Finite Group Databasebeta

← D53×#2235 #27538852

Subgroups — D53×#2235 #27538852

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#467×D53C211yesmaximal, characteristic
2C2×#107×D53C211yesmaximal, characteristic
3#20276914C212yesmaximal, complemented
4#20276913C211yesmaximal, characteristic, complemented
5#20276912C211yesmaximal, characteristic, complemented
6#20276911C212yesmaximal, complemented
7#20276910C211yesmaximal, characteristic, complemented
8#367×D53C212yesmaximal, complemented
9#358×D53C212yesmaximal, complemented
10C22⋊C16×D53C211yesmaximal, characteristic
11C53×#2235C211yesmaximal, characteristic, complemented
12C2×Q16×D53C2211yescharacteristic
13C22×C8×D53C2211yescharacteristic
14C22⋊Q8×D53—21no—
15C2×C4⋊C4×D53—21no—
16C2×C16×D53C2212yes—
17#16810865C2211yescharacteristic
18#16810864C2211yescharacteristic
19#16810863—21no—
20#16810862—21no—
21#16810861C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133150—21no—
24#107×D53—21no—
25#107×D53—21no—
26#107×D53C2211yescharacteristic
27#107×D53C2212yes—
28#107×D53C2212yes—
29#103×D53—21no—
30#16810451C2212yescomplemented
31#16810450C2212yescomplemented
32#16810450C2212yescomplemented
33#16810428C2212yescomplemented
34#16810427C2212yescomplemented
35#16810426C2212yescomplemented
36C22⋊#11719908C2211yescharacteristic
37#16810402C2212yes—
38C53×#467C2211yescharacteristic
39C2×#107×C53C2211yescharacteristic
40C53×#367C2212yescomplemented
41C53×#358C2212yescomplemented
42C53×C22⋊C16C2211yescharacteristic
43C2×Q8×D53—21no—
44C22×C4×D53—41no—
45C22×C4×D53D411yescharacteristic
46C2×Dic212C2311yescharacteristic
47C2×#11719925—21no—
48Q16×D53—24no—
49C22×#88322D411yescharacteristic
50C2×C8×D53—24no—
51C2×C8×D53D411yescharacteristic
52C2×C8×D53D411yescharacteristic
53C2×C8×D53C2311yescharacteristic
54C2×C8×D53C2311yescharacteristic
55#13133197—21no—
56#13133196—21no—
57#13133195—41no—
58#13133194—41no—
59#13133193—21no—
60C2×C4⋊Dic53—21no—
61C2×#11719919—41no—
62C4⋊C4×D53—22no—
63C4⋊C4×D53—22no—
64C4⋊C4×D53—21no—
65C4⋊C4×D53—41no—
66C4⋊C4×D53—41no—
67C4⋊C4×D53—41no—
68C22⋊C4×D53—41no—
69C2×#11719908C2312yes—
70C16×D53—24no—
71#13133151—21no—
72#13133151C2311yescharacteristic
73#13133151C2312yes—
74#13133150—21no—
75#13133150—22no—
76#13133150—22no—
77#13133142—21no—
78#13133141—21no—
79#13133140—21no—
80C2×Q16×C53C2311yescharacteristic
81C22×C8×C53C2311yescharacteristic
82C22⋊Q8×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C16×C53C2312yes—
85#107×C53—21no—
86#107×C53C2311yescharacteristic
87#107×C53C2312yes—
88#103×C53—21no—
89C23×D53D811yescharacteristic
90C2×Q8×C53—21no—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×Dic53—41no—
94C22×Dic53D811yescharacteristic
95Q8×D53—42no—
96Q8×D53—42no—
97C2×C4×D53—22no—
98C2×C4×D53—22no—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×C4×D53—42no—
102C2×C4×D53—42no—
103C2×C4×D53—84no—
104C2×C4×D53D811yescharacteristic
105C2×C4×D53D811yescharacteristic
106C2×C4×D53C2×D411yescharacteristic
107C2×C4×D53C2×D411yescharacteristic
108C2×Dic106—21no—
109C2×Dic106—41no—
110Q16×C53—22no—
111C2×C8×C53—22no—
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2411yescharacteristic
114C4⋊C4×C53—21no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C22⋊Dic53—41no—
120#11719925—42no—
121#11719921—41no—
122#11719921—41no—
123C4⋊Dic53—21no—
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127#11719919—41no—
128#11719919—42no—
129#11719919—42no—
130#11719919—41no—
131#11719919—41no—
132C2×#88322—22no—
133C2×#88322C2×D411yescharacteristic
134C2×#88322C2×D411yescharacteristic
135Dic212—22no—
136C8×D53—22no—
137C8×D53—22no—
138C8×D53—22no—
139C8×D53#4114yes—
140C16×C53—22no—
141#11719908—22no—
142C23×C53C2×D811yescharacteristic
143C22×D53—22no—
144C22×D53—22no—
145C22×D53SD3214yes—
146C22×D53C2×D811yescharacteristic
147C22×D53C2×D811yescharacteristic
148Q8×C53—41no—
149Q8×C53—41no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—82no—
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C22×D411yescharacteristic
158C2×Dic53—22no—
159C2×Dic53—41no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53—82no—
163C2×Dic53C2×D811yescharacteristic
164C2×Dic53C2×D811yescharacteristic
165C4×D53—22no—
166C4×D53—21no—
167C4×D53—21no—
168C4×D53—22no—
169C4×D53—82no—
170C4×D53—84no—
171C4×D53—84no—
172C4×D53—82no—
173C4×D53#12312yes—
174C4×D53#13712yes—
175Dic106—41no—
176Dic106—41no—
177Dic106—82no—
178C8×C53—21no—
179C8×C53C2×#4112yes—
180#88322—21no—
181#88322#12312yes—
182C2×#2235—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53C2×SD3212yes—
186C22×C53C22×D811yescharacteristic
187C2×D53—21no—
188C2×D53—21no—
189C2×D53—28no—
190C2×D53#48012yes—
191C2×D53C2×SD3212yes—
192C2×D53C2×SD3212yes—
193C2×D53#51012yes—
194C4×C53—21no—
195C4×C53—81no—
196C4×C53—82no—
197C4×C53—81no—
198C4×C53C2×#12311yescharacteristic
199C4×C53C2×#13711yescharacteristic
200Dic53—21no—
201Dic53—81no—
202Dic53—82no—
203Dic53—81no—
204Dic53#48012yes—
205C2×#467—531no—
206C22×#107—531no—
207#2235—531no—
208#2235—531no—
209#2235—532no—
210#2235—532no—
211#2235—531no—
212#2235D5311yescharacteristic, direct product factor
213C2×#367—532no—
214C2×#358—532no—
215C2×C22⋊C16—531no—
216C2×C53—22no—
217C2×C53C2×#48011yescharacteristic
218C2×C53C22×SD3211yescharacteristic
219C2×C53C2×#51011yescharacteristic
220D53—22no—
221D53#223514yesdirect product factor
222C22×Q16—531no—
223C23×C8—531no—
224C2×C22⋊Q8—1061no—
225C22×C4⋊C4—1061no—
226C22×C16—532no—
227#467—531no—
228#467—531no—
229#467—531no—
230#467—1061no—
231#467—1061no—
232#467C2×D5311yescharacteristic
233C2×#107—531no—
234C2×#107—532no—
235C2×#107—532no—
236C2×#107—531no—
237C2×#107—1061no—
238C2×#107—1061no—
239C2×#107—1061no—
240C2×#107C2×D5311yescharacteristic
241C2×#103—1061no—
242#367—532no—
243#367—532no—
244#367—532no—
245#367C2×D5312yescomplemented
246#358—532no—
247#358—532no—
248#358—532no—
249#358C2×D5312yescomplemented
250C22⋊C16—531no—
251C22⋊C16—532no—
252C22⋊C16C2×D5311yescharacteristic
253C53C2×#223511yescharacteristic, complemented, Sylow
254C22×Q8—1061no—
255C23×C4—531no—
256C23×C4—2121no—
257C2×Q16—531no—
258C2×Q16—1064no—
259C2×Q16—1061no—
260C2×Q16C22×D5311yescharacteristic
261C22×C8—531no—
262C22×C8—531no—
263C22×C8—531no—
264C22×C8—531no—
265C22×C8—531no—
266C22×C8—1064no—
267C22×C8C22×D5311yescharacteristic
268C22⋊Q8—21no—
269C22⋊Q8—1061no—
270C22⋊Q8—1061no—
271C22⋊Q8—1061no—
272C22⋊Q8—2121no—
273C22⋊Q8—2121no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—1062no—
276C2×C4⋊C4—1062no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4—2121no—
280C2×C4⋊C4—2121no—
281C2×C4⋊C4—2121no—
282C2×C4⋊C4—2121no—
283C2×C22⋊C4—2121no—
284C2×C16—532no—
285C2×C16—1064no—
286C2×C16C22×D5312yes—
287#107—21no—
288#107—532no—
289#107—531no—
290#107—1062no—
291#107—1062no—
292#107—1061no—
293#107—1061no—
294#107C22×D5312yes—
295#107C22×D5311yescharacteristic
296#103—21no—
297#103—1061no—
298#103—1061no—
299#103—1061no—
300C24—531no—
301C2×Q8—21no—
302C2×Q8—1061no—
303C2×Q8—2121no—
304C2×Q8—2122no—
305C2×Q8—2122no—
306C22×C4—41no—
307C22×C4—531no—
308C22×C4—531no—
309C22×C4—531no—
310C22×C4—531no—
311C22×C4—531no—
312C22×C4—1062no—
313C22×C4—1062no—
314C22×C4—4244no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4—2122no—
318C22×C4—2122no—
319C22×C4—2121no—
320C22×C4D4×D5311yescharacteristic
321Q16—22no—
322Q16—1062no—
323Q16—2122no—
324C2×C8—22no—
325C2×C8—531no—
326C2×C8—534no—
327C2×C8—531no—
328C2×C8—1062no—
329C2×C8—1062no—
330C2×C8—1062no—
331C2×C8—1062no—
332C2×C8D4×D5311yescharacteristic
333C2×C8C23×D5311yescharacteristic
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—1061no—
339C4⋊C4—1062no—
340C4⋊C4—2122no—
341C4⋊C4—2122no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C4⋊C4—2121no—
345C4⋊C4—2121no—
346C4⋊C4—2121no—
347C22⋊C4—41no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C16—22no—
352C16—1062no—
353C23—531no—
354C23—534no—
355C23—531no—
356C23—1062no—
357C23—1062no—
358C23D8×D5311yescharacteristic
359Q8—41no—
360Q8—41no—
361Q8—4242no—
362Q8—2121no—
363Q8—2121no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—531no—
371C2×C4—532no—
372C2×C4—532no—
373C2×C4—531no—
374C2×C4—1062no—
375C2×C4—1062no—
376C2×C4—1062no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—4244no—
380C2×C4—4244no—
381C2×C4—4242no—
382C2×C4—4242no—
383C2×C4—4242no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4—2122no—
387C2×C4D8×D5311yescharacteristic
388C2×C4C2×D4×D5311yescharacteristic
389C8—21no—
390C8—532no—
391C8—1061no—
392C8#41×D5312yes—
393C22—21no—
394C22—21no—
395C22—532no—
396C22—532no—
397C22—532no—
398C22—532no—
399C22—1068no—
400C22—1061no—
401C22—1061no—
402C22SD32×D5312yes—
403C22C2×D8×D5311yescharacteristic
404C4—21no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—532no—
409C4—1061no—
410C4—4242no—
411C4—4241no—
412C4—4241no—
413C4#123×D5311yescharacteristic
414C4#137×D5311yescharacteristic
415C2—22no—
416C2—534no—
417C2—1062no—
418C2#480×D5311yescharacteristic
419C2C2×SD32×D5311yescharacteristic
420C2#510×D5311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2235C2
socleC22×C53C22×D8
prime radicalC23×D53D8
hypercenter#2235D53
hypocenterC53C2×#2235
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2235

Complements

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

NH ≅ G/Ntype
3. #20276914C2semidirect
4. #20276913C2semidirect
5. #20276912C2semidirect
6. #20276911C2semidirect
7. #20276910C2semidirect
8. #367×D53C2semidirect
9. #358×D53C2semidirect
11. C53×#2235C2semidirect
30. #16810451C22semidirect
31. #16810450C22semidirect
32. #16810450C22semidirect
33. #16810428C22semidirect
34. #16810427C22semidirect
35. #16810426C22semidirect
40. C53×#367C22semidirect
41. C53×#358C22semidirect
212. #2235D53direct (N × H)
221. D53#2235direct (N × H)
245. #367C2×D53semidirect
249. #358C2×D53semidirect
253. C53C2×#2235semidirect

Maximal subgroups

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

subgroupindex
#467×D532
C2×#107×D532
#202769142
#202769132
#202769122
#202769112
#202769102
#367×D532
#358×D532
C22⋊C16×D532
C53×#22352
C2×#223553