Finite Group Databasebeta

← #27277993

Subgroups — #27277993

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283828C211yesmaximal, characteristic
2#20274831C211yesmaximal, characteristic, complemented
3C4×#13133143C211yesmaximal, characteristic
4#20273100C211yesmaximal, characteristic
5#20272566C211yesmaximal, characteristic, complemented
6#20272547C211yesmaximal, characteristic, complemented
7C53×#1565C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810753—21no—
10#16810702—21no—
11#16810691—22no—
12#16810679—21no—
13#16810676C2211yescharacteristic
14#16810645—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133143C411yescharacteristic
17C2×#13133143C411yescharacteristic
18C42×Dic53C2211yescharacteristic
19#16810384C411yescharacteristic
20#16810384C411yescharacteristic
21#16810377—22no—
22#16810353—21no—
23#16810350—21no—
24C53×#431C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C4×M16×C53C2211yescharacteristic
27C53×#335—21no—
28C53×#334—21no—
29C2×#11719945C2×C411yescharacteristic
30C2×#11719912—22no—
31C22×#88322D412yes—
32C4×Dic106D411yescharacteristic
33C4×Dic106Q811yescharacteristic
34C4×D212D411yescharacteristic
35C4×D212Q811yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53—21no—
38#13133176C2×C411yescharacteristic
39#13133174—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44C42×D53—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133146—22no—
49#13133143—22no—
50#13133143—22no—
51C22⋊#88322—21no—
52#13133129—22no—
53#13133127—22no—
54C4×#88322—21no—
55C2×M16×C53C2×C411yescharacteristic
56C2×M16×C53C2×C411yescharacteristic
57C22×C8×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53D411yescharacteristic
62C4⋊C8×C53D411yescharacteristic
63C22⋊C8×C53—21no—
64#97×C53—21no—
65C4×C8×C53—21no—
66C4×C8×C53—21no—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4111yescharacteristic
70C2×C22⋊D53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73#11719945—21no—
74#11719945—21no—
75#11719945—21no—
76#11719945—21no—
77C2×D212C4⋊C411yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×Dic106C4⋊C411yescharacteristic
82M16×C53—21no—
83M16×C53—21no—
84M16×C53—21no—
85M16×C53M1612yescomplemented
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—42no—
91C2×C8×C53—41no—
92C42×C53C2×D411yescharacteristic
93C42×C53C2×D411yescharacteristic
94C42×C53C2×Q811yescharacteristic
95C42×C53#4111yescharacteristic
96C22⋊Dic53—21no—
97#11719921—21no—
98#11719921—21no—
99C4⋊Dic53C4⋊C411yescharacteristic
100C4⋊Dic53C4⋊C411yescharacteristic
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53—21no—
109C4×Dic53—42no—
110C4×Dic53—41no—
111C4×Dic53—41no—
112C4×Dic53M1612yescomplemented
113C2×#88322—22no—
114C2×#88322—42no—
115C2×#88322—41no—
116#11719912—82no—
117C23×C53C4×D411yescharacteristic
118C22×D53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53C2×C4⋊C411yescharacteristic
123C2×C4×C53C4×D411yescharacteristic
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53C22⋊Q811yescharacteristic
126C2×C4×C53C22⋊Q811yescharacteristic
127C2×C4×C53#12311yescharacteristic
128C2×C4×C53C2×M1611yescharacteristic
129C2×C4×C53C2×M1611yescharacteristic
130C2×C4×C53#13111yescharacteristic
131C2×C4×C53#13111yescharacteristic
132C22⋊D53—41no—
133C22⋊D53—41no—
134C2×Dic53—22no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—22no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143D212—21no—
144D212—21no—
145C4×D53—41no—
146C4×D53—41no—
147C4×D53—41no—
148C4×D53—41no—
149C4×D53—81no—
150Dic106—21no—
151Dic106—21no—
152C8×C53—22no—
153C8×C53—41no—
154C8×C53—41no—
155C8×C53—41no—
156C8×C53—81no—
157#88322—42no—
158#88322—81no—
159#8842—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53#38511yescharacteristic
163C22×C53#43511yescharacteristic
164C22×C53#43611yescharacteristic
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53#42311yescharacteristic
172C4×C53#42411yescharacteristic
173C4×C53#43511yescharacteristic
174C4×C53#43611yescharacteristic
175C4×C53#45411yescharacteristic
176C4×C53#45511yescharacteristic
177Dic53—42no—
178Dic53—42no—
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182#2989—531no—
183#1831—531no—
184C4×#104—531no—
185#1565D5311yescharacteristic, complemented
186#1477—531no—
187#1283—531no—
188#1276—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53#185111yescharacteristic
192C2×C53#186011yescharacteristic
193C2×C53#186111yescharacteristic
194D53—81no—
195C4×#41—531no—
196#446—1061no—
197#436—1061no—
198#435—1062no—
199#432—1061no—
200#431—531no—
201#431C2×D5311yescharacteristic
202#423—1061no—
203C2×C4⋊C8—531no—
204C2×C4⋊C8C2×D5311yescharacteristic
205C2×#104—531no—
206C2×#104—531no—
207C4×M16C2×D5311yescharacteristic
208C43—531no—
209#339—531no—
210#339—531no—
211#336—1062no—
212#335—21no—
213#334—21no—
214Q8⋊C8—1061no—
215D4⋊C8—1061no—
216C53#884211yescharacteristic, complemented, Sylow
217C2×#41—531no—
218C2×M16—1062no—
219C2×M16C4×D5311yescharacteristic
220C2×M16C4×D5311yescharacteristic
221C22×C8—21no—
222C22×C8—532no—
223C4×Q8—531no—
224C4×Q8—531no—
225C4×D4—531no—
226C4×D4—531no—
227C4×D4—1061no—
228C4×D4—1061no—
229#117—531no—
230#117—1061no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—1061no—
234C2×C42—1061no—
235C2×C42—1061no—
236C2×C42C22×D5311yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—1062no—
240C4⋊C8—1061no—
241C4⋊C8—1061no—
242C4⋊C8—1061no—
243C4⋊C8C22⋊D5311yescharacteristic
244C4⋊C8C22⋊D5311yescharacteristic
245#104—1062no—
246#104—1062no—
247C22⋊C8—21no—
248C22⋊C8—1062no—
249C22⋊C8—1061no—
250#97—21no—
251#97—1062no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—1061no—
255#41—1061no—
256#41—1061no—
257#41—1061no—
258#41—1061no—
259C2×Q8—531no—
260C2×D4—531no—
261C2×D4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4C2×C4×D5311yescharacteristic
268C22×C4D4×D5311yescharacteristic
269C22×C4#1171994911yescharacteristic
270M16—21no—
271M16—21no—
272M16—21no—
273M16—4242no—
274M16#1171991212yescomplemented
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—1062no—
282C2×C8—2121no—
283C2×C8—2122no—
284C4⋊C4—531no—
285C4⋊C4—531no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C42—532no—
292C42—1061no—
293C42—1061no—
294C42—1061no—
295C42—1061no—
296C42—1061no—
297C42—1061no—
298C42—2121no—
299C42—2121no—
300C42—2122no—
301C42#1171994511yescharacteristic
302C42D4×D5311yescharacteristic
303C42Q8×D5311yescharacteristic
304C42C2×C22⋊D5311yescharacteristic
305C23—1061no—
306C23#1313317911yescharacteristic
307Q8—1061no—
308Q8—1061no—
309D4—1061no—
310D4—1061no—
311D4—2121no—
312D4—2121no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—4241no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4C4⋊C4×D5311yescharacteristic
331C2×C4C4×C22⋊D5311yescharacteristic
332C2×C4#1313318511yescharacteristic
333C2×C4#1313319311yescharacteristic
334C2×C4#1313319511yescharacteristic
335C2×C4#1313320211yescharacteristic
336C2×C4M16×D5311yescharacteristic
337C2×C4C2×#1171991211yescharacteristic
338C2×C4#1313322311yescharacteristic
339C2×C4#1313322511yescharacteristic
340C8—22no—
341C8—41no—
342C8—41no—
343C8—41no—
344C8—81no—
345C8—4241no—
346C8—2122no—
347C22—21no—
348C22—21no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22#1681049411yescharacteristic
353C22#1681069311yescharacteristic
354C22#1681070011yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—2122no—
359C4—2122no—
360C4—2121no—
361C4—2121no—
362C4—2121no—
363C4#1681064511yescharacteristic
364C4#1681064811yescharacteristic
365C4#1681069411yescharacteristic
366C4#1681069911yescharacteristic
367C4#1681079411yescharacteristic
368C4#1681079911yescharacteristic
369C2—21no—
370C2—21no—
371C2—4241no—
372C2#2027489611yescharacteristic
373C2#2027494511yescharacteristic
374C2#2027495211yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1565C2
socleC22×C53#385
prime radicalC2×#11719945C2×C4
cosocleC2×C42×C53C23
hypercenter#1565D53
hypocenterC53#8842
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
2. #20274831C2semidirect
5. #20272566C2semidirect
6. #20272547C2semidirect
7. C53×#1565C2semidirect
85. M16×C53M16semidirect
112. C4×Dic53M16semidirect
185. #1565D53semidirect
216. C53#8842semidirect
274. M16#11719912semidirect

Maximal subgroups

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

subgroupindex
#202838282
#202748312
C4×#131331432
#202731002
#202725662
#202725472
C53×#15652
#884253