Finite Group Databasebeta

← #20285830

Subgroups — #20285830

Order 6784.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811246C211yesmaximal, characteristic, complemented
2#16811197C211yesmaximal, characteristic, complemented
3#16811016C211yesmaximal, characteristic, complemented
4#16811008C211yesmaximal, characteristic, complemented
5#16810984C211yesmaximal, characteristic, complemented
6#16810972C211yesmaximal, characteristic, complemented
7#16810847C211yesmaximal, characteristic, complemented
8#16810841C211yesmaximal, characteristic, complemented
9#16810831C211yesmaximal, characteristic, complemented
10#16810827C211yesmaximal, characteristic, complemented
11#16810783C211yesmaximal, characteristic, complemented
12#16810770C211yesmaximal, characteristic
13#16810762C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#451C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C2×D424C2211yescharacteristic
23#13133216C2211yescharacteristic
24#13133215C2211yescharacteristic
25#13133215C2211yescharacteristic
26#13133212—21no—
27#13133206C2211yescharacteristic
28#13133206C2211yescharacteristic
29#13133206C2211yescharacteristic
30#13133203—21no—
31#13133200—21no—
32#13133200—21no—
33#13133199—21no—
34#13133197—21no—
35#13133193—21no—
36#13133193—21no—
37#13133191C2211yescharacteristic
38Q8×Dic53C2211yescharacteristic
39#13133179C2211yescharacteristic
40#13133176C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C22×Q8×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×Q16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×Q8×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69#11719945—41no—
70#11719945—41no—
71C2×D212C2311yescharacteristic
72C2×D212C2311yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77Q16×C53—41no—
78Q16×C53—41no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊Dic53D411yescharacteristic
85C22⋊Dic53D411yescharacteristic
86#11719925—41no—
87#11719924—41no—
88#11719924—41no—
89#11719924—41no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53D411yescharacteristic
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112D424—41no—
113#11719913—41no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122Q8×C53—21no—
123Q8×C53—42no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—42no—
134C22⋊D53—81no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141D212—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145C4×D53—42no—
146C4×D53—81no—
147Dic106—21no—
148Dic106—21no—
149Dic106—81no—
150C8×C53—21no—
151C8×C53—21no—
152#88322—21no—
153#88322—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53C22×D411yescharacteristic
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53H(5,𝔽2)11yescharacteristic
170C4×C53H(5,𝔽2)11yescharacteristic
171Dic53—21no—
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177#3225—531nomaximal, Sylow
178C2×C53—41no—
179C2×C53#53511yescharacteristic
180C2×C53#57711yescharacteristic
181C2×C53#57811yescharacteristic
182D53—81no—
183D53—81no—
184#542—531no—
185#536—531no—
186#497—531no—
187#495—531no—
188#489—531no—
189#487—531no—
190#464—531no—
191#463—531no—
192#462—531no—
193#461—531no—
194#452—531no—
195#451D5311yescharacteristic, complemented
196#449—531no—
197#448—531no—
198#433—531no—
199C53#322511yescharacteristic, complemented, Sylow
200C2×#41—531no—
201C22×Q8C2×D5311yescharacteristic
202C2×Q16—531no—
203C2×Q16C2×D5311yescharacteristic
204C2×Q16C2×D5311yescharacteristic
205C2×SD16—531no—
206C2×SD16—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×D8—531no—
210C4⋊Q8—531no—
211C4⋊Q8—531no—
212C4⋊Q8—531no—
213#126—1061no—
214#124—531no—
215#124—531no—
216#124—531no—
217#124—1061no—
218#123—1061no—
219#123—1061no—
220#123—1061no—
221C22⋊Q8—1061no—
222C22⋊Q8—1061no—
223C22⋊Q8—1061no—
224C22⋊Q8C2×D5311yescharacteristic
225#121—531no—
226C4×Q8—531no—
227C4×D4—531no—
228#117—531no—
229C4⋊C8—531no—
230C4⋊C8—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103C2×D5311yescharacteristic
236#103C2×D5311yescharacteristic
237#102—531no—
238#102—531no—
239C22⋊C8—531no—
240C22⋊C8C2×D5311yescharacteristic
241#97—531no—
242C4×C8—531no—
243#41—2121no—
244#41—2121no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—531no—
249C2×Q8—1061no—
250C2×Q8C22×D5311yescharacteristic
251C2×Q8C22×D5311yescharacteristic
252C2×Q8C22×D5311yescharacteristic
253C2×D4—531no—
254C2×D4—531no—
255C2×D4—1061no—
256C2×D4—1061no—
257C22×C4—21no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4C22×D5311yescharacteristic
261Q16—41no—
262Q16—41no—
263Q16—2121no—
264SD16—2121no—
265SD16—2121no—
266SD16—2121no—
267SD16—2121no—
268D8—2121no—
269C2×C8—531no—
270C2×C8—531no—
271C2×C8C22×D5311yescharacteristic
272C2×C8C22×D5311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4C22×D5311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—531no—
286C22⋊C4—531no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C42—531no—
296C42—531no—
297C42—1061no—
298C23—1061no—
299C23—1061no—
300C23D4×D5311yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—1061no—
309Q8—1061no—
310Q8—4241no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—4241no—
316D4—2122no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—4241no—
331C2×C4—2122no—
332C2×C4D4×D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1061no—
337C8—1061no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—1061no—
354C4—1061no—
355C4—1061no—
356C4—1061no—
357C4—2121no—
358C4—2121no—
359C4#1313326211yescharacteristic
360C4#1313326211yescharacteristic
361C2—41no—
362C2—4241no—
363C2—4241no—
364C2#1681118811yescharacteristic
365C2#1681143211yescharacteristic
366C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#451C2
socleC22×C53C22×D4
hypercenter#451D53
hypocenterC53#3225
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #16811246C2semidirect
2. #16811197C2semidirect
3. #16811016C2semidirect
4. #16811008C2semidirect
5. #16810984C2semidirect
6. #16810972C2semidirect
7. #16810847C2semidirect
8. #16810841C2semidirect
9. #16810831C2semidirect
10. #16810827C2semidirect
11. #16810783C2semidirect
13. #16810762C2semidirect
14. #16810681C2semidirect
15. C53×#451C2semidirect
195. #451D53semidirect
199. C53#3225semidirect