Finite Group Databasebeta

← #27299061

Subgroups — #27299061

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810597C211yesmaximal, characteristic
2#20277206C211yesmaximal, characteristic
3#20272839C214yesmaximal, complemented
4C2×C53×#410C211yesmaximal, characteristic
5C2×#13133176—21no—
6C2×#13133176C411yescharacteristic, complemented
7C22×C4×Dic53C411yescharacteristic, complemented
8#16810597—22no—
9#16810597C414yescomplemented
10C2×#13133132—21no—
11C2×#13133132C2212yes—
12#16810530C412yescomplemented
13#16810526C2212yes—
14#16810503—21no—
15#16810503—21no—
16C4×C22⋊Dic53—21no—
17C4×C22⋊Dic53—21no—
18C22×#41×C53C2211yescharacteristic
19C2×#117×C53—21no—
20C53×#410—22no—
21C53×#410C414yescomplemented
22C2×#99×C53—21no—
23C2×#99×C53C2212yes—
24C23×Dic53—21no—
25#13133176—24no—
26#13133176—42no—
27#13133176—41no—
28#13133176—41no—
29C2×C4⋊Dic53—41no—
30C2×C4⋊Dic53C2×C412yes—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—42no—
38C2×C4×Dic53—41no—
39C2×C4×Dic53C2×C412yes—
40#13133132—24no—
41#13133132—22no—
42#13133132C2×C414yescomplemented
43#13133124—22no—
44#13133124—22no—
45C2×#41×C53—41no—
46C2×#41×C53—41no—
47C2×#41×C53—42no—
48C2×#41×C53C2×C412yes—
49C2×#41×C53C2×C412yes—
50C2×#41×C53D412yes—
51C2×#41×C53Q812yes—
52C22×Q8×C53C2×C411yescharacteristic
53C22×D4×C53D411yescharacteristic
54C22×D4×C53D411yescharacteristic
55C22×D4×C53C2311yescharacteristic
56C23×C4×C53C2×C411yescharacteristic
57C23×C4×C53D411yescharacteristic
58C23×C4×C53D411yescharacteristic
59#117×C53—24no—
60#117×C53—42no—
61C2×C4⋊C4×C53—41no—
62C2×C22⋊C4×C53—22no—
63C2×C42×C53—41no—
64#99×C53—24no—
65#99×C53—22no—
66#99×C53C2×C414yescomplemented
67C24×C53C22⋊C411yescharacteristic
68C24×C53C2×D411yescharacteristic
69C24×C53C2×D411yescharacteristic
70#41×C53—42no—
71#41×C53—42no—
72#41×C53—42no—
73#41×C53—42no—
74#41×C53—42no—
75#41×C53—42no—
76#41×C53—42no—
77#41×C53—42no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53C4212yes—
81C2×Q8×C53C4⋊C412yes—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53C22⋊C412yes—
89C2×D4×C53C22⋊C412yes—
90C2×D4×C53C4⋊C412yes—
91C2×D4×C53C4⋊C412yes—
92C2×D4×C53C22×C411yescharacteristic
93C2×D4×C53C22×C411yescharacteristic
94C2×D4×C53C2×D411yescharacteristic
95C2×D4×C53C2×Q811yescharacteristic
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—42no—
106C22×C4×C53C4212yes—
107C22×C4×C53C22⋊C411yescharacteristic
108C22×C4×C53C22⋊C411yescharacteristic
109C22×C4×C53C22⋊C412yes—
110C22×C4×C53C22⋊C412yes—
111C22×C4×C53C22⋊C411yescharacteristic
112C22×C4×C53C4⋊C412yes—
113C22×C4×C53C4⋊C412yes—
114C22×C4×C53C4⋊C412yes—
115C22×C4×C53C22×C411yescharacteristic
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—44no—
119C22×Dic53—42no—
120C22×Dic53—42no—
121C4⋊C4×C53—44no—
122C22⋊C4×C53—24no—
123C22⋊C4×C53—42no—
124C42×C53—44no—
125C22⋊Dic53—24no—
126C22⋊Dic53—44no—
127C22⋊Dic53—44no—
128C22⋊Dic53—42no—
129C22⋊Dic53—42no—
130C4⋊Dic53—44no—
131C4⋊Dic53—42no—
132C4×Dic53—44no—
133C4×Dic53—42no—
134C4×Dic53—41no—
135C4×Dic53—41no—
136C4×Dic53—42no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53—42no—
146C23×C53#9512yes—
147C23×C53C2×C22⋊C411yescharacteristic
148C23×C53C2×C22⋊C411yescharacteristic
149C23×C53C2×C22⋊C411yescharacteristic
150C23×C53C2×C4⋊C411yescharacteristic
151C23×C53C2×C4⋊C411yescharacteristic
152Q8×C53—42no—
153Q8×C53—42no—
154D4×C53—42no—
155D4×C53—42no—
156D4×C53—42no—
157D4×C53—42no—
158D4×C53—41no—
159D4×C53—41no—
160D4×C53—41no—
161D4×C53—41no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—22no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—44no—
181C2×C4×C53—84no—
182C2×C4×C53#9512yes—
183C2×C4×C53#9512yes—
184C2×C4×C53#9512yes—
185C2×C4×C53#9912yes—
186C2×C4×C53#9911yescharacteristic
187C2×C4×C53#9911yescharacteristic
188C2×C4×C53C2×C4211yescharacteristic
189C2×C4×C53C2×C4⋊C411yescharacteristic
190C2×Dic53—24no—
191C2×Dic53—42no—
192C2×Dic53—48no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—44no—
197C2×Dic53—42no—
198C2×Dic53—84no—
199#12581—531nomaximal, Sylow
200C22×C53—22no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—22no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—42no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53—42no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53C2×#9511yescharacteristic
218C22×C53C2×#9911yescharacteristic
219C22×C53C2×#9911yescharacteristic
220C4×C53—22no—
221C4×C53—22no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—84no—
225C4×C53#34212yes—
226C4×C53#34212yes—
227Dic53—44no—
228Dic53—84no—
229Dic53—82no—
230C2×#410—531no—
231C2×#410D5311yescharacteristic
232#2297—531no—
233#1393—534no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—41no—
237C2×C53—41no—
238C2×C53—42no—
239C2×C53—41no—
240C2×C53—41no—
241C2×C53C2×#34211yescharacteristic
242C2×C53#212612yes—
243C22×#41C2×D5311yescharacteristic
244C2×#117—21no—
245C2×#117—531no—
246C2×#117—1061no—
247C22×C42—531no—
248#410—22no—
249#410—534no—
250#410—1062no—
251#410Dic5314yescomplemented
252C2×#99—21no—
253C2×#99—532no—
254C2×#99—1061no—
255C2×#99C2×D5312yes—
256#391—532no—
257#390—532no—
258#386—1061no—
259#386—1061no—
260C4×C22⋊C4—1061no—
261C4×C22⋊C4—1061no—
262C53#1258111yescharacteristic, complemented, Sylow
263C2×#41—42no—
264C2×#41—41no—
265C2×#41—41no—
266C2×#41Dic10612yes—
267C2×#41C4×D5312yes—
268C2×#41C2×Dic5312yes—
269C2×#41C22⋊D5312yes—
270C22×Q8C4×D5311yescharacteristic
271C22×D4D21211yescharacteristic
272C22×D4C22⋊D5311yescharacteristic
273C22×D4C22×D5311yescharacteristic
274C23×C4—1061no—
275C23×C4C4×D5311yescharacteristic
276C23×C4D21211yescharacteristic
277C23×C4C22⋊D5311yescharacteristic
278#117—24no—
279#117—42no—
280#117—1064no—
281#117—2122no—
282#117—2121no—
283#117—2121no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—532no—
286C2×C4⋊C4—2121no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—1062no—
289C2×C22⋊C4—1062no—
290C2×C22⋊C4—1062no—
291C2×C22⋊C4—1061no—
292C2×C42—41no—
293C2×C42—532no—
294C2×C42—1061no—
295C2×C42—1061no—
296C2×C42—2121no—
297C2×C42—2122no—
298#99—22no—
299#99—24no—
300#99—534no—
301#99—1062no—
302#99—1064no—
303#99C2×Dic5314yescomplemented
304#95—1062no—
305#95—1062no—
306C24#1171992111yescharacteristic
307C24C2×D21211yescharacteristic
308C24C2×C22⋊D5311yescharacteristic
309#41—42no—
310#41—42no—
311#41—42no—
312#41—42no—
313#41—42no—
314#41—42no—
315#41—42no—
316#41—42no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8C4×Dic5312yes—
320C2×Q8#1171991912yes—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4#1171991912yes—
328C2×D4C4⋊Dic5312yes—
329C2×D4#1171992112yes—
330C2×D4C22⋊Dic5312yes—
331C2×D4C2×Dic10611yescharacteristic
332C2×D4C2×C4×D5311yescharacteristic
333C2×D4C22×Dic5311yescharacteristic
334C2×D4C2×C22⋊D5311yescharacteristic
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—1062no—
346C22×C4—1062no—
347C22×C4—2124no—
348C22×C4—2122no—
349C22×C4—2122no—
350C22×C4C4×Dic5312yes—
351C22×C4#1171991912yes—
352C22×C4#1171991912yes—
353C22×C4C4⋊Dic5312yes—
354C22×C4#1171992111yescharacteristic
355C22×C4#1171992111yescharacteristic
356C22×C4#1171992112yes—
357C22×C4#1171992111yescharacteristic
358C22×C4C22⋊Dic5312yes—
359C22×C4C2×C4×D5311yescharacteristic
360C4⋊C4—44no—
361C4⋊C4—2122no—
362C4⋊C4—2124no—
363C22⋊C4—24no—
364C22⋊C4—42no—
365C22⋊C4—1064no—
366C22⋊C4—2124no—
367C22⋊C4—2124no—
368C22⋊C4—2122no—
369C22⋊C4—2122no—
370C42—44no—
371C42—2122no—
372C42—2124no—
373C42—2122no—
374C42—2121no—
375C42—2121no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23#1313312412yes—
386C23C2×#1171992111yescharacteristic
387C23C2×#1171992111yescharacteristic
388C23C2×C22⋊Dic5311yescharacteristic
389C23C2×#1171991911yescharacteristic
390C23C2×C4⋊Dic5311yescharacteristic
391Q8—42no—
392Q8—42no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—42no—
398D4—42no—
399D4—42no—
400D4—42no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—44no—
420C2×C4—84no—
421C2×C4—1064no—
422C2×C4—4244no—
423C2×C4—2124no—
424C2×C4—2122no—
425C2×C4—2122no—
426C2×C4—2122no—
427C2×C4—2122no—
428C2×C4—2128no—
429C2×C4—2122no—
430C2×C4#1313312412yes—
431C2×C4#1313312412yes—
432C2×C4#1313312412yes—
433C2×C4#1313313111yescharacteristic
434C2×C4#1313313112yes—
435C2×C4#1313313111yescharacteristic
436C2×C4C2×C4×Dic5311yescharacteristic
437C2×C4C2×#1171991911yescharacteristic
438C22—22no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—21no—
443C22—21no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22C2×#1313312411yescharacteristic
456C22C2×#1313313111yescharacteristic
457C22C2×#1313313111yescharacteristic
458C4—22no—
459C4—22no—
460C4—21no—
461C4—21no—
462C4—84no—
463C4—4244no—
464C4—4242no—
465C4—2124no—
466C4#1681038912yes—
467C4#1681038912yes—
468C2—21no—
469C2—21no—
470C2—41no—
471C2—41no—
472C2—42no—
473C2—41no—
474C2—41no—
475C2C2×#1681038911yescharacteristic
476C2#2027637812yes—
subgroupG/H
centerC2×C4#13133131
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC2×C53×#410C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#410D53
hypocenterC53#12581
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
3. #20272839C2semidirect
6. C2×#13133176C4semidirect
7. C22×C4×Dic53C4semidirect
9. #16810597C4semidirect
12. #16810530C4semidirect
21. C53×#410C4semidirect
42. #13133132C2×C4semidirect
66. #99×C53C2×C4semidirect
251. #410Dic53semidirect
262. C53#12581semidirect
303. #99C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105972
#202772062
#202728392
C2×C53×#4102
#1258153