Finite Group Databasebeta

← #27294653

Subgroups — #27294653

Order 13568.

544 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133132C211yesmaximal, characteristic, complemented
2#20273053C211yesmaximal, characteristic, complemented
3#20272715C212yesmaximal, complemented
4#20272712C212yesmaximal, complemented
5C53×#2535C211yesmaximal, characteristic
6C2×#13133132—21no—
7#16810587—21no—
8C4×C22⋊Dic53—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×#121×C53—21no—
13C2×C4×D4×C53C411yescharacteristic, complemented
14C2×C4×D4×C53C2211yescharacteristic
15C53×#397C411yescharacteristic, complemented
16C53×#397C2211yescharacteristic
17C53×#392C411yescharacteristic, complemented
18C53×#392C2211yescharacteristic
19C53×#386—21no—
20C53×#386—21no—
21C53×#382C411yescharacteristic, complemented
22C53×#380—21no—
23C2×#11719917—41no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—41no—
26#13133132—24no—
27C22⋊#88322—22no—
28C22⋊#88322—41no—
29#13133124—41no—
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53C2×C411yescharacteristic
33C22×D4×C53D411yescharacteristic
34C23×C4×C53—21no—
35C23×C4×C53D411yescharacteristic
36C23×C4×C53D411yescharacteristic
37#121×C53—44no—
38C4×D4×C53—41no—
39C4×D4×C53—41no—
40C4×D4×C53—41no—
41C4×D4×C53—41no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53C2311yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53—21no—
59C24×C53C22⋊C411yescharacteristic
60C24×C53C2×D411yescharacteristic
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—42no—
66C2×D4×C53—42no—
67C2×D4×C53—42no—
68C2×D4×C53—42no—
69C2×D4×C53—42no—
70C2×D4×C53—81no—
71C2×D4×C53—81no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53—42no—
82C22×C4×C53—81no—
83C22×C4×C53—81no—
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22×C411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×Dic53—42no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—81no—
100C22⋊C4×C53—81no—
101C42×C53—21no—
102C42×C53—21no—
103C22⋊Dic53—24no—
104C22⋊Dic53—42no—
105C4×Dic53—82no—
106#11719917—82no—
107C2×#88322—42no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—22no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53AGL(1,ℤ/8)12yes—
126D4×C53—44no—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—84no—
130D4×C53—84no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53—84no—
154C2×C4×C53#9911yescharacteristic
155C2×C4×C53#9911yescharacteristic
156C2×C4×C53#10412yes—
157C2×Dic53—44no—
158C2×Dic53—82no—
159C2×Dic53—82no—
160#88322—82no—
161#11774—531nomaximal, Sylow
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—22no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—42no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53C2×#9911yescharacteristic
182C22×C53#41811yescharacteristic
183C22×C53C2×#10411yescharacteristic
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—22no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53—81no—
191C4×C53—81no—
192C4×C53—81no—
193Dic53—84no—
194#2535D5311yescharacteristic
195C4×#99—531no—
196#1465—531no—
197#1344—532no—
198#1343—532no—
199C2×C53—21no—
200C2×C53—21no—
201C2×C53—42no—
202C2×C53—81no—
203C2×C53—82no—
204C2×C53#150711yescharacteristic
205C2×C53#212011yescharacteristic
206C2×C53#212711yescharacteristic
207C2×#121—21no—
208C2×C4×D4Dic5311yescharacteristic, complemented
209C2×C4×D4C2×D5311yescharacteristic
210C2×#99—1061no—
211#407—1061no—
212#397Dic5311yescharacteristic, complemented
213#397C2×D5311yescharacteristic
214#392Dic5311yescharacteristic, complemented
215#392C2×D5311yescharacteristic
216#386—21no—
217#386—21no—
218#382Dic5311yescharacteristic, complemented
219#380—21no—
220C4×C22⋊C4—1061no—
221#342—532no—
222#324—1061no—
223#323—532no—
224C53#1177411yescharacteristic, complemented, Sylow
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C2×Dic5311yescharacteristic
228C22×D4C22⋊D5311yescharacteristic
229C23×C4—21no—
230C23×C4C22⋊D5311yescharacteristic
231C23×C4C22⋊D5311yescharacteristic
232C2×M16—2121no—
233#121—44no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C2×Dic5311yescharacteristic
240C2×C4⋊C4C22⋊D5311yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—1062no—
247C2×C22⋊C4C2×Dic5311yescharacteristic
248C2×C22⋊C4C22×D5311yescharacteristic
249C2×C42—2121no—
250C2×C42C2×Dic5311yescharacteristic
251#99—1064no—
252C22⋊C8—1062no—
253C22⋊C8—2121no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—2121no—
258#95C2×Dic5311yescharacteristic
259#95C2×Dic5311yescharacteristic
260C24—21no—
261C24C22⋊Dic5311yescharacteristic
262C24C2×C22⋊D5311yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—2122no—
287C22×C4C22⋊Dic5311yescharacteristic
288C22×C4C22⋊Dic5311yescharacteristic
289C22×C4C22⋊Dic5311yescharacteristic
290C22×C4C22×Dic5311yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292M16—4242no—
293C2×C8—2122no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C22⋊C4—1064no—
306C22⋊C4—2122no—
307C42—21no—
308C42—21no—
309C42—4242no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23C2×C22⋊Dic5311yescharacteristic
327C23#1313324312yes—
328D4—44no—
329D4—42no—
330D4—42no—
331D4—84no—
332D4—84no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—84no—
356C2×C4—4242no—
357C2×C4—4242no—
358C2×C4—2124no—
359C2×C4#1313313211yescharacteristic
360C2×C4#1313313211yescharacteristic
361C2×C4#1313314412yes—
362C8—4242no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22C2×#1313313211yescharacteristic
383C22#1681062811yescharacteristic
384C22C2×#1313314411yescharacteristic
385C4—22no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—4244no—
395C2—21no—
396C2—21no—
397C2—42no—
398C2—81no—
399C2—82no—
400C2#2027323111yescharacteristic
401C2#2027635611yescharacteristic
402C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
Fitting, prime radicalC53×#2535C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2535D53
hypocenterC53#11774
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C4×#13133132C2semidirect
2. #20273053C2semidirect
3. #20272715C2semidirect
4. #20272712C2semidirect
13. C2×C4×D4×C53C4semidirect
15. C53×#397C4semidirect
17. C53×#392C4semidirect
21. C53×#382C4semidirect
208. C2×C4×D4Dic53semidirect
212. #397Dic53semidirect
214. #392Dic53semidirect
218. #382Dic53semidirect
224. C53#11774semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730532
#202727152
#202727122
C53×#25352
#1177453