Finite Group Databasebeta

← #27538893

Subgroups — #27538893

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
2C2×#16810661C211yesmaximal, characteristic, complemented
3#20276924C212yesmaximal, complemented
4#20276918C211yesmaximal, characteristic, complemented
5#20276913C211yesmaximal, characteristic, complemented
6#20276907C212yesmaximal, complemented
7#20276901C211yesmaximal, characteristic, complemented
8#20276896C211yesmaximal, characteristic, complemented
9#20276499C212yesmaximal, complemented
10#20276423C212yesmaximal, complemented
11C2×C53×C22⋊C16C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13#16811134—21no—
14C2×#13133176—21no—
15#16810864C2211yescharacteristic
16#16810864C2211yescharacteristic
17#16810859C2211yescharacteristic
18#16810859C2211yescharacteristic
19#16810852—21no—
20#16810852—21no—
21#16810661—21no—
22#16810661—21no—
23#16810661C2211yescharacteristic
24#16810661C2212yes—
25#16810661C2212yes—
26C2×#13133151C2211yescharacteristic
27C2×#13133151C2211yescharacteristic
28C2×#13133149—21no—
29#16810620—21no—
30#16810451C2212yescomplemented
31#16810451C2212yescomplemented
32#16810449C2212yescomplemented
33#16810449C2212yescomplemented
34#16810428C2212yescomplemented
35#16810428C2212yescomplemented
36#16810425C2212yescomplemented
37#16810425C2212yescomplemented
38C23×C8×C53C2211yescharacteristic
39C22×C16×C53C2212yes—
40C53×C22⋊C16C2211yescharacteristic
41C53×C22⋊C16C2211yescharacteristic
42C53×C22⋊C16C2212yes—
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—22no—
57#13133176—22no—
58#13133176—21no—
59#13133176—41no—
60#13133176—41no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C22⋊Dic53—41no—
64C2×C4×Dic53—41no—
65#13133151—21no—
66#13133151—21no—
67#13133151C2311yescharacteristic
68#13133151C2311yescharacteristic
69#13133151C2312yes—
70#13133151C2312yes—
71#13133149—21no—
72#13133149—22no—
73#13133149—22no—
74#13133142—21no—
75#13133142—21no—
76#13133139—21no—
77#13133139—21no—
78C23×C4×C53D411yescharacteristic
79C22×C8×C53—24no—
80C22×C8×C53D411yescharacteristic
81C22×C8×C53D411yescharacteristic
82C22×C8×C53D411yescharacteristic
83C22×C8×C53C2311yescharacteristic
84C22×C8×C53C2311yescharacteristic
85C22×C8×C53C2311yescharacteristic
86C2×C16×C53—24no—
87C2×C16×C53C2312yes—
88C2×C16×C53C2312yes—
89C24×C53D811yescharacteristic
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53D811yescharacteristic
93C22×C4×C53D811yescharacteristic
94C22×C4×C53D811yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C2×C22⋊D53—41no—
99C2×C22⋊D53—41no—
100C2×C22⋊D53—41no—
101C22×Dic53—41no—
102C22×Dic53—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
118C22⋊Dic53—42no—
119C22⋊Dic53—42no—
120C22⋊Dic53—41no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123#11719921—41no—
124#11719921—41no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53—22no—
128C4⋊Dic53—22no—
129C4⋊Dic53—41no—
130C4⋊Dic53—41no—
131#11719919—41no—
132#11719919—41no—
133C4×Dic53—42no—
134C4×Dic53—42no—
135C4×Dic53—41no—
136Dic212—22no—
137D424—22no—
138#11719913—42no—
139C16×C53—22no—
140C16×C53—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#12312yes—
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C2×D811yescharacteristic
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53#13712yes—
160C2×C4×C53C22×D411yescharacteristic
161C22⋊D53—82no—
162C22⋊D53—82no—
163C22⋊D53—82no—
164C2×Dic53—41no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—82no—
170C2×Dic53—82no—
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#48012yes—
198C4×C53C2×#12311yescharacteristic
199C4×C53C2×#13711yescharacteristic
200Dic53—81no—
201Dic53—82no—
202Dic53—81no—
203Dic53—82no—
204Dic53—81no—
205#3050—531no—
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⋊C16D5311yescharacteristic, complemented
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—1061no—
227C22×C16C2×D5312yes—
228#467—531no—
229#467—531no—
230#466—531no—
231#466—531no—
232#465—1061no—
233#465—1061no—
234#427—532no—
235#427—532no—
236#427—531no—
237#427—1061no—
238#427—1061no—
239C2×#107—531no—
240C2×#107—531no—
241C2×#106—1061no—
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⋊C16C2×D5311yescharacteristic
252C22⋊C16C2×D5311yescharacteristic
253C22⋊C16C2×D5312yes—
254C53#3364511yescharacteristic, complemented, Sylow
255C2×#41—1061no—
256C23×C4D21211yescharacteristic
257#135—1064no—
258C2×Q16—531no—
259C2×SD16—1061no—
260C2×D8—531no—
261C22×C8—24no—
262C22×C8D21211yescharacteristic
263C22×C8D21211yescharacteristic
264C22×C8D21211yescharacteristic
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—1062no—
277#117—1062no—
278#117—1061no—
279#117—2121no—
280#117—2121no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1061no—
283C2×C22⋊C4—2121no—
284C2×C42—2121no—
285C2×C16—24no—
286C2×C16C22×D5312yes—
287C2×C16C22×D5312yes—
288#107—532no—
289#107—532no—
290#107—531no—
291#107—531no—
292#107—1061no—
293#107—1061no—
294#106—1062no—
295#106—1062no—
296#106—1061no—
297#103—1061no—
298#103—1061no—
299#102—1061no—
300#102—1061no—
301C24D42411yescharacteristic
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—2121no—
312C22×C4—2121no—
313C22×C4—2121no—
314C22×C4D42411yescharacteristic
315C22×C4D42411yescharacteristic
316C22×C4D42411yescharacteristic
317C22×C4C2×D21211yescharacteristic
318C22×C4C2×D21211yescharacteristic
319C22×C4C2×D21211yescharacteristic
320Q16—1062no—
321SD16—2122no—
322D8—1062no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×D21211yescharacteristic
329C2×C8C2×D21211yescharacteristic
330C2×C8C2×D21211yescharacteristic
331C2×C8#1171994714yes—
332C2×C8C23×D5311yescharacteristic
333C4⋊C4—1061no—
334C4⋊C4—1061no—
335C4⋊C4—1062no—
336C4⋊C4—1062no—
337C4⋊C4—2121no—
338C4⋊C4—2121no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C22⋊C4—2122no—
342C22⋊C4—2122no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C42—2122no—
349C42—2122no—
350C42—2121no—
351C16—22no—
352C16—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—2121no—
357C23C2×D42411yescharacteristic
358C23C2×D42411yescharacteristic
359C23C2×D42411yescharacteristic
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—4242no—
375C2×C4—4242no—
376C2×C4—4242no—
377C2×C4—2122no—
378C2×C4—2122no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4#1313320112yes—
383C2×C4C2×D42411yescharacteristic
384C2×C4C2×D42411yescharacteristic
385C2×C4C2×D42411yescharacteristic
386C2×C4#1313325012yes—
387C2×C4C22×D21211yescharacteristic
388C8—21no—
389C8—21no—
390C8#1313320112yes—
391C8C2×#1171994712yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—4241no—
401C22—4241no—
402C22—4241no—
403C22#1681093712yes—
404C22#1681105512yes—
405C22C22×D42411yescharacteristic
406C4—21no—
407C4—21no—
408C4—4242no—
409C4—4241no—
410C4—4242no—
411C4—4241no—
412C4—4241no—
413C4#1681093712yes—
414C4C2×#1313320111yescharacteristic
415C4C2×#1313325011yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—8481no—
420C2C2×#1681093711yescharacteristic
421C2C2×#1681105511yescharacteristic
422C2#2028785111yescharacteristic
subgroupG/H
centerC2×C4C2×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×C22⋊C16C2
socleC22×C53C22×D8
prime radical#20284296C2
hypercenterC2×C22⋊C16D53
hypocenterC53#33645
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#16810661C2semidirect
3. #20276924C2semidirect
4. #20276918C2semidirect
5. #20276913C2semidirect
6. #20276907C2semidirect
7. #20276901C2semidirect
8. #20276896C2semidirect
9. #20276499C2semidirect
10. #20276423C2semidirect
11. C2×C53×C22⋊C16C2semidirect
30. #16810451C22semidirect
31. #16810451C22semidirect
32. #16810449C22semidirect
33. #16810449C22semidirect
34. #16810428C22semidirect
35. #16810428C22semidirect
36. #16810425C22semidirect
37. #16810425C22semidirect
215. C2×C22⋊C16D53semidirect
254. C53#33645semidirect

Maximal subgroups

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

subgroupindex
#202842962
C2×#168106612
#202769242
#202769182
#202769132
#202769072
#202769012
#202768962
#202764992
#202764232
C2×C53×C22⋊C162
#3364553