Finite Group Databasebeta

← #20284383

Subgroups — #20284383

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133238C211yesmaximal, characteristic, complemented
2C2×#13133225C211yesmaximal, characteristic, complemented
3#16811203C211yesmaximal, characteristic, complemented
4#16811202C211yesmaximal, characteristic, complemented
5#16810870C211yesmaximal, characteristic, complemented
6#16810869C211yesmaximal, characteristic, complemented
7#16810866C211yesmaximal, characteristic, complemented
8#16810863C211yesmaximal, characteristic, complemented
9#16810862C211yesmaximal, characteristic, complemented
10#16810858C211yesmaximal, characteristic, complemented
11#16810857C211yesmaximal, characteristic, complemented
12#107×D53C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#16810621C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719947C2211yescharacteristic
19#13133238—24no—
20C2×#11719925C2211yescharacteristic
21C2×#11719913C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23#13133225—24no—
24C2×#11719912C2211yescharacteristic
25C2×#11719912C2211yescharacteristic
26C22×#88322C2211yescharacteristic
27C2×C8×D53C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic
29#13133197C2211yescharacteristic
30#13133196C2211yescharacteristic
31#13133195—21no—
32#13133194—21no—
33#13133193C2211yescharacteristic
34#13133192—21no—
35#13133192C2211yescharacteristic
36#13133191C2211yescharacteristic
37#13133190—21no—
38#13133190—21no—
39#13133190—21no—
40#13133189—21no—
41#13133188—21no—
42#13133188C2211yescharacteristic
43C4⋊C4×D53C2211yescharacteristic
44C4⋊C4×D53C2211yescharacteristic
45#13133151C2211yescharacteristic
46#13133150C2211yescharacteristic
47#13133150C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133141C2211yescharacteristic
50#13133140C2211yescharacteristic
51#13133139C2211yescharacteristic
52#13133138C2211yescharacteristic
53#13133137C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C2×M16×C53C2211yescharacteristic
56C22⋊Q8×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58#107×C53C2211yescharacteristic
59#103×C53C2211yescharacteristic
60#102×C53C2211yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53D411yescharacteristic
69C2×C22⋊D53D411yescharacteristic
70C22×Dic53—21no—
71#11719949—22no—
72#11719949—22no—
73#11719947—22no—
74#11719947—22no—
75#11719945—22no—
76#11719945—22no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212D411yescharacteristic
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53C2311yescharacteristic
85C2×C4×D53C2311yescharacteristic
86C2×Dic106D411yescharacteristic
87C2×Dic106C2311yescharacteristic
88SD16×C53—22no—
89M16×C53—22no—
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C4⋊C4×C53—21no—
93C4⋊C4×C53C2311yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99#11719925—22no—
100#11719922—22no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53C2311yescharacteristic
107C4⋊Dic53C2311yescharacteristic
108#11719919—21no—
109#11719919—21no—
110#11719919—21no—
111#11719919—21no—
112C2×#88322—22no—
113C2×#88322C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115#11719913—22no—
116#11719912—22no—
117#11719912—22no—
118C8×D53—22no—
119C8×D53D414yescomplemented
120C23×C53—21no—
121C23×C53C2×D411yescharacteristic
122C22×D53—21no—
123C22×D53C2×D411yescharacteristic
124C22×D53C2×D411yescharacteristic
125Q8×C53—21no—
126Q8×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—81no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2411yescharacteristic
137C22⋊D53—22no—
138C22⋊D53—22no—
139C22⋊D53—42no—
140C22⋊D53—81no—
141C22⋊D53—81no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—42no—
146C2×Dic53C2×D411yescharacteristic
147C2×Dic53C2×D411yescharacteristic
148D212—22no—
149D212—21no—
150D212—21no—
151D212—42no—
152D212—81no—
153C4×D53—21no—
154C4×D53—21no—
155C4×D53—42no—
156C4×D53—81no—
157C4×D53—81no—
158C4×D53C2×D412yes—
159C4×D53#4112yes—
160Dic106—22no—
161Dic106—21no—
162Dic106—21no—
163C8×C53—21no—
164C8×C53C2×D412yes—
165#88322—21no—
166#88322C2×D412yes—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53C22×D411yescharacteristic
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53#12112yes—
183C4×C53—21no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53C22×D411yescharacteristic
188C4×C53C2×#4111yescharacteristic
189Dic53—21no—
190Dic53—41no—
191Dic53—41no—
192Dic53—41no—
193Dic53#12112yes—
194#3061—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53—81no—
197C2×C53C2×#12111yescharacteristic
198C2×C53#57611yescharacteristic
199C2×C53#57811yescharacteristic
200D53—22no—
201D53—41no—
202D53—81no—
203C2×#135—531no—
204C2×#131—531no—
205#537—531no—
206#537—531no—
207#468—531no—
208#468—531no—
209#468—531no—
210#468D5311yescharacteristic, complemented
211#467—531no—
212#467—531no—
213#466—531no—
214#466—531no—
215C2×#107—531no—
216#417—531no—
217#417—531no—
218C53#306111yescharacteristic, complemented, Sylow
219C2×#41—531no—
220C2×#41—531no—
221C2×#41—531no—
222#135—1064no—
223C2×Q16—531no—
224C2×SD16—531no—
225C2×SD16C2×D5311yescharacteristic
226C2×D8—531no—
227#131—1064no—
228C2×M16—531no—
229C2×M16—531no—
230C2×M16C2×D5311yescharacteristic
231C22×C8—531no—
232C22×C8—531no—
233C22×C8—531no—
234C22⋊Q8—531no—
235C22⋊Q8—531no—
236C22⋊Q8—531no—
237C22⋊Q8—1061no—
238C22⋊Q8—1061no—
239C22⋊Q8C2×D5311yescharacteristic
240#121—531no—
241#121—531no—
242#121—531no—
243#121—1061no—
244#121—1061no—
245#121—1061no—
246#121—1061no—
247#121—1061no—
248#121—1061no—
249#121C2×D5311yescharacteristic
250C2×C4⋊C4—531no—
251C2×C4⋊C4—531no—
252#107—531no—
253#107—531no—
254#107—531no—
255#107C2×D5311yescharacteristic
256#103—531no—
257#103—531no—
258#103—531no—
259#103C2×D5311yescharacteristic
260#102—531no—
261#102—531no—
262#102—531no—
263#102C2×D5311yescharacteristic
264#41—1062no—
265#41—1062no—
266#41—1062no—
267#41—1062no—
268#41—1062no—
269#41—1062no—
270C2×Q8—531no—
271C2×Q8—531no—
272C2×Q8C22×D5311yescharacteristic
273C2×D4—21no—
274C2×D4—531no—
275C2×D4—531no—
276C2×D4—531no—
277C2×D4—531no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4C22×D5311yescharacteristic
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4C22×D5311yescharacteristic
291Q16—1062no—
292SD16—22no—
293SD16—1062no—
294D8—1062no—
295M16—22no—
296M16—1062no—
297M16—1062no—
298C2×C8—531no—
299C2×C8—534no—
300C2×C8—531no—
301C2×C8—1062no—
302C2×C8—1062no—
303C2×C8C22×D5311yescharacteristic
304C2×C8C22×D5311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—531no—
307C4⋊C4—531no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4C22×D5311yescharacteristic
314C4⋊C4C22×D5311yescharacteristic
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C23—21no—
324C23—531no—
325C23—531no—
326C23—1061no—
327C23D4×D5311yescharacteristic
328Q8—21no—
329Q8—21no—
330Q8—1062no—
331Q8—1061no—
332Q8—1061no—
333D4—21no—
334D4—21no—
335D4—81no—
336D4—1062no—
337D4—1062no—
338D4—1062no—
339D4—1061no—
340D4—1061no—
341D4—4241no—
342D4—4241no—
343D4—4241no—
344D4—2122no—
345D4—2122no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—531no—
352C2×C4—532no—
353C2×C4—532no—
354C2×C4—531no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—4241no—
361C2×C4—4241no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4D4×D5311yescharacteristic
365C2×C4C23×D5311yescharacteristic
366C8—21no—
367C8—532no—
368C8—1061no—
369C8D4×D5312yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—532no—
377C22—1061no—
378C22—1061no—
379C22—1061no—
380C22—1061no—
381C22—1061no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22C2×D4×D5311yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—532no—
391C4—1061no—
392C4—2121no—
393C4—2121no—
394C4—2121no—
395C4C2×D4×D5311yescharacteristic
396C4#41×D5311yescharacteristic
397C2—41no—
398C2—81no—
399C2—1062no—
400C2—4241no—
401C2—2121no—
402C2#121×D5311yescharacteristic
403C2#1681142711yescharacteristic
404C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3061
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#13133238C2semidirect
2. C2×#13133225C2semidirect
3. #16811203C2semidirect
4. #16811202C2semidirect
5. #16810870C2semidirect
6. #16810869C2semidirect
7. #16810866C2semidirect
8. #16810863C2semidirect
9. #16810862C2semidirect
10. #16810858C2semidirect
11. #16810857C2semidirect
12. #107×D53C2semidirect
13. #16810623C2semidirect
14. #16810621C2semidirect
15. C53×#468C2semidirect
119. C8×D53D4semidirect
210. #468D53semidirect
218. C53#3061semidirect