Finite Group Databasebeta

← #27294884

Subgroups — #27294884

Order 13568.

414 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280632C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20272737C212yesmaximal, complemented
4#20272716C212yesmaximal, complemented
5C53×#1466C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic, complemented
7C2×#13133132—21no—
8#16810553C411yescharacteristic, complemented
9#16810551—21no—
10#16810548C2211yescharacteristic
11#16810543C411yescharacteristic, complemented
12#16810537—21no—
13#16810535C2211yescharacteristic
14#16810507—21no—
15#16810496—21no—
16#16810486C411yescharacteristic, complemented
17#16810483—21no—
18#16810390C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic
24#13133196—41no—
25#13133196—41no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36#13133132—42no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—41no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×D4×C53D411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C4×D4×C53—41no—
49C4×D4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53D411yescharacteristic
54C4⋊C8×C53—41no—
55C22⋊C8×C53—22no—
56#97×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—81no—
68C22×C4×C53—81no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—42no—
79C2×Dic106—41no—
80C2×Dic106—81no—
81C2×C8×C53—42no—
82C4⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C42×C53—21no—
86C42×C53—21no—
87C22⋊Dic53—42no—
88C22⋊Dic53—41no—
89C22⋊Dic53—41no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—81no—
93C22⋊Dic53—81no—
94C4⋊Dic53—41no—
95C4⋊Dic53—82no—
96#11719919—42no—
97#11719919—41no—
98#11719919—41no—
99#11719919—81no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53AGL(1,ℤ/8)11yescharacteristic
111C23×C53#13711yescharacteristic
112D4×C53—42no—
113D4×C53—42no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#9911yescharacteristic
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—82no—
143C2×Dic53—82no—
144C2×Dic53—82no—
145Dic106—82no—
146C8×C53—82no—
147#11826—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53C2×#9911yescharacteristic
163C22×C53#41711yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171Dic53—81no—
172Dic53—81no—
173Dic53—81no—
174Dic53—81no—
175Dic53—162no—
176#2638—531no—
177#1799—531no—
178#1466D5311yescharacteristic
179#1351—532no—
180#1344—532no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—41no—
184C2×C53—41no—
185C2×C53—81no—
186C2×C53#151311yescharacteristic
187C2×C53#212111yescharacteristic
188C2×C53#212911yescharacteristic
189C2×C22⋊Q8—531no—
190C2×C4×D4C2×D5311yescharacteristic
191#432—21no—
192C2×#99—1061no—
193#398—531no—
194#397—531no—
195#397—1061no—
196#395—531no—
197#394—1061no—
198#393—531no—
199#387—1061no—
200#385—1061no—
201#381—531no—
202#380—1061no—
203#342—532no—
204#323—21no—
205#323C2×D5312yes—
206C53#1182611yescharacteristic, complemented, Sylow
207C22×Q8—531no—
208C22×D4D21211yescharacteristic
209C23×C4C4×D5311yescharacteristic
210C23×C4D21211yescharacteristic
211C22⋊Q8—2121no—
212C22⋊Q8—2121no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—2121no—
220C2×C4⋊C4C4×D5311yescharacteristic
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1062no—
226C2×C22⋊C4C22⋊D5311yescharacteristic
227C2×C22⋊C4C22×D5311yescharacteristic
228C2×C42C22⋊D5311yescharacteristic
229C4⋊C8—41no—
230#99—2122no—
231C22⋊C8—22no—
232#97—41no—
233#95—531no—
234#95—531no—
235#95—1061no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—2121no—
241C24#1171992111yescharacteristic
242C24C2×D21211yescharacteristic
243C2×Q8—4241no—
244C2×Q8—2121no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—2122no—
260C22×C4#1171992111yescharacteristic
261C22×C4#1171992111yescharacteristic
262C22×C4#1171992111yescharacteristic
263C22×C4C2×C4×D5311yescharacteristic
264C22×C4C2×C22⋊D5311yescharacteristic
265C2×C8—42no—
266C4⋊C4—41no—
267C4⋊C4—4242no—
268C4⋊C4—4241no—
269C4⋊C4—2121no—
270C4⋊C4—2122no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—4241no—
276C22⋊C4—4241no—
277C22⋊C4—2122no—
278C22⋊C4—2122no—
279C22⋊C4—2122no—
280C22⋊C4—2121no—
281C22⋊C4—2121no—
282C42—21no—
283C42—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23C2×#1171992111yescharacteristic
294C23#1313324511yescharacteristic
295C23#1313325011yescharacteristic
296Q8—4242no—
297D4—42no—
298D4—42no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—4242no—
314C2×C4—4242no—
315C2×C4—4242no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4#1313313111yescharacteristic
329C2×C4#1313313111yescharacteristic
330C8—82no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#1313313111yescharacteristic
346C22#1681062511yescharacteristic
347C22#1681063911yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—8482no—
355C4—4241no—
356C4—4241no—
357C4—4241no—
358C4—4241no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#2027325911yescharacteristic
365C2#2027636211yescharacteristic
366C2#2027638911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1466C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1466D53
hypocenterC53#11826
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #20272737C2semidirect
4. #20272716C2semidirect
6. C2×#13133196C4semidirect
8. #16810553C4semidirect
11. #16810543C4semidirect
16. #16810486C4semidirect
206. C53#11826semidirect

Maximal subgroups

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

subgroupindex
#202806322
#202747032
#202727372
#202727162
C53×#14662
#1182653