic
201C22×C8—531no—
202C4⋊Q8—531no—
203#126—1061no—
204#126—1061no—
205#124—531no—
206#124—1061no—
207#124—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212#121—531no—
213#121—1061no—
214C4×Q8—531no—
215C4×Q8—531no—
216C4×Q8—1061no—
217C4×D4—531no—
218C4×D4—1061no—
219#117—531no—
220#117C2×D5311yescharacteristic
221#107—531no—
222#107—531no—
223#106—531no—
224C4⋊C8—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103C2×D5311yescharacteristic
229#103C2×D5311yescharacteristic
230#103C2×D5311yescharacteristic
231#103C2×D5311yescharacteristic
232#102—531no—
233#102—531no—
234#102—531no—
235C22⋊C8—531no—
236C22⋊C8—531no—
237#97—531no—
238#41—1062no—
239#41—2121no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—531no—
244C2×Q8—1061no—
245C2×Q8C22×D5311yescharacteristic
246C2×Q8C22×D5311yescharacteristic
247C2×D4—531no—
248C2×D4—531no—
249C2×D4—1061no—
250C2×D4—1061no—
251C22×C4—21no—
252C22×C4—531no—
253C22×C4—1061no—
254C22×C4C22×D5311yescharacteristic
255Q16—1062no—
256Q16—2121no—
257SD16—2121no—
258M16—41no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—2121no—
262C2×C8C22×D5311yescharacteristic
263C2×C8C22×D5311yescharacteristic
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4C22×D5311yescharacteristic
274C4⋊C4C22×D5311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—531no—
277C22⋊C4—531no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C42—21no—
286C42—531no—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C23—531no—
291C23—1061no—
292C23D4×D5311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—1061no—
298Q8—1061no—
299Q8—2122no—
300Q8#1171994512yes—
301D4—1061no—
302D4—1061no—
303D4—2122no—
304D4—2121no—
305D4—2122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—2122no—
322C2×C4D4×D5311yescharacteristic
323C2×C4C23×D5311yescharacteristic
324C8—21no—
325C8—21no—
326C8—1061no—
327C8—1061no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—1061no—
332C22—1061no—
333C22—1061no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—1061no—
344C4—1061no—
345C4—1061no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4C2×#1171994511yescharacteristic
350C4#1313326211yescharacteristic
351C2—41no—
352C2—4241no—
353C2—2121no—
354C2#1681126311yescharacteristic
355C2#1681142211yescharacteristic
356C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#419C2
socleC22×C53C22×D4
hypercenter#419D53
hypocenterC53#3313
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #16811341C2semidirect
3. #16810984C2semidirect
4. #16810963C2semidirect
5. #16810947C2semidirect
6. #16810916C2semidirect
7. #16810869C2semidirect
8. #16810863C2semidirect
9. #16810848C2semidirect
10. #16810788C2semidirect
11. #16810779C2semidirect
12. #16810727C2semidirect
13. #16810692C2semidirect
14. #16810618C2semidirect
15. C53×#419C2semidirect
192. #419D53semidirect
194. C53#3313semidirect