Finite Group Databasebeta

← #27276220

Subgroups — #27276220

Order 13568.

1330 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1330 conjugacy classes of subgroups — too many to draw. Only its 147 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#20274636C211yesmaximal, characteristic, complemented
2#20273219C214yesmaximal, complemented
3#20272956C211yesmaximal, characteristic, complemented
4C53×#3484C211yesmaximal, characteristic
5C22×C22⋊Dic53—22no—
6C2×#13133134—22no—
7#16810587C2212yescomplemented
8C2×#13133124C2212yescomplemented
9#16810360—24no—
10#16810357—24no—
11C2×H(5,𝔽2)×C53C411yescharacteristic, complemented
12C23×D4×C53C2211yescharacteristic
13C53×#546—24no—
14D42×C53—24no—
15C53×#535C412yescomplemented
16C53×#534—22no—
17C53×#525C2212yes—
18C2×#120×C53—22no—
19C53×#518C411yescharacteristic, complemented
20C23×Dic53—22no—
21C2×#11719917—22no—
22C2×C22⋊Dic53—48no—
23C2×C22⋊Dic53—44no—
24#13133134—44no—
25C22⋊#88322—24no—
26#13133124—24no—
27C25×C53D412yes—
28H(5,𝔽2)×C53—44no—
29C2×#41×C53—22no—
30C2×#41×C53C2×C412yes—
31C22×D4×C53—22no—
32C22×D4×C53—24no—
33C22×D4×C53—22no—
34C22×D4×C53—24no—
35C22×D4×C53—44no—
36C22×D4×C53C2×C411yescharacteristic
37C22×D4×C53C2×C411yescharacteristic
38C22×D4×C53D412yes—
39C23×C4×C53C2311yescharacteristic
40#127×C53—22no—
41#124×C53—22no—
42#123×C53—24no—
43#123×C53—22no—
44C22⋊Q8×C53D414yes—
45#121×C53—28no—
46#121×C53—24no—
47#121×C53D414yes—
48#120×C53—28no—
49#120×C53—24no—
50#120×C53—24no—
51#120×C53—22no—
52C4×D4×C53—24no—
53C4×D4×C53—24no—
54#117×C53C2×C412yes—
55C2×C22⋊C4×C53—24no—
56C2×C22⋊C4×C53—22no—
57C24×C53—22no—
58C24×C53—24no—
59C24×C53—24no—
60C24×C53—42no—
61C24×C53—44no—
62C24×C53—44no—
63C24×C53C22⋊C412yes—
64C24×C53C22⋊C412yes—
65C24×C53C2×D411yescharacteristic
66#41×C53—44no—
67#41×C53—48no—
68C2×Q8×C53C22⋊C412yescomplemented
69C2×D4×C53—24no—
70C2×D4×C53—24no—
71C2×D4×C53—24no—
72C2×D4×C53—22no—
73C2×D4×C53—24no—
74C2×D4×C53—28no—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—24no—
78C2×D4×C53—42no—
79C2×D4×C53—42no—
80C2×D4×C53—416no—
81C2×D4×C53—48no—
82C2×D4×C53—44no—
83C2×D4×C53—44no—
84C2×D4×C53—84no—
85C2×D4×C53—84no—
86C2×D4×C53C22⋊C412yescomplemented
87C22×C4×C53—22no—
88C22×C4×C53—24no—
89C22×C4×C53—22no—
90C22×C4×C53—42no—
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D414yes—
94C22×Dic53—24no—
95C22×Dic53—44no—
96C22×Dic53—44no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53C22⋊C414yescomplemented
101C22⋊C4×C53—24no—
102C22⋊C4×C53—28no—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—24no—
105C22⋊C4×C53—24no—
106C42×C53—22no—
107C22⋊Dic53—416no—
108#11719917—44no—
109C2×#88322—42no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—22no—
118C23×C53—22no—
119C23×C53—48no—
120C23×C53—44no—
121C23×C53—44no—
122C23×C53—48no—
123C23×C53—44no—
124C23×C53—44no—
125C23×C53—44no—
126C23×C53—44no—
127C23×C53—44no—
128C23×C53—84no—
129C23×C53—82no—
130C23×C53—84no—
131C23×C53—82no—
132C23×C53—84no—
133C23×C53#9914yes—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53#12012yes—
136Q8×C53—42no—
137D4×C53—28no—
138D4×C53—48no—
139D4×C53—42no—
140D4×C53—44no—
141D4×C53—48no—
142D4×C53—44no—
143D4×C53—44no—
144D4×C53—44no—
145D4×C53—88no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—24no—
150C2×C4×C53—44no—
151C2×C4×C53—44no—
152C2×C4×C53—44no—
153C2×C4×C53—84no—
154C2×C4×C53C2×C22⋊C412yes—
155C2×C4×C53#12012yes—
156C2×Dic53—44no—
157C2×Dic53—48no—
158C2×Dic53—84no—
159C2×Dic53—82no—
160C2×Dic53—82no—
161#88322—82no—
162#8453—531nomaximal, Sylow
163C22×C53—22no—
164C22×C53—24no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—24no—
170C22×C53—44no—
171C22×C53—44no—
172C22×C53—44no—
173C22×C53—44no—
174C22×C53—44no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—82no—
178C22×C53—88no—
179C22×C53—88no—
180C22×C53—82no—
181C22×C53—164no—
182C22×C53#37911yescharacteristic
183C22×C53C2×#9912yes—
184C22×C53#45312yes—
185C22×C53#45512yes—
186C4×C53—22no—
187C4×C53—22no—
188C4×C53—44no—
189C4×C53—42no—
190Dic53—84no—
191#3484D5311yescharacteristic
192#1783—531no—
193#1504—534no—
194#1438—531no—
195C2×C53—22no—
196C2×C53—22no—
197C2×C53—22no—
198C2×C53—44no—
199C2×C53—82no—
200C2×C53—82no—
201C2×C53#178111yescharacteristic
202C2×C53#179212yes—
203C2×H(5,𝔽2)Dic5311yescharacteristic, complemented
204C23×D4C2×D5311yescharacteristic
205#546—24no—
206D42—24no—
207#535Dic5312yescomplemented
208#534—22no—
209#525C2×D5312yes—
210C2×#120—22no—
211#518Dic5311yescharacteristic, complemented
212C22×C22⋊C4—1062no—
213C2×#100—1062no—
214#407—532no—
215C2×#95—532no—
216#328—1064no—
217#327—1064no—
218C53#845311yescharacteristic, complemented, Sylow
219C25C22⋊D5312yes—
220H(5,𝔽2)—44no—
221C2×#41—22no—
222C2×#41C2×Dic5312yes—
223C22×D4—22no—
224C22×D4—24no—
225C22×D4—22no—
226C22×D4—24no—
227C22×D4—44no—
228C22×D4C2×Dic5311yescharacteristic
229C22×D4C2×Dic5311yescharacteristic
230C22×D4C22⋊D5312yes—
231C23×C4—1062no—
232C23×C4C22×D5311yescharacteristic
233C2×M16—1062no—
234#127—22no—
235#124—22no—
236#123—24no—
237#123—22no—
238C22⋊Q8C22⋊D5314yes—
239#121—28no—
240#121—24no—
241#121C22⋊D5314yes—
242#120—28no—
243#120—24no—
244#120—22no—
245#120—24no—
246C4×D4—24no—
247C4×D4—24no—
248#117C2×Dic5312yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—2128no—
252C2×C22⋊C4—2124no—
253#100—2124no—
254C22⋊C8—1064no—
255#95—1064no—
256C24—22no—
257C24—24no—
258C24—24no—
259C24—42no—
260C24—44no—
261C24—44no—
262C24C22⋊Dic5312yes—
263C24C22⋊Dic5312yes—
264C24C2×C22⋊D5311yescharacteristic
265#41—44no—
266#41—48no—
267C2×Q8C22⋊Dic5312yescomplemented
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—28no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—416no—
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—84no—
284C2×D4—84no—
285C2×D4C22⋊Dic5312yescomplemented
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—1064no—
291C22×C4—2124no—
292C22×C4—2124no—
293C22×C4—2122no—
294C22×C4—2122no—
295C22×C4C22×Dic5311yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297C22×C4C2×C22⋊D5314yes—
298M16—2124no—
299C2×C8—2122no—
300C4⋊C4—24no—
301C4⋊C4C22⋊Dic5314yescomplemented
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—22no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—21216no—
308C42—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—48no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—48no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—84no—
328C23—84no—
329C23—82no—
330C23—82no—
331C23—84no—
332C23#1313313214yes—
333C23C2×C22⋊Dic5311yescharacteristic
334C23#1313318612yes—
335Q8—42no—
336D4—28no—
337D4—48no—
338D4—42no—
339D4—44no—
340D4—48no—
341D4—44no—
342D4—44no—
343D4—44no—
344D4—88no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—84no—
353C2×C4—4244no—
354C2×C4—4242no—
355C2×C4—4242no—
356C2×C4—2128no—
357C2×C4—2124no—
358C2×C4C2×C22⋊Dic5312yes—
359C2×C4#1313318612yes—
360C8—4242no—
361C22—24no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—24no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22—44no—
374C22—44no—
375C22—82no—
376C22—88no—
377C22—88no—
378C22—82no—
379C22—164no—
380C22#1681047911yescharacteristic
381C22C2×#1313313212yes—
382C22#1681079112yes—
383C22#1681080112yes—
384C4—22no—
385C4—22no—
386C4—42no—
387C4—44no—
388C4—4244no—
389C2—22no—
390C2—22no—
391C2—22no—
392C2—44no—
393C2—82no—
394C2—82no—
395C2#2027463011yescharacteristic
396C2#2027466912yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3484C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#3484D53
hypocenterC53#8453
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3484

Complements

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

NH ≅ G/Ntype
1. #20274636C2semidirect
2. #20273219C2semidirect
3. #20272956C2semidirect
7. #16810587C22semidirect
8. C2×#13133124C22semidirect
11. C2×H(5,𝔽2)×C53C4semidirect
15. C53×#535C4semidirect
19. C53×#518C4semidirect
68. C2×Q8×C53C22⋊C4semidirect
86. C2×D4×C53C22⋊C4semidirect
100. C4⋊C4×C53C22⋊C4semidirect
203. C2×H(5,𝔽2)Dic53semidirect
207. #535Dic53semidirect
211. #518Dic53semidirect
218. C53#8453semidirect
267. C2×Q8C22⋊Dic53semidirect
285. C2×D4C22⋊Dic53semidirect
301. C4⋊C4C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
#202746362
#202732192
#202729562
C53×#34842
#845353