Finite Group Databasebeta

← #20285819

Subgroups — #20285819

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#16811242C211yesmaximal, characteristic, complemented
2#16811198C211yesmaximal, characteristic, complemented
3#16811005C211yesmaximal, characteristic, complemented
4#16810983C211yesmaximal, characteristic, complemented
5#16810973C211yesmaximal, characteristic, complemented
6#16810847C211yesmaximal, characteristic, complemented
7#16810840C211yesmaximal, characteristic, complemented
8#16810831C211yesmaximal, characteristic, complemented
9#16810826C211yesmaximal, characteristic, complemented
10#16810784C211yesmaximal, characteristic, complemented
11#16810777C211yesmaximal, characteristic, complemented
12#16810767C211yesmaximal, characteristic
13#16810760C211yesmaximal, characteristic, complemented
14#16810685C211yesmaximal, characteristic, complemented
15C53×#497C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22#13133217C2211yescharacteristic
23#13133215C2211yescharacteristic
24#13133210—21no—
25#13133206C2211yescharacteristic
26#13133205C2211yescharacteristic
27#13133203—21no—
28#13133202—21no—
29#13133202—21no—
30#13133201—21no—
31#13133195—21no—
32#13133195—21no—
33#13133194—21no—
34#13133193C2211yescharacteristic
35#13133188C2211yescharacteristic
36Q8×Dic53C2211yescharacteristic
37D4×Dic53C2211yescharacteristic
38#13133174C2211yescharacteristic
39#13133146C2211yescharacteristic
40#13133146C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51C4×#88322C2211yescharacteristic
52C2×Q16×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C2×SD16×C53C2211yescharacteristic
56C4⋊Q8×C53C2211yescharacteristic
57#124×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C22×Dic53—21no—
65#11719949—41no—
66#11719949—41no—
67Q8×D53—41no—
68Q8×D53—41no—
69C2×D212—21no—
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77Q16×C53—41no—
78Q16×C53—41no—
79SD16×C53—41no—
80SD16×C53—41no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90#11719925—41no—
91#11719924—41no—
92#11719923—41no—
93#11719922—41no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921D411yescharacteristic
100#11719921D411yescharacteristic
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108#11719919D411yescharacteristic
109#11719919D411yescharacteristic
110C4×Dic53—21no—
111C4×Dic53C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C22×D53—21no—
116C22×D53C2×D411yescharacteristic
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122Q8×C53—21no—
123D4×C53—21no—
124D4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—81no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—42no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—42no—
145C4×D53—42no—
146Dic106—21no—
147Dic106—21no—
148Dic106—42no—
149Dic106—81no—
150C8×C53—21no—
151C8×C53—21no—
152#88322—21no—
153#88322—21no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—21no—
165C4×C53—21no—
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—41no—
175Dic53—41no—
176Dic53—41no—
177#3225—531nomaximal, Sylow
178C2×C53—81no—
179C2×C53#53511yescharacteristic
180C2×C53#57711yescharacteristic
181C2×C53#57811yescharacteristic
182D53—41no—
183D53—81no—
184#542—531no—
185#536—531no—
186#497D5311yescharacteristic, complemented
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#451—531no—
196#449—531no—
197#448—531no—
198#433—531no—
199C53#322511yescharacteristic, complemented, Sylow
200C2×#41—531no—
201C22×Q8—531no—
202C2×Q16—531no—
203C2×Q16C2×D5311yescharacteristic
204C2×Q16C2×D5311yescharacteristic
205C2×SD16—531no—
206C2×SD16—531no—
207C2×SD16C2×D5311yescharacteristic
208C2×SD16C2×D5311yescharacteristic
209C2×D8—531no—
210C4⋊Q8—531no—
211C4⋊Q8—531no—
212C4⋊Q8C2×D5311yescharacteristic
213#126—1061no—
214#124—531no—
215#124—531no—
216#124—1061no—
217#124C2×D5311yescharacteristic
218#123—1061no—
219#123—1061no—
220#123—1061no—
221C22⋊Q8—531no—
222C22⋊Q8—1061no—
223C22⋊Q8—1061no—
224C22⋊Q8—1061no—
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#103—531no—
236#103—531no—
237#102—531no—
238#102—531no—
239C22⋊C8—531no—
240C22⋊C8—531no—
241#97C2×D5311yescharacteristic
242C4×C8—531no—
243#41—2121no—
244#41—2121no—
245C2×Q8—531no—
246C2×Q8—1061no—
247C2×Q8—1061no—
248C2×Q8—2121no—
249C2×Q8—2121no—
250C2×Q8C22×D5311yescharacteristic
251C2×Q8C22×D5311yescharacteristic
252C2×Q8C22×D5311yescharacteristic
253C2×D4—531no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4C22×D5311yescharacteristic
257C22×C4—531no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261Q16—41no—
262Q16—41no—
263Q16—2121no—
264SD16—41no—
265SD16—41no—
266SD16—2121no—
267SD16—2121no—
268D8—2121no—
269C2×C8—531no—
270C2×C8—531no—
271C2×C8C22×D5311yescharacteristic
272C2×C8C22×D5311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—531no—
287C22⋊C4—531no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C42—531no—
296C42—1061no—
297C42C22×D5311yescharacteristic
298C23—21no—
299C23—531no—
300C23—1061no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—1061no—
308Q8—1061no—
309Q8—4241no—
310Q8—2122no—
311D4—21no—
312D4—21no—
313D4—1061no—
314D4—1061no—
315D4—4241no—
316D4—2122no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—531no—
321C2×C4—531no—
322C2×C4—531no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—4241no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4D4×D5311yescharacteristic
332C2×C4D4×D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1061no—
337C8—1061no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—1061no—
342C22—1061no—
343C22—1061no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—1061no—
354C4—1061no—
355C4—1061no—
356C4—2121no—
357C4—2121no—
358C4—2121no—
359C4#1313326211yescharacteristic
360C4#1313326211yescharacteristic
361C2—81no—
362C2—4241no—
363C2—2121no—
364C2#1681118611yescharacteristic
365C2#1681143211yescharacteristic
366C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#497C2
socleC22×C53C22×D4
hypercenter#497D53
hypocenterC53#3225
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #16811242C2semidirect
2. #16811198C2semidirect
3. #16811005C2semidirect
4. #16810983C2semidirect
5. #16810973C2semidirect
6. #16810847C2semidirect
7. #16810840C2semidirect
8. #16810831C2semidirect
9. #16810826C2semidirect
10. #16810784C2semidirect
11. #16810777C2semidirect
13. #16810760C2semidirect
14. #16810685C2semidirect
15. C53×#497C2semidirect
186. #497D53semidirect
199. C53#3225semidirect