Finite Group Databasebeta

← #27294573

Subgroups — #27294573

Order 13568.

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

normal subgroups only

This group has 486 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#20272736C212yesmaximal, complemented
4#20272712C212yesmaximal
5C53×#2610C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810680—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×#124×C53C411yescharacteristic, complemented
13C2×#117×C53C2211yescharacteristic
14C53×#395C411yescharacteristic, complemented
15C53×#394C411yescharacteristic, complemented
16C53×#393C2211yescharacteristic
17C53×#392C2211yescharacteristic
18C53×#390—21no—
19C53×#388—21no—
20C53×#388—21no—
21C53×#386—21no—
22C53×#376C411yescharacteristic, 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×Q8×C53C2×C411yescharacteristic
32C22×D4×C53—21no—
33C22×D4×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35#124×C53—41no—
36#124×C53—41no—
37#117×C53—41no—
38#117×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53—21no—
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×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
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×Q8×C53—41no—
61C2×Q8×C53—81no—
62C2×D4×C53—42no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C2×D4×C53—81no—
67C2×D4×C53—81no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—81no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—81no—
94C22⋊C4×C53—81no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—21no—
99C42×C53—42no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102C4⋊Dic53—44no—
103C4×Dic53—44no—
104C2×#88322—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—41no—
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
119Q8×C53—82no—
120D4×C53—82no—
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—22no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
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#9911yescharacteristic
149C2×C4×C53#9911yescharacteristic
150C2×C4×C53#10412yes—
151C2×Dic53—44no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154#88322—82no—
155#11758—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53C2×#9911yescharacteristic
173C22×C53#41711yescharacteristic
174C22×C53C2×#10411yescharacteristic
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—22no—
178C4×C53—42no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—81no—
184C4×C53—81no—
185Dic53—84no—
186#2610D5311yescharacteristic
187#1740—531no—
188#1464—531no—
189#1351—532no—
190#1343—532no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53—161no—
196C2×C53#150411yescharacteristic
197C2×C53#212211yescharacteristic
198C2×C53#212611yescharacteristic
199C2×#124Dic5311yescharacteristic, complemented
200C2×#117—1061no—
201C2×#117C2×D5311yescharacteristic
202#432—1061no—
203#395Dic5311yescharacteristic, complemented
204#394Dic5311yescharacteristic, complemented
205#393C2×D5311yescharacteristic
206#392C2×D5311yescharacteristic
207#390—21no—
208#388—21no—
209#388—21no—
210#386—21no—
211#376Dic5311yescharacteristic, complemented
212#342—532no—
213#342—1061no—
214#324—1061no—
215#323—532no—
216C53#1175811yescharacteristic, complemented, Sylow
217C22×Q8C2×Dic5311yescharacteristic
218C22×D4—21no—
219C22×D4C2×Dic5311yescharacteristic
220C23×C4C22⋊D5311yescharacteristic
221#124—41no—
222#124—41no—
223#117—41no—
224#117—41no—
225#117—1064no—
226#117—2122no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—2121no—
229C2×C4⋊C4C2×Dic5311yescharacteristic
230C2×C4⋊C4C22⋊D5311yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—1062no—
238C2×C22⋊C4C22⋊D5311yescharacteristic
239C2×C22⋊C4C22×D5311yescharacteristic
240C2×C42—21no—
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#95C2×Dic5311yescharacteristic
251#95C2×Dic5311yescharacteristic
252C24—21no—
253C24C2×C22⋊D5311yescharacteristic
254C2×Q8—41no—
255C2×Q8—81no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C2×D4—81no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—2122no—
271C22×C4C22⋊Dic5311yescharacteristic
272C22×C4C22⋊Dic5311yescharacteristic
273C22×C4C22⋊Dic5311yescharacteristic
274C22×C4C22⋊Dic5311yescharacteristic
275C22×C4C22×Dic5311yescharacteristic
276C22×C4C2×C22⋊D5311yescharacteristic
277C2×C8—2122no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—2124no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—1064no—
292C22⋊C4—2122no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—42no—
298C42—2124no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C2×C22⋊Dic5311yescharacteristic
311C23#1313324211yescharacteristic
312C23#1313324711yescharacteristic
313Q8—82no—
314D4—82no—
315D4—82no—
316D4—82no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
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—4242no—
343C2×C4—4242no—
344C2×C4—2124no—
345C2×C4#1313313211yescharacteristic
346C2×C4#1313313211yescharacteristic
347C2×C4#1313314412yes—
348C8—4242no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C2×#1313313211yescharacteristic
366C22#1681062411yescharacteristic
367C22C2×#1313314411yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—4244no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2#2027321911yescharacteristic
385C2#2027636511yescharacteristic
386C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2610C2
socleC22×C53C2×#99
prime radicalC53×#392C22
cosocleC2×C22⋊C4×C53C23
hypercenter#2610D53
hypocenterC53#11758
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #20274448C2semidirect
2. #20273049C2semidirect
3. #20272736C2semidirect
12. C2×#124×C53C4semidirect
14. C53×#395C4semidirect
15. C53×#394C4semidirect
22. C53×#376C4semidirect
199. C2×#124Dic53semidirect
203. #395Dic53semidirect
204. #394Dic53semidirect
211. #376Dic53semidirect
216. C53#11758semidirect

Maximal subgroups

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

subgroupindex
#202744482
#202730492
#202727362
#202727122
C53×#26102
#1175853