Finite Group Databasebeta

← #28307044

Subgroups — #28307044

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719940C211yesmaximal, characteristic
2C2×#16811503C212yesmaximal
3C2×#16811503C212yesmaximal
4#20290378C211yesmaximal, characteristic
5C2×#16810520C211yesmaximal, characteristic
6C22×#11719940C2212yes—
7C22×#11719940C2212yes—
8C4×#11719940—24no—
9C2×#13133272—21no—
10C2×C4×#88324—21no—
11#16811503—24no—
12#16811503C418yescomplemented
13#16811489—21no—
14#16811489—21no—
15C22×C4⋊Dic53C411yescharacteristic
16C22×#11719919C411yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20#16810520—21no—
21#16810520—21no—
22#16810520—21no—
23#16810520—21no—
24C2×#13133124C411yescharacteristic
25C2×#95×C53C411yescharacteristic
26C2×#11719940—22no—
27C2×#11719940—22no—
28C2×#11719940—22no—
29C2×#11719940—22no—
30C2×#11719940—22no—
31C2×#11719940—22no—
32C2×#11719940—22no—
33C2×#11719940—22no—
34C2×#11719940C2×C414yes—
35C2×#11719940C2×C414yes—
36C2×#11719940D414yes—
37C2×#11719940Q814yes—
38C22×#88324—22no—
39C22×#88324—22no—
40#13133272—24no—
41C4×#88324—24no—
42C23×Dic53C2×C411yescharacteristic
43C23×Dic53C2×C411yescharacteristic
44C23×Dic53C2×C411yescharacteristic
45C23×Dic53C2311yescharacteristic
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—42no—
49C2×#11719919—21no—
50C2×#11719919—21no—
51C2×#11719919—21no—
52C2×#11719919—21no—
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—42no—
59C2×#11719919C2×C414yes—
60C2×C4×Dic53—22no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53D411yescharacteristic
63C2×C4×Dic53D411yescharacteristic
64C2×C4×Dic53D411yescharacteristic
65C2×C4×Dic53D411yescharacteristic
66#13133124—21no—
67#13133124—21no—
68C23×C4×C53C2×C411yescharacteristic
69C23×C4×C53C2×C411yescharacteristic
70C23×C4×C53C2×C411yescharacteristic
71#95×C53—21no—
72#95×C53—21no—
73C24×C53C22×C411yescharacteristic
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53C4212yes—
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—22no—
91C22×Dic53—22no—
92C22×Dic53—21no—
93C22×Dic53—22no—
94C22×Dic53—41no—
95C22×Dic53—41no—
96C22×Dic53—41no—
97C22×Dic53—41no—
98C22×Dic53—41no—
99C22×Dic53C4212yes—
100C22×Dic53C22⋊C411yescharacteristic
101C22×Dic53C22⋊C411yescharacteristic
102C22×Dic53C22×C412yes—
103C22×Dic53C2×D412yes—
104C22×Dic53C2×D411yescharacteristic
105C22×Dic53C2×D411yescharacteristic
106C22×Dic53C2×Q812yes—
107C22×Dic53#4112yes—
108C22×Dic53#4112yes—
109C22×Dic53#4112yes—
110#11719940—24no—
111#11719940—24no—
112#11719940—24no—
113#11719940—24no—
114#11719940—24no—
115#11719940—42no—
116#11719940—42no—
117#11719940C4⋊C418yescomplemented
118C2×#88324—24no—
119C2×#88324—24no—
120C2×#88324—24no—
121C2×#88324—24no—
122C2×#88324—42no—
123C2×#88324—42no—
124C4⋊Dic53—42no—
125C4⋊Dic53—42no—
126#11719919—24no—
127#11719919—24no—
128#11719919—42no—
129#11719919—42no—
130#11719919—41no—
131#11719919—42no—
132#11719919—41no—
133C4×Dic53—21no—
134C4×Dic53—21no—
135C4×Dic53—22no—
136C4×Dic53—22no—
137C4×Dic53—21no—
138C4×Dic53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53C2×C4211yescharacteristic
144C23×C53C2×C22⋊C411yescharacteristic
145C23×C53#11711yescharacteristic
146C23×C53C4×D412yes—
147C23×C53C4×Q812yes—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—22no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—44no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—82no—
163C2×C4×C53#10112yes—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—22no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—44no—
171C2×Dic53—42no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—42no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—82no—
179C2×Dic53#10012yes—
180C2×Dic53C2×C4⋊C412yes—
181C2×Dic53#11712yes—
182C2×Dic53C4×D412yes—
183C2×Dic53C4×D412yes—
184C2×Dic53C4×D412yes—
185C2×Dic53C4×Q812yes—
186C2×Dic53#12011yescharacteristic
187C2×Dic53#12112yes—
188C2×Dic53C22⋊Q812yes—
189C2×Dic53C22⋊Q812yes—
190C2×Dic53#12312yes—
191C2×Dic53#12311yescharacteristic
192C2×Dic53#12411yescharacteristic
193C2×Dic53#12512yes—
194C2×Dic53#12612yes—
195C2×Dic53C4⋊Q811yescharacteristic
196#88324—28no—
197#88324—44no—
198#10913—531nomaximal, Sylow
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—22no—
202C22×C53—22no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53C4×C22⋊C411yescharacteristic
208C22×C53C4×C4⋊C412yes—
209C22×C53#38611yescharacteristic
210C22×C53#38712yes—
211C22×C53#39111yescharacteristic
212C22×C53C2×#10011yescharacteristic
213C22×C53C2×#10111yescharacteristic
214C22×C53#41311yescharacteristic
215C22×C53#41311yescharacteristic
216C4×C53—22no—
217C4×C53—41no—
218C4×C53—42no—
219C4×C53—81no—
220Dic53—22no—
221Dic53—41no—
222Dic53—42no—
223Dic53—81no—
224Dic53#38512yes—
225Dic53#38712yes—
226Dic53#38812yes—
227Dic53#38912yes—
228C2×C4×M16—531no—
229C2×#389—531no—
230C2×#343—532no—
231C2×#343—532no—
232#1728—531no—
233C2×C53—22no—
234C2×C53—22no—
235C2×C53#143611yescharacteristic
236C2×C53C4×#10011yescharacteristic
237C2×C53C4×#10111yescharacteristic
238C2×C53#190011yescharacteristic
239C2×C53#190111yescharacteristic
240C2×C53#199711yescharacteristic
241C2×C53#199811yescharacteristic
242C22×M16—532no—
243C22×M16—532no—
244C22×C4⋊C4—531no—
245C22×C4⋊C4—531no—
246C22×C4⋊C4—531no—
247C22×C4⋊C4—531no—
248C22×C42—531no—
249C4×M16—1064no—
250C2×#97—1061no—
251C2×C4×C8—1061no—
252#389—1061no—
253#389—1061no—
254#389—1061no—
255#389—1061no—
256C2×#95—531no—
257C2×#95#615711yescharacteristic
258#343—538no—
259#343—1064no—
260#336—1061no—
261#336—1061no—
262C53#1091311yescharacteristic, complemented, Sylow
263C23×C4—531no—
264C23×C4—531no—
265C23×C4—531no—
266C23×C4—531no—
267C23×C4C2×#615711yescharacteristic
268C23×C4C2×#615711yescharacteristic
269C23×C4C2×#615711yescharacteristic
270C2×M16—534no—
271C2×M16—534no—
272C2×M16—534no—
273C2×M16—534no—
274C2×M16—1062no—
275C2×M16—1062no—
276C2×M16—1062no—
277C2×M16—1062no—
278C2×M16—1062no—
279C2×M16—1062no—
280C2×M16—1062no—
281C2×M16—1062no—
282C22×C8—1062no—
283C22×C8—1062no—
284C2×C4⋊C4—534no—
285C2×C4⋊C4—1062no—
286C2×C4⋊C4—1061no—
287C2×C4⋊C4—1061no—
288C2×C4⋊C4—1062no—
289C2×C4⋊C4—1062no—
290C2×C4⋊C4—1061no—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4—1061no—
293C2×C4⋊C4—1061no—
294C2×C4⋊C4—1061no—
295C2×C4⋊C4—1061no—
296C2×C4⋊C4—2122no—
297C2×C4⋊C4—2122no—
298C2×C42—531no—
299C2×C42—531no—
300C2×C42—531no—
301C2×C42—531no—
302C2×C42—1062no—
303C2×C42—1062no—
304#97—1064no—
305C4×C8—1064no—
306#95—21no—
307#95—21no—
308#95—1061no—
309#95—1061no—
310C24C22×#615711yescharacteristic
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—532no—
323C22×C4—531no—
324C22×C4—531no—
325C22×C4—532no—
326C22×C4—532no—
327C22×C4—532no—
328C22×C4—532no—
329C22×C4—532no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—532no—
333C22×C4—1062no—
334C22×C4—1061no—
335C22×C4—1061no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—1061no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4C4×#615712yes—
345C22×C4C22⋊#615711yescharacteristic
346C22×C4C22⋊#615711yescharacteristic
347M16—538no—
348M16—1064no—
349M16—1064no—
350M16—1064no—
351M16—1064no—
352M16—1064no—
353M16—2122no—
354M16—2122no—
355C2×C8—1064no—
356C2×C8—1064no—
357C2×C8—1064no—
358C2×C8—1064no—
359C2×C8—2122no—
360C2×C8—2122no—
361C4⋊C4—1064no—
362C4⋊C4—1064no—
363C4⋊C4—2121no—
364C4⋊C4—2122no—
365C4⋊C4—2121no—
366C4⋊C4—2122no—
367C4⋊C4—2122no—
368C4⋊C4—2122no—
369C4⋊C4—2122no—
370C42—1061no—
371C42—1061no—
372C42—1062no—
373C42—1061no—
374C42—1062no—
375C42—1061no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23C2×C4×#615711yescharacteristic
381C23C2×C22⋊#615711yescharacteristic
382C23#1313329611yescharacteristic
383C23D4×#615712yes—
384C23Q8×#615712yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—532no—
401C2×C4—532no—
402C2×C4—532no—
403C2×C4—532no—
404C2×C4—532no—
405C2×C4—532no—
406C2×C4—531no—
407C2×C4—532no—
408C2×C4—531no—
409C2×C4—532no—
410C2×C4—532no—
411C2×C4—532no—
412C2×C4—532no—
413C2×C4—532no—
414C2×C4—531no—
415C2×C4—531no—
416C2×C4—532no—
417C2×C4—1061no—
418C2×C4—1061no—
419C2×C4—1062no—
420C2×C4—1061no—
421C2×C4—1061no—
422C2×C4—1061no—
423C2×C4—4242no—
424C2×C4—2124no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2122no—
429C2×C4—2121no—
430C2×C4—2121no—
431C2×C4—2122no—
432C2×C4#1313327812yes—
433C8—1068no—
434C8—2124no—
435C22—22no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22C4×C22⋊#615711yescharacteristic
444C22C4⋊C4×#615712yes—
445C22#1681157111yescharacteristic
446C22#1681157212yes—
447C22(C2×#6157)⋊Q811yescharacteristic
448C22D53⋊#10011yescharacteristic
449C22C2×#1313327811yescharacteristic
450C22#1681160911yescharacteristic
451C22#1681161011yescharacteristic
452C4—22no—
453C4—42no—
454C4—41no—
455C4—81no—
456C4—532no—
457C4—532no—
458C4—532no—
459C4—532no—
460C4—1062no—
461C4—4241no—
462C4—2122no—
463C4—2121no—
464C2—22no—
465C2—22no—
466C2#2028989411yescharacteristic
467C2#2029046511yescharacteristic
468C2C4×#1313327811yescharacteristic
469C2#2029074111yescharacteristic
470C2#2029074211yescharacteristic
471C2#2029090811yescharacteristic
472C2#2029090911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC2×#95×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC23×Dic53C23
hypercenterC2×#95#6157
hypocenterC53#10913
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
12. #16811503C4semidirect
117. #11719940C4⋊C4semidirect
262. C53#10913semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199402
C2×#168115032
#202903782
C2×#168105202
#1091353