Finite Group Databasebeta

← #27294553

Subgroups — #27294553

Order 13568.

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

normal subgroups only

This group has 500 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
1#20274448C211yesmaximal, characteristic, complemented
2#20273049C211yesmaximal, characteristic, complemented
3#20272739C212yesmaximal, complemented
4#20272712C212yesmaximal
5C53×#2598C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810680—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×C4×D4×C53C411yescharacteristic, complemented
13C2×#117×C53C2211yescharacteristic
14C53×#400C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#394C411yescharacteristic, complemented
17C53×#392C2211yescharacteristic
18C53×#388—21no—
19C53×#386—21no—
20C53×#386—21no—
21C53×#385—21no—
22C53×#383C411yescharacteristic, complemented
23#13133176—24no—
24#13133176—42no—
25C2×C4⋊Dic53—41no—
26C2×C22⋊Dic53—22no—
27C2×C4×Dic53—41no—
28C4⋊#88322—41no—
29C22⋊#88322—22no—
30#13133128—41no—
31C22×D4×C53—21no—
32C22×D4×C53C2×C411yescharacteristic
33C23×C4×C53—21no—
34C23×C4×C53D411yescharacteristic
35C4×D4×C53—41no—
36C4×D4×C53—41no—
37#117×C53—41no—
38#117×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53C2311yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53D411yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53—21no—
59C24×C53C2×D411yescharacteristic
60C2×D4×C53—41no—
61C2×D4×C53—42no—
62C2×D4×C53—42no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—81no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×Dic53—42no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—81no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—81no—
95C22⋊C4×C53—81no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—21no—
99C42×C53—21no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102C4⋊Dic53—44no—
103C4×Dic53—44no—
104C2×#88322—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—22no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53AGL(1,ℤ/8)11yescharacteristic
118C23×C53#13711yescharacteristic
119D4×C53—42no—
120D4×C53—42no—
121D4×C53—82no—
122D4×C53—82no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
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—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
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—84no—
149C2×C4×C53#9911yescharacteristic
150C2×C4×C53#9911yescharacteristic
151C2×C4×C53#10412yes—
152C2×Dic53—44no—
153C2×Dic53—82no—
154C2×Dic53—82no—
155#88322—82no—
156#11754—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—42no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53C2×#9911yescharacteristic
174C22×C53#41711yescharacteristic
175C22×C53C2×#10411yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—22no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—81no—
184C4×C53—81no—
185C4×C53—81no—
186Dic53—84no—
187#2598D5311yescharacteristic
188#1740—531no—
189#1464—531no—
190#1352—532no—
191#1343—532no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—41no—
195C2×C53—41no—
196C2×C53—82no—
197C2×C53#150411yescharacteristic
198C2×C53#212011yescharacteristic
199C2×C53#212811yescharacteristic
200C2×C4×D4Dic5311yescharacteristic, complemented
201C2×#117—1061no—
202C2×#117C2×D5311yescharacteristic
203#432—1061no—
204#400Dic5311yescharacteristic, complemented
205#399C2×D5311yescharacteristic
206#394Dic5311yescharacteristic, complemented
207#392C2×D5311yescharacteristic
208#388—21no—
209#386—21no—
210#386—21no—
211#385—21no—
212#383Dic5311yescharacteristic, complemented
213#342—532no—
214#342—1061no—
215#324—1061no—
216#323—532no—
217C53#1175411yescharacteristic, complemented, Sylow
218C22×D4—21no—
219C22×D4C2×Dic5311yescharacteristic
220C23×C4—21no—
221C23×C4C22⋊D5311yescharacteristic
222C4×D4—41no—
223C4×D4—41no—
224#117—41no—
225#117—41no—
226#117—1064no—
227#117—2122no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—2121no—
230C2×C4⋊C4C2×Dic5311yescharacteristic
231C2×C4⋊C4C2×Dic5311yescharacteristic
232C2×C4⋊C4C22⋊D5311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—1062no—
239C2×C22⋊C4C22⋊D5311yescharacteristic
240C2×C22⋊C4C22×D5311yescharacteristic
241C2×C42—2121no—
242C2×C42C2×Dic5311yescharacteristic
243C2×C42C22⋊D5311yescharacteristic
244C4⋊C8—2121no—
245C22⋊C8—1062no—
246#97—2121no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95C2×Dic5311yescharacteristic
252#95C2×Dic5311yescharacteristic
253C24—21no—
254C24C2×C22⋊D5311yescharacteristic
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—2122no—
273C22×C4C22⋊Dic5311yescharacteristic
274C22×C4C22⋊Dic5311yescharacteristic
275C22×C4C22⋊Dic5311yescharacteristic
276C22×C4C22⋊Dic5311yescharacteristic
277C22×C4C22×Dic5311yescharacteristic
278C22×C4C2×C22⋊D5311yescharacteristic
279C2×C8—2122no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—2124no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—1064no—
294C22⋊C4—2122no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—2124no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23C2×C22⋊Dic5311yescharacteristic
312C23#1313324211yescharacteristic
313C23#1313324711yescharacteristic
314D4—42no—
315D4—42no—
316D4—82no—
317D4—82no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—84no—
344C2×C4—4242no—
345C2×C4—4242no—
346C2×C4—2124no—
347C2×C4#1313313211yescharacteristic
348C2×C4#1313313211yescharacteristic
349C2×C4#1313314412yes—
350C8—4242no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C2×#1313313211yescharacteristic
368C22#1681062411yescharacteristic
369C22C2×#1313314411yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—4244no—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—82no—
386C2#2027321911yescharacteristic
387C2#2027635611yescharacteristic
388C2#2027638711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2598C2
socleC22×C53C2×#99
prime radicalC53×#392C22
cosocleC2×C22⋊C4×C53C23
hypercenter#2598D53
hypocenterC53#11754
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2598

Complements

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

NH ≅ G/Ntype
1. #20274448C2semidirect
2. #20273049C2semidirect
3. #20272739C2semidirect
12. C2×C4×D4×C53C4semidirect
14. C53×#400C4semidirect
16. C53×#394C4semidirect
22. C53×#383C4semidirect
200. C2×C4×D4Dic53semidirect
204. #400Dic53semidirect
206. #394Dic53semidirect
212. #383Dic53semidirect
217. C53#11754semidirect

Maximal subgroups

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

subgroupindex
#202744482
#202730492
#202727392
#202727122
C53×#25982
#1175453