Finite Group Databasebeta

← #27539659

Subgroups — #27539659

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#20285285C211yesmaximal, characteristic, complemented
2#20276998C211yesmaximal, characteristic, complemented
3#20276979C211yesmaximal, characteristic, complemented
4#20276956C211yesmaximal, characteristic, complemented
5#20276954C211yesmaximal, characteristic, complemented
6#20276826C212yesmaximal, complemented
7#20276816C212yesmaximal, complemented
8C4×#13133161C212yesmaximal, complemented
9C4×#13133158C212yesmaximal, complemented
10C4×#13133157C211yesmaximal, characteristic, complemented
11C53×#3169C211yesmaximal, characteristic, complemented
12#16811157—21no—
13C2×#13133161C2212yes—
14C2×#13133158C2212yes—
15C2×#13133157C2212yes—
16#16811025C2211yescharacteristic
17#16811007C2211yescharacteristic
18#16810978—21no—
19#16810864C2211yescharacteristic
20#16810859C2211yescharacteristic
21#16810852—21no—
22C4×Dic212C2211yescharacteristic
23C4×#11719913—21no—
24C4×D424C2211yescharacteristic
25#16810450C2212yes—
26#16810448C2212yes—
27#16810426C2212yes—
28#16810424C2212yes—
29#16810424C2212yes—
30#16810423C2212yes—
31#16810399C2211yescharacteristic
32#16810399C2211yescharacteristic
33C4×#11719908C2212yes—
34C53×#532—21no—
35C53×#500C2211yescharacteristic
36C53×#493C2211yescharacteristic
37C53×#486—21no—
38C53×#466—21no—
39C53×#466C2212yes—
40C4×D8×C53—21no—
41C4×D8×C53C2212yes—
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#13133161—24no—
57#13133158—24no—
58#13133157—24no—
59C2×#11719908C2312yes—
60C2×#11719908C2312yes—
61#13133151C2311yescharacteristic
62#13133151C2311yescharacteristic
63#13133149—21no—
64#13133142—21no—
65#13133142—21no—
66#13133139—21no—
67#13133139—21no—
68C2×#41×C53—41no—
69C2×D8×C53—21no—
70C2×D8×C53C2312yes—
71C22×C8×C53D411yescharacteristic
72C22×C8×C53C2311yescharacteristic
73C4⋊Q8×C53—21no—
74#127×C53—21no—
75#124×C53—41no—
76#121×C53—22no—
77#121×C53—41no—
78C4×D4×C53—22no—
79C4×D4×C53—41no—
80C2×C42×C53D411yescharacteristic
81#107×C53—21no—
82#107×C53C2312yes—
83#102×C53—22no—
84#102×C53—22no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89#41×C53—84no—
90C2×Q8×C53—41no—
91C2×D4×C53—22no—
92C2×D4×C53—42no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C22×C4×C53—42no—
96C22×C4×C53D811yescharacteristic
97C22×C4×C53D811yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C2×C22⋊D53—41no—
100C2×D212—21no—
101C2×C4×D53—41no—
102C2×Dic106—21no—
103D8×C53—22no—
104D8×C53—22no—
105D8×C53—42no—
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—22no—
113C4⋊C4×C53—41no—
114C22⋊C4×C53—42no—
115C22⋊C4×C53—42no—
116C42×C53D811yescharacteristic
117C42×C53D811yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C22⋊Dic53—41no—
121#11719921—41no—
122#11719921—41no—
123#11719921—41no—
124#11719921—41no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127#11719919—41no—
128#11719919—41no—
129#11719919—41no—
130#11719919—41no—
131C4×Dic53—41no—
132Dic212—22no—
133D424—22no—
134#11719913—42no—
135#11719908—22no—
136#11719908#4114yescomplemented
137C23×C53—42no—
138C23×C53C2×D811yescharacteristic
139C22×D53—41no—
140Q8×C53—82no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—82no—
144D4×C53—82no—
145D4×C53—84no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—84no—
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C22×D411yescharacteristic
158C22⋊D53—82no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162D212—41no—
163D212—41no—
164C4×D53—82no—
165Dic106—41no—
166Dic106—41no—
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—82no—
176C22×C53—82no—
177C22×C53—82no—
178C22×C53C22×D811yescharacteristic
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—82no—
187C4×C53#50912yes—
188C4×C53#53211yescharacteristic
189C4×C53C2×#13511yescharacteristic
190Dic53—81no—
191Dic53—81no—
192Dic53—81no—
193#3169D5311yescharacteristic, complemented
194#3168—531no—
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—162no—
206C2×C53#316911yescharacteristic
207C2×C53C2×#50911yescharacteristic
208C2×C53#341411yescharacteristic
209D53—161no—
210#532—21no—
211#529—1061no—
212C2×SD32—532no—
213C2×D16—532no—
214C2×M32—532no—
215#500C2×D5311yescharacteristic
216#499—531no—
217#495—531no—
218#493C2×D5311yescharacteristic
219#488—1061no—
220#486—21no—
221#467—531no—
222#466—21no—
223#466—531no—
224#466C2×D5312yes—
225#465—1061no—
226C4×Q16—531no—
227C4×SD16—1061no—
228C4×D8—21no—
229C4×D8—531no—
230C4×D8C2×D5312yes—
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—41no—
243C2×Q16—531no—
244C2×SD16—1061no—
245C2×D8—21no—
246C2×D8—531no—
247C2×D8C22×D5312yes—
248C22×C8C22⋊D5311yescharacteristic
249C22×C8C22×D5311yescharacteristic
250C4⋊Q8—21no—
251#127—21no—
252#126—2121no—
253#125—1061no—
254#124—41no—
255#124—1061no—
256#123—2121no—
257C22⋊Q8—1061no—
258#121—22no—
259#121—41no—
260#121—1061no—
261C4×Q8—1061no—
262C4×D4—22no—
263C4×D4—41no—
264C4×D4—1061no—
265C4×D4—2121no—
266#117—2121no—
267C2×C42C22⋊D5311yescharacteristic
268SD32—1064no—
269D16—1064no—
270M32—1064no—
271C2×C16—532no—
272C2×C16—532no—
273#107—21no—
274#107—531no—
275#107—531no—
276#107C22×D5312yes—
277#106—1061no—
278#103—1061no—
279#103—1061no—
280#102—22no—
281#102—22no—
282#102—1061no—
283#102—1061no—
284C4×C8C22⋊D5311yescharacteristic
285C4×C8C22⋊D5311yescharacteristic
286C4×C8C22×D5311yescharacteristic
287C4×C8C22×D5311yescharacteristic
288#41—84no—
289C2×Q8—41no—
290C2×Q8—1061no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—1061no—
296C2×D4—2121no—
297C22×C4—42no—
298C22×C4—2121no—
299C22×C4#1171992211yescharacteristic
300C22×C4#1171992211yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302Q16—1062no—
303SD16—2122no—
304D8—22no—
305D8—22no—
306D8—42no—
307D8—1062no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8C2×C22⋊D5311yescharacteristic
311C2×C8C2×C22⋊D5311yescharacteristic
312C2×C8C2×C22⋊D5311yescharacteristic
313C2×C8C23×D5311yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—2121no—
325C22⋊C4—2121no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C22⋊C4—2121no—
329C42—2121no—
330C42#1171992211yescharacteristic
331C42#1171992211yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C42C2×C22⋊D5311yescharacteristic
334C16—534no—
335C16—1062no—
336C23—42no—
337C23—2121no—
338C23C2×#1171992211yescharacteristic
339Q8—82no—
340Q8—2121no—
341Q8—2121no—
342D4—42no—
343D4—42no—
344D4—82no—
345D4—82no—
346D4—84no—
347D4—4242no—
348D4—2121no—
349D4—2121no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—84no—
356C2×C4—4242no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
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—82no—
374C22—82no—
375C22—82no—
376C22—4241no—
377C22—4241no—
378C22—4241no—
379C22C22×#1171992211yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—4241no—
386C4—4241no—
387C4—4241no—
388C4#1681105912yes—
389C4#1681116911yescharacteristic
390C4C2×#1313323711yescharacteristic
391C2—41no—
392C2—162no—
393C2—8481no—
394C2#2028529011yescharacteristic
395C2C2×#1681105911yescharacteristic
396C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3169C2
socleC22×C53C22×D8
hypercenter#3169D53
hypocenterC53#33737
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #20285285C2semidirect
2. #20276998C2semidirect
3. #20276979C2semidirect
4. #20276956C2semidirect
5. #20276954C2semidirect
6. #20276826C2semidirect
7. #20276816C2semidirect
8. C4×#13133161C2semidirect
9. C4×#13133158C2semidirect
10. C4×#13133157C2semidirect
11. C53×#3169C2semidirect
136. #11719908#41semidirect
193. #3169D53semidirect
241. C53#33737semidirect

Maximal subgroups

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

subgroupindex
#202852852
#202769982
#202769792
#202769562
#202769542
#202768262
#202768162
C4×#131331612
C4×#131331582
C4×#131331572
C53×#31692
#3373753