Finite Group Databasebeta

← #20285403

Subgroups — #20285403

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#16811247C211yesmaximal, characteristic, complemented
2#16811229C211yesmaximal, characteristic, complemented
3#16811012C211yesmaximal, characteristic, complemented
4#16810980C211yesmaximal, characteristic, complemented
5#16810971C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810870C211yesmaximal, characteristic, complemented
8#16810860C211yesmaximal, characteristic, complemented
9#16810853C211yesmaximal, characteristic, complemented
10#16810734C211yesmaximal, characteristic, complemented
11#16810722C211yesmaximal, characteristic, complemented
12#16810710C211yesmaximal, characteristic, complemented
13#16810705C211yesmaximal, characteristic, complemented
14#16810580C211yesmaximal, characteristic, complemented
15C53×#501C211yesmaximal, characteristic, complemented
16C2×#11719913C2211yescharacteristic
17C2×#11719913C2211yescharacteristic
18C2×D424C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133214C2211yescharacteristic
23#13133212—21no—
24#13133206—21no—
25#13133205C2211yescharacteristic
26#13133202—21no—
27#13133202—21no—
28#13133202—21no—
29#13133197C2211yescharacteristic
30#13133193—21no—
31#13133192—21no—
32#13133192—21no—
33#13133192C2211yescharacteristic
34#13133192C2211yescharacteristic
35#13133192C2211yescharacteristic
36#13133182C2211yescharacteristic
37#13133179C2211yescharacteristic
38#13133179C2211yescharacteristic
39#13133179C2211yescharacteristic
40#13133174C2211yescharacteristic
41#13133140C2211yescharacteristic
42#13133140C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133127C2211yescharacteristic
50C4×#88322C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C4⋊Q8×C53C2211yescharacteristic
53#125×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic
57#106×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×D212—21no—
61C2×D212—21no—
62C2×D212—21no—
63C2×D212—21no—
64C2×D212—21no—
65C2×D212C2311yescharacteristic
66C2×D212C2311yescharacteristic
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53C2311yescharacteristic
72C2×Dic106C2311yescharacteristic
73C2×C8×C53C2311yescharacteristic
74C2×C8×C53C2311yescharacteristic
75C4⋊C4×C53—21no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C42×C53C2311yescharacteristic
83#11719921—21no—
84#11719921—21no—
85#11719921—21no—
86#11719921—21no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921D411yescharacteristic
93#11719921D411yescharacteristic
94C4⋊Dic53—21no—
95#11719919—21no—
96#11719919D411yescharacteristic
97#11719919D411yescharacteristic
98C4×Dic53—21no—
99C4×Dic53C2311yescharacteristic
100C2×#88322C2311yescharacteristic
101C2×#88322C2311yescharacteristic
102D424—22no—
103D424—22no—
104#11719913—22no—
105#11719913—22no—
106#11719912—22no—
107C8×D53—22no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53C2×D411yescharacteristic
112Q8×C53—81no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53C2×D411yescharacteristic
118C2×C4×C53C2×D411yescharacteristic
119C2×C4×C53C2411yescharacteristic
120C2×Dic53—21no—
121C2×Dic53C2×D411yescharacteristic
122C2×Dic53C2×D411yescharacteristic
123C2×Dic53C2×D411yescharacteristic
124D212—21no—
125D212—21no—
126D212—21no—
127D212—21no—
128D212—21no—
129D212—21no—
130D212—81no—
131D212—81no—
132D212—81no—
133D212—81no—
134D212—81no—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—81no—
138C4×D53—81no—
139C4×D53—81no—
140Dic106—21no—
141Dic106—21no—
142C8×C53#4112yes—
143C8×C53#4112yes—
144#88322—21no—
145#88322—21no—
146C22×C53C22×D411yescharacteristic
147C2×D53—21no—
148C2×D53—21no—
149C2×D53—21no—
150C2×D53—41no—
151C2×D53—41no—
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53C2×#4111yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—21no—
170Dic53—41no—
171#3179—531nomaximal, Sylow
172C2×C53#53211yescharacteristic
173C2×C53#57611yescharacteristic
174C2×C53#57711yescharacteristic
175D53—41no—
176D53—81no—
177D53—81no—
178D53—81no—
179#542—531no—
180#540—531no—
181#501D5311yescharacteristic, complemented
182#496—531no—
183#488—531no—
184#486—531no—
185#469—531no—
186#468—531no—
187#466—531no—
188#465—531no—
189#442—531no—
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×D8—531no—
199C2×M16—531no—
200C22×C8—531no—
201C4⋊Q8C2×D5311yescharacteristic
202#127—531no—
203#126—1061no—
204#125C2×D5311yescharacteristic
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×D4—531no—
220C4×D4—531no—
221#117—531no—
222#107C2×D5311yescharacteristic
223#107C2×D5311yescharacteristic
224#106C2×D5311yescharacteristic
225#106C2×D5311yescharacteristic
226#103—531no—
227#103—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#97—531no—
235#97C2×D5311yescharacteristic
236C4×C8—531no—
237C4×C8—531no—
238C2×Q8—21no—
239C2×Q8—531no—
240C2×D4—531no—
241C2×D4—531no—
242C2×D4—531no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4—1061no—
248C22×C4—531no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1061no—
252SD16—1062no—
253SD16—1062no—
254D8—1062no—
255D8—1062no—
256M16—1062no—
257C2×C8—531no—
258C2×C8—531no—
259C2×C8—1062no—
260C2×C8C22×D5311yescharacteristic
261C2×C8C22×D5311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4C22×D5311yescharacteristic
270C4⋊C4C22×D5311yescharacteristic
271C4⋊C4C22×D5311yescharacteristic
272C4⋊C4C22×D5311yescharacteristic
273C22⋊C4—531no—
274C22⋊C4—531no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
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—531no—
288C23—1061no—
289C23—1061no—
290C23—1061no—
291Q8—81no—
292Q8—1061no—
293Q8—1061no—
294D4—1061no—
295D4—1061no—
296D4—1061no—
297D4—1061no—
298D4—1061no—
299D4—1061no—
300D4—4241no—
301D4—4241no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—4241no—
318C2×C4D4×D5311yescharacteristic
319C2×C4D4×D5311yescharacteristic
320C2×C4C23×D5311yescharacteristic
321C8—1061no—
322C8—1061no—
323C8#1171994912yes—
324C8#1171994912yes—
325C22—1061no—
326C22—1061no—
327C22—1061no—
328C22—2121no—
329C22—2121no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—1061no—
345C4—1061no—
346C4—1061no—
347C4—2121no—
348C4C2×#1171994911yescharacteristic
349C4C2×#1171994911yescharacteristic
350C2—4241no—
351C2—4241no—
352C2—4241no—
353C2—2121no—
354C2#1681117411yescharacteristic
355C2#1681142711yescharacteristic
356C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#501C2
socleC22×C53C22×D4
hypercenter#501D53
hypocenterC53#3179
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #16811247C2semidirect
2. #16811229C2semidirect
3. #16811012C2semidirect
4. #16810980C2semidirect
5. #16810971C2semidirect
6. #16810877C2semidirect
7. #16810870C2semidirect
8. #16810860C2semidirect
9. #16810853C2semidirect
10. #16810734C2semidirect
11. #16810722C2semidirect
12. #16810710C2semidirect
13. #16810705C2semidirect
14. #16810580C2semidirect
15. C53×#501C2semidirect
181. #501D53semidirect
194. C53#3179semidirect