Finite Group Databasebeta

← #20285616

Subgroups — #20285616

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811203C211yesmaximal, characteristic, complemented
2#16811201C211yesmaximal, characteristic, complemented
3#16810957C211yesmaximal, characteristic, complemented
4#16810956C211yesmaximal, characteristic, complemented
5#16810937C211yesmaximal, characteristic, complemented
6#16810935C211yesmaximal, characteristic, complemented
7#16810872C211yesmaximal, characteristic, complemented
8#16810866C211yesmaximal, characteristic, complemented
9#16810861C211yesmaximal, characteristic, complemented
10#16810858C211yesmaximal, characteristic, complemented
11#16810784C211yesmaximal, characteristic, complemented
12#16810768C211yesmaximal, characteristic, complemented
13#16810767C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133196—21no—
23#13133195—21no—
24#13133193C2211yescharacteristic
25#13133193C2211yescharacteristic
26#13133193C2211yescharacteristic
27#13133192—21no—
28#13133192C2211yescharacteristic
29#13133191—21no—
30#13133191C2211yescharacteristic
31#13133190—21no—
32#13133190—21no—
33#13133189—21no—
34#13133189—21no—
35#13133188C2211yescharacteristic
36#13133188C2211yescharacteristic
37C2×C4⋊Dic53C2211yescharacteristic
38C4⋊C4×D53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133151C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133150C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59#41×C53—41no—
60#41×C53—41no—
61C2×Q8×C53C2311yescharacteristic
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53D411yescharacteristic
71C2×C22⋊D53D411yescharacteristic
72C22×Dic53—21no—
73#11719949—41no—
74#11719949—41no—
75#11719945—41no—
76#11719945—41no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212D411yescharacteristic
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53C2311yescharacteristic
84C2×C4×D53C2311yescharacteristic
85C2×Dic106D411yescharacteristic
86Q16×C53—41no—
87SD16×C53—41no—
88C2×C8×C53C2311yescharacteristic
89C2×C8×C53C2311yescharacteristic
90C4⋊C4×C53—21no—
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95#11719924—41no—
96#11719922—41no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108#11719919—21no—
109#11719919—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112#11719912—41no—
113C8×D53—41no—
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53—21no—
117C22×D53C2×D411yescharacteristic
118C22×D53C2×D411yescharacteristic
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122Q8×C53—21no—
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—41no—
135C22⋊D53—41no—
136C22⋊D53—81no—
137C22⋊D53—81no—
138C22⋊D53—81no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—81no—
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145D212—21no—
146D212—21no—
147D212—41no—
148D212—42no—
149D212—81no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—42no—
155C4×D53—81no—
156Dic106—41no—
157C8×C53—21no—
158C8×C53—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53C22×D411yescharacteristic
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—21no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53H(5,𝔽2)11yescharacteristic
182C4×C53H(5,𝔽2)11yescharacteristic
183Dic53—21no—
184Dic53—21no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188#3203—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53—81no—
191C2×C53#53411yescharacteristic
192C2×C53#57611yescharacteristic
193C2×C53#57811yescharacteristic
194D53—41no—
195D53—41no—
196D53—81no—
197#537—531no—
198#537—531no—
199#484—531no—
200#484—531no—
201#480—531no—
202#480—531no—
203#468—531no—
204#468—531no—
205#467—531no—
206#466—531no—
207#452—531no—
208#452D5311yescharacteristic, complemented
209#449—531no—
210#449—531no—
211#408—531no—
212C53#320311yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C2×#41—531no—
215C2×#41C2×D5311yescharacteristic
216C2×Q16C2×D5311yescharacteristic
217C2×SD16—531no—
218C2×SD16C2×D5311yescharacteristic
219C2×D8—531no—
220C2×M16—531no—
221C22×C8—531no—
222C22⋊Q8—531no—
223C22⋊Q8—531no—
224C22⋊Q8—531no—
225C22⋊Q8—1061no—
226C22⋊Q8—1061no—
227C22⋊Q8C2×D5311yescharacteristic
228#121—531no—
229#121—531no—
230#121—531no—
231#121—531no—
232#121—1061no—
233#121—1061no—
234#121—1061no—
235#121—1061no—
236#121—1061no—
237#121—1061no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240#107—531no—
241#107—531no—
242#107—531no—
243#107—531no—
244#103—531no—
245#103—531no—
246#103—531no—
247#103C2×D5311yescharacteristic
248#102—531no—
249#102—531no—
250#102—531no—
251#102C2×D5311yescharacteristic
252C22⋊C8—531no—
253C22⋊C8—531no—
254C22⋊C8—531no—
255C22⋊C8C2×D5311yescharacteristic
256#41—41no—
257#41—41no—
258#41—2121no—
259#41—2121no—
260#41—2121no—
261#41—2121no—
262C2×Q8—531no—
263C2×Q8C22×D5311yescharacteristic
264C2×Q8C22×D5311yescharacteristic
265C2×D4—21no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—531no—
269C2×D4—531no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4C22×D5311yescharacteristic
276C22×C4—21no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4C22×D5311yescharacteristic
283Q16—41no—
284SD16—41no—
285SD16—2121no—
286D8—2121no—
287M16—2121no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8—2121no—
291C2×C8C22×D5311yescharacteristic
292C2×C8C22×D5311yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—531no—
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4C22×D5311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C23—21no—
312C23—531no—
313C23—531no—
314C23—1061no—
315C23D4×D5311yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—2121no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—1061no—
325D4—1061no—
326D4—4241no—
327D4—4241no—
328D4—4241no—
329D4—4241no—
330D4—2121no—
331D4—2122no—
332D4—2121no—
333D4—2121no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—531no—
341C2×C4—531no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—4241no—
350C2×C4—4241no—
351C2×C4—2122no—
352C2×C4D4×D5311yescharacteristic
353C2×C4C23×D5311yescharacteristic
354C8—21no—
355C8—21no—
356C8—1061no—
357C8—1061no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—1061no—
365C22—1061no—
366C22—1061no—
367C22—1061no—
368C22—1061no—
369C22—1061no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C2×D4×D5311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—1061no—
379C4—1061no—
380C4—2121no—
381C4—2121no—
382C4—2121no—
383C4#1313326211yescharacteristic
384C4#1313326211yescharacteristic
385C2—41no—
386C2—81no—
387C2—4241no—
388C2—2121no—
389C2—2121no—
390C2#1681118311yescharacteristic
391C2#1681142511yescharacteristic
392C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3203
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #16811203C2semidirect
2. #16811201C2semidirect
3. #16810957C2semidirect
4. #16810956C2semidirect
5. #16810937C2semidirect
6. #16810935C2semidirect
7. #16810872C2semidirect
8. #16810866C2semidirect
9. #16810861C2semidirect
10. #16810858C2semidirect
11. #16810784C2semidirect
12. #16810768C2semidirect
13. #16810767C2semidirect
14. #16810588C2semidirect
15. C53×#452C2semidirect
208. #452D53semidirect
212. C53#3203semidirect