Finite Group Databasebeta

← #27538891

Subgroups — #27538891

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
1#20284296C211yesmaximal, characteristic
2#20276923C212yesmaximal, complemented
3#20276917C211yesmaximal, characteristic, complemented
4#20276912C211yesmaximal, characteristic, complemented
5#20276906C212yesmaximal, complemented
6#20276900C211yesmaximal, characteristic, complemented
7#20276895C211yesmaximal, characteristic, complemented
8#20276497C212yesmaximal, complemented
9#20276420C212yesmaximal, complemented
10C2×C22⋊#11719908C211yesmaximal, characteristic, complemented
11C2×C53×#427C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13#16811134—21no—
14C22×#11719908C2212yes—
15#16810864C2211yescharacteristic
16#16810864C2211yescharacteristic
17#16810859C2211yescharacteristic
18#16810859C2211yescharacteristic
19#16810852—21no—
20#16810852—21no—
21#16810661C2211yescharacteristic
22#16810620—21no—
23#16810450C2212yescomplemented
24#16810450C2212yescomplemented
25#16810448C2212yescomplemented
26#16810448C2212yescomplemented
27#16810426C2212yescomplemented
28#16810426C2212yescomplemented
29#16810423C2212yescomplemented
30#16810423C2212yescomplemented
31C22⋊#11719908C2211yescharacteristic
32C22⋊#11719908C2211yescharacteristic
33C22⋊#11719908C2212yes—
34C23×C8×C53C2211yescharacteristic
35C2×#117×C53—21no—
36C53×#427—21no—
37C53×#427—21no—
38C53×#427C2212yes—
39C53×#427C2212yes—
40C2×#107×C53C2211yescharacteristic
41C2×#107×C53C2211yescharacteristic
42C2×#106×C53—21no—
43C2×#11719945—21no—
44#13133239—24no—
45C2×Dic212C2311yescharacteristic
46C2×#11719913—21no—
47C2×D424C2311yescharacteristic
48#13133200—41no—
49#13133196—21no—
50#13133196—21no—
51#13133191—21no—
52#13133191—21no—
53#13133186—41no—
54C4×C22⋊D53—41no—
55C4×C22⋊D53—41no—
56#13133176—21no—
57C2×#11719908—24no—
58C2×#11719908C2312yes—
59C2×#11719908C2312yes—
60#13133151C2311yescharacteristic
61#13133151C2311yescharacteristic
62#13133149—21no—
63#13133142—21no—
64#13133142—21no—
65#13133139—21no—
66#13133139—21no—
67C23×C4×C53D411yescharacteristic
68C22×C8×C53—24no—
69C22×C8×C53D411yescharacteristic
70C22×C8×C53D411yescharacteristic
71C22×C8×C53D411yescharacteristic
72C22×C8×C53C2311yescharacteristic
73C22×C8×C53C2311yescharacteristic
74C22×C8×C53C2311yescharacteristic
75#117×C53—22no—
76#117×C53—22no—
77#117×C53—41no—
78#117×C53—41no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C22⋊C4×C53—41no—
82C2×C42×C53—41no—
83#107×C53—21no—
84#107×C53—21no—
85#107×C53C2312yes—
86#107×C53C2312yes—
87#106×C53—22no—
88#106×C53—22no—
89C24×C53D811yescharacteristic
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53D811yescharacteristic
95C22×C4×C53D811yescharacteristic
96C22×C4×C53D811yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C2×C22⋊D53—41no—
101C2×C22⋊D53—41no—
102C2×C22⋊D53—41no—
103#11719945—42no—
104#11719945—42no—
105C2×D212—21no—
106C2×C4×D53—41no—
107C2×Dic106—21no—
108C2×C8×C53—22no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—22no—
112C2×C8×C53—22no—
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53#4114yes—
117C2×C8×C53C2411yescharacteristic
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C22⋊C4×C53—42no—
123C22⋊C4×C53—42no—
124C42×C53—42no—
125C42×C53—42no—
126C22⋊Dic53—41no—
127C22⋊Dic53—41no—
128C22⋊Dic53—41no—
129#11719921—41no—
130#11719921—41no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133#11719919—41no—
134#11719919—41no—
135C4×Dic53—41no—
136Dic212—22no—
137D424—22no—
138#11719913—42no—
139#11719908—22no—
140#11719908—22no—
141C23×C53—22no—
142C23×C53—22no—
143C23×C53—41no—
144C23×C53C2×D811yescharacteristic
145C23×C53C2×D811yescharacteristic
146C23×C53C2×D811yescharacteristic
147C22×D53—41no—
148C2×C4×C53—22no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53#12312yes—
160C2×C4×C53C2×D811yescharacteristic
161C2×C4×C53C2×D811yescharacteristic
162C2×C4×C53C2×D811yescharacteristic
163C2×C4×C53#13712yes—
164C2×C4×C53C22×D411yescharacteristic
165C22⋊D53—82no—
166C22⋊D53—82no—
167C22⋊D53—82no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171D212—41no—
172D212—41no—
173C4×D53—82no—
174Dic106—41no—
175Dic106—41no—
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#48012yes—
190C22×C53#50812yes—
191C22×C53C22×D811yescharacteristic
192C2×D53—81no—
193C2×D53—81no—
194C2×D53—81no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—82no—
198C4×C53—82no—
199C4×C53#48012yes—
200C4×C53C2×#12311yescharacteristic
201C4×C53C2×#13711yescharacteristic
202Dic53—81no—
203Dic53—81no—
204Dic53—81no—
205#3050—531no—
206C2×#427D5311yescharacteristic, complemented
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×C53C2×#48011yescharacteristic
220C2×C53C2×#50811yescharacteristic
221C2×C53#341411yescharacteristic
222D53—161no—
223C2×#135—531no—
224C23×C8C2×D5311yescharacteristic
225#525—1061no—
226C2×#117—21no—
227C22×C16—532no—
228#467—531no—
229#467—531no—
230#466—531no—
231#466—531no—
232#465—1061no—
233#465—1061no—
234#427—21no—
235#427—21no—
236#427—531no—
237#427C2×D5312yes—
238#427C2×D5312yes—
239C2×#107C2×D5311yescharacteristic
240C2×#107C2×D5311yescharacteristic
241C2×#106—21no—
242#416—1061no—
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—1061no—
256C23×C4C22⋊D5311yescharacteristic
257#135—1064no—
258C2×Q16—531no—
259C2×SD16—1061no—
260C2×D8—531no—
261C22×C8—24no—
262C22×C8C22⋊D5311yescharacteristic
263C22×C8C22⋊D5311yescharacteristic
264C22×C8C22⋊D5311yescharacteristic
265C22×C8C22×D5311yescharacteristic
266C22×C8C22×D5311yescharacteristic
267C22×C8C22×D5311yescharacteristic
268#123—2121no—
269C22⋊Q8—1061no—
270C22⋊Q8—1061no—
271#121—1061no—
272#121—1061no—
273#120—2121no—
274C4×D4—2121no—
275C4×D4—2121no—
276#117—22no—
277#117—22no—
278#117—41no—
279#117—41no—
280#117—1061no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C22⋊C4—41no—
284C2×C42—41no—
285C2×C16—532no—
286C2×C16—532no—
287C2×C16—1064no—
288#107—21no—
289#107—21no—
290#107—531no—
291#107—531no—
292#107C22×D5312yes—
293#107C22×D5312yes—
294#106—22no—
295#106—22no—
296#106—1061no—
297#103—1061no—
298#103—1061no—
299#102—1061no—
300#102—1061no—
301C24#1171992211yescharacteristic
302#41—2122no—
303#41—2122no—
304C2×Q8—1061no—
305C2×D4—1061no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4—2121no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
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—1062no—
321SD16—2122no—
322D8—1062no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8#1171994514yes—
329C2×C8C2×C22⋊D5311yescharacteristic
330C2×C8C2×C22⋊D5311yescharacteristic
331C2×C8C2×C22⋊D5311yescharacteristic
332C2×C8C23×D5311yescharacteristic
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—1061no—
338C4⋊C4—1061no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C42—42no—
349C42—42no—
350C42—2121no—
351C16—1062no—
352C16—1062no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—2121no—
357C23C2×#1171992211yescharacteristic
358C23C2×#1171992211yescharacteristic
359C23C2×#1171992211yescharacteristic
360Q8—2121no—
361Q8—2121no—
362D4—4242no—
363D4—4242no—
364D4—4242no—
365D4—2121no—
366D4—2121no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—4242no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4#1313320012yes—
383C2×C4C2×#1171992211yescharacteristic
384C2×C4C2×#1171992211yescharacteristic
385C2×C4C2×#1171992211yescharacteristic
386C2×C4#1313324812yes—
387C2×C4C22×C22⋊D5311yescharacteristic
388C8—21no—
389C8—21no—
390C8#1313320012yes—
391C8C2×#1171994512yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—4241no—
401C22—4241no—
402C22—4241no—
403C22#1681093612yes—
404C22#1681105312yes—
405C22C22×#1171992211yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—82no—
410C4—4241no—
411C4—4241no—
412C4—4241no—
413C4#1681093612yes—
414C4C2×#1313320011yescharacteristic
415C4C2×#1313324811yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—8481no—
420C2C2×#1681093611yescharacteristic
421C2C2×#1681105311yescharacteristic
422C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#427C2
socleC22×C53C22×D8
prime radical#20284296C2
hypercenterC2×#427D53
hypocenterC53#33645
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #20276923C2semidirect
3. #20276917C2semidirect
4. #20276912C2semidirect
5. #20276906C2semidirect
6. #20276900C2semidirect
7. #20276895C2semidirect
8. #20276497C2semidirect
9. #20276420C2semidirect
10. C2×C22⋊#11719908C2semidirect
11. C2×C53×#427C2semidirect
23. #16810450C22semidirect
24. #16810450C22semidirect
25. #16810448C22semidirect
26. #16810448C22semidirect
27. #16810426C22semidirect
28. #16810426C22semidirect
29. #16810423C22semidirect
30. #16810423C22semidirect
206. C2×#427D53semidirect
254. C53#33645semidirect

Maximal subgroups

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

subgroupindex
#202842962
#202769232
#202769172
#202769122
#202769062
#202769002
#202768952
#202764972
#202764202
C2×C22⋊#117199082
C2×C53×#4272
#3364553