Finite Group Databasebeta

← #20285404

Subgroups — #20285404

Order 6784.

374 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#16811238C211yesmaximal, characteristic, complemented
2#16811224C211yesmaximal, characteristic, complemented
3#16811030C211yesmaximal, characteristic, complemented
4#16811011C211yesmaximal, characteristic, complemented
5#16810976C211yesmaximal, characteristic, complemented
6#16810968C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810868C211yesmaximal, characteristic, complemented
9#16810856C211yesmaximal, characteristic, complemented
10#16810853C211yesmaximal, characteristic, complemented
11#16810725C211yesmaximal, characteristic, complemented
12C4×#11719923C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810580C211yesmaximal, characteristic, complemented
15C53×#442C211yesmaximal, characteristic, complemented
16C2×#11719913C2211yescharacteristic
17C2×#11719923C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133213C2211yescharacteristic
23#13133210—21no—
24#13133208C2211yescharacteristic
25#13133204C2211yescharacteristic
26#13133203—21no—
27#13133200—21no—
28#13133200—21no—
29#13133199—21no—
30#13133197C2211yescharacteristic
31#13133196—21no—
32#13133192C2211yescharacteristic
33#13133191—21no—
34#13133190—21no—
35#13133188C2211yescharacteristic
36#13133188C2211yescharacteristic
37C4×Dic106C2211yescharacteristic
38C4×D212C2211yescharacteristic
39#13133174C2211yescharacteristic
40#13133151C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133127C2211yescharacteristic
50C4×#88322C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×D212—21no—
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53C2311yescharacteristic
71C2×Dic106—21no—
72C2×Dic106C2311yescharacteristic
73D8×C53—22no—
74C2×C8×C53C2311yescharacteristic
75C2×C8×C53C2311yescharacteristic
76C4⋊C4×C53C2311yescharacteristic
77C4⋊C4×C53C2311yescharacteristic
78C22⋊C4×C53—21no—
79C22⋊C4×C53—21no—
80C42×C53C2311yescharacteristic
81C22⋊Dic53—21no—
82C22⋊Dic53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85#11719923—22no—
86#11719922—22no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921D411yescharacteristic
91#11719921D411yescharacteristic
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919D411yescharacteristic
100#11719919D411yescharacteristic
101C4×Dic53—21no—
102C4×Dic53C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105#11719913—22no—
106#11719912—22no—
107C8×D53—22no—
108C23×C53—21no—
109C23×C53—21no—
110C22×D53—21no—
111C22×D53C2×D411yescharacteristic
112D4×C53—21no—
113D4×C53—21no—
114D4×C53—21no—
115D4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C22⋊D53—81no—
124C22⋊D53—81no—
125C22⋊D53—81no—
126C22⋊D53—81no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133D212—21no—
134D212—21no—
135D212—81no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—81no—
139Dic106—21no—
140Dic106—21no—
141Dic106—81no—
142C8×C53—21no—
143C8×C53#4112yes—
144#88322—21no—
145#88322#4112yes—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53C22×D411yescharacteristic
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53C2×#4111yescharacteristic
164C4×C53C2×#4111yescharacteristic
165Dic53—21no—
166Dic53—21no—
167Dic53—21no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#3179—531nomaximal, Sylow
172C2×C53—81no—
173C2×C53—81no—
174C2×C53#53211yescharacteristic
175C2×C53#57611yescharacteristic
176C2×C53#57711yescharacteristic
177D53—41no—
178D53—81no—
179#542—531no—
180#540—531no—
181#501—531no—
182#496—531no—
183#488—531no—
184#486—531no—
185#469—531no—
186#468—531no—
187#466—531no—
188#465—531no—
189#442D5311yescharacteristic, complemented
190#440—531no—
191C4×SD16—531no—
192C4×D8—531no—
193#405—531no—
194C53#317911yescharacteristic, complemented, Sylow
195C2×SD16—531no—
196C2×SD16—531no—
197C2×D8—531no—
198C2×D8C2×D5311yescharacteristic
199C2×M16—531no—
200C22×C8—531no—
201C4⋊Q8—531no—
202#127—531no—
203#126—1061no—
204#125—531no—
205#124—531no—
206#124—1061no—
207#123—1061no—
208#123—1061no—
209#123—1061no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212#121—531no—
213#121—531no—
214#121—531no—
215#121—1061no—
216#121—1061no—
217C4×Q8—531no—
218C4×D4—531no—
219C4×D4C2×D5311yescharacteristic
220C4×D4C2×D5311yescharacteristic
221#117—531no—
222#107—531no—
223#107—531no—
224#106—531no—
225#106C2×D5311yescharacteristic
226#103—531no—
227#103—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102C2×D5311yescharacteristic
233#102C2×D5311yescharacteristic
234#97—531no—
235#97C2×D5311yescharacteristic
236C4×C8—531no—
237C4×C8—531no—
238C2×Q8—531no—
239C2×Q8—1061no—
240C2×D4—531no—
241C2×D4—1061no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4—1061no—
246C2×D4C22×D5311yescharacteristic
247C2×D4C22×D5311yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—531no—
251C22×C4—1061no—
252SD16—1062no—
253SD16—1062no—
254D8—22no—
255D8—1062no—
256M16—1062no—
257C2×C8—531no—
258C2×C8—531no—
259C2×C8—1062no—
260C2×C8C22×D5311yescharacteristic
261C2×C8C22×D5311yescharacteristic
262C4⋊C4—531no—
263C4⋊C4—531no—
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4C22×D5311yescharacteristic
272C4⋊C4C22×D5311yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—531no—
276C22⋊C4—531no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C42—531no—
285C42—1061no—
286C42C22×D5311yescharacteristic
287C23—21no—
288C23—21no—
289C23—531no—
290C23—1061no—
291Q8—1061no—
292Q8—1061no—
293Q8—4241no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—1061no—
299D4—1061no—
300D4—4241no—
301D4—4241no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—4241no—
318C2×C4D4×D5311yescharacteristic
319C2×C4D4×D5311yescharacteristic
320C2×C4C23×D5311yescharacteristic
321C8—21no—
322C8—532no—
323C8—1061no—
324C8#1171994512yes—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—1061no—
332C22—1061no—
333C22—1061no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—1061no—
343C4—1061no—
344C4—1061no—
345C4—2121no—
346C4—2121no—
347C4—2121no—
348C4#41×D5311yescharacteristic
349C4C2×#1171994511yescharacteristic
350C2—81no—
351C2—81no—
352C2—4241no—
353C2—2121no—
354C2#1681117011yescharacteristic
355C2#1681142711yescharacteristic
356C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3179
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16811238C2semidirect
2. #16811224C2semidirect
3. #16811030C2semidirect
4. #16811011C2semidirect
5. #16810976C2semidirect
6. #16810968C2semidirect
7. #16810875C2semidirect
8. #16810868C2semidirect
9. #16810856C2semidirect
10. #16810853C2semidirect
11. #16810725C2semidirect
12. C4×#11719923C2semidirect
13. C4×#11719922C2semidirect
14. #16810580C2semidirect
15. C53×#442C2semidirect
189. #442D53semidirect
194. C53#3179semidirect