Finite Group Databasebeta

← #27294774

Subgroups — #27294774

Order 13568.

758 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 95 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#20274795C211yesmaximal, characteristic, complemented
2#20273045C211yesmaximal, characteristic, complemented
3#20272718C214yesmaximal, complemented
4C2×C53×#394C211yesmaximal, characteristic
5#16810587—21no—
6#16810587—21no—
7#16810392—21no—
8#16810392C2212yescomplemented
9#16810349—21no—
10#16810349C2212yescomplemented
11C23×D4×C53C411yescharacteristic
12C22×C4⋊C4×C53C2211yescharacteristic
13C22×C22⋊C4×C53—21no—
14C22×C22⋊C4×C53—21no—
15C53×#394—21no—
16C53×#394—21no—
17C53×#394C412yes—
18C53×#394C2212yes—
19C2×#95×C53C411yescharacteristic
20C2×#11719917—22no—
21C2×#11719917—41no—
22C22⋊#88322—22no—
23C22⋊#88322—41no—
24C22⋊#88322—41no—
25C22⋊#88322—41no—
26C25×C53—21no—
27C22×D4×C53—21no—
28C22×D4×C53—41no—
29C22×D4×C53—41no—
30C22×D4×C53—44no—
31C22×D4×C53C2×C412yes—
32C23×C4×C53—21no—
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C2×C4⋊C4×C53—41no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—44no—
47C2×C22⋊C4×C53—44no—
48#95×C53—21no—
49#95×C53C2×C412yes—
50C24×C53—24no—
51C24×C53—22no—
52C24×C53—42no—
53C24×C53—42no—
54C24×C53—42no—
55C24×C53—42no—
56C24×C53—42no—
57C24×C53—41no—
58C24×C53—41no—
59C24×C53C22⋊C411yescharacteristic
60C2×D4×C53—42no—
61C2×D4×C53—42no—
62C2×D4×C53—44no—
63C2×D4×C53—48no—
64C2×D4×C53—44no—
65C2×D4×C53—44no—
66C2×D4×C53—44no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—41no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—44no—
93C22⋊C4×C53—44no—
94C22⋊C4×C53—44no—
95#11719917—42no—
96#11719917—42no—
97#11719917—82no—
98C2×#88322—42no—
99C2×#88322—42no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—24no—
105C23×C53—42no—
106C23×C53—44no—
107C23×C53—48no—
108C23×C53—48no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—44no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—84no—
116C23×C53—84no—
117C23×C53—82no—
118C23×C53—82no—
119C23×C53—82no—
120C23×C53—82no—
121C23×C53#9911yescharacteristic
122C23×C53#9911yescharacteristic
123C23×C53C2×C22⋊C411yescharacteristic
124D4×C53—44no—
125D4×C53—48no—
126D4×C53—44no—
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—21no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—81no—
149C2×C4×C53—84no—
150C2×C4×C53AGL(1,ℤ/8)12yes—
151#88322—82no—
152#88322—82no—
153#11802—531nomaximal, Sylow
154C22×C53—24no—
155C22×C53—24no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—44no—
163C22×C53—44no—
164C22×C53—44no—
165C22×C53—82no—
166C22×C53—82no—
167C22×C53—88no—
168C22×C53—88no—
169C22×C53—82no—
170C22×C53—82no—
171C22×C53AΣL(1,𝔽16)12yes—
172C22×C53#35512yes—
173C22×C53C2×#9911yescharacteristic
174C22×C53#41811yescharacteristic
175C22×C53#42111yescharacteristic
176C4×C53—22no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—82no—
183C2×#394D5311yescharacteristic
184#1821—531no—
185#1463—531no—
186#1345—534no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—22no—
190C2×C53—22no—
191C2×C53—84no—
192C2×C53#151011yescharacteristic
193C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
194C2×C53C2×#35511yescharacteristic
195C23×D4Dic5311yescharacteristic
196C22×C4⋊C4C2×D5311yescharacteristic
197C22×C22⋊C4—21no—
198C22×C22⋊C4—21no—
199#407—1061no—
200#407—1061no—
201#394—21no—
202#394—21no—
203#394Dic5312yes—
204#394C2×D5312yes—
205C2×#95Dic5311yescharacteristic
206#343—532no—
207#343—1061no—
208#324—532no—
209#324—1061no—
210C53#1180211yescharacteristic, complemented, Sylow
211C25—21no—
212C22×D4—21no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—44no—
216C22×D4C2×Dic5312yes—
217C23×C4—21no—
218C23×C4C2×Dic5311yescharacteristic
219C23×C4C22⋊D5311yescharacteristic
220C23×C4C22⋊D5311yescharacteristic
221C2×M16—1062no—
222C2×M16—2121no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C2×Dic5311yescharacteristic
226C2×C4⋊C4C22⋊D5311yescharacteristic
227C2×C4⋊C4C22⋊D5311yescharacteristic
228C2×C4⋊C4C22×D5311yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C22⋊C8—1062no—
236C22⋊C8—2121no—
237C22⋊C8—2121no—
238C22⋊C8—2121no—
239#95—21no—
240#95C2×Dic5312yes—
241C24—24no—
242C24—22no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24C22⋊Dic5311yescharacteristic
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—44no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C22⋊Dic5311yescharacteristic
273C22×C4C22⋊Dic5311yescharacteristic
274C22×C4C22⋊Dic5311yescharacteristic
275C22×C4C22×Dic5311yescharacteristic
276C22×C4C2×C22⋊D5311yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278M16—4242no—
279M16—2122no—
280M16—2122no—
281C2×C8—2122no—
282C2×C8—2122no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—24no—
296C23—42no—
297C23—44no—
298C23—48no—
299C23—48no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—44no—
306C23—84no—
307C23—84no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23#1313313211yescharacteristic
313C23#1313313211yescharacteristic
314C23C2×C22⋊Dic5311yescharacteristic
315D4—44no—
316D4—44no—
317D4—48no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—84no—
341C2×C4#1313324312yes—
342C8—4242no—
343C8—4242no—
344C22—24no—
345C22—24no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—82no—
356C22—82no—
357C22—88no—
358C22—88no—
359C22—82no—
360C22—82no—
361C22#1681041012yes—
362C22#1681042012yes—
363C22C2×#1313313211yescharacteristic
364C22#1681062811yescharacteristic
365C22#1681064011yescharacteristic
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—82no—
373C2—21no—
374C2—21no—
375C2—22no—
376C2—22no—
377C2—84no—
378C2#2027324411yescharacteristic
379C2C2×#1681041011yescharacteristic
380C2C2×#1681042011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#394C2
socleC22×C53C2×#99
prime radicalC23×D4×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#394D53
hypocenterC53#11802
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. #20274795C2semidirect
2. #20273045C2semidirect
3. #20272718C2semidirect
8. #16810392C22semidirect
10. #16810349C22semidirect
210. C53#11802semidirect

Maximal subgroups

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

subgroupindex
#202747952
#202730452
#202727182
C2×C53×#3942
#1180253