Finite Group Databasebeta

← #27294973

Subgroups — #27294973

Order 13568.

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

normal subgroups only

This group has 432 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#20280402C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic, complemented
3#20272735C212yesmaximal, complemented
4#20272711C212yesmaximal
5C53×#1820C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic, complemented
7#16810676—21no—
8#16810543C411yescharacteristic, complemented
9#16810540C411yescharacteristic, complemented
10#16810535C2211yescharacteristic
11#16810532C2211yescharacteristic
12#16810523—21no—
13#16810514—21no—
14#16810513—21no—
15#16810502—21no—
16#16810469C411yescharacteristic, complemented
17#16810349—21no—
18#16810347C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#381—21no—
21C53×#342—21no—
22C53×#342C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26#13133205—41no—
27#13133205—41no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×#11719921—21no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—41no—
37C22⋊#88322—22no—
38C4×#88322—41no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45#117×C53—41no—
46#117×C53—41no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53C2311yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#95×C53—41no—
55#95×C53—41no—
56C24×C53C2×D411yescharacteristic
57C23×D53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—41no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C2×C22⋊D53—42no—
70C2×C22⋊D53—42no—
71C2×C22⋊D53—81no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C2×D212—41no—
76C2×D212—81no—
77C2×Dic106—41no—
78C2×Dic106—81no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—81no—
90#11719921—42no—
91#11719921—42no—
92#11719921—42no—
93#11719921—41no—
94#11719921—41no—
95#11719921—81no—
96#11719919—81no—
97C4×Dic53—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53#13512yes—
105C22×D53—41no—
106C22×D53—41no—
107C22×D53—41no—
108C22×D53—41no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53—41no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
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—21no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—82no—
128C2×C4×C53—82no—
129C2×C4×C53—82no—
130C2×C4×C53#9911yescharacteristic
131C2×C4×C53#9911yescharacteristic
132C22⋊D53—84no—
133C2×Dic53—22no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142D212—82no—
143Dic106—82no—
144#88322—82no—
145#11844—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—22no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—81no—
153C22×C53C2×#9911yescharacteristic
154C22×C53#41611yescharacteristic
155C22×C53#42111yescharacteristic
156C2×D53—81no—
157C2×D53—81no—
158C2×D53—81no—
159C2×D53—81no—
160C2×D53—81no—
161C2×D53—81no—
162C2×D53—81no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—162no—
171Dic53—42no—
172Dic53—81no—
173Dic53—81no—
174#2610—531no—
175#1820D5311yescharacteristic, complemented
176#1458—531no—
177#1351—532no—
178#1343—532no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—42no—
182C2×C53#151611yescharacteristic
183C2×C53#212211yescharacteristic
184C2×C53#212611yescharacteristic
185D53—161no—
186C2×#124—531no—
187C2×#117C2×D5311yescharacteristic
188#431—1061no—
189#395—531no—
190#394—531no—
191#393—531no—
192#392—531no—
193#390—1061no—
194#388—1061no—
195#388—1061no—
196#386—1061no—
197#381—21no—
198#376—531no—
199#342—21no—
200#342C2×D5312yes—
201#324—1061no—
202#323—532no—
203C53#1184411yescharacteristic, complemented, Sylow
204C22×Q8—531no—
205C22×D4—531no—
206C22×D4—1061no—
207C23×C4C22⋊D5311yescharacteristic
208#124—2121no—
209#124—2121no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4C4×D5311yescharacteristic
214C2×C4⋊C4D21211yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1061no—
222C2×C22⋊C4C22⋊D5311yescharacteristic
223C2×C22⋊C4C22×D5311yescharacteristic
224C2×C42—1061no—
225C2×C42C4×D5311yescharacteristic
226C2×C42D21211yescharacteristic
227C4⋊C8—2121no—
228C22⋊C8—1062no—
229C4×C8—2121no—
230#95—41no—
231#95—41no—
232#95—531no—
233#95—531no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237C24—1061no—
238C24C2×C22⋊D5311yescharacteristic
239C2×Q8—4241no—
240C2×Q8—2121no—
241C2×D4—4241no—
242C2×D4—4241no—
243C2×D4—2121no—
244C2×D4—2122no—
245C2×D4—2122no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4#1171992111yescharacteristic
257C22×C4#1171992111yescharacteristic
258C22×C4C2×C4×D5311yescharacteristic
259C22×C4C2×D21211yescharacteristic
260C2×C8—2122no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—4241no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—4241no—
269C22⋊C4—4241no—
270C22⋊C4—2122no—
271C22⋊C4—2122no—
272C22⋊C4—2122no—
273C22⋊C4—2121no—
274C22⋊C4—2121no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—2122no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—2121no—
285C23—2121no—
286C23—2121no—
287C23—2121no—
288C23—2121no—
289C23—2121no—
290C23—2121no—
291C23C2×#1171992111yescharacteristic
292C23#1313323712yes—
293Q8—4242no—
294D4—4242no—
295D4—4244no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—1062no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4#1313313111yescharacteristic
324C2×C4#1313313111yescharacteristic
325C8—4242no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—4241no—
334C22—4241no—
335C22—4241no—
336C22—4241no—
337C22—4241no—
338C22—4241no—
339C22—4241no—
340C22C2×#1313313111yescharacteristic
341C22#1681061711yescharacteristic
342C22#1681064111yescharacteristic
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—162no—
351C4—4241no—
352C4—4241no—
353C4—2122no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—8481no—
358C2#2027326911yescharacteristic
359C2#2027636411yescharacteristic
360C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1820C2
socleC22×C53C2×#99
prime radical#16810532C22
cosocleC2×C22⋊C4×C53C23
hypercenter#1820D53
hypocenterC53#11844
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #20273029C2semidirect
3. #20272735C2semidirect
5. C53×#1820C2semidirect
6. C2×#13133205C4semidirect
8. #16810543C4semidirect
9. #16810540C4semidirect
16. #16810469C4semidirect
175. #1820D53semidirect
203. C53#11844semidirect

Maximal subgroups

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

subgroupindex
#202804022
#202730292
#202727352
#202727112
C53×#18202
#1184453