Finite Group Databasebeta

← #20285831

Subgroups — #20285831

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#16811245C211yesmaximal, characteristic, complemented
2#16811198C211yesmaximal, characteristic, complemented
3#16811016C211yesmaximal, characteristic, complemented
4#16811008C211yesmaximal, characteristic, complemented
5#16810984C211yesmaximal, characteristic, complemented
6#16810972C211yesmaximal, characteristic, complemented
7#16810845C211yesmaximal, characteristic, complemented
8#16810839C211yesmaximal, characteristic, complemented
9#16810825C211yesmaximal, characteristic, complemented
10#16810784C211yesmaximal, characteristic, complemented
11#16810777C211yesmaximal, characteristic, complemented
12#16810771C211yesmaximal, characteristic
13#16810763C211yesmaximal, characteristic, complemented
14#16810683C211yesmaximal, characteristic, complemented
15C53×#462C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C2×#11719924C2211yescharacteristic
23C2×D424C2211yescharacteristic
24#13133216C2211yescharacteristic
25#13133215C2211yescharacteristic
26#13133211—21no—
27#13133206C2211yescharacteristic
28#13133206C2211yescharacteristic
29#13133205—21no—
30#13133205C2211yescharacteristic
31#13133202—21no—
32#13133202—21no—
33#13133202—21no—
34#13133195—21no—
35#13133195—21no—
36#13133195—21no—
37#13133193C2211yescharacteristic
38#13133192C2211yescharacteristic
39C4×D212C2211yescharacteristic
40#13133174C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×Q16×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C4⋊Q8×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62#11719949—41no—
63#11719949—41no—
64Q8×D53—41no—
65Q8×D53—41no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212C2311yescharacteristic
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
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—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C42×C53—21no—
86C42×C53C2311yescharacteristic
87#11719925—41no—
88#11719924—41no—
89#11719924—41no—
90#11719924—41no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921D411yescharacteristic
101#11719921D411yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107#11719919D411yescharacteristic
108#11719919D411yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112D424—41no—
113#11719913—41no—
114C22×D53—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—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—21no—
136D212—21no—
137D212—21no—
138D212—21no—
139D212—42no—
140D212—81no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—42no—
144C4×D53—42no—
145C4×D53—81no—
146Dic106—21no—
147Dic106—21no—
148Dic106—42no—
149Dic106—81no—
150C8×C53—21no—
151C8×C53—21no—
152#88322—21no—
153#88322—21no—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
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×C53—41no—
170C4×C53H(5,𝔽2)11yescharacteristic
171C4×C53H(5,𝔽2)11yescharacteristic
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177#3225—531nomaximal, Sylow
178C2×C53#53511yescharacteristic
179C2×C53#57711yescharacteristic
180C2×C53#57811yescharacteristic
181D53—41no—
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#462D5311yescharacteristic, complemented
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×SD16—531no—
208C2×SD16—531no—
209C2×D8—531no—
210C4⋊Q8—531no—
211C4⋊Q8—531no—
212C4⋊Q8C2×D5311yescharacteristic
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—531no—
222C22⋊Q8—1061no—
223C22⋊Q8—1061no—
224C22⋊Q8—1061no—
225#121—531no—
226C4×Q8C2×D5311yescharacteristic
227C4×D4—531no—
228#117—531no—
229C4⋊C8—531no—
230C4⋊C8C2×D5311yescharacteristic
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⋊C8—531no—
241#97—531no—
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—531no—
255C2×D4—1061no—
256C2×D4—1061no—
257C22×C4—531no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
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—21no—
275C4⋊C4—21no—
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4C22×D5311yescharacteristic
284C22⋊C4—531no—
285C22⋊C4—531no—
286C22⋊C4—1061no—
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—21no—
296C42—531no—
297C42C22×D5311yescharacteristic
298C23—531no—
299C23—1061no—
300C23—1061no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—1061no—
308Q8—1061no—
309Q8—4241no—
310Q8—2122no—
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—531no—
322C2×C4—531no—
323C2×C4—531no—
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—1061no—
339C22—1061no—
340C22—1061no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—1061no—
355C4—1061no—
356C4—1061no—
357C4—2121no—
358C4—2121no—
359C4#1313326211yescharacteristic
360C4#1313326311yescharacteristic
361C2—4241no—
362C2—4241no—
363C2—2121no—
364C2#1681118911yescharacteristic
365C2#1681143211yescharacteristic
366C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#462C2
socleC22×C53C22×D4
hypercenter#462D53
hypocenterC53#3225
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #16811245C2semidirect
2. #16811198C2semidirect
3. #16811016C2semidirect
4. #16811008C2semidirect
5. #16810984C2semidirect
6. #16810972C2semidirect
7. #16810845C2semidirect
8. #16810839C2semidirect
9. #16810825C2semidirect
10. #16810784C2semidirect
11. #16810777C2semidirect
13. #16810763C2semidirect
14. #16810683C2semidirect
15. C53×#462C2semidirect
192. #462D53semidirect
199. C53#3225semidirect