Finite Group Databasebeta

← #20286296

Subgroups — #20286296

Order 6784.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133247C211yesmaximal, characteristic, complemented
2C2×#13133237C211yesmaximal, characteristic, complemented
3#16811282C211yesmaximal, characteristic, complemented
4#16811010C211yesmaximal, characteristic, complemented
5#16810874C211yesmaximal, characteristic, complemented
6#16810850C211yesmaximal, characteristic, complemented
7#16810845C211yesmaximal, characteristic, complemented
8#16810820C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810768C211yesmaximal, characteristic, complemented
11#16810766C211yesmaximal, characteristic, complemented
12#16810761C211yesmaximal, characteristic, complemented
13#16810728C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15C53×#550C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18#13133247—24no—
19#13133237—24no—
20C2×#11719925C2211yescharacteristic
21C2×#11719925C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719923C2211yescharacteristic
24C2×#11719923C2211yescharacteristic
25C2×#11719923C2211yescharacteristic
26C2×#11719924C2211yescharacteristic
27C2×#11719922C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719917C2211yescharacteristic
30C22×#88322C2211yescharacteristic
31#13133205C2211yescharacteristic
32#13133203—21no—
33#13133196C2211yescharacteristic
34#13133195—21no—
35#13133191C2211yescharacteristic
36#13133189—21no—
37C4×Dic106C2211yescharacteristic
38#13133177—21no—
39#13133148C2211yescharacteristic
40C4⋊#88322C2211yescharacteristic
41#13133141C2211yescharacteristic
42#13133140C2211yescharacteristic
43#13133140C2211yescharacteristic
44#13133138C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133137C2211yescharacteristic
47C22⋊#88322C2211yescharacteristic
48C22⋊#88322C2211yescharacteristic
49#13133128C2211yescharacteristic
50C2×#41×C53C2211yescharacteristic
51C2×#41×C53C2211yescharacteristic
52#127×C53—21no—
53#127×C53C2211yescharacteristic
54#121×C53—21no—
55#121×C53—21no—
56#121×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58C4×Q8×C53C2211yescharacteristic
59C4×D4×C53—21no—
60C4×D4×C53C2211yescharacteristic
61#41×C53—22no—
62#41×C53—22no—
63#41×C53—41no—
64#41×C53—41no—
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53D411yescharacteristic
73C2×D4×C53C2311yescharacteristic
74C2×D4×C53C2311yescharacteristic
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C2311yescharacteristic
78C22×C4×C53C2311yescharacteristic
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81C22×Dic53—21no—
82#11719945—22no—
83#11719945—41no—
84C2×D212C2311yescharacteristic
85C2×C4×D53—21no—
86C2×Dic106—22no—
87C2×Dic106—21no—
88C2×Dic106—41no—
89C2×Dic106C2311yescharacteristic
90C2×Dic106C2311yescharacteristic
91C4⋊C4×C53—21no—
92C4⋊C4×C53D411yescharacteristic
93C4⋊C4×C53C2311yescharacteristic
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53D411yescharacteristic
97C22⋊C4×C53D411yescharacteristic
98C42×C53—21no—
99C42×C53C2311yescharacteristic
100C22⋊Dic53—21no—
101#11719925—22no—
102#11719925D414yescomplemented
103#11719924—22no—
104#11719923—22no—
105#11719923—22no—
106#11719923—22no—
107#11719923—41no—
108#11719922—22no—
109#11719922—41no—
110#11719921—21no—
111#11719921—21no—
112#11719921—21no—
113C4⋊Dic53C2311yescharacteristic
114#11719919—21no—
115#11719919—21no—
116C4×Dic53—21no—
117#11719917—22no—
118C2×#88322—22no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122C2×#88322C2311yescharacteristic
123C23×C53—21no—
124C23×C53—21no—
125C23×C53C2×D411yescharacteristic
126C23×C53C2×D411yescharacteristic
127C22×D53—21no—
128Q8×C53—21no—
129Q8×C53C2×D412yes—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—41no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—81no—
139D4×C53—81no—
140D4×C53—81no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53C2×D411yescharacteristic
150C2×C4×C53C2×D411yescharacteristic
151C2×C4×C53C2×D411yescharacteristic
152C2×C4×C53C2×D411yescharacteristic
153C2×C4×C53C2411yescharacteristic
154C22⋊D53—42no—
155C22⋊D53—42no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—42no—
160D212—21no—
161D212—21no—
162C4×D53—42no—
163Dic106—21no—
164Dic106—21no—
165Dic106—21no—
166Dic106—42no—
167Dic106C2×D412yes—
168#88322—21no—
169#88322—21no—
170#88322—21no—
171#88322C2×D412yes—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53C22×D411yescharacteristic
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—22no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—41no—
194C4×C53C22×D411yescharacteristic
195C4×C53H(5,𝔽2)11yescharacteristic
196Dic53—22no—
197Dic53—41no—
198Dic53—41no—
199#3271—531nomaximal, Sylow
200C2×C53—41no—
201C2×C53—41no—
202C2×C53—81no—
203C2×C53—81no—
204C2×C53D4211yescharacteristic
205C2×C53#57511yescharacteristic
206C2×C53#57711yescharacteristic
207D53—81no—
208C2×#137—531no—
209C2×#135—531no—
210#550D5311yescharacteristic, complemented
211#548—531no—
212#496—531no—
213#469—531no—
214#465—531no—
215#464—531no—
216#460—531no—
217#452—531no—
218#449—531no—
219#449—531no—
220#448—531no—
221#441—531no—
222#435—531no—
223C53#327111yescharacteristic, complemented, Sylow
224C2×#41—531no—
225C2×#41C2×D5311yescharacteristic
226C2×#41C2×D5311yescharacteristic
227C22×Q8—531no—
228#137—1064no—
229#135—1064no—
230C2×Q16—531no—
231C2×Q16—531no—
232C2×SD16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—531no—
236C2×SD16—531no—
237C2×D8—531no—
238C2×D8—531no—
239C2×M16—531no—
240C22×C8—531no—
241#127—21no—
242#127C2×D5311yescharacteristic
243#124—531no—
244#124—1061no—
245C22⋊Q8—531no—
246C22⋊Q8—1061no—
247#121—21no—
248#121—21no—
249#121—531no—
250#121—1061no—
251#121C2×D5311yescharacteristic
252#121C2×D5311yescharacteristic
253C4×Q8—531no—
254C4×Q8C2×D5311yescharacteristic
255C4×D4—21no—
256C4×D4—1061no—
257C4×D4C2×D5311yescharacteristic
258#106—531no—
259C4⋊C8—531no—
260#103—531no—
261#103—531no—
262#103—531no—
263#102—531no—
264#102—531no—
265#102—531no—
266C22⋊C8—531no—
267C22⋊C8—531no—
268#97—531no—
269#41—22no—
270#41—22no—
271#41—41no—
272#41—41no—
273#41—1062no—
274#41—2121no—
275C2×Q8—531no—
276C2×Q8—531no—
277C2×Q8—1062no—
278C2×Q8—1061no—
279C2×Q8—2121no—
280C2×Q8C22×D5311yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—531no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4C22⋊D5311yescharacteristic
291C2×D4C22×D5311yescharacteristic
292C2×D4C22×D5311yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4C22×D5311yescharacteristic
298C22×C4C22×D5311yescharacteristic
299Q16—534no—
300Q16—1062no—
301SD16—1062no—
302SD16—1062no—
303SD16—1062no—
304SD16—1062no—
305SD16—2121no—
306D8—1062no—
307D8—2121no—
308M16—1062no—
309C2×C8—531no—
310C2×C8—531no—
311C2×C8—531no—
312C2×C8—531no—
313C2×C8—1062no—
314C4⋊C4—21no—
315C4⋊C4—531no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4C22⋊D5311yescharacteristic
319C4⋊C4C22×D5311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4C22⋊D5311yescharacteristic
327C22⋊C4C22⋊D5311yescharacteristic
328C42—21no—
329C42—1061no—
330C42C22×D5311yescharacteristic
331C23—21no—
332C23—21no—
333C23—1061no—
334C23C2×C22⋊D5311yescharacteristic
335C23C2×C22⋊D5311yescharacteristic
336Q8—21no—
337Q8—532no—
338Q8—1061no—
339Q8—1061no—
340Q8—1061no—
341Q8—2122no—
342Q8D4×D5312yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—1061no—
355D4—1061no—
356D4—2122no—
357D4—2122no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4C2×C22⋊D5311yescharacteristic
372C2×C4C2×C22⋊D5311yescharacteristic
373C2×C4C2×C22⋊D5311yescharacteristic
374C2×C4C2×C22⋊D5311yescharacteristic
375C2×C4C23×D5311yescharacteristic
376C8—532no—
377C8—1061no—
378C8—1061no—
379C8—1061no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22C22×C22⋊D5311yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—22no—
401C4—41no—
402C4—1062no—
403C4—2121no—
404C4—2121no—
405C4C2×D4×D5311yescharacteristic
406C4#1313326211yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—81no—
411C2—4241no—
412C2D4×C22⋊D5311yescharacteristic
413C2#1681142111yescharacteristic
414C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#550C2
socleC22×C53C22×D4
hypercenter#550D53
hypocenterC53#3271
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#13133247C2semidirect
2. C2×#13133237C2semidirect
3. #16811282C2semidirect
4. #16811010C2semidirect
5. #16810874C2semidirect
6. #16810850C2semidirect
7. #16810845C2semidirect
8. #16810820C2semidirect
9. #16810785C2semidirect
10. #16810768C2semidirect
11. #16810766C2semidirect
12. #16810761C2semidirect
13. #16810728C2semidirect
14. #16810695C2semidirect
15. C53×#550C2semidirect
102. #11719925D4semidirect
210. #550D53semidirect
223. C53#3271semidirect