Finite Group Databasebeta

← #27294663

Subgroups — #27294663

Order 13568.

652 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279661C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic, complemented
3#20272713C214yesmaximal, complemented
4C53×#1465C211yesmaximal, characteristic, complemented
5C2×#13133214C411yescharacteristic, complemented
6C2×#13133213—21no—
7C2×#13133191—21no—
8C2×#13133191—21no—
9C2×#13133132—21no—
10#16810532C412yescomplemented
11#16810532C2212yes—
12#16810524—21no—
13C4×C4⋊Dic53C411yescharacteristic, complemented
14C4×C22⋊Dic53—21no—
15#16810390C2212yes—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324—21no—
19C53×#323C2212yes—
20C22×D212—21no—
21C22×D212—21no—
22C22×D212C2×C412yes—
23C22×C22⋊D53—22no—
24C22×C22⋊D53—22no—
25#13133214—41no—
26#13133214—41no—
27#13133213—44no—
28#13133191—44no—
29#13133191—44no—
30C2×C4⋊Dic53—21no—
31C2×C22⋊Dic53—22no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—41no—
36#13133132—24no—
37#13133124—41no—
38#13133124C2×C412yes—
39C22×D4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C2×M16×C53—41no—
43C4×D4×C53—41no—
44C4×D4×C53—41no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53C2311yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49C22⋊C8×C53—22no—
50C22⋊C8×C53—41no—
51C24×C53C22⋊C411yescharacteristic
52C24×C53C2×D411yescharacteristic
53C23×D53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53—41no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C2×C22⋊D53—44no—
69C2×C22⋊D53—44no—
70C2×C22⋊D53—44no—
71C2×C22⋊D53—44no—
72C2×C22⋊D53—82no—
73C2×C22⋊D53—82no—
74C22×Dic53—22no—
75C22×Dic53—42no—
76C2×D212—42no—
77C2×D212—42no—
78C2×D212—42no—
79C2×D212—42no—
80C2×D212—42no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—42no—
84C2×D212—82no—
85M16×C53—82no—
86C2×C8×C53—42no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—21no—
91C42×C53—21no—
92C22⋊Dic53—24no—
93C22⋊Dic53—42no—
94#11719921—44no—
95#11719921—44no—
96C4⋊Dic53—24no—
97C4×Dic53—42no—
98C4×Dic53—82no—
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×C53C2×C22⋊C411yescharacteristic
109C23×C53AGL(1,ℤ/8)12yes—
110C22×D53—42no—
111C22×D53—42no—
112C22×D53—42no—
113C22×D53—42no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—42no—
117D4×C53—44no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53#9911yescharacteristic
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#10412yes—
134C22⋊D53—88no—
135C22⋊D53—88no—
136C2×Dic53—24no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—44no—
140C2×Dic53—82no—
141C2×Dic53—82no—
142D212—82no—
143D212—84no—
144D212—84no—
145D212—82no—
146C8×C53—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×C53C2×#9911yescharacteristic
161C22×C53#41811yescharacteristic
162C22×C53C2×#10411yescharacteristic
163C2×D53—82no—
164C2×D53—82no—
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—82no—
168C2×D53—82no—
169C2×D53—82no—
170C4×C53—21no—
171C4×C53—22no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53#35312yes—
176Dic53—44no—
177Dic53—84no—
178#2541—531no—
179C4×#99—531no—
180#1465D5311yescharacteristic, complemented
181#1343—534no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—42no—
185C2×C53—81no—
186C2×C53#150711yescharacteristic
187C2×C53#212211yescharacteristic
188C2×C53C2×#35311yescharacteristic
189D53—162no—
190C2×#127—531no—
191C2×#127—1061no—
192C2×#121—1061no—
193C2×#121—1061no—
194C2×C4×D4C2×D5311yescharacteristic
195C2×#99—1061no—
196#407—21no—
197#392—532no—
198#392—532no—
199#390—1061no—
200C4×C4⋊C4—531no—
201C4×C22⋊C4—1061no—
202#342—532no—
203#324—21no—
204#323C2×D5312yes—
205C53#1177611yescharacteristic, complemented, Sylow
206C22×D4—532no—
207C22×D4—1062no—
208C22×D4—1062no—
209C22×D4—1061no—
210C22×D4—1061no—
211C22×D4D21211yescharacteristic
212C23×C4D21211yescharacteristic
213C23×C4C22⋊D5311yescharacteristic
214C2×M16—41no—
215#127—2121no—
216#127—2121no—
217#127—2124no—
218#121—2124no—
219#121—2124no—
220C4×D4—41no—
221C4×D4—41no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4C22⋊D5311yescharacteristic
224C2×C22⋊C4—1062no—
225C2×C22⋊C4—1062no—
226C2×C22⋊C4—1062no—
227C2×C22⋊C4C4×D5311yescharacteristic
228C2×C22⋊C4C22×D5311yescharacteristic
229C2×C42—1061no—
230C2×C42—2121no—
231C2×C42C4×D5311yescharacteristic
232#99—1064no—
233C22⋊C8—22no—
234C22⋊C8—41no—
235#95—532no—
236#95—2121no—
237C24—1062no—
238C24#1171992111yescharacteristic
239C24C2×D21211yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—4242no—
244C2×D4—4242no—
245C2×D4—4242no—
246C2×D4—2122no—
247C2×D4—2122no—
248C2×D4—2122no—
249C2×D4—2122no—
250C2×D4—2122no—
251C2×D4—2124no—
252C2×D4—2124no—
253C2×D4—2124no—
254C2×D4—2124no—
255C2×D4—2121no—
256C2×D4—2121no—
257C2×D4—2122no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—1062no—
265C22×C4—2122no—
266C22×C4#1171992111yescharacteristic
267C22×C4#1171992111yescharacteristic
268C22×C4#1171992111yescharacteristic
269C22×C4C2×C4×D5311yescharacteristic
270C22×C4C2×C22⋊D5311yescharacteristic
271M16—82no—
272C2×C8—42no—
273C4⋊C4—41no—
274C4⋊C4—1064no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—1064no—
278C22⋊C4—2124no—
279C22⋊C4—2124no—
280C22⋊C4—2122no—
281C42—21no—
282C42—21no—
283C42—4242no—
284C42—2122no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—2122no—
295C23—2122no—
296C23—2122no—
297C23—2122no—
298C23—2122no—
299C23—2122no—
300C23—2122no—
301C23C2×#1171992111yescharacteristic
302C23#1313324512yes—
303D4—44no—
304D4—4244no—
305D4—4242no—
306D4—4244no—
307D4—4248no—
308D4—4248no—
309D4—4242no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—1064no—
324C2×C4—4242no—
325C2×C4—4242no—
326C2×C4—2124no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4#1313313111yescharacteristic
330C2×C4#1313313111yescharacteristic
331C2×C4#1313314312yes—
332C8—82no—
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—4242no—
346C22—4242no—
347C22—4242no—
348C22—4242no—
349C22—4242no—
350C22—4242no—
351C22—4242no—
352C22C2×#1313313111yescharacteristic
353C22#1681062911yescharacteristic
354C22C2×#1313314311yescharacteristic
355C4—22no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—4244no—
361C4—2124no—
362C4#1681041512yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—8482no—
368C2#2027323211yescharacteristic
369C2#2027636611yescharacteristic
370C2C2×#1681041511yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1465C2
socleC22×C53C2×#99
prime radical#20279661C2
cosocleC2×C22⋊C4×C53C23
hypercenter#1465D53
hypocenterC53#11776
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. C4×#13133132C2semidirect
3. #20272713C2semidirect
4. C53×#1465C2semidirect
5. C2×#13133214C4semidirect
10. #16810532C4semidirect
13. C4×C4⋊Dic53C4semidirect
180. #1465D53semidirect
205. C53#11776semidirect

Maximal subgroups

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

subgroupindex
#202796612
C4×#131331322
#202727132
C53×#14652
#1177653