Finite Group Databasebeta

← #27277995

Subgroups — #27277995

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#20283833C211yesmaximal, characteristic
2C4×#13133145C211yesmaximal, characteristic
3#20273465C211yesmaximal, characteristic, complemented
4#20273100C211yesmaximal, characteristic
5#20272565C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C53×#1831C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810750—21no—
10#16810700—21no—
11#16810694—22no—
12#16810679—21no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810645—21no—
16C2×#13133145C411yescharacteristic
17C2×#13133145C411yescharacteristic
18C4×#11719917C2211yescharacteristic
19#16810383C411yescharacteristic
20#16810383C411yescharacteristic
21#16810375—21no—
22#16810373—21no—
23#16810353—21no—
24#16810350—21no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C43×C53C2211yescharacteristic
28C53×#336—22no—
29C2×#11719945C2×C411yescharacteristic
30C2×#11719917C2×C411yescharacteristic
31C2×#11719917C2×C411yescharacteristic
32C2×#11719912—22no—
33C4×Dic106D411yescharacteristic
34C4×Dic106Q811yescharacteristic
35C4×D212D411yescharacteristic
36C4×D212Q811yescharacteristic
37C4×C22⋊D53—21no—
38C4×C22⋊D53—21no—
39#13133176C2×C411yescharacteristic
40#13133174—21no—
41C42×D53—21no—
42C4⋊#88322—21no—
43C4⋊#88322D411yescharacteristic
44C4⋊#88322D411yescharacteristic
45#13133146—22no—
46#13133145—22no—
47#13133145—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50#13133129—22no—
51#13133128—21no—
52#13133127—22no—
53C4×#88322—21no—
54C4×#88322—21no—
55C4×#88322—21no—
56C22×C8×C53—21no—
57C22×C8×C53D412yes—
58C2×C42×C53—21no—
59C2×C42×C53—21no—
60C2×C42×C53C2×C411yescharacteristic
61C2×C42×C53C2×C411yescharacteristic
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53#4111yescharacteristic
72C2×C22⋊D53—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
82C2×C8×C53—22no—
83C2×C8×C53—42no—
84C2×C8×C53—42no—
85C2×C8×C53—41no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—41no—
90C42×C53—41no—
91C42×C53—42no—
92C42×C53M1612yescomplemented
93C42×C53C2×D411yescharacteristic
94C42×C53C2×D411yescharacteristic
95C42×C53C2×Q811yescharacteristic
96C42×C53#4111yescharacteristic
97C22⋊Dic53—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53C4⋊C411yescharacteristic
101C4⋊Dic53C4⋊C411yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107#11719917—21no—
108#11719917—21no—
109#11719917—21no—
110#11719917M1612yescomplemented
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322—41no—
116#11719912—82no—
117C23×C53C4×D411yescharacteristic
118C22×D53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53C2×C4⋊C411yescharacteristic
129C2×C4×C53C4×D411yescharacteristic
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53C22⋊Q811yescharacteristic
132C2×C4×C53C22⋊Q811yescharacteristic
133C2×C4×C53#12311yescharacteristic
134C2×C4×C53C2×M1611yescharacteristic
135C2×C4×C53C2×M1611yescharacteristic
136C2×C4×C53#13111yescharacteristic
137C2×C4×C53#13111yescharacteristic
138C22⋊D53—41no—
139C22⋊D53—41no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
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—42no—
153C8×C53—81no—
154#88322—22no—
155#88322—41no—
156#88322—41no—
157#88322—41no—
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—42no—
172C4×C53—42no—
173C4×C53#42311yescharacteristic
174C4×C53#42411yescharacteristic
175C4×C53#43511yescharacteristic
176C4×C53#43611yescharacteristic
177C4×C53#45411yescharacteristic
178C4×C53#45511yescharacteristic
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182#2989—531no—
183#1831D5311yescharacteristic, complemented
184C4×#104—531no—
185#1565—531no—
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#431—531no—
202#423—1061no—
203C2×C4⋊C8C2×D5311yescharacteristic
204C2×C4⋊C8C2×D5311yescharacteristic
205C2×#104—531no—
206C2×#104—531no—
207C4×M16—531no—
208C43C2×D5311yescharacteristic
209#339—531no—
210#339—531no—
211#336—22no—
212#335—1061no—
213#334—1061no—
214Q8⋊C8—1061no—
215D4⋊C8—1061no—
216C53#884211yescharacteristic, complemented, Sylow
217C2×#41—531no—
218C2×M16—531no—
219C2×M16—531no—
220C2×M16—1062no—
221C22×C8—21no—
222C22×C8C22⋊D5312yes—
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—21no—
232C2×C42—21no—
233C2×C42—1061no—
234C2×C42C4×D5311yescharacteristic
235C2×C42C4×D5311yescharacteristic
236C2×C42C22×D5311yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—531no—
242C4⋊C8—531no—
243C4⋊C8—1062no—
244C4⋊C8—1061no—
245#104—1062no—
246#104—1062no—
247C22⋊C8—1062no—
248C22⋊C8—1061no—
249C22⋊C8—1061no—
250#97—1062no—
251#97—1061no—
252C4×C8—1061no—
253C4×C8—1061no—
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—21no—
263C22×C4—21no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4C2×C4×D5311yescharacteristic
268C22×C4#1171994511yescharacteristic
269C22×C4C2×C22⋊D5311yescharacteristic
270M16—532no—
271M16—1061no—
272M16—1061no—
273M16—1061no—
274M16—4242no—
275C2×C8—22no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—1061no—
280C2×C8—1061no—
281C2×C8—1061no—
282C2×C8—1061no—
283C2×C8—2121no—
284C4⋊C4—531no—
285C4⋊C4—531no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—41no—
295C42—41no—
296C42—42no—
297C42—1061no—
298C42—1061no—
299C42—1061no—
300C42#1171991212yescomplemented
301C42D4×D5311yescharacteristic
302C42D4×D5311yescharacteristic
303C42Q8×D5311yescharacteristic
304C42#1171994911yescharacteristic
305C23—1061no—
306C23C4×C22⋊D5311yescharacteristic
307Q8—1061no—
308Q8—1061no—
309D4—1061no—
310D4—1061no—
311D4—2121no—
312D4—2121no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—4241no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4C4⋊C4×D5311yescharacteristic
331C2×C4#1313317911yescharacteristic
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—42no—
341C8—81no—
342C8—1062no—
343C8—4241no—
344C8—2121no—
345C8—2121no—
346C8—2121no—
347C22—21no—
348C22—21no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22#1681049411yescharacteristic
353C22#1681069411yescharacteristic
354C22#1681069911yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—2121no—
361C4—2121no—
362C4—2121no—
363C4#1681064511yescharacteristic
364C4#1681064811yescharacteristic
365C4#1681069311yescharacteristic
366C4#1681070011yescharacteristic
367C4#1681079511yescharacteristic
368C4#1681080011yescharacteristic
369C2—21no—
370C2—21no—
371C2—4241no—
372C2#2027489611yescharacteristic
373C2#2027494711yescharacteristic
374C2#2027495411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1831C2
socleC22×C53#385
prime radicalC2×#11719945C2×C4
cosocleC2×C42×C53C23
hypercenter#1831D53
hypocenterC53#8842
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1831

Complements

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

NH ≅ G/Ntype
3. #20273465C2semidirect
5. #20272565C2semidirect
6. #20272548C2semidirect
7. C53×#1831C2semidirect
92. C42×C53M16semidirect
110. #11719917M16semidirect
183. #1831D53semidirect
216. C53#8842semidirect
300. C42#11719912semidirect

Maximal subgroups

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

subgroupindex
#202838332
C4×#131331452
#202734652
#202731002
#202725652
#202725482
C53×#18312
#884253