Finite Group Databasebeta

← #27287133

Subgroups — #27287133

Order 13568.

536 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275642C211yesmaximal, characteristic
2#20275642C211yesmaximal, characteristic
3#20275520C211yesmaximal, characteristic
4#20272598C211yesmaximal, characteristic, complemented
5#20272556C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2552C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×C4⋊#88322C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133137—22no—
12C2×#13133137—21no—
13C2×#13133137—21no—
14C2×#13133137—21no—
15C2×#13133137—21no—
16C2×C4×#88322C2211yescharacteristic
17C2×#125×C53—21no—
18C53×#398—22no—
19C53×#389—21no—
20C53×#389—21no—
21C53×#384—21no—
22C53×#384C2211yescharacteristic
23C4×C4⋊C4×C53C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C22×D212—21no—
26C22×D212—21no—
27C22×D212—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133214—21no—
32#13133214—21no—
33#13133214—21no—
34#13133214Q812yes—
35C4⋊#88322—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39#13133137—44no—
40#13133137—42no—
41#13133137—42no—
42#13133137—42no—
43#13133137—42no—
44C4×#88322—21no—
45C4×#88322—21no—
46#125×C53—42no—
47#125×C53—42no—
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53—22no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53D412yes—
58C2×C42×C53—21no—
59C2×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61#95×C53—22no—
62#95×C53—21no—
63#95×C53—21no—
64C23×D53—41no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—42no—
77C2×D212—42no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—44no—
96C4⋊C4×C53—44no—
97C4⋊C4×C53—42no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—82no—
103C4⋊C4×C53D812yes—
104C4⋊C4×C53SD1612yes—
105C42×C53—41no—
106C42×C53—41no—
107C42×C53—41no—
108C42×C53—41no—
109C42×C53C2×Q811yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53#4111yescharacteristic
112C42×C53#4111yescharacteristic
113C2×#88322—22no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C23×C53#12011yescharacteristic
123C22×D53—42no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—81no—
129C22×D53—81no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53C22⋊Q811yescharacteristic
148C2×C4×C53C22⋊Q811yescharacteristic
149C2×C4×C53C22⋊Q811yescharacteristic
150C2×C4×C53#12411yescharacteristic
151C2×C4×C53#12411yescharacteristic
152C2×C4×C53#12411yescharacteristic
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53C2×SD1611yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53#13711yescharacteristic
161D212—44no—
162D212—42no—
163D212—42no—
164D212—42no—
165D212—42no—
166D212—82no—
167D212—81no—
168D212—81no—
169D212—81no—
170D212—81no—
171#88322—42no—
172#88322—81no—
173#88322—81no—
174#10354—531nomaximal, Sylow
175C22×C53#39311yescharacteristic
176C22×C53#44711yescharacteristic
177C22×C53#44811yescharacteristic
178C22×C53#44911yescharacteristic
179C22×C53#44911yescharacteristic
180C22×C53#45311yescharacteristic
181C22×C53#45411yescharacteristic
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—82no—
185C2×D53—161no—
186C2×D53—161no—
187C2×D53—161no—
188C2×D53—161no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—42no—
192C4×C53—42no—
193C4×C53—81no—
194C4×C53—82no—
195C4×C53—81no—
196C4×C53#47411yescharacteristic
197C4×C53#47611yescharacteristic
198C4×C53#47811yescharacteristic
199C4×C53#47811yescharacteristic
200C4×C53#48611yescharacteristic
201C4×C53#48811yescharacteristic
202C4×C53#48911yescharacteristic
203C4×C53#49011yescharacteristic
204#2552D5311yescharacteristic, complemented
205#1982—531no—
206#1982—531no—
207#1957—531no—
208#1296—531no—
209#1279—531no—
210#1279—531no—
211C2×C53#162111yescharacteristic
212C2×C53#162211yescharacteristic
213C2×C53#162511yescharacteristic
214C2×C53#162611yescharacteristic
215C2×C53#202311yescharacteristic
216C2×C53#202511yescharacteristic
217C2×C53#202711yescharacteristic
218D53—162no—
219C2×#127—531no—
220C2×#125—21no—
221C2×C4⋊C8—531no—
222C2×C4⋊C8—531no—
223C2×#102—1062no—
224C2×#102—1061no—
225C2×#102—1061no—
226C2×#102—1061no—
227C2×#102—1061no—
228C2×C4×C8—531no—
229#398—22no—
230#389—21no—
231#389—21no—
232#384—21no—
233#384C2×D5311yescharacteristic
234C4×C4⋊C4C2×D5311yescharacteristic
235C4×C4⋊C4C2×D5311yescharacteristic
236C53#1035411yescharacteristic, complemented, Sylow
237C22×D4—1061no—
238C22×D4—1061no—
239C22×D4—1061no—
240C22×C8—1061no—
241C22×C8—1061no—
242C22×C8—1061no—
243#127—532no—
244#127—1061no—
245#127—1061no—
246#127—1061no—
247#125—42no—
248#125—42no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C22⋊D5311yescharacteristic
255C2×C4⋊C4C22⋊D5311yescharacteristic
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C4⋊C4C22⋊D5311yescharacteristic
258C2×C4⋊C4C22⋊D5312yes—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C22×D5311yescharacteristic
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266#102—2122no—
267#102—2122no—
268#102—2124no—
269#102—2122no—
270#102—2122no—
271C4×C8—1061no—
272C4×C8—1061no—
273#95—22no—
274#95—21no—
275#95—21no—
276C24—2121no—
277C2×D4—1062no—
278C2×D4—1062no—
279C2×D4—1062no—
280C2×D4—2122no—
281C2×D4—2122no—
282C2×D4—2121no—
283C2×D4—2121no—
284C2×D4—2121no—
285C2×D4—2121no—
286C2×D4—2121no—
287C2×D4—2121no—
288C2×D4—2121no—
289C2×D4—2121no—
290C2×D4—2121no—
291C2×D4—2121no—
292C2×D4—2121no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4C2×C22⋊D5311yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300C22×C4C2×C22⋊D5311yescharacteristic
301C2×C8—1062no—
302C2×C8—2121no—
303C2×C8—2121no—
304C2×C8—2121no—
305C2×C8—2121no—
306C2×C8—2121no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8—2121no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—82no—
324C4⋊C4#1171992212yes—
325C4⋊C4#1171992412yes—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42Q8×D5311yescharacteristic
331C42#1171994911yescharacteristic
332C42#1171994911yescharacteristic
333C42#1171994911yescharacteristic
334C23—4241no—
335C23—4241no—
336C23—4241no—
337C23—4241no—
338C23—4241no—
339C23—4241no—
340C23—2122no—
341C23#1313318611yescharacteristic
342D4—4242no—
343D4—4241no—
344D4—4241no—
345D4—4241no—
346D4—4241no—
347D4—2124no—
348D4—2122no—
349D4—2122no—
350D4—2122no—
351D4—2122no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4#1313319311yescharacteristic
370C2×C4#1313319311yescharacteristic
371C2×C4#1313319311yescharacteristic
372C2×C4#1313320611yescharacteristic
373C2×C4#1313320611yescharacteristic
374C2×C4#1313320611yescharacteristic
375C2×C4C2×#1171992211yescharacteristic
376C2×C4C2×#1171992411yescharacteristic
377C2×C4#1313323711yescharacteristic
378C2×C4#1313323711yescharacteristic
379C2×C4#1313324211yescharacteristic
380C2×C4#1313324211yescharacteristic
381C2×C4#1313324311yescharacteristic
382C2×C4#1313324811yescharacteristic
383C8—4241no—
384C8—4241no—
385C8—2122no—
386C22—8481no—
387C22—8481no—
388C22—8481no—
389C22—8481no—
390C22—4242no—
391C22—4242no—
392C22—4242no—
393C22#1681053411yescharacteristic
394C22#1681075711yescharacteristic
395C22#1681076211yescharacteristic
396C22#1681076811yescharacteristic
397C22#1681076811yescharacteristic
398C22#1681079111yescharacteristic
399C22#1681079611yescharacteristic
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4#1681090011yescharacteristic
408C4#1681091011yescharacteristic
409C4#1681092011yescharacteristic
410C4#1681092011yescharacteristic
411C4#1681097111yescharacteristic
412C4#1681098011yescharacteristic
413C4#1681098511yescharacteristic
414C4#1681098911yescharacteristic
415C2—8482no—
416C2#2027378011yescharacteristic
417C2#2027378711yescharacteristic
418C2#2027380411yescharacteristic
419C2#2027381111yescharacteristic
420C2#2027584811yescharacteristic
421C2#2027585811yescharacteristic
422C2#2027586811yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2552C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#2552D53
hypocenterC53#10354
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2552

Complements

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

NH ≅ G/Ntype
4. #20272598C2semidirect
5. #20272556C2semidirect
6. #20272556C2semidirect
7. C53×#2552C2semidirect
204. #2552D53semidirect
236. C53#10354semidirect

Maximal subgroups

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

subgroupindex
#202756422
#202755202
#202725982
#202725562
C53×#25522
#1035453