Finite Group Databasebeta

← #27538887

Subgroups — #27538887

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810661C211yesmaximal, characteristic, complemented
2#20276920C212yesmaximal, complemented
3#20276915C211yesmaximal, characteristic, complemented
4#20276910C211yesmaximal, characteristic, complemented
5#20276903C212yesmaximal, complemented
6#20276898C211yesmaximal, characteristic, complemented
7#20276893C211yesmaximal, characteristic, complemented
8#20276497C212yesmaximal, complemented
9#20276422C212yesmaximal, complemented
10C2×C22⋊#11719908C211yesmaximal, characteristic, complemented
11C53×#3050C211yesmaximal, characteristic
12C2×#13133176—21no—
13C22×#11719908C2212yes—
14#16810661—21no—
15#16810661—21no—
16#16810661C2212yes—
17#16810661C2212yes—
18C2×#13133151C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149—21no—
21#16810450C2212yescomplemented
22#16810450C2212yescomplemented
23#16810448C2212yescomplemented
24#16810448C2212yescomplemented
25#16810427C2212yescomplemented
26#16810427C2212yescomplemented
27#16810424C2212yescomplemented
28#16810424C2212yescomplemented
29C22⋊#11719908C2211yescharacteristic
30C22⋊#11719908C2211yescharacteristic
31C22⋊#11719908C2212yes—
32C2×#135×C53C2211yescharacteristic
33C23×C8×C53C2211yescharacteristic
34C53×#525—21no—
35C53×#467C2211yescharacteristic
36C53×#467C2211yescharacteristic
37C53×#466C2211yescharacteristic
38C53×#466C2211yescharacteristic
39C53×#465—21no—
40C53×#465—21no—
41C53×#427C2211yescharacteristic
42C53×#416—21no—
43#13133176—22no—
44#13133176—22no—
45#13133176—41no—
46#13133176—41no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C22⋊Dic53—41no—
50C2×C4×Dic53—41no—
51C2×#11719908—24no—
52C2×#11719908C2312yes—
53C2×#11719908C2312yes—
54#13133151—21no—
55#13133151—21no—
56#13133151C2312yes—
57#13133151C2312yes—
58#13133149—22no—
59#13133149—22no—
60C2×#41×C53—21no—
61C23×C4×C53D411yescharacteristic
62#135×C53—24no—
63C2×Q16×C53C2311yescharacteristic
64C2×SD16×C53—21no—
65C2×D8×C53C2311yescharacteristic
66C22×C8×C53—24no—
67C22×C8×C53D411yescharacteristic
68C22×C8×C53D411yescharacteristic
69C22×C8×C53D411yescharacteristic
70C22×C8×C53C2311yescharacteristic
71C22×C8×C53C2311yescharacteristic
72C22×C8×C53C2311yescharacteristic
73#123×C53—41no—
74C22⋊Q8×C53—21no—
75C22⋊Q8×C53—21no—
76#121×C53—21no—
77#121×C53—21no—
78#120×C53—41no—
79C4×D4×C53—41no—
80C4×D4×C53—41no—
81#117×C53—21no—
82#107×C53C2311yescharacteristic
83#107×C53C2311yescharacteristic
84#106×C53—21no—
85#103×C53—21no—
86#103×C53—21no—
87#102×C53—21no—
88#102×C53—21no—
89C24×C53D811yescharacteristic
90#41×C53—42no—
91#41×C53—42no—
92C2×Q8×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—41no—
100C22×C4×C53D811yescharacteristic
101C22×C4×C53D811yescharacteristic
102C22×C4×C53D811yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×Dic53—41no—
107C22×Dic53—41no—
108Q16×C53—22no—
109SD16×C53—42no—
110D8×C53—22no—
111C2×C8×C53—22no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53—22no—
115C2×C8×C53—22no—
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53#4114yes—
120C2×C8×C53C2411yescharacteristic
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C22⋊C4×C53—41no—
130C42×C53—41no—
131C22⋊Dic53—42no—
132C22⋊Dic53—42no—
133C4⋊Dic53—22no—
134C4⋊Dic53—22no—
135C4⋊Dic53—41no—
136C4⋊Dic53—41no—
137C4×Dic53—42no—
138C4×Dic53—42no—
139#11719908—22no—
140#11719908—22no—
141C23×C53—22no—
142C23×C53—22no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53C2×D811yescharacteristic
146C23×C53C2×D811yescharacteristic
147C23×C53C2×D811yescharacteristic
148Q8×C53—41no—
149Q8×C53—41no—
150D4×C53—41no—
151D4×C53—41no—
152D4×C53—82no—
153D4×C53—82no—
154D4×C53—82no—
155C2×C4×C53—22no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—82no—
166C2×C4×C53#12312yes—
167C2×C4×C53C2×D811yescharacteristic
168C2×C4×C53C2×D811yescharacteristic
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53#13712yes—
171C2×C4×C53C22×D411yescharacteristic
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—82no—
175C2×Dic53—82no—
176C8×C53—21no—
177C8×C53—21no—
178C8×C53#12312yes—
179C8×C53C2×#4112yes—
180#33645—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—44no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53#48012yes—
193C22×C53#50812yes—
194C22×C53C22×D811yescharacteristic
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—81no—
198C4×C53—81no—
199C4×C53—81no—
200C4×C53#48012yes—
201C4×C53C2×#12311yescharacteristic
202C4×C53C2×#13711yescharacteristic
203Dic53—82no—
204Dic53—82no—
205#3050D5311yescharacteristic
206C2×#427—531no—
207#2237—532no—
208#2236—531no—
209#2235—531no—
210#2234—532no—
211#2233—531no—
212#2232—531no—
213#2158—532no—
214#2138—532no—
215C2×C22⋊C16—531no—
216C2×C53—22no—
217C2×C53—22no—
218C2×C53—41no—
219C2×C53—161no—
220C2×C53C2×#48011yescharacteristic
221C2×C53C2×#50811yescharacteristic
222C2×C53#341411yescharacteristic
223C2×#135C2×D5311yescharacteristic
224C23×C8C2×D5311yescharacteristic
225#525—21no—
226C2×#117—1061no—
227C22×C16—532no—
228#467C2×D5311yescharacteristic
229#467C2×D5311yescharacteristic
230#466C2×D5311yescharacteristic
231#466C2×D5311yescharacteristic
232#465—21no—
233#465—21no—
234#427—532no—
235#427—532no—
236#427—1061no—
237#427—1061no—
238#427C2×D5311yescharacteristic
239C2×#107—531no—
240C2×#107—531no—
241C2×#106—1061no—
242#416—21no—
243#367—532no—
244#367—532no—
245#366—532no—
246#366—532no—
247#358—532no—
248#358—532no—
249#357—532no—
250#357—532no—
251C22⋊C16—531no—
252C22⋊C16—531no—
253C22⋊C16—532no—
254C53#3364511yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C23×C4C22⋊D5311yescharacteristic
257#135—24no—
258C2×Q16C22×D5311yescharacteristic
259C2×SD16—21no—
260C2×D8C22×D5311yescharacteristic
261C22×C8—24no—
262C22×C8C22⋊D5311yescharacteristic
263C22×C8C22⋊D5311yescharacteristic
264C22×C8C22⋊D5311yescharacteristic
265C22×C8C22×D5311yescharacteristic
266C22×C8C22×D5311yescharacteristic
267C22×C8C22×D5311yescharacteristic
268#123—41no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271#121—21no—
272#121—21no—
273#120—41no—
274C4×D4—41no—
275C4×D4—41no—
276#117—21no—
277#117—1062no—
278#117—1062no—
279#117—2121no—
280#117—2121no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1061no—
283C2×C22⋊C4—2121no—
284C2×C42—2121no—
285C2×C16—532no—
286C2×C16—532no—
287C2×C16—1064no—
288#107—532no—
289#107—532no—
290#107—1061no—
291#107—1061no—
292#107C22×D5311yescharacteristic
293#107C22×D5311yescharacteristic
294#106—21no—
295#106—1062no—
296#106—1062no—
297#103—21no—
298#103—21no—
299#102—21no—
300#102—21no—
301C24#1171992211yescharacteristic
302#41—42no—
303#41—42no—
304C2×Q8—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—2121no—
313C22×C4—2121no—
314C22×C4#1171992211yescharacteristic
315C22×C4#1171992211yescharacteristic
316C22×C4#1171992211yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320Q16—22no—
321SD16—42no—
322D8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8#1171994714yes—
329C2×C8C2×C22⋊D5311yescharacteristic
330C2×C8C2×C22⋊D5311yescharacteristic
331C2×C8C2×C22⋊D5311yescharacteristic
332C2×C8C23×D5311yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—1062no—
338C4⋊C4—1062no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—2122no—
347C22⋊C4—2122no—
348C42—41no—
349C42—2122no—
350C42—2122no—
351C16—1062no—
352C16—1062no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23C2×#1171992211yescharacteristic
358C23C2×#1171992211yescharacteristic
359C23C2×#1171992211yescharacteristic
360Q8—41no—
361Q8—41no—
362D4—41no—
363D4—41no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—82no—
378C2×C4—4242no—
379C2×C4—4242no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4#1313319812yes—
383C2×C4C2×#1171992211yescharacteristic
384C2×C4C2×#1171992211yescharacteristic
385C2×C4C2×#1171992211yescharacteristic
386C2×C4#1313324712yes—
387C2×C4C22×C22⋊D5311yescharacteristic
388C8—21no—
389C8—21no—
390C8#1313319812yes—
391C8C2×#1171994712yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22#1681093412yes—
404C22#1681105312yes—
405C22C22×#1171992211yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—4242no—
412C4—4242no—
413C4#1681093412yes—
414C4C2×#1313319811yescharacteristic
415C4C2×#1313324711yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—161no—
420C2C2×#1681093411yescharacteristic
421C2C2×#1681105311yescharacteristic
422C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3050C2
socleC22×C53C22×D8
hypercenter#3050D53
hypocenterC53#33645
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#16810661C2semidirect
2. #20276920C2semidirect
3. #20276915C2semidirect
4. #20276910C2semidirect
5. #20276903C2semidirect
6. #20276898C2semidirect
7. #20276893C2semidirect
8. #20276497C2semidirect
9. #20276422C2semidirect
10. C2×C22⋊#11719908C2semidirect
21. #16810450C22semidirect
22. #16810450C22semidirect
23. #16810448C22semidirect
24. #16810448C22semidirect
25. #16810427C22semidirect
26. #16810427C22semidirect
27. #16810424C22semidirect
28. #16810424C22semidirect
254. C53#33645semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168106612
#202769202
#202769152
#202769102
#202769032
#202768982
#202768932
#202764972
#202764222
C2×C22⋊#117199082
C53×#30502
#3364553