Finite Group Databasebeta

← #27367201

Subgroups — #27367201

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285642C211yesmaximal, characteristic
2#20285640C211yesmaximal, characteristic
3C2×#16811026C211yesmaximal, characteristic, complemented
4#20283426C211yesmaximal, characteristic
5#20274227C214yesmaximal, complemented
6#20274103C214yesmaximal, complemented
7#20273461C211yesmaximal, characteristic, complemented
8#20273401C211yesmaximal, characteristic, complemented
9C53×#1549C211yesmaximal, characteristic, complemented
10C2×#13133225—21no—
11#16811173C2211yescharacteristic
12C2×#13133217C2211yescharacteristic
13#16811026—21no—
14#16811026—21no—
15#16811026C2212yes—
16#16811026C2212yes—
17#16810997—21no—
18#16810996—21no—
19#16810974—21no—
20#16810973—21no—
21#16810829C2214yes—
22#16810825C2214yes—
23#16810764—22no—
24#16810762—22no—
25#16810700—21no—
26#16810700C2212yes—
27#16810688—21no—
28#16810688C2212yes—
29#16810676C2211yescharacteristic
30C2×#13133150—21no—
31C2×#13133150—21no—
32#16810591—21no—
33C2×C4×#88322C2211yescharacteristic
34#16810375C2212yes—
35#16810373C2212yes—
36#16810368C2214yes—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C53×#431C2211yescharacteristic
39C53×#423C2211yescharacteristic
40C53×#332C2214yes—
41C2×#11719945—21no—
42C22×Dic106—21no—
43C2×#11719913—24no—
44C2×#11719924—24no—
45#13133225—44no—
46C2×#11719912—22no—
47C22×#88322D411yescharacteristic
48C22×#88322D411yescharacteristic
49C2×C8×D53—22no—
50#13133217—21no—
51#13133217—21no—
52#13133217—21no—
53#13133217D411yescharacteristic
54#13133217C2312yes—
55#13133214D411yescharacteristic
56#13133205—21no—
57#13133191—21no—
58#13133191D412yes—
59C4×D212—21no—
60C4×D212C2312yes—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63#13133150—22no—
64#13133150—22no—
65#13133150—22no—
66#13133150—22no—
67C4⋊#88322C2312yes—
68#13133142—24no—
69#13133141—24no—
70C22⋊#88322—21no—
71#13133129—22no—
72#13133129—22no—
73C4×#88322—21no—
74C4×#88322D411yescharacteristic
75C4×#88322D411yescharacteristic
76C4×#88322C2311yescharacteristic
77C4×#88322C2311yescharacteristic
78C22×Q8×C53—21no—
79C2×M16×C53—21no—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53C2312yes—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C42×C53C2311yescharacteristic
87C4⋊C8×C53—21no—
88C4⋊C8×C53C2312yes—
89C4⋊C8×C53C2312yes—
90C22⋊C8×C53—21no—
91C4×C8×C53—21no—
92C2×Q8×C53—22no—
93C2×Q8×C53—41no—
94C2×Q8×C53—82no—
95C22×C4×C53—41no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C2×C22⋊D53—22no—
100C22×Dic53—41no—
101#11719945—42no—
102#11719945—42no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212C2×D412yes—
106C2×C4×D53—22no—
107C2×Dic106—21no—
108C2×Dic106—22no—
109C2×Dic106—41no—
110C2×Dic106—82no—
111M16×C53—42no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53C2411yescharacteristic
122#11719924—84no—
123#11719921—22no—
124#11719921—22no—
125C4⋊Dic53—21no—
126C4⋊Dic53—22no—
127C4⋊Dic53—22no—
128C4⋊Dic53—41no—
129C4⋊Dic53—41no—
130C4⋊Dic53C2×D412yes—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322SD1614yescomplemented
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136C2×#88322C2×D411yescharacteristic
137C2×#88322C2×D411yescharacteristic
138#11719913—84no—
139#11719912—44no—
140C8×D53—44no—
141C23×C53#12111yescharacteristic
142C22×D53—22no—
143Q8×C53—44no—
144Q8×C53—82no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—42no—
148C2×C4×C53—82no—
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53C2×SD1612yes—
153C2×C4×C53#13712yes—
154C2×C4×C53C22×D411yescharacteristic
155C2×C4×C53C22×D411yescharacteristic
156C2×C4×C53C2×#4111yescharacteristic
157C22⋊D53—44no—
158C2×Dic53—22no—
159C2×Dic53—42no—
160C2×Dic53—82no—
161D212—24no—
162D212—42no—
163D212—42no—
164C4×D53—42no—
165C4×D53—42no—
166Dic106—42no—
167Dic106—44no—
168Dic106—82no—
169C8×C53—42no—
170C8×C53—42no—
171#88322—21no—
172#88322—21no—
173#88322—42no—
174#88322C2×SD1612yes—
175#88322C2×SD1612yes—
176#19192—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53#46112yes—
180C22×C53C2×#12111yescharacteristic
181C2×D53—42no—
182C2×D53—42no—
183C2×D53—42no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—82no—
187C4×C53#46112yes—
188C4×C53#47312yes—
189C4×C53C22×SD1611yescharacteristic
190C4×C53C2×#13711yescharacteristic
191C4×C53#57612yes—
192Dic53—42no—
193Dic53—82no—
194#3206—531no—
195#3206—531no—
196C2×#500—531no—
197#2931—531no—
198#1693—534no—
199#1673—534no—
200#1564—531no—
201#1549—531no—
202#1549D5311yescharacteristic, complemented
203C2×C53—22no—
204C2×C53C2×#46111yescharacteristic
205C2×C53#305811yescharacteristic
206C2×C53C2×#47311yescharacteristic
207D53—82no—
208C2×#131—1061no—
209#532—531no—
210C2×C4⋊Q8—531no—
211C2×C4⋊Q8C2×D5311yescharacteristic
212#500—532no—
213#500—532no—
214#500—1061no—
215#500—1061no—
216#492—1061no—
217#492—1061no—
218#487—1061no—
219#487—1061no—
220#461—534no—
221#461—534no—
222#448—1062no—
223#448—1062no—
224#436—532no—
225#436—1061no—
226C8×D4—532no—
227C8×D4—1061no—
228#431—531no—
229#431C2×D5311yescharacteristic
230C2×#107—1061no—
231C2×#107—1061no—
232#423C2×D5311yescharacteristic
233#408—1061no—
234C2×C4×C8—531no—
235#335—532no—
236#334—532no—
237#332—534no—
238#332C2×D5314yes—
239C53#1919211yescharacteristic, complemented, Sylow
240C2×#41—1061no—
241C22×Q8—21no—
242C22×Q8—1061no—
243C2×SD16—1064no—
244C2×SD16—1064no—
245#131—2124no—
246C2×M16—21no—
247C2×M16—1062no—
248C22×C8—531no—
249C22×C8—531no—
250C22×C8—1062no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—531no—
255C4⋊Q8—532no—
256C4⋊Q8—1061no—
257C4⋊Q8—1061no—
258C4⋊Q8—1061no—
259C4⋊Q8C22×D5312yes—
260#127—531no—
261#124—1061no—
262#121—532no—
263#121—1061no—
264C4×D4—532no—
265C4×D4—1061no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1061no—
270C2×C42C22×D5311yescharacteristic
271#107—1062no—
272#107—1062no—
273#107—1062no—
274#107—1062no—
275C4⋊C8—21no—
276C4⋊C8—532no—
277C4⋊C8C22×D5312yes—
278C4⋊C8C22×D5312yes—
279#103—1064no—
280#103—1064no—
281C22⋊C8—21no—
282C22⋊C8—1062no—
283C22⋊C8—1062no—
284C22⋊C8—1061no—
285C4×C8—21no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—531no—
289C4×C8—531no—
290C4×C8—1061no—
291#41—2122no—
292#41—2122no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8—1061no—
297C2×Q8—1062no—
298C2×Q8—4242no—
299C2×Q8—2121no—
300C2×D4—532no—
301C2×D4—1062no—
302C2×D4—1061no—
303C2×D4—1061no—
304C22×C4—41no—
305C22×C4—1062no—
306C22×C4—2121no—
307C22×C4#1171994511yescharacteristic
308C22×C4D4×D5311yescharacteristic
309C22×C4D4×D5311yescharacteristic
310SD16—4244no—
311SD16—4244no—
312M16—42no—
313M16—2124no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—534no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—531no—
320C2×C8—531no—
321C2×C8—1062no—
322C2×C8—1062no—
323C2×C8—2124no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—532no—
329C4⋊C4—1061no—
330C4⋊C4—1062no—
331C4⋊C4—1062no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C22⋊C4—1062no—
335C22⋊C4—1062no—
336C42#1171994511yescharacteristic
337C42D4×D5311yescharacteristic
338C42D4×D5311yescharacteristic
339C42C23×D5311yescharacteristic
340C23—1062no—
341C23#1313319011yescharacteristic
342Q8—44no—
343Q8—82no—
344Q8—4242no—
345Q8—2124no—
346Q8—2122no—
347D4—1064no—
348D4—2124no—
349D4—2122no—
350D4—2122no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—1062no—
356C2×C4—4242no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4#1313319011yescharacteristic
361C2×C4#1313319011yescharacteristic
362C2×C4#1313319011yescharacteristic
363C2×C4SD16×D5312yes—
364C2×C4#1313324912yes—
365C2×C4C2×D4×D5311yescharacteristic
366C2×C4C2×D4×D5311yescharacteristic
367C2×C4C2×#1171994511yescharacteristic
368C8—42no—
369C8—42no—
370C8—532no—
371C8—532no—
372C8—1061no—
373C8—1061no—
374C8—2122no—
375C22—21no—
376C22—21no—
377C22—2122no—
378C22—2122no—
379C22—2122no—
380C22#1681082712yes—
381C22C2×#1313319011yescharacteristic
382C4—21no—
383C4—21no—
384C4—82no—
385C4—4242no—
386C4—2122no—
387C4#1681082712yes—
388C4#1681089712yes—
389C4C2×SD16×D5311yescharacteristic
390C4C2×#1313324911yescharacteristic
391C4#1681142712yes—
392C2—22no—
393C2—4242no—
394C2C2×#1681082711yescharacteristic
395C2#2028436211yescharacteristic
396C2C2×#1681089711yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1549C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#1549D53
hypocenterC53#19192
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C2×#16811026C2semidirect
5. #20274227C2semidirect
6. #20274103C2semidirect
7. #20273461C2semidirect
8. #20273401C2semidirect
9. C53×#1549C2semidirect
133. C2×#88322SD16semidirect
202. #1549D53semidirect
239. C53#19192semidirect

Maximal subgroups

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

subgroupindex
#202856422
#202856402
C2×#168110262
#202834262
#202742272
#202741032
#202734612
#202734012
C53×#15492
#1919253