Finite Group Databasebeta

← C4×#16810410 #27299224

Subgroups — C4×#16810410 #27299224

Order 13568.

844 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810410C212yesmaximal
2C4×#13133134C211yesmaximal, characteristic, complemented
3C4×#13133132C211yesmaximal, characteristic, complemented
4#20272833C212yesmaximal, complemented
5C4×#120×C53C211yesmaximal, characteristic
6C2×#13133134C2212yes—
7C2×#13133132C2212yes—
8C4×#11719917—21no—
9C4×C22⋊Dic53—21no—
10#16810410C418yesdirect product factor
11#16810392—21no—
12#16810390—21no—
13C24×C4×C53C411yescharacteristic, complemented
14C2×#120×C53C2211yescharacteristic
15C2×C4×D4×C53—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#386—21no—
18C53×#386C2211yescharacteristic
19C53×#385—21no—
20C53×#385C411yescharacteristic, complemented
21C53×#379C411yescharacteristic, complemented
22C4×C22⋊C4×C53—21no—
23C4×C22⋊C4×C53C411yescharacteristic, complemented
24C2×#11719917—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—41no—
27#13133134—22no—
28#13133134C2×C414yescomplemented
29#13133132—22no—
30#13133132C2×C414yescomplemented
31#13133128—41no—
32C4×#88322—41no—
33#13133124—41no—
34C25×C53C2×C411yescharacteristic
35C22×D4×C53—21no—
36C22×D4×C53C2311yescharacteristic
37C23×C4×C53—21no—
38C23×C4×C53—21no—
39C23×C4×C53—44no—
40C23×C4×C53—41no—
41C23×C4×C53—41no—
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45#120×C53—22no—
46#120×C53C2×C414yes—
47C4×D4×C53—41no—
48C4×D4×C53—41no—
49C4×D4×C53—44no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53D411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53D411yescharacteristic
62#95×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65C24×C53—21no—
66C24×C53—21no—
67C24×C53—42no—
68C24×C53—41no—
69C24×C53—42no—
70C24×C53—41no—
71C24×C53C4212yescomplemented
72C24×C53C22×C411yescharacteristic
73C24×C53C2×D411yescharacteristic
74C2×D4×C53—24no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—42no—
78C2×D4×C53—42no—
79C2×D4×C53C22×C412yes—
80C2×D4×C53#4112yes—
81C22×C4×C53—24no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—42no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53—44no—
97C22×C4×C53—44no—
98C22×C4×C53—44no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—44no—
103C22×C4×C53C22⋊C411yescharacteristic
104C22×C4×C53C22⋊C411yescharacteristic
105C22×C4×C53C22⋊C411yescharacteristic
106C22×C4×C53C22⋊C411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×Dic53—42no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—42no—
111C22⋊C4×C53—21no—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—42no—
116C22⋊C4×C53—42no—
117C22⋊C4×C53—42no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53C4212yescomplemented
120C42×C53—21no—
121C42×C53—21no—
122C42×C53—42no—
123C22⋊Dic53—24no—
124C22⋊Dic53—42no—
125C4×Dic53—82no—
126#11719917—24no—
127#11719917—42no—
128C2×#88322—42no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—41no—
137C23×C53—44no—
138C23×C53—42no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—44no—
143C23×C53—42no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—42no—
147C23×C53—42no—
148C23×C53—42no—
149C23×C53—48no—
150C23×C53—42no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53C2×C4211yescharacteristic
154C23×C53C2×C22⋊C411yescharacteristic
155C23×C53#11711yescharacteristic
156C23×C53C4×D412yes—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—48no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—44no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—48no—
174C2×C4×C53—42no—
175C2×C4×C53—41no—
176C2×C4×C53—44no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—81no—
187C2×C4×C53#9911yescharacteristic
188C2×C4×C53#9911yescharacteristic
189C2×C4×C53C4×D412yes—
190C2×Dic53—44no—
191C2×Dic53—82no—
192C2×Dic53—82no—
193#88322—44no—
194C4×AΣL(1,𝔽16)—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—24no—
198C22×C53—22no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—42no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—42no—
205C22×C53—42no—
206C22×C53—42no—
207C22×C53—42no—
208C22×C53—44no—
209C22×C53—44no—
210C22×C53—44no—
211C22×C53—44no—
212C22×C53—41no—
213C22×C53—42no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—42no—
217C22×C53—41no—
218C22×C53—42no—
219C22×C53—81no—
220C22×C53C4×C22⋊C411yescharacteristic
221C22×C53C2×#9911yescharacteristic
222C22×C53#41011yescharacteristic
223C4×C53—21no—
224C4×C53—22no—
225C4×C53—42no—
226C4×C53—42no—
227C4×C53—41no—
228C4×C53—41no—
229C4×C53—41no—
230C4×C53—81no—
231C4×C53AΣL(1,𝔽16)12yescomplemented
232Dic53—84no—
233C4×#120D5311yescharacteristic
234C2×AΣL(1,𝔽16)—532no—
235C4×#100—531no—
236C4×#99—531no—
237#1390—532no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53—42no—
241C2×C53—44no—
242C2×C53—41no—
243C2×C53—41no—
244C2×C53C4×#9911yescharacteristic
245C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
246C2×C53#212011yescharacteristic
247C24×C4Dic5311yescharacteristic, complemented
248C2×#120C2×D5311yescharacteristic
249C2×C4×D4—21no—
250C2×C4×D4C2×D5311yescharacteristic
251C2×#100—532no—
252C2×#99—532no—
253C4×M16—1061no—
254#386—21no—
255#386C2×D5311yescharacteristic
256#385—21no—
257#385Dic5311yescharacteristic, complemented
258#379Dic5311yescharacteristic, complemented
259C4×C22⋊C4—21no—
260C4×C22⋊C4—1061no—
261C4×C22⋊C4Dic5311yescharacteristic, complemented
262AΣL(1,𝔽16)—538no—
263#343—1061no—
264#342—1061no—
265C53C4×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
266C25C2×Dic5311yescharacteristic
267C22×D4—21no—
268C22×D4C22×D5311yescharacteristic
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—44no—
272C23×C4—41no—
273C23×C4—41no—
274C23×C4C2×Dic5311yescharacteristic
275C23×C4C2×Dic5311yescharacteristic
276C23×C4C22⋊D5311yescharacteristic
277C2×M16—1062no—
278#120—22no—
279#120C4×D5314yes—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C22⋊D5311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—1062no—
290C2×C22⋊C4C2×Dic5311yescharacteristic
291C2×C22⋊C4C2×Dic5311yescharacteristic
292C2×C22⋊C4C2×Dic5311yescharacteristic
293C2×C22⋊C4C22⋊D5311yescharacteristic
294C2×C42—21no—
295C2×C42—2121no—
296C2×C42C22⋊D5311yescharacteristic
297#100—534no—
298#100—1062no—
299#99—534no—
300#99—1062no—
301#97—2121no—
302C4×C8—2121no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—2121no—
307C24—21no—
308C24—21no—
309C24—42no—
310C24—41no—
311C24—42no—
312C24—41no—
313C24C4×Dic5312yescomplemented
314C24C22×Dic5311yescharacteristic
315C24C2×C22⋊D5311yescharacteristic
316C2×D4—24no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4C2×C4×D5312yes—
322C2×D4#1171994512yes—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—44no—
345C22×C4—2122no—
346C22×C4C22⋊Dic5311yescharacteristic
347C22×C4C22⋊Dic5311yescharacteristic
348C22×C4C22⋊Dic5311yescharacteristic
349C22×C4C22⋊Dic5311yescharacteristic
350C22×C4C2×C22⋊D5311yescharacteristic
351M16—1064no—
352M16—2122no—
353C2×C8—2122no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C22⋊C4—21no—
357C22⋊C4—24no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—41no—
364C22⋊C4—1064no—
365C22⋊C4—2122no—
366C22⋊C4C4×Dic5312yescomplemented
367C42—21no—
368C42—21no—
369C42—42no—
370C42—4242no—
371C23—22no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—21no—
378C23—41no—
379C23—41no—
380C23—44no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—44no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—48no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23C2×C4×Dic5311yescharacteristic
396C23C2×C22⋊Dic5311yescharacteristic
397C23#1313317611yescharacteristic
398C23C4×C22⋊D5312yes—
399D4—42no—
400D4—42no—
401D4—48no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—81no—
429C2×C4—4242no—
430C2×C4—4242no—
431C2×C4—2124no—
432C2×C4#1313313211yescharacteristic
433C2×C4#1313313211yescharacteristic
434C2×C4C4×C22⋊D5312yes—
435C8—2124no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—41no—
444C22—41no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—42no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—42no—
459C22—41no—
460C22—81no—
461C22C4×C22⋊Dic5311yescharacteristic
462C22C2×#1313313211yescharacteristic
463C22#1681059711yescharacteristic
464C4—22no—
465C4—21no—
466C4—42no—
467C4—42no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—4244no—
473C4#1681041012yesdirect product factor
474C2—21no—
475C2—21no—
476C2—42no—
477C2—44no—
478C2—41no—
479C2—41no—
480C2C4×#1313313211yescharacteristic
481C2C2×#1681041011yescharacteristic
482C2#2027635611yescharacteristic
subgroupG/H
centerC2×C4#13133132
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#120×C53C2
socleC22×C53C2×#99
prime radicalC2×#120×C53C22
cosocleC22×D4×C53C23
hypercenterC4×#120D53
hypocenterC53C4×AΣL(1,𝔽16)
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
2. C4×#13133134C2semidirect
3. C4×#13133132C2semidirect
4. #20272833C2semidirect
10. #16810410C4direct (N × H)
13. C24×C4×C53C4semidirect
20. C53×#385C4semidirect
21. C53×#379C4semidirect
23. C4×C22⋊C4×C53C4semidirect
28. #13133134C2×C4semidirect
30. #13133132C2×C4semidirect
71. C24×C53C42semidirect
119. C22⋊C4×C53C42semidirect
231. C4×C53AΣL(1,𝔽16)semidirect
247. C24×C4Dic53semidirect
257. #385Dic53semidirect
258. #379Dic53semidirect
261. C4×C22⋊C4Dic53semidirect
265. C53C4×AΣL(1,𝔽16)semidirect
313. C24C4×Dic53semidirect
366. C22⋊C4C4×Dic53semidirect
473. C4#16810410direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168104102
C4×#131331342
C4×#131331322
#202728332
C4×#120×C532
C4×AΣL(1,𝔽16)53