Finite Group Databasebeta

← #27539187

Subgroups — #27539187

Order 13568.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285285C211yesmaximal, characteristic
2#20276917C211yesmaximal, characteristic, complemented
3#20276900C211yesmaximal, characteristic
4#20276879C212yesmaximal, complemented
5#20276777C212yesmaximal, complemented
6#20276649C211yesmaximal, characteristic, complemented
7#20276632C211yesmaximal, characteristic
8#20276584C212yesmaximal, complemented
9#20276577C212yesmaximal, complemented
10#20276518C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C53C211yesmaximal, characteristic, complemented
12#16811157—21no—
13C2×#13133163C2212yes—
14C2×#13133160C2212yes—
15#16811025C2211yescharacteristic
16#16811007C2211yescharacteristic
17#16810978—21no—
18#16810864C2211yescharacteristic
19#16810859C2211yescharacteristic
20#16810852—21no—
21C4×Dic212C2211yescharacteristic
22C4×#11719913—21no—
23C4×D424C2211yescharacteristic
24#16810450C2212yes—
25#16810448C2212yes—
26C4⋊#11719908C2212yes—
27#16810427C2212yes—
28#16810427C2212yes—
29#16810426C2212yes—
30#16810423C2212yes—
31C22⋊#11719908C2211yescharacteristic
32C22⋊#11719908C2211yescharacteristic
33#16810399C2212yes—
34C22×Q16×C53C2211yescharacteristic
35C2×C4×Q8×C53—21no—
36C4×Q16×C53—21no—
37C4×Q16×C53—21no—
38C4×Q16×C53C2212yes—
39C4×Q16×C53C2212yes—
40C2×#107×C53C2211yescharacteristic
41C2×#103×C53—21no—
42C2×C4×C8×C53C2211yescharacteristic
43C2×Dic212C2311yescharacteristic
44C2×#11719913—21no—
45C2×D424C2311yescharacteristic
46#13133211—41no—
47#13133209—21no—
48#13133205—21no—
49#13133200—41no—
50#13133196—21no—
51#13133191—21no—
52C4×Dic106—21no—
53C4×D212—21no—
54C4×C22⋊D53—41no—
55#13133174—41no—
56#13133163—24no—
57#13133160—24no—
58C2×#11719908C2312yes—
59C2×#11719908C2312yes—
60#13133151C2311yescharacteristic
61#13133151C2311yescharacteristic
62#13133149—21no—
63#13133142—21no—
64#13133142—21no—
65#13133139—21no—
66#13133139—21no—
67C22×Q8×C53—21no—
68C2×Q16×C53—24no—
69C2×Q16×C53—21no—
70C2×Q16×C53C2312yes—
71C22×C8×C53D411yescharacteristic
72C22×C8×C53C2311yescharacteristic
73C4×Q8×C53—22no—
74C4×Q8×C53—22no—
75C4×Q8×C53—41no—
76C4×Q8×C53—41no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—41no—
79C2×C42×C53—41no—
80C2×C42×C53D411yescharacteristic
81#107×C53—21no—
82#107×C53C2312yes—
83#103×C53—22no—
84#103×C53—22no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C2×Q8×C53—24no—
90C2×Q8×C53—22no—
91C2×Q8×C53—42no—
92C2×Q8×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53D811yescharacteristic
96C22×C4×C53D811yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C2×C22⋊D53—41no—
99C2×D212—21no—
100C2×C4×D53—41no—
101C2×Dic106—21no—
102Q16×C53—24no—
103Q16×C53—22no—
104Q16×C53#4114yes—
105C2×C8×C53—22no—
106C2×C8×C53—41no—
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2411yescharacteristic
111C4⋊C4×C53—22no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—42no—
114C4⋊C4×C53—42no—
115C42×C53—42no—
116C42×C53—42no—
117C42×C53D811yescharacteristic
118C42×C53D811yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C22⋊Dic53—41no—
122#11719921—41no—
123#11719921—41no—
124#11719921—41no—
125#11719921—41no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128#11719919—41no—
129#11719919—41no—
130#11719919—41no—
131#11719919—41no—
132C4×Dic53—41no—
133Dic212—22no—
134D424—22no—
135#11719913—42no—
136#11719908—22no—
137#11719908—22no—
138C23×C53C2×D811yescharacteristic
139C22×D53—41no—
140Q8×C53—24no—
141Q8×C53—44no—
142Q8×C53—42no—
143C2×C4×C53—22no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—41no—
147C2×C4×C53—44no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C22×D411yescharacteristic
157C22⋊D53—82no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161D212—41no—
162D212—41no—
163C4×D53—82no—
164Dic106—41no—
165Dic106—41no—
166C8×C53—21no—
167C8×C53—21no—
168C8×C53—21no—
169C8×C53C2×#4112yes—
170#33677—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53#51012yes—
174C22×C53C22×D811yescharacteristic
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—44no—
184C4×C53—82no—
185C4×C53#52511yescharacteristic
186C4×C53#57511yescharacteristic
187Dic53—81no—
188Dic53—81no—
189Dic53—81no—
190#3168—531no—
191C2×C4×Q16D5311yescharacteristic, complemented
192#2236—531no—
193#2233—531no—
194#2229—532no—
195#2211—532no—
196#2188—531no—
197#2185—531no—
198#2176—532no—
199#2175—532no—
200#2163—531no—
201C2×C53—22no—
202C2×C53#310011yescharacteristic
203C2×C53C2×#51011yescharacteristic
204C2×C53#341411yescharacteristic
205D53—161no—
206C22×Q16C2×D5311yescharacteristic
207#529—1061no—
208C2×C4×Q8—21no—
209C2×Q32—532no—
210C2×SD32—532no—
211#499—531no—
212#495—531no—
213#488—1061no—
214#467—531no—
215#466—531no—
216#465—1061no—
217C4×Q16—21no—
218C4×Q16—21no—
219C4×Q16—531no—
220C4×Q16C2×D5312yes—
221C4×Q16C2×D5312yes—
222C4×SD16—1061no—
223C4×D8—531no—
224C2×#107C2×D5311yescharacteristic
225C2×#103—21no—
226C2×C4×C8C2×D5311yescharacteristic
227#367—532no—
228#366—532no—
229C4⋊C16—532no—
230#358—532no—
231#358—532no—
232#358—532no—
233#357—532no—
234C22⋊C16—531no—
235C22⋊C16—531no—
236#346—532no—
237C53#3367711yescharacteristic, complemented, Sylow
238C22×Q8—21no—
239C2×Q16—24no—
240C2×Q16—21no—
241C2×Q16—531no—
242C2×Q16C22×D5312yes—
243C2×SD16—1061no—
244C2×D8—531no—
245C22×C8C22⋊D5311yescharacteristic
246C22×C8C22×D5311yescharacteristic
247#126—2121no—
248#125—1061no—
249#124—1061no—
250#123—2121no—
251C22⋊Q8—1061no—
252#121—1061no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8—1061no—
258C4×D4—1061no—
259C4×D4—2121no—
260#117—2121no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—41no—
263C2×C42—41no—
264C2×C42C22⋊D5311yescharacteristic
265Q32—1064no—
266SD32—1064no—
267C2×C16—532no—
268C2×C16—532no—
269#107—21no—
270#107—531no—
271#107—531no—
272#107C22×D5312yes—
273#106—1061no—
274#103—22no—
275#103—22no—
276#103—1061no—
277#103—1061no—
278#102—1061no—
279#102—1061no—
280C4×C8C22⋊D5311yescharacteristic
281C4×C8C22⋊D5311yescharacteristic
282C4×C8C22×D5311yescharacteristic
283C4×C8C22×D5311yescharacteristic
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—1061no—
289C2×D4—1061no—
290C2×D4—2121no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—2121no—
294C22×C4#1171992211yescharacteristic
295C22×C4#1171992211yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297Q16—24no—
298Q16—22no—
299Q16—1062no—
300Q16#1171994514yes—
301SD16—2122no—
302D8—1062no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8C2×C22⋊D5311yescharacteristic
306C2×C8C2×C22⋊D5311yescharacteristic
307C2×C8C2×C22⋊D5311yescharacteristic
308C2×C8C23×D5311yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C22⋊C4—2121no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C42—42no—
325C42—42no—
326C42—2121no—
327C42#1171992211yescharacteristic
328C42#1171992211yescharacteristic
329C42C2×C22⋊D5311yescharacteristic
330C42C2×C22⋊D5311yescharacteristic
331C16—1062no—
332C16—1062no—
333C23—2121no—
334C23C2×#1171992211yescharacteristic
335Q8—24no—
336Q8—44no—
337Q8—42no—
338Q8—2121no—
339Q8—2121no—
340D4—4242no—
341D4—2121no—
342D4—2121no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—4242no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4C2×#1171992211yescharacteristic
356C2×C4C2×#1171992211yescharacteristic
357C2×C4C2×#1171992211yescharacteristic
358C2×C4C2×#1171992211yescharacteristic
359C2×C4C2×#1171992211yescharacteristic
360C2×C4C22×C22⋊D5311yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8C2×#1171994512yes—
365C22—21no—
366C22—21no—
367C22—4241no—
368C22—4241no—
369C22—4241no—
370C22#1681106412yes—
371C22C22×#1171992211yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—44no—
378C4—82no—
379C4—4241no—
380C4—4241no—
381C4—4241no—
382C4#1681113411yescharacteristic
383C4#1681142111yescharacteristic
384C2—22no—
385C2—8481no—
386C2#2028471511yescharacteristic
387C2C2×#1681106411yescharacteristic
388C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C4×Q16×C53C2
socleC22×C53C22×D8
prime radical#16810859C22
hypercenterC2×C4×Q16D53
hypocenterC53#33677
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. #20276917C2semidirect
4. #20276879C2semidirect
5. #20276777C2semidirect
6. #20276649C2semidirect
8. #20276584C2semidirect
9. #20276577C2semidirect
10. #20276518C2semidirect
11. C2×C4×Q16×C53C2semidirect
191. C2×C4×Q16D53semidirect
237. C53#33677semidirect

Maximal subgroups

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

subgroupindex
#202852852
#202769172
#202769002
#202768792
#202767772
#202766492
#202766322
#202765842
#202765772
#202765182
C2×C4×Q16×C532
#3367753