Finite Group Databasebeta

← #27277549

Subgroups — #27277549

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277068C211yesmaximal, characteristic
2C2×#16810347C211yesmaximal, characteristic
3#20272562C212yesmaximal, complemented
4#20272561C212yesmaximal, complemented
5C2×C53×#323C211yesmaximal, characteristic
6C22×C22⋊Dic53C411yescharacteristic
7C22×C22⋊Dic53C411yescharacteristic
8C2×C22⋊#88322C2212yes—
9C22⋊C4×Dic53—21no—
10C22⋊C4×Dic53—21no—
11C22⋊C4×Dic53—21no—
12C22⋊C4×Dic53—21no—
13C4×C22⋊Dic53C411yescharacteristic
14C4×C22⋊Dic53C411yescharacteristic
15C4×C22⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17#16810347—21no—
18#16810347—21no—
19C22×C22⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53C2212yes—
21C53×#323—21no—
22C53×#323—21no—
23C23×Dic53—21no—
24C23×Dic53—21no—
25C22×#88322—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53C814yescomplemented
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53C2×C411yescharacteristic
34C2×C22⋊Dic53C2×C411yescharacteristic
35C2×C22⋊Dic53C2×C411yescharacteristic
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53—21no—
40C22⋊#88322—22no—
41C22⋊#88322—22no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C25×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53C2311yescharacteristic
49C22×C8×C53—22no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—22no—
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—21no—
61C24×C53—21no—
62C24×C53—81no—
63C24×C53—81no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C22⋊C411yescharacteristic
66C24×C53C2×C812yes—
67C24×C53M1612yes—
68C24×C53C22×C411yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4112yes—
78C22×C4×C53#4112yes—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—24no—
84C22×Dic53—24no—
85C22×Dic53—81no—
86C22×Dic53—81no—
87C22×Dic53—81no—
88C22×Dic53—81no—
89C2×C8×C53—42no—
90C2×C8×C53—82no—
91C22⋊C4×C53—22no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—22no—
95C22⋊Dic53—24no—
96C22⋊Dic53—24no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101C22⋊Dic53C2×C812yes—
102C22⋊Dic53C2×C812yes—
103C22⋊Dic53M1612yes—
104C22⋊Dic53M1612yes—
105C4×Dic53—81no—
106C4×Dic53—81no—
107C4×Dic53—81no—
108C4×Dic53—81no—
109C2×#88322—42no—
110C2×#88322—82no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—22no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—81no—
126C23×C53—81no—
127C23×C53—81no—
128C23×C53—81no—
129C23×C53—82no—
130C23×C53—82no—
131C23×C53—82no—
132C23×C53—82no—
133C23×C53C22⋊C812yes—
134C23×C53C22⋊C812yes—
135C23×C53C2×C22⋊C411yescharacteristic
136C23×C53#11711yescharacteristic
137C23×C53#11711yescharacteristic
138C23×C53C22×C811yescharacteristic
139C23×C53C2×M1611yescharacteristic
140C23×C53C2×M1611yescharacteristic
141C23×C53C2×M1611yescharacteristic
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—81no—
151C2×C4×C53—81no—
152C2×C4×C53#12312yes—
153C2×C4×C53#12312yes—
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—84no—
167C2×Dic53—84no—
168C8×C53—82no—
169#88322—82no—
170#8734—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—22no—
178C22×C53—22no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—42no—
184C22×C53—42no—
185C22×C53—42no—
186C22×C53—42no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—82no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53#38111yescharacteristic
196C22×C53C2×C22⋊C811yescharacteristic
197C22×C53#40711yescharacteristic
198C22×C53#43111yescharacteristic
199C22×C53#43111yescharacteristic
200C22×C53#43211yescharacteristic
201C22×C53#43211yescharacteristic
202C4×C53—22no—
203C4×C53—22no—
204C4×C53—41no—
205C4×C53—41no—
206Dic53—42no—
207Dic53—42no—
208Dic53—42no—
209Dic53—42no—
210#2273—531no—
211C2×#323—531no—
212C2×#323D5311yescharacteristic
213#1282—532no—
214#1282—532no—
215C2×C53—21no—
216C2×C53—21no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—21no—
220C2×C53—21no—
221C2×C53—42no—
222C2×C53—42no—
223C2×C53#181711yescharacteristic
224C2×C53#182811yescharacteristic
225C2×C53#182911yescharacteristic
226C22×C22⋊C4—531no—
227C22×C22⋊C4—531no—
228C22×C22⋊C4C2×D5311yescharacteristic
229C2×C22⋊C8—532no—
230C2×C22⋊C8C2×D5312yes—
231C4×C22⋊C4—531no—
232C4×C22⋊C4—531no—
233C4×C22⋊C4—531no—
234C4×C22⋊C4—531no—
235C4×C22⋊C4—1061no—
236C4×C22⋊C4—1061no—
237C4×C22⋊C4—1061no—
238C4×C22⋊C4—1061no—
239#323—21no—
240#323—21no—
241#323—1061no—
242#323—1061no—
243C53#873411yescharacteristic, complemented, Sylow
244C25C4×D5311yescharacteristic
245C23×C4—1061no—
246C23×C4—1061no—
247C23×C4C4×D5311yescharacteristic
248C23×C4C22×D5311yescharacteristic
249C22×C8—22no—
250C22×C8—1062no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—531no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4—531no—
257C2×C22⋊C4—534no—
258C2×C22⋊C4—1062no—
259C2×C22⋊C4—1062no—
260C2×C22⋊C4—1062no—
261C2×C22⋊C4—1061no—
262C2×C22⋊C4—1061no—
263C2×C22⋊C4D21211yescharacteristic
264C2×C22⋊C4D21211yescharacteristic
265C2×C22⋊C4C22⋊D5311yescharacteristic
266C2×C22⋊C4C22⋊D5311yescharacteristic
267C2×C42—1061no—
268C2×C42—1061no—
269C2×C42—1061no—
270C2×C42—1061no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—1062no—
274C22⋊C8—1062no—
275#95—1061no—
276#95—1061no—
277#95—1061no—
278#95—1061no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—81no—
284C24—81no—
285C24C8×D5312yes—
286C24#1171991212yes—
287C24#1171992111yescharacteristic
288C24#1171992111yescharacteristic
289C24C2×C4×D5311yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—1064no—
295C22×C4—1064no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—1061no—
300C22×C4—4241no—
301C22×C4—4241no—
302C22×C4—4241no—
303C22×C4—4241no—
304C22×C4#1171992111yescharacteristic
305C22×C4#1171992111yescharacteristic
306C22×C4C2×D21211yescharacteristic
307C22×C4#1171994712yes—
308C22×C4#1171994712yes—
309C22×C4C2×C22⋊D5311yescharacteristic
310C2×C8—42no—
311C2×C8—82no—
312C2×C8—4242no—
313C2×C8—2122no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—532no—
319C22⋊C4—532no—
320C22⋊C4—532no—
321C22⋊C4—532no—
322C22⋊C4—1064no—
323C22⋊C4—1064no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C42—4241no—
329C42—4241no—
330C42—4241no—
331C42—4241no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—82no—
351C23—82no—
352C23—82no—
353C23—82no—
354C23#1313312912yes—
355C23#1313312912yes—
356C23C2×#1171992111yescharacteristic
357C23#1313317311yescharacteristic
358C23#1313317311yescharacteristic
359C23C2×C8×D5311yescharacteristic
360C23M16×D5311yescharacteristic
361C23M16×D5311yescharacteristic
362C23C2×#1171991211yescharacteristic
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—1062no—
374C2×C4—1062no—
375C2×C4—1062no—
376C2×C4—1062no—
377C2×C4—4244no—
378C2×C4—4244no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4—2121no—
387C2×C4#1313319812yes—
388C2×C4#1313320112yes—
389C8—82no—
390C8—4242no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22#1681048411yescharacteristic
416C22C2×#1313312911yescharacteristic
417C22#1681058611yescharacteristic
418C22#1681067311yescharacteristic
419C22#1681067311yescharacteristic
420C22#1681067711yescharacteristic
421C22#1681067711yescharacteristic
422C4—22no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—2122no—
427C4—2122no—
428C4—2122no—
429C4—2122no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—42no—
437C2—42no—
438C2#2027478111yescharacteristic
439C2#2027482111yescharacteristic
440C2#2027482411yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC2×C53×#323C2
socleC22×C53#381
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenterC2×#323D53
hypocenterC53#8734
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20272562C2semidirect
4. #20272561C2semidirect
31. C2×C22⋊Dic53C8semidirect
243. C53#8734semidirect

Maximal subgroups

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

subgroupindex
#202770682
C2×#168103472
#202725622
#202725612
C2×C53×#3232
#873453