Finite Group Databasebeta

← #27281892

Subgroups — #27281892

Order 13568.

958 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277068C211yesmaximal, characteristic, complemented
2#20274659C212yesmaximal, complemented
3#20273042C211yesmaximal, characteristic, complemented
4#20272567C212yesmaximal, complemented
5C53×#2404C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7C2×#13133144—24no—
8#16810587C2212yes—
9C22⋊C4×Dic53—24no—
10C4×C22⋊Dic53—21no—
11C4×C22⋊Dic53C2212yes—
12#16810347—22no—
13C22×#41×C53C2211yescharacteristic
14C2×#120×C53—21no—
15C22×C22⋊C4×C53C2211yescharacteristic
16C53×#393C412yescomplemented
17C53×#392C412yescomplemented
18C53×#386—21no—
19C53×#386—22no—
20C53×#381C2211yescharacteristic
21C23×Dic53—22no—
22C2×#11719917—22no—
23C2×C22⋊Dic53—28no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—22no—
27C2×C4×Dic53D414yescomplemented
28#13133144—48no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124—22no—
33C25×C53D411yescharacteristic
34C2×#41×C53—22no—
35C2×#41×C53—22no—
36C2×#41×C53—44no—
37C22×Q8×C53C2×C411yescharacteristic, complemented
38C22×D4×C53—21no—
39C22×D4×C53—22no—
40C22×D4×C53C2×C411yescharacteristic, complemented
41C23×C4×C53—21no—
42C23×C4×C53D411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44#120×C53—44no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53C2×C412yes—
51C2×C22⋊C4×C53D412yes—
52#95×C53—22no—
53#95×C53—21no—
54#95×C53C2×C412yescomplemented
55C24×C53—21no—
56C24×C53—24no—
57C24×C53—21no—
58C24×C53—21no—
59C24×C53—42no—
60C24×C53—81no—
61C24×C53C22⋊C412yes—
62C24×C53C2×D411yescharacteristic
63#41×C53—28no—
64#41×C53—44no—
65#41×C53—44no—
66#41×C53—44no—
67C2×Q8×C53—22no—
68C2×Q8×C53—42no—
69C2×D4×C53—24no—
70C2×D4×C53—22no—
71C2×D4×C53—42no—
72C2×D4×C53—42no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—44no—
76C2×D4×C53—44no—
77C2×D4×C53—82no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53—81no—
87C22×C4×C53C22⋊C412yes—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4112yes—
92C22×Dic53—24no—
93C22×Dic53—28no—
94C22×Dic53—82no—
95C22×Dic53—82no—
96C22⋊C4×C53—24no—
97C22⋊C4×C53—24no—
98C22⋊C4×C53—44no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—82no—
101C22⋊Dic53—28no—
102C22⋊Dic53—28no—
103C4×Dic53—44no—
104C4×Dic53—82no—
105#11719917—44no—
106C2×#88322—42no—
107C23×C53—22no—
108C23×C53—22no—
109C23×C53—24no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—41no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—41no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—41no—
123C23×C53—42no—
124C23×C53—81no—
125C23×C53—81no—
126C23×C53—88no—
127C23×C53#10414yescomplemented
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53#12011yescharacteristic
130C23×C53#12311yescharacteristic
131Q8×C53—24no—
132Q8×C53—42no—
133D4×C53—24no—
134D4×C53—42no—
135D4×C53—44no—
136D4×C53—44no—
137D4×C53—88no—
138C2×C4×C53—24no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—48no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—81no—
153C2×C4×C53C4×D412yes—
154C2×C4×C53#12112yes—
155C2×Dic53—28no—
156C2×Dic53—44no—
157C2×Dic53—44no—
158C2×Dic53—88no—
159#88322—82no—
160#9475—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—24no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—44no—
170C22×C53—44no—
171C22×C53—44no—
172C22×C53—44no—
173C22×C53—44no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—82no—
178C22×C53—82no—
179C22×C53—84no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53#38611yescharacteristic
183C22×C53C2×#10412yes—
184C22×C53B(4,𝔽2)12yes—
185C4×C53—22no—
186C4×C53—22no—
187C4×C53—42no—
188C4×C53—42no—
189C4×C53—82no—
190Dic53—48no—
191#2404D5311yescharacteristic
192#2273—531no—
193#1789—532no—
194#1462—531no—
195#1284—532no—
196C2×C53—22no—
197C2×C53—22no—
198C2×C53—41no—
199C2×C53—44no—
200C2×C53—82no—
201C2×C53#188911yescharacteristic
202C2×C53#189611yescharacteristic
203C2×C53#191111yescharacteristic
204C22×#41C2×D5311yescharacteristic
205C2×#120—21no—
206C22×C22⋊C4—1061no—
207C22×C22⋊C4C2×D5311yescharacteristic
208C2×#104—1064no—
209#407—532no—
210#393Dic5312yescomplemented
211#392Dic5312yescomplemented
212#386—21no—
213#386—22no—
214#381C2×D5311yescharacteristic
215C4×C22⋊C4—532no—
216C4×C22⋊C4—1064no—
217C4×C22⋊C4—1061no—
218#323—1062no—
219C53#947511yescharacteristic, complemented, Sylow
220C25C22⋊D5311yescharacteristic
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—44no—
224C22×Q8C2×Dic5311yescharacteristic, complemented
225C22×D4—21no—
226C22×D4—22no—
227C22×D4C2×Dic5311yescharacteristic, complemented
228C23×C4—21no—
229C23×C4—1062no—
230C23×C4C22⋊D5311yescharacteristic
231C23×C4C22×D5311yescharacteristic
232C2×M16—1062no—
233#120—44no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—1068no—
240C2×C22⋊C4—1062no—
241C2×C22⋊C4—1062no—
242C2×C22⋊C4C2×Dic5312yes—
243C2×C22⋊C4C22⋊D5312yes—
244C2×C42—534no—
245C2×C42—1062no—
246#104—2128no—
247C22⋊C8—1062no—
248C22⋊C8—1062no—
249#95—21no—
250#95—22no—
251#95—1062no—
252#95—1062no—
253#95C2×Dic5312yescomplemented
254C24—21no—
255C24—24no—
256C24—21no—
257C24—21no—
258C24—42no—
259C24—81no—
260C24C22⋊Dic5312yes—
261C24C2×C22⋊D5311yescharacteristic
262#41—28no—
263#41—44no—
264#41—44no—
265#41—44no—
266C2×Q8—22no—
267C2×Q8—42no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—82no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—81no—
286C22×C4—1068no—
287C22×C4—1064no—
288C22×C4—4242no—
289C22×C4—4242no—
290C22×C4C22⋊Dic5312yes—
291C22×C4D4×D5312yes—
292C22×C4#1171994712yes—
293C22×C4C22×Dic5311yescharacteristic
294C22×C4C2×C22⋊D5311yescharacteristic
295M16—2124no—
296C2×C8—2122no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—44no—
300C22⋊C4—42no—
301C22⋊C4—82no—
302C22⋊C4—1068no—
303C22⋊C4—1068no—
304C42—4242no—
305C42—2124no—
306C23—22no—
307C23—22no—
308C23—24no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—42no—
321C23—41no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—88no—
326C23#1313314414yescomplemented
327C23C2×C22⋊Dic5311yescharacteristic
328C23#1313318511yescharacteristic
329C23#1313319811yescharacteristic
330Q8—24no—
331Q8—42no—
332D4—24no—
333D4—42no—
334D4—44no—
335D4—44no—
336D4—88no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—1068no—
353C2×C4—4248no—
354C2×C4—2124no—
355C2×C4—2124no—
356C2×C4D4×Dic5312yes—
357C2×C4#1313318912yes—
358C8—4242no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—24no—
366C22—21no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—82no—
376C22—82no—
377C22—84no—
378C22—81no—
379C22—81no—
380C22#1681050311yescharacteristic
381C22C2×#1313314412yes—
382C22#1681080912yes—
383C4—22no—
384C4—22no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—2128no—
389C2—22no—
390C2—22no—
391C2—41no—
392C2—44no—
393C2—82no—
394C2#2027512411yescharacteristic
395C2#2027515911yescharacteristic
396C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2404C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#2404D53
hypocenterC53#9475
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. #20277068C2semidirect
2. #20274659C2semidirect
3. #20273042C2semidirect
4. #20272567C2semidirect
16. C53×#393C4semidirect
17. C53×#392C4semidirect
27. C2×C4×Dic53D4semidirect
37. C22×Q8×C53C2×C4semidirect
40. C22×D4×C53C2×C4semidirect
54. #95×C53C2×C4semidirect
127. C23×C53#104semidirect
210. #393Dic53semidirect
211. #392Dic53semidirect
219. C53#9475semidirect
224. C22×Q8C2×Dic53semidirect
227. C22×D4C2×Dic53semidirect
253. #95C2×Dic53semidirect
326. C23#13133144semidirect

Maximal subgroups

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

subgroupindex
#202770682
#202746592
#202730422
#202725672
C53×#24042
#947553