Finite Group Databasebeta

← #27539666

Subgroups — #27539666

Order 13568.

554 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#20285294C211yesmaximal, characteristic, complemented
2#20277002C211yesmaximal, characteristic, complemented
3#20276982C211yesmaximal, characteristic, complemented
4#20276958C211yesmaximal, characteristic, complemented
5#20276951C211yesmaximal, characteristic, complemented
6#20276830C212yesmaximal, complemented
7#20276816C212yesmaximal, complemented
8C4×#13133160C212yesmaximal, complemented
9C4×#13133158C212yesmaximal, complemented
10C4×#13133157C211yesmaximal, characteristic, complemented
11C53×#3168C211yesmaximal, characteristic, complemented
12#16811173—21no—
13C2×#13133160C2212yes—
14C2×#13133158C2212yes—
15C2×#13133157C2212yes—
16#16811028C2211yescharacteristic
17#16811000C2211yescharacteristic
18#16810970—21no—
19#16810859—21no—
20#16810859C2212yes—
21C4×D424—21no—
22C4×D424C2212yes—
23#16810450C2212yes—
24#16810448C2212yes—
25#16810427C2212yes—
26#16810424C2212yes—
27#16810423C2212yes—
28#16810423C2212yes—
29#16810399C2211yescharacteristic
30#16810399C2211yescharacteristic
31C4×#11719908C2212yes—
32C53×#529—21no—
33C53×#499C2211yescharacteristic
34C53×#495C2211yescharacteristic
35C53×#488—21no—
36C53×#467C2211yescharacteristic
37C53×#466C2211yescharacteristic
38C53×#465—21no—
39C4×Q16×C53C2211yescharacteristic
40C4×SD16×C53—21no—
41C4×D8×C53C2211yescharacteristic
42C2×C4×C8×C53C2211yescharacteristic
43C2×#11719945—41no—
44C2×D424—21no—
45C2×D424C2312yes—
46#13133217—21no—
47#13133214—21no—
48#13133205—41no—
49#13133191—22no—
50#13133191—41no—
51C4×D212—22no—
52C4×D212—41no—
53#13133160—24no—
54#13133158—24no—
55#13133157—24no—
56C2×#11719908C2312yes—
57C2×#11719908C2312yes—
58#13133151—21no—
59#13133151C2312yes—
60#13133139—22no—
61#13133139—22no—
62C2×Q16×C53C2311yescharacteristic
63C2×SD16×C53—21no—
64C2×D8×C53C2311yescharacteristic
65C22×C8×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67#126×C53—41no—
68#125×C53—21no—
69#124×C53—21no—
70#123×C53—41no—
71C22⋊Q8×C53—21no—
72#121×C53—21no—
73C4×Q8×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—41no—
76#117×C53—41no—
77C2×C42×C53D411yescharacteristic
78#107×C53C2311yescharacteristic
79#107×C53C2311yescharacteristic
80#106×C53—21no—
81#103×C53—21no—
82#103×C53—21no—
83#102×C53—21no—
84#102×C53—21no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C2×Q8×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53D811yescharacteristic
94C22×C4×C53D811yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C2×C22⋊D53—42no—
97#11719945—84no—
98C2×D212—22no—
99C2×D212—41no—
100C2×D212—41no—
101C2×C4×D53—42no—
102C2×Dic106—41no—
103Q16×C53—22no—
104SD16×C53—42no—
105D8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2411yescharacteristic
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C42×C53—41no—
124C42×C53D811yescharacteristic
125C42×C53D811yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128#11719921—42no—
129#11719921—42no—
130C4⋊Dic53—22no—
131C4⋊Dic53—41no—
132D424—22no—
133D424—22no—
134D424—42no—
135#11719908—22no—
136#11719908#4114yescomplemented
137C23×C53—41no—
138C23×C53C2×D811yescharacteristic
139C22×D53—42no—
140Q8×C53—41no—
141Q8×C53—41no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—82no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—82no—
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C2×D811yescharacteristic
158C2×C4×C53C22×D411yescharacteristic
159C22⋊D53—84no—
160C2×Dic53—42no—
161D212—42no—
162D212—42no—
163D212—82no—
164D212—82no—
165C4×D53—84no—
166Dic106—82no—
167C8×C53—21no—
168C8×C53—21no—
169C8×C53#13512yes—
170C8×C53C2×#4112yes—
171#33737—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53C22×D811yescharacteristic
179C2×D53—82no—
180C2×D53—82no—
181C2×D53—82no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—81no—
187C4×C53—81no—
188C4×C53—81no—
189C4×C53#50912yes—
190C4×C53#53211yescharacteristic
191C4×C53C2×#13511yescharacteristic
192Dic53—82no—
193#3169—531no—
194#3168D5311yescharacteristic, complemented
195#2255—531no—
196#2250—531no—
197#2244—531no—
198#2243—531no—
199#2220—532no—
200#2218—532no—
201C4×SD32—532no—
202C4×D16—532no—
203C4×M32—531no—
204C2×C53—41no—
205C2×C53—161no—
206C2×C53#316911yescharacteristic
207C2×C53C2×#50911yescharacteristic
208C2×C53#341411yescharacteristic
209D53—162no—
210#532—1061no—
211#529—21no—
212C2×SD32—532no—
213C2×D16—532no—
214C2×M32—532no—
215#500—531no—
216#499C2×D5311yescharacteristic
217#495C2×D5311yescharacteristic
218#493—531no—
219#488—21no—
220#486—1061no—
221#467C2×D5311yescharacteristic
222#466—532no—
223#466—1061no—
224#466C2×D5311yescharacteristic
225#465—21no—
226C4×Q16C2×D5311yescharacteristic
227C4×SD16—21no—
228C4×D8—532no—
229C4×D8—1061no—
230C4×D8C2×D5311yescharacteristic
231C2×C4×C8C2×D5311yescharacteristic
232#367—532no—
233#366—532no—
234#358—532no—
235#357—532no—
236#357—532no—
237#357—532no—
238#346—531no—
239#346—531no—
240C4×C16—532no—
241C53#3373711yescharacteristic, complemented, Sylow
242C2×#41—2121no—
243C2×Q16C22×D5311yescharacteristic
244C2×SD16—21no—
245C2×D8—532no—
246C2×D8—1061no—
247C2×D8C22×D5311yescharacteristic
248C22×C8C22⋊D5311yescharacteristic
249C22×C8C22×D5311yescharacteristic
250C4⋊Q8—1061no—
251#127—1061no—
252#126—41no—
253#125—21no—
254#124—21no—
255#124—2121no—
256#123—41no—
257C22⋊Q8—21no—
258#121—21no—
259#121—1062no—
260#121—2121no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—41no—
264C4×D4—1062no—
265C4×D4—2121no—
266#117—41no—
267C2×C42C22⋊D5311yescharacteristic
268SD32—1064no—
269D16—1064no—
270M32—1064no—
271C2×C16—532no—
272C2×C16—532no—
273#107—532no—
274#107—1061no—
275#107C22×D5311yescharacteristic
276#107C22×D5311yescharacteristic
277#106—21no—
278#103—21no—
279#103—21no—
280#102—21no—
281#102—21no—
282#102—1062no—
283#102—1062no—
284C4×C8C22⋊D5311yescharacteristic
285C4×C8C22⋊D5311yescharacteristic
286C4×C8C22×D5311yescharacteristic
287C4×C8C22×D5311yescharacteristic
288#41—4244no—
289C2×Q8—21no—
290C2×Q8—2121no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—1062no—
294C2×D4—2122no—
295C2×D4—2121no—
296C2×D4—2121no—
297C22×C4—41no—
298C22×C4—2122no—
299C22×C4#1171992211yescharacteristic
300C22×C4#1171992211yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302Q16—22no—
303SD16—42no—
304D8—22no—
305D8—1062no—
306D8—1062no—
307D8—2122no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8C2×C22⋊D5311yescharacteristic
311C2×C8C2×C22⋊D5311yescharacteristic
312C2×C8C2×C22⋊D5311yescharacteristic
313C2×C8C23×D5311yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—1062no—
321C4⋊C4—2121no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—2122no—
328C22⋊C4—2122no—
329C42—41no—
330C42#1171992211yescharacteristic
331C42#1171992211yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C42C2×C22⋊D5311yescharacteristic
334C16—534no—
335C16—1062no—
336C23—41no—
337C23—2122no—
338C23C2×#1171992211yescharacteristic
339Q8—41no—
340Q8—41no—
341Q8—4242no—
342D4—41no—
343D4—41no—
344D4—82no—
345D4—4242no—
346D4—4242no—
347D4—4244no—
348D4—2122no—
349D4—2122no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—4244no—
359C2×C4—2122no—
360C2×C4C2×#1171992211yescharacteristic
361C2×C4C2×#1171992211yescharacteristic
362C2×C4C2×#1171992211yescharacteristic
363C2×C4C2×#1171992211yescharacteristic
364C2×C4C2×#1171992211yescharacteristic
365C2×C4C22×C22⋊D5311yescharacteristic
366C8—21no—
367C8—21no—
368C8#1313323712yes—
369C8#41×D5312yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—4242no—
377C22—4242no—
378C22—4242no—
379C22C22×#1171992211yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—4242no—
388C4#1681105812yes—
389C4#1681116911yescharacteristic
390C4C2×#1313323711yescharacteristic
391C2—41no—
392C2—161no—
393C2—8482no—
394C2#2028529011yescharacteristic
395C2C2×#1681105811yescharacteristic
396C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3168C2
socleC22×C53C22×D8
hypercenter#3168D53
hypocenterC53#33737
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #20285294C2semidirect
2. #20277002C2semidirect
3. #20276982C2semidirect
4. #20276958C2semidirect
5. #20276951C2semidirect
6. #20276830C2semidirect
7. #20276816C2semidirect
8. C4×#13133160C2semidirect
9. C4×#13133158C2semidirect
10. C4×#13133157C2semidirect
11. C53×#3168C2semidirect
136. #11719908#41semidirect
194. #3168D53semidirect
241. C53#33737semidirect

Maximal subgroups

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

subgroupindex
#202852942
#202770022
#202769822
#202769582
#202769512
#202768302
#202768162
C4×#131331602
C4×#131331582
C4×#131331572
C53×#31682
#3373753