Finite Group Databasebeta

← #27276771

Subgroups — #27276771

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810507C211yesmaximal, characteristic
2#20274695C211yesmaximal, characteristic
3C2×#16810348C212yesmaximal
4#20272557C212yesmaximal
5C53×#1797C211yesmaximal, characteristic
6C22×#11719919C411yescharacteristic
7C22×#11719919C2211yescharacteristic
8C2×C4⋊#88322—21no—
9C2×C22⋊#88322C2212yes—
10#16810507—21no—
11#16810507—21no—
12#16810507—21no—
13#16810507—21no—
14C2×#13133124C411yescharacteristic
15C2×#13133124C411yescharacteristic
16C2×#13133124C411yescharacteristic
17C2×#13133124C2211yescharacteristic
18#16810378—21no—
19#16810348—24no—
20C22×C42×C53C2211yescharacteristic
21C2×C4⋊C8×C53—21no—
22C2×C22⋊C8×C53C2212yes—
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—21no—
34C2×#11719919—21no—
35C2×#11719919—42no—
36C2×#11719919D412yes—
37C2×#11719919Q812yes—
38C4⋊#88322—42no—
39C22⋊#88322—22no—
40C22⋊#88322—22no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53C2311yescharacteristic
52C22×C8×C53—22no—
53C2×C42×C53—42no—
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C4⋊C8×C53—42no—
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C24×C53C22×C411yescharacteristic
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×Q811yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—22no—
89C22×Dic53—22no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C22×Dic53—41no—
95C22×Dic53—41no—
96C22×Dic53C4⋊C412yes—
97C22×Dic53C4⋊C412yes—
98C2×C8×C53—42no—
99C2×C8×C53—42no—
100C2×C8×C53—42no—
101C42×C53—22no—
102C42×C53—22no—
103C42×C53—41no—
104C42×C53—41no—
105#11719919—22no—
106#11719919—22no—
107#11719919—22no—
108#11719919—22no—
109#11719919—42no—
110#11719919—42no—
111C2×#88322—42no—
112C2×#88322—42no—
113C2×#88322—42no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53C2×C22⋊C411yescharacteristic
119C23×C53C2×C4⋊C411yescharacteristic
120C23×C53#11711yescharacteristic
121C23×C53#13112yes—
122C23×C53#13112yes—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—44no—
142C2×C4×C53—44no—
143C2×C4×C53#9912yes—
144C2×C4×C53#10112yes—
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53C22⋊Q811yescharacteristic
148C2×C4×C53C22⋊Q811yescharacteristic
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—24no—
152C2×Dic53—42no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—82no—
166C2×Dic53—82no—
167C8×C53—82no—
168#88322—82no—
169#8562—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—22no—
178C22×C53#38011yescharacteristic
179C22×C53#40811yescharacteristic
180C22×C53#40811yescharacteristic
181C22×C53C2×#9911yescharacteristic
182C22×C53#41011yescharacteristic
183C22×C53C2×#10111yescharacteristic
184C22×C53#41311yescharacteristic
185C22×C53#42412yes—
186C22×C53#43312yes—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—22no—
192C4×C53—22no—
193C4×C53—41no—
194C4×C53—41no—
195Dic53—42no—
196Dic53—42no—
197Dic53—81no—
198Dic53—81no—
199C2×#387—531no—
200#1797—531no—
201#1797D5311yescharacteristic
202C2×#324—532no—
203#1280—532no—
204C2×C53—22no—
205C2×C53—22no—
206C2×C53#146811yescharacteristic
207C2×C53#147011yescharacteristic
208C2×C53#147211yescharacteristic
209C2×C53#147311yescharacteristic
210C2×C53#179811yescharacteristic
211C2×C53#179911yescharacteristic
212C2×C53#180111yescharacteristic
213C22×C4⋊C4—531no—
214C22×C4⋊C4—531no—
215C22×C42C2×D5311yescharacteristic
216C2×C4⋊C8—21no—
217C2×C4⋊C8—1061no—
218C2×C22⋊C8—532no—
219C2×C22⋊C8C2×D5312yes—
220#387—1061no—
221#387—1061no—
222#387—1061no—
223#387—1061no—
224C2×#95—531no—
225C2×#95—531no—
226C2×#95—531no—
227C2×#95—531no—
228#336—21no—
229#336—1061no—
230#324—1064no—
231C53#856211yescharacteristic, complemented, Sylow
232C23×C4—531no—
233C23×C4—531no—
234C23×C4—531no—
235C23×C4—531no—
236C23×C4C4×D5311yescharacteristic
237C23×C4C4×D5311yescharacteristic
238C23×C4C22×D5311yescharacteristic
239C22×C8—22no—
240C22×C8—1062no—
241C2×C4⋊C4—532no—
242C2×C4⋊C4—532no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—2122no—
250C2×C42—42no—
251C2×C42D21211yescharacteristic
252C2×C42D21211yescharacteristic
253C2×C42C22⋊D5311yescharacteristic
254C2×C42C22⋊D5311yescharacteristic
255C4⋊C8—42no—
256C4⋊C8—2122no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—1062no—
260C22⋊C8—1062no—
261#95—1061no—
262#95—1061no—
263#95—1061no—
264#95—1061no—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268#95—1061no—
269C24C2×C4×D5311yescharacteristic
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—532no—
279C22×C4—532no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1062no—
285C22×C4—1062no—
286C22×C4—1062no—
287C22×C4—1062no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4#1171992111yescharacteristic
297C22×C4#1171992111yescharacteristic
298C22×C4#1171992111yescharacteristic
299C22×C4#1171992111yescharacteristic
300C22×C4C2×D21211yescharacteristic
301C22×C4D4×D5311yescharacteristic
302C22×C4#1171994711yescharacteristic
303C22×C4Q8×D5311yescharacteristic
304C22×C4#1171994911yescharacteristic
305C22×C4C2×C22⋊D5311yescharacteristic
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—2122no—
310C2×C8—2122no—
311C2×C8—2122no—
312C4⋊C4—1062no—
313C4⋊C4—1062no—
314C4⋊C4—1062no—
315C4⋊C4—1062no—
316C4⋊C4—2122no—
317C4⋊C4—2122no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42—41no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23C2×#1171992111yescharacteristic
327C23C4⋊C4×D5311yescharacteristic
328C23#1313317511yescharacteristic
329C23#1313322312yes—
330C23#1313322512yes—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—1064no—
352C2×C4—1062no—
353C2×C4—1062no—
354C2×C4—4242no—
355C2×C4—4242no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4#1313313112yes—
370C2×C4#1313313512yes—
371C2×C4#1313318811yescharacteristic
372C2×C4#1313319211yescharacteristic
373C2×C4#1313319311yescharacteristic
374C2×C4#1313319711yescharacteristic
375C8—82no—
376C8—4242no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22#1681048011yescharacteristic
386C22#1681058911yescharacteristic
387C22#1681059111yescharacteristic
388C22C2×#1313313111yescharacteristic
389C22#1681059611yescharacteristic
390C22C2×#1313313511yescharacteristic
391C22#1681060511yescharacteristic
392C22#1681064812yes—
393C22#1681068312yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—4241no—
403C4—4241no—
404C4—2122no—
405C4—2122no—
406C2—22no—
407C2—22no—
408C2#2027306411yescharacteristic
409C2#2027307211yescharacteristic
410C2#2027307811yescharacteristic
411C2#2027308211yescharacteristic
412C2#2027469611yescharacteristic
413C2#2027470011yescharacteristic
414C2#2027470811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenter#1797D53
hypocenterC53#8562
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
231. C53#8562semidirect

Maximal subgroups

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

subgroupindex
C2×#168105072
#202746952
C2×#168103482
#202725572
C53×#17972
#856253