Finite Group Databasebeta

← #27280029

Subgroups — #27280029

Order 13568.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277625C211yesmaximal, characteristic, complemented
2#20273275C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic, complemented
4#20273106C211yesmaximal, characteristic, complemented
5#20272616C211yesmaximal, characteristic, complemented
6#20272575C211yesmaximal, characteristic, complemented
7C53×#3036C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9#16810676C2211yescharacteristic
10#16810676C2211yescharacteristic
11#16810522—21no—
12#16810498—22no—
13#16810489C2211yescharacteristic
14#16810483—21no—
15C4×C4⋊Dic53C2211yescharacteristic
16C2×#13133124—21no—
17#16810357—21no—
18#16810357—21no—
19#16810357—22no—
20D4⋊#88322—22no—
21C2×AGL(1,ℤ/8)×C53—21no—
22C53×#532C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#463—22no—
25C53×#461C412yescomplemented
26C53×#459C412yescomplemented
27C53×#423C2211yescharacteristic
28C53×#417—21no—
29C23×Dic53—22no—
30D4×Dic53—42no—
31D4×Dic53—42no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×C4×Dic53—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C4×#88322—21no—
42C4×#88322—21no—
43#13133124—22no—
44#13133124—21no—
45#13133124D412yes—
46#13133124D411yescharacteristic
47#13133124D411yescharacteristic
48C2×#41×C53—21no—
49C22×D4×C53D411yescharacteristic
50C23×C4×C53—21no—
51AGL(1,ℤ/8)×C53—42no—
52AGL(1,ℤ/8)×C53—42no—
53C2×SD16×C53—22no—
54C2×D8×C53—22no—
55C2×M16×C53—21no—
56C4⋊Q8×C53C2×C411yescharacteristic, complemented
57#127×C53C2×C411yescharacteristic, complemented
58#124×C53—21no—
59#121×C53—21no—
60#121×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53C2×C412yes—
66C4×D4×C53D412yes—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53C2×C412yescomplemented
72#103×C53—22no—
73#102×C53—22no—
74C24×C53—21no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79C2×Q8×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C2×D4×C53C22⋊C412yes—
88C2×D4×C53D812yes—
89C2×D4×C53SD1612yes—
90C22×C4×C53—21no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—41no—
94C22×C4×C53—82no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×Dic53—21no—
99C22×Dic53—22no—
100C22×Dic53—21no—
101C22×Dic53—44no—
102C22×Dic53—44no—
103C22×Dic53—82no—
104SD16×C53—44no—
105D8×C53—44no—
106M16×C53—41no—
107M16×C53—41no—
108C2×C8×C53—22no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53C22⋊C412yes—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C42×C53C22×C411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120C42×C53#4111yescharacteristic
121C22⋊Dic53—44no—
122C4⋊Dic53—42no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C4⋊Dic53—82no—
126C4×Dic53—41no—
127C4×Dic53—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53—41no—
136C23×C53#12011yescharacteristic
137Q8×C53—41no—
138Q8×C53—41no—
139D4×C53—22no—
140D4×C53—22no—
141D4×C53—22no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—81no—
145D4×C53—81no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53#10214yes—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—82no—
155C2×C4×C53—82no—
156C2×C4×C53—81no—
157C2×C4×C53—81no—
158C2×C4×C53C2×C22⋊C411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12311yescharacteristic
164C2×C4×C53C2×D811yescharacteristic
165C2×C4×C53C2×SD1611yescharacteristic
166C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C53#13711yescharacteristic
168C2×Dic53—22no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—41no—
177C2×Dic53—88no—
178C8×C53—42no—
179#88322—81no—
180#88322—81no—
181#9179—531nomaximal, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—24no—
185C22×C53—44no—
186C22×C53—44no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53#38611yescharacteristic
193C22×C53#44711yescharacteristic
194C22×C53#44811yescharacteristic
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—42no—
200C4×C53—81no—
201C4×C53C2×#10211yescharacteristic
202C4×C53#41711yescharacteristic
203C4×C53#48011yescharacteristic
204C4×C53#48111yescharacteristic
205Dic53—42no—
206Dic53—81no—
207Dic53—82no—
208#3036D5311yescharacteristic
209#2349—531no—
210#1517—531no—
211#1502—531no—
212#1478—531no—
213#1303—531no—
214#1286—531no—
215C2×C53—21no—
216C2×C53—21no—
217C2×C53—44no—
218C2×C53—161no—
219C2×C53#187411yescharacteristic
220C2×C53#190411yescharacteristic
221C2×C53#190611yescharacteristic
222C2×AGL(1,ℤ/8)—21no—
223#532C2×D5311yescharacteristic
224C2×C4×D4—1061no—
225C2×C4×D4C2×D5311yescharacteristic
226#463—22no—
227#461Dic5312yescomplemented
228#459Dic5312yescomplemented
229#431—531no—
230#431—531no—
231#423C2×D5311yescharacteristic
232#417—21no—
233#389—1061no—
234#385—1062no—
235#382—531no—
236#380—1061no—
237C4×C4⋊C4—531no—
238C2×#95—1061no—
239#327—1061no—
240#327—1061no—
241#327—1062no—
242D4⋊C8—1062no—
243C53#917911yescharacteristic, complemented, Sylow
244C2×#41—21no—
245C22×D4C22⋊D5311yescharacteristic
246C23×C4—21no—
247C23×C4—1062no—
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—42no—
250C2×SD16—22no—
251C2×D8—22no—
252C2×M16—21no—
253C4⋊Q8C2×Dic5311yescharacteristic, complemented
254#127C2×Dic5311yescharacteristic, complemented
255#124—21no—
256#121—21no—
257#121—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—2122no—
263C4×D4—2122no—
264C4×D4C2×Dic5312yes—
265C4×D4C22⋊D5312yes—
266C2×C4⋊C4—1062no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4C22⋊D5311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1062no—
272C2×C42—1061no—
273C2×C42C22×D5311yescharacteristic
274C4⋊C8—21no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8C2×Dic5312yescomplemented
278#103—22no—
279#102—22no—
280C22⋊C8—1061no—
281C22⋊C8—1061no—
282C4×C8—1061no—
283C4×C8—1061no—
284#95—531no—
285#95—531no—
286#95—532no—
287#95—1062no—
288#95—1061no—
289C24—21no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4#1171992212yes—
303C2×D4#1171992312yes—
304C2×D4C22⋊Dic5312yes—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—82no—
310C22×C4—1061no—
311C22×C4—1062no—
312C22×C4—1061no—
313C22×C4—4242no—
314C22×C4—2124no—
315C22×C4—2124no—
316C22×C4D4×D5311yescharacteristic
317C22×C4D4×D5311yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319SD16—44no—
320D8—44no—
321M16—41no—
322M16—41no—
323C2×C8—22no—
324C2×C8—2121no—
325C2×C8—2121no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—4242no—
330C4⋊C4—2122no—
331C4⋊C4—2121no—
332C4⋊C4—2121no—
333C4⋊C4C22⋊Dic5312yes—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—2124no—
339C42—2121no—
340C42—2121no—
341C42#1171994711yescharacteristic
342C42#1171994711yescharacteristic
343C42C22×Dic5311yescharacteristic
344C42C2×C22⋊D5311yescharacteristic
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—41no—
351C23#1313318511yescharacteristic
352Q8—41no—
353Q8—41no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—41no—
358D4—41no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4#1313313814yes—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—1062no—
374C2×C4—4248no—
375C2×C4—2121no—
376C2×C4—2122no—
377C2×C4—2121no—
378C2×C4—2122no—
379C2×C4—2121no—
380C2×C4—2122no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4C2×C22⋊Dic5311yescharacteristic
384C2×C4D4×Dic5311yescharacteristic
385C2×C4D4×Dic5311yescharacteristic
386C2×C4#1313318911yescharacteristic
387C2×C4#1313318911yescharacteristic
388C2×C4#1313319811yescharacteristic
389C2×C4C2×#1171992211yescharacteristic
390C2×C4C2×#1171992311yescharacteristic
391C2×C4#1313324211yescharacteristic
392C2×C4#1313324711yescharacteristic
393C8—42no—
394C8—4241no—
395C8—4241no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1681050311yescharacteristic
407C22#1681075611yescharacteristic
408C22#1681076111yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—81no—
415C4—4242no—
416C4—4241no—
417C4—2122no—
418C4C2×#1313313811yescharacteristic
419C4#1681062411yescharacteristic
420C4#1681093411yescharacteristic
421C4#1681093911yescharacteristic
422C2—21no—
423C2—21no—
424C2—44no—
425C2—161no—
426C2#2027503311yescharacteristic
427C2#2027520411yescharacteristic
428C2#2027521811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3036C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3036D53
hypocenterC53#9179
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #20277625C2semidirect
2. #20273275C2semidirect
3. #20273210C2semidirect
4. #20273106C2semidirect
5. #20272616C2semidirect
6. #20272575C2semidirect
25. C53×#461C4semidirect
26. C53×#459C4semidirect
56. C4⋊Q8×C53C2×C4semidirect
57. #127×C53C2×C4semidirect
71. C4⋊C8×C53C2×C4semidirect
227. #461Dic53semidirect
228. #459Dic53semidirect
243. C53#9179semidirect
253. C4⋊Q8C2×Dic53semidirect
254. #127C2×Dic53semidirect
277. C4⋊C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202776252
#202732752
#202732102
#202731062
#202726162
#202725752
C53×#30362
#917953