Finite Group Databasebeta

← #27294662

Subgroups — #27294662

Order 13568.

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

normal subgroups only

This group has 652 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
1C4×#13133132C211yesmaximal, characteristic, complemented
2#20273053C211yesmaximal, characteristic, complemented
3#20272712C214yesmaximal, complemented
4C53×#2541C211yesmaximal, characteristic
5C2×#13133132—21no—
6#16810587—21no—
7C4×C22⋊Dic53—21no—
8#16810390C2212yes—
9#16810349—21no—
10#16810347C2212yes—
11C2×#127×C53—21no—
12C2×#127×C53C411yescharacteristic, complemented
13C2×#121×C53—21no—
14C2×#121×C53—21no—
15C2×C4×D4×C53C2211yescharacteristic
16C53×#392C412yescomplemented
17C53×#392C2212yes—
18C53×#390—21no—
19C4×C4⋊C4×C53C411yescharacteristic, complemented
20C2×#11719917—41no—
21C2×C22⋊Dic53—22no—
22C2×C4×Dic53—41no—
23#13133132—24no—
24C22⋊#88322—22no—
25C22⋊#88322—41no—
26#13133124—41no—
27C22×D4×C53—22no—
28C22×D4×C53—22no—
29C22×D4×C53—21no—
30C22×D4×C53—21no—
31C22×D4×C53C2×C412yes—
32C22×D4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C23×C4×C53D411yescharacteristic
35#127×C53—44no—
36#127×C53—41no—
37#127×C53—41no—
38#121×C53—44no—
39#121×C53—44no—
40C4×D4×C53—41no—
41C4×D4×C53—41no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53C2311yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53C2×C411yescharacteristic
50#95×C53C2×C412yes—
51C24×C53—22no—
52C24×C53C22⋊C411yescharacteristic
53C24×C53C2×D411yescharacteristic
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53—41no—
57C2×D4×C53—42no—
58C2×D4×C53—42no—
59C2×D4×C53—44no—
60C2×D4×C53—44no—
61C2×D4×C53—42no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C2×D4×C53—44no—
65C2×D4×C53—44no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C2×D4×C53—42no—
69C2×D4×C53—82no—
70C2×D4×C53—82no—
71C2×D4×C53—82no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—42no—
85C4⋊C4×C53—24no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—44no—
90C22⋊C4×C53—44no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—42no—
94C22⋊Dic53—24no—
95C22⋊Dic53—42no—
96C4×Dic53—82no—
97#11719917—82no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53AGL(1,ℤ/8)12yes—
117D4×C53—44no—
118D4×C53—82no—
119D4×C53—84no—
120D4×C53—88no—
121D4×C53—88no—
122D4×C53—84no—
123D4×C53—82no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—22no—
127C2×C4×C53—24no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53#9911yescharacteristic
141C2×C4×C53#9911yescharacteristic
142C2×C4×C53#10412yes—
143C2×Dic53—44no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146#88322—82no—
147#11776—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—22no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53—82no—
161C22×C53—82no—
162C22×C53—82no—
163C22×C53—82no—
164C22×C53—82no—
165C22×C53—82no—
166C22×C53—82no—
167C22×C53C2×#9911yescharacteristic
168C22×C53#41811yescharacteristic
169C22×C53C2×#10411yescharacteristic
170C4×C53—21no—
171C4×C53—22no—
172C4×C53—44no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53#35312yes—
177Dic53—84no—
178#2541D5311yescharacteristic
179C4×#99—531no—
180#1465—531no—
181#1343—534no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—42no—
185C2×C53—81no—
186C2×C53—162no—
187C2×C53#150711yescharacteristic
188C2×C53#212211yescharacteristic
189C2×C53C2×#35311yescharacteristic
190C2×#127—21no—
191C2×#127Dic5311yescharacteristic, complemented
192C2×#121—21no—
193C2×#121—21no—
194C2×C4×D4C2×D5311yescharacteristic
195C2×#99—1061no—
196#407—1061no—
197#392Dic5312yescomplemented
198#392C2×D5312yes—
199#390—21no—
200C4×C4⋊C4Dic5311yescharacteristic, complemented
201C4×C22⋊C4—1061no—
202#342—532no—
203#324—1061no—
204#323—532no—
205C53#1177611yescharacteristic, complemented, Sylow
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2×Dic5312yes—
211C22×D4C22⋊D5311yescharacteristic
212C23×C4C22⋊D5311yescharacteristic
213C23×C4C22⋊D5311yescharacteristic
214C2×M16—2121no—
215#127—41no—
216#127—41no—
217#127—44no—
218#121—44no—
219#121—44no—
220C4×D4—41no—
221C4×D4—41no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22⋊D5311yescharacteristic
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—1062no—
227C2×C22⋊C4C2×Dic5311yescharacteristic
228C2×C22⋊C4C22×D5311yescharacteristic
229C2×C42—21no—
230C2×C42—2121no—
231C2×C42C2×Dic5311yescharacteristic
232#99—1064no—
233C22⋊C8—1062no—
234C22⋊C8—2121no—
235#95—2121no—
236#95C2×Dic5312yes—
237C24—22no—
238C24C22⋊Dic5311yescharacteristic
239C24C2×C22⋊D5311yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—2122no—
266C22×C4C22⋊Dic5311yescharacteristic
267C22×C4C22⋊Dic5311yescharacteristic
268C22×C4C22⋊Dic5311yescharacteristic
269C22×C4C22×Dic5311yescharacteristic
270C22×C4C2×C22⋊D5311yescharacteristic
271M16—4242no—
272C2×C8—2122no—
273C4⋊C4—24no—
274C4⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—1064no—
280C22⋊C4—2122no—
281C42—21no—
282C42—21no—
283C42—42no—
284C42—4242no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23C2×C22⋊Dic5311yescharacteristic
302C23#1313324312yes—
303D4—44no—
304D4—84no—
305D4—82no—
306D4—88no—
307D4—88no—
308D4—84no—
309D4—82no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—4242no—
327C2×C4—4242no—
328C2×C4—2124no—
329C2×C4#1313313211yescharacteristic
330C2×C4#1313313211yescharacteristic
331C2×C4#1313314412yes—
332C8—4242no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22C2×#1313313211yescharacteristic
353C22#1681062811yescharacteristic
354C22C2×#1313314411yescharacteristic
355C4—22no—
356C4—21no—
357C4—44no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—4244no—
362C4#1681041612yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—162no—
368C2#2027323111yescharacteristic
369C2#2027636511yescharacteristic
370C2C2×#1681041611yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
Fitting, prime radicalC53×#2541C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2541D53
hypocenterC53#11776
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C4×#13133132C2semidirect
2. #20273053C2semidirect
3. #20272712C2semidirect
12. C2×#127×C53C4semidirect
16. C53×#392C4semidirect
19. C4×C4⋊C4×C53C4semidirect
191. C2×#127Dic53semidirect
197. #392Dic53semidirect
200. C4×C4⋊C4Dic53semidirect
205. C53#11776semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730532
#202727122
C53×#25412
#1177653