Finite Group Databasebeta

← #27274607

Subgroups — #27274607

Order 13568.

698 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719919C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic
3C2×#16810348C212yesmaximal
4#20272557C212yesmaximal
5C53×#1749C211yesmaximal, characteristic
6C22×#11719919C411yescharacteristic
7C22×#11719919C2211yescharacteristic
8C22×C4×Dic53C411yescharacteristic
9C22×C4×Dic53C411yescharacteristic
10C2×C22⋊#88322C2212yes—
11C2×#13133128—21no—
12C4×#11719919—21no—
13C4×#11719919—21no—
14C4×#11719919—21no—
15C4×#11719919—21no—
16C2×#13133124C411yescharacteristic
17C2×#13133124C2211yescharacteristic
18#16810378—21no—
19#16810348C418yescomplemented
20C22×C42×C53C2211yescharacteristic
21C2×C22⋊C8×C53C2212yes—
22C2×#97×C53—21no—
23C53×#336—21no—
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28C22×#88322—22no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919C2×C414yes—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—22no—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—42no—
44C22⋊#88322—22no—
45C22⋊#88322C2×C414yes—
46#13133128—42no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C23×C4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C22×C8×C53—22no—
55C2×C42×C53—42no—
56C2×C42×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C22⋊C8×C53—22no—
61C22⋊C8×C53C2×C414yes—
62#97×C53—42no—
63C24×C53C22×C411yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C412yes—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4112yes—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C22×Dic53—41no—
95C22×Dic53—41no—
96C22×Dic53—41no—
97C22×Dic53—41no—
98C22×Dic53C4212yes—
99C22×Dic53C4212yes—
100C22×Dic53M1612yes—
101C22×Dic53M1612yes—
102C22×Dic53M1612yes—
103C22×Dic53M1612yes—
104C2×C8×C53—24no—
105C2×C8×C53—42no—
106C2×C8×C53—42no—
107C42×C53—22no—
108C42×C53—22no—
109C42×C53—41no—
110C42×C53—41no—
111#11719919—24no—
112#11719919—24no—
113#11719919—24no—
114#11719919—24no—
115C4×Dic53—24no—
116C4×Dic53—24no—
117C4×Dic53—42no—
118C4×Dic53—42no—
119C2×#88322—24no—
120C2×#88322—42no—
121C2×#88322—42no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53C2×C4211yescharacteristic
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#11711yescharacteristic
129C23×C53C2×M1611yescharacteristic
130C23×C53C2×M1611yescharacteristic
131C23×C53C2×M1611yescharacteristic
132C23×C53C2×M1611yescharacteristic
133C2×C4×C53—24no—
134C2×C4×C53—24no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53#9912yes—
154C2×C4×C53#10112yes—
155C2×C4×C53C4×D412yes—
156C2×C4×C53C4×D412yes—
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—22no—
162C2×Dic53—22no—
163C2×Dic53—22no—
164C2×Dic53—22no—
165C2×Dic53—22no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53#9714yescomplemented
179C8×C53—44no—
180#88322—44no—
181#8153—531nomaximal, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53C4×C22⋊C411yescharacteristic
195C22×C53C2×#9711yescharacteristic
196C22×C53C4×M1611yescharacteristic
197C22×C53#40711yescharacteristic
198C22×C53#40711yescharacteristic
199C22×C53C2×#9911yescharacteristic
200C22×C53#41011yescharacteristic
201C22×C53C2×#10111yescharacteristic
202C22×C53#41311yescharacteristic
203C22×C53#43211yescharacteristic
204C22×C53#43211yescharacteristic
205C4×C53—22no—
206C4×C53—22no—
207C4×C53—22no—
208C4×C53—22no—
209C4×C53—41no—
210C4×C53—41no—
211Dic53—24no—
212Dic53—42no—
213Dic53—42no—
214Dic53—42no—
215C2×C4×C4⋊C4—531no—
216#1749—531no—
217#1749D5311yescharacteristic
218C2×#324—532no—
219#1280—532no—
220C2×C53—21no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53#146311yescharacteristic
225C2×C53#146411yescharacteristic
226C2×C53#146511yescharacteristic
227C2×C53#146711yescharacteristic
228C2×C53#174911yescharacteristic
229C2×C53C4×#9911yescharacteristic
230C2×C53C4×#10111yescharacteristic
231C22×C4⋊C4—531no—
232C22×C4⋊C4—531no—
233C22×C42—531no—
234C22×C42—531no—
235C22×C42C2×D5311yescharacteristic
236C2×C22⋊C8—532no—
237C2×C22⋊C8C2×D5312yes—
238C2×#97—21no—
239C2×#97—1061no—
240C4×C4⋊C4—1061no—
241C4×C4⋊C4—1061no—
242C4×C4⋊C4—1061no—
243C4×C4⋊C4—1061no—
244C2×#95—531no—
245C2×#95—531no—
246#336—21no—
247#336—1061no—
248#324—538no—
249C53#815311yescharacteristic, complemented, Sylow
250C23×C4—531no—
251C23×C4—531no—
252C23×C4—531no—
253C23×C4—531no—
254C23×C4C4×D5311yescharacteristic
255C23×C4C4×D5311yescharacteristic
256C23×C4C22×D5311yescharacteristic
257C22×C8—22no—
258C22×C8—1062no—
259C2×C4⋊C4—534no—
260C2×C4⋊C4—1062no—
261C2×C4⋊C4—1062no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1062no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4—1061no—
267C2×C42—42no—
268C2×C42—1062no—
269C2×C42—1062no—
270C2×C42—1061no—
271C2×C42—1061no—
272C2×C42—1061no—
273C2×C42—1061no—
274C2×C42—2122no—
275C2×C42D21211yescharacteristic
276C2×C42D21211yescharacteristic
277C2×C42C22⋊D5311yescharacteristic
278C2×C42C22⋊D5311yescharacteristic
279C22⋊C8—22no—
280C22⋊C8—534no—
281C22⋊C8—1062no—
282C22⋊C8C4×D5314yes—
283#97—42no—
284#97—2122no—
285#95—1061no—
286#95—1061no—
287#95—1061no—
288#95—1061no—
289C24C2×C4×D5311yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—532no—
301C22×C4—532no—
302C22×C4—532no—
303C22×C4—532no—
304C22×C4—532no—
305C22×C4—532no—
306C22×C4—1061no—
307C22×C4—1061no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—2121no—
317C22×C4—2121no—
318C22×C4—2121no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4—2121no—
322C22×C4#1171992111yescharacteristic
323C22×C4#1171992111yescharacteristic
324C22×C4#1171992111yescharacteristic
325C22×C4#1171992111yescharacteristic
326C22×C4C2×C4×D5312yes—
327C22×C4C2×D21211yescharacteristic
328C22×C4#1171994512yes—
329C22×C4C2×C22⋊D5311yescharacteristic
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—1064no—
334C2×C8—2122no—
335C2×C8—2122no—
336C4⋊C4—1064no—
337C4⋊C4—1064no—
338C4⋊C4—1064no—
339C4⋊C4—1064no—
340C42—22no—
341C42—22no—
342C42—41no—
343C42—41no—
344C42—1064no—
345C42—1064no—
346C42—2122no—
347C42—2122no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23C42×D5311yescharacteristic
353C23C2×#1171992111yescharacteristic
354C23#1313317411yescharacteristic
355C23M16×D5311yescharacteristic
356C23M16×D5311yescharacteristic
357C23M16×D5311yescharacteristic
358C23M16×D5311yescharacteristic
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—534no—
380C2×C4—1062no—
381C2×C4—1062no—
382C2×C4—1062no—
383C2×C4—1062no—
384C2×C4—1062no—
385C2×C4—1062no—
386C2×C4—1062no—
387C2×C4—1062no—
388C2×C4—1062no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4—2122no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4#1313313112yes—
402C2×C4#1313313512yes—
403C2×C4C4×C22⋊D5312yes—
404C2×C4C4×D21212yes—
405C8—44no—
406C8—2124no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22C4×#1171992111yescharacteristic
420C22#97×D5311yescharacteristic
421C22#1681057511yescharacteristic
422C22#1681058611yescharacteristic
423C22#1681058611yescharacteristic
424C22C2×#1313313111yescharacteristic
425C22#1681059611yescharacteristic
426C22C2×#1313313511yescharacteristic
427C22#1681060511yescharacteristic
428C22#1681067811yescharacteristic
429C22#1681067811yescharacteristic
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—1064no—
437C4—2122no—
438C4—2122no—
439C4—2122no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2#2027304411yescharacteristic
445C2#2027304711yescharacteristic
446C2#2027305111yescharacteristic
447C2#2027305811yescharacteristic
448C2#2027447811yescharacteristic
449C2C4×#1313313111yescharacteristic
450C2C4×#1313313511yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1749C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC23×C4×C53C23
hypercenter#1749D53
hypocenterC53#8153
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
19. #16810348C4semidirect
178. C2×Dic53#97semidirect
249. C53#8153semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199192
#202744772
C2×#168103482
#202725572
C53×#17492
#815353