Finite Group Databasebeta

← #27282032

Subgroups — #27282032

Order 13568.

880 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283744C211yesmaximal, characteristic, complemented
2#20274636C212yesmaximal, complemented
3#20273053C211yesmaximal, characteristic, complemented
4#20272621C212yesmaximal, complemented
5C53×#2408C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8C2×#13133134—24no—
9#16810587C2212yes—
10#16810587C2212yes—
11C2×C22⋊#88322—21no—
12#16810349—21no—
13#16810347—21no—
14C23×D4×C53C2211yescharacteristic
15C2×#121×C53—21no—
16C2×#120×C53C412yes—
17C2×C4×D4×C53C2211yescharacteristic
18C53×#394—21no—
19C53×#392—21no—
20C53×#390—21no—
21C53×#380C2211yescharacteristic
22C53×#379C412yes—
23C2×#11719917—22no—
24C2×#11719917D414yescomplemented
25C22×#88322—22no—
26C4⋊#88322—22no—
27#13133134—48no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133128—22no—
34C25×C53C2×C412yes—
35C22×D4×C53—22no—
36C22×D4×C53—21no—
37C22×D4×C53—22no—
38C22×D4×C53—21no—
39C22×D4×C53—22no—
40C22×D4×C53—44no—
41C22×D4×C53C2×C411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44#121×C53—44no—
45#120×C53—44no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53—22no—
53C2×C22⋊C4×C53C2×C412yes—
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57#95×C53—21no—
58C24×C53—22no—
59C24×C53—24no—
60C24×C53—44no—
61C24×C53—42no—
62C24×C53—42no—
63C24×C53—82no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C22×C411yescharacteristic
66C2×D4×C53—24no—
67C2×D4×C53—28no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—41no—
71C2×D4×C53—42no—
72C2×D4×C53—44no—
73C2×D4×C53—44no—
74C2×D4×C53—48no—
75C2×D4×C53—44no—
76C2×D4×C53—44no—
77C2×D4×C53—41no—
78C2×D4×C53—42no—
79C2×D4×C53—41no—
80C2×D4×C53—82no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4112yes—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—42no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—44no—
98C22⋊C4×C53—44no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—82no—
101C42×C53—21no—
102C42×C53—21no—
103#11719917—44no—
104#11719917—44no—
105C2×#88322—24no—
106C2×#88322—42no—
107C2×#88322—82no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—24no—
113C23×C53—28no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—42no—
119C23×C53—44no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53—84no—
126C23×C53—84no—
127C23×C53—88no—
128C23×C53—88no—
129C23×C53#10012yescomplemented
130C23×C53C2×C22⋊C411yescharacteristic
131C23×C53C4×D412yes—
132D4×C53—48no—
133D4×C53—44no—
134D4×C53—44no—
135D4×C53—44no—
136D4×C53—88no—
137C2×C4×C53—22no—
138C2×C4×C53—24no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—81no—
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12112yes—
155C2×C4×C53#12311yescharacteristic
156#88322—44no—
157#88322—82no—
158#9503—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—44no—
169C22×C53—44no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—44no—
173C22×C53—81no—
174C22×C53—82no—
175C22×C53—82no—
176C22×C53—88no—
177C22×C53—88no—
178C22×C53—82no—
179C22×C53—82no—
180C22×C53#38611yescharacteristic
181C22×C53C2×#10011yescharacteristic
182C22×C53#41311yescharacteristic
183C4×C53—22no—
184C4×C53—21no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—82no—
189C4×C53B(4,𝔽2)12yes—
190#2973—531no—
191#2408D5311yescharacteristic
192#1783—532no—
193#1465—531no—
194#1305—532no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—42no—
198C2×C53—42no—
199C2×C53—84no—
200C2×C53#188911yescharacteristic
201C2×C53#190011yescharacteristic
202C2×C53#191411yescharacteristic
203C23×D4C2×D5311yescharacteristic
204C2×#121—21no—
205C2×#120Dic5312yes—
206C2×C4×D4C2×D5311yescharacteristic
207#435—1064no—
208#432—1061no—
209C2×#100—1064no—
210#407—532no—
211#407—532no—
212C2×C22⋊C8—1061no—
213#394—21no—
214#392—21no—
215#390—21no—
216#380C2×D5311yescharacteristic
217#379Dic5312yes—
218#324—1061no—
219#323—1061no—
220C53#950311yescharacteristic, complemented, Sylow
221C25C2×Dic5312yes—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4—44no—
228C22×D4C2×Dic5311yescharacteristic
229C23×C4C22⋊D5311yescharacteristic
230C23×C4C22×D5311yescharacteristic
231C2×M16—534no—
232C2×M16—1062no—
233C22×C8—1062no—
234#121—44no—
235#120—44no—
236C4×D4—22no—
237C4×D4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22⋊D5311yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4C2×Dic5311yescharacteristic
244C2×C22⋊C4C2×Dic5312yes—
245C2×C22⋊C4C22⋊D5311yescharacteristic
246C2×C42C22⋊D5311yescharacteristic
247C4⋊C8—1062no—
248#100—2128no—
249C22⋊C8—1062no—
250C22⋊C8—1062no—
251C22⋊C8—1062no—
252C22⋊C8—1062no—
253C22⋊C8—1062no—
254#97—1062no—
255#95—21no—
256C24—22no—
257C24—24no—
258C24—44no—
259C24—42no—
260C24—42no—
261C24—82no—
262C24C22⋊Dic5311yescharacteristic
263C24C22×Dic5311yescharacteristic
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—82no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C22⋊Dic5311yescharacteristic
285C22×C4C22⋊Dic5311yescharacteristic
286C22×C4C22⋊Dic5311yescharacteristic
287C22×C4D4×D5312yes—
288C22×C4#1171994712yes—
289C22×C4C2×C22⋊D5311yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291M16—2124no—
292M16—2124no—
293C2×C8—1064no—
294C2×C8—4242no—
295C2×C8—2122no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—82no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—24no—
311C23—28no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—84no—
324C23—84no—
325C23—88no—
326C23—88no—
327C23#1313313412yescomplemented
328C23C2×C22⋊Dic5311yescharacteristic
329C23D4×Dic5312yes—
330D4—48no—
331D4—44no—
332D4—44no—
333D4—44no—
334D4—88no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4#1313318511yescharacteristic
352C2×C4#1313318912yes—
353C2×C4#1313319811yescharacteristic
354C8—4242no—
355C8—2124no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—44no—
366C22—44no—
367C22—41no—
368C22—41no—
369C22—44no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—88no—
374C22—88no—
375C22—82no—
376C22—82no—
377C22#1681050311yescharacteristic
378C22C2×#1313313411yescharacteristic
379C22#1681060711yescharacteristic
380C4—22no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4#1681080912yes—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—84no—
392C2#2027512411yescharacteristic
393C2#2027517911yescharacteristic
394C2#2027526311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2408C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#2408D53
hypocenterC53#9503
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #20283744C2semidirect
2. #20274636C2semidirect
3. #20273053C2semidirect
4. #20272621C2semidirect
24. C2×#11719917D4semidirect
129. C23×C53#100semidirect
220. C53#9503semidirect
327. C23#13133134semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202746362
#202730532
#202726212
C53×#24082
#950353