Finite Group Databasebeta

← #27294991

Subgroups — #27294991

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#20280281C211yesmaximal, characteristic
2#20274798C211yesmaximal, characteristic, complemented
3#20272743C212yesmaximal, complemented
4#20272719C212yesmaximal, complemented
5C53×#1458C211yesmaximal, characteristic, complemented
6C2×C4×D212C411yescharacteristic
7#16810587—21no—
8#16810562C2211yescharacteristic
9#16810559C411yescharacteristic
10#16810540C2211yescharacteristic
11#16810532C411yescharacteristic
12#16810514—21no—
13#16810502—21no—
14#16810499—21no—
15#16810494—21no—
16#16810491C411yescharacteristic
17#16810392—21no—
18#16810392C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#431—21no—
21C53×#324C2212yes—
22C53×#323—21no—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C4×D53—21no—
26C2×#11719917—22no—
27C4×D212—41no—
28C4×D212—41no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C22⋊#88322—41no—
37C22⋊#88322—41no—
38#13133124—21no—
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×C53D411yescharacteristic
48C2×C4⋊C4×C53C2311yescharacteristic
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C4⋊C8×C53—41no—
54C22⋊C8×C53—22no—
55C4×C8×C53—41no—
56C24×C53C22⋊C411yescharacteristic
57C23×D53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—41no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C2×C22⋊D53—42no—
69C2×C22⋊D53—42no—
70C2×C22⋊D53—81no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C2×D212—41no—
75C2×D212—42no—
76C2×C4×D53—42no—
77C2×C4×D53—81no—
78C2×C4×D53—81no—
79C2×C8×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C22⋊Dic53—81no—
89#11719921—42no—
90#11719921—41no—
91#11719921—41no—
92#11719921—42no—
93#11719921—81no—
94C4⋊Dic53—41no—
95#11719919—81no—
96#11719917—42no—
97#11719917—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C22×D53—22no—
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—42no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53#13512yes—
129C22⋊D53—84no—
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—
139D212—42no—
140D212—42no—
141C4×D53—84no—
142C8×C53—82no—
143#88322—82no—
144#11847—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—42no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—81no—
152C22×C53C2×#9911yescharacteristic
153C22×C53#41611yescharacteristic
154C22×C53#42111yescharacteristic
155C2×D53—42no—
156C2×D53—42no—
157C2×D53—42no—
158C2×D53—81no—
159C2×D53—81no—
160C2×D53—81no—
161C2×D53—81no—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169Dic53—81no—
170Dic53—81no—
171Dic53—81no—
172#2598—531no—
173#1822—531no—
174#1458D5311yescharacteristic, complemented
175#1353—532no—
176#1345—532no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—81no—
180C2×C53#151611yescharacteristic
181C2×C53#212011yescharacteristic
182C2×C53#213411yescharacteristic
183D53—82no—
184C2×C4×D4—531no—
185C2×#117C2×D5311yescharacteristic
186#431—21no—
187#407—1061no—
188#400—531no—
189#399—531no—
190#394—531no—
191#392—531no—
192#388—1061no—
193#386—1061no—
194#386—1061no—
195#385—1061no—
196#383—531no—
197#343—532no—
198#343—1061no—
199#324C2×D5312yes—
200#323—21no—
201C53#1184711yescharacteristic, complemented, Sylow
202C22×D4—531no—
203C22×D4—1061no—
204C23×C4—1061no—
205C23×C4D21211yescharacteristic
206C2×M16—1062no—
207C4×D4—2121no—
208C4×D4—2121no—
209#117—41no—
210#117—41no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4D21211yescharacteristic
214C2×C4⋊C4C22×D5311yescharacteristic
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4C4×D5311yescharacteristic
221C2×C22⋊C4C22⋊D5311yescharacteristic
222C2×C42C4×D5311yescharacteristic
223C2×C42C22⋊D5311yescharacteristic
224C4⋊C8—41no—
225C22⋊C8—22no—
226C22⋊C8—2121no—
227C22⋊C8—2121no—
228C4×C8—41no—
229#95—531no—
230#95—531no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235C24—1061no—
236C24#1171992111yescharacteristic
237C2×D4—4241no—
238C2×D4—2121no—
239C2×D4—2122no—
240C2×D4—2122no—
241C2×D4—2122no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—4241no—
250C22×C4—4241no—
251C22×C4—2122no—
252C22×C4#1171992111yescharacteristic
253C22×C4#1171992111yescharacteristic
254C22×C4#1171992111yescharacteristic
255C22×C4C2×C4×D5311yescharacteristic
256C22×C4C2×D21211yescharacteristic
257C22×C4C2×C22⋊D5311yescharacteristic
258M16—2122no—
259M16—2122no—
260C2×C8—42no—
261C2×C8—2122no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—4241no—
265C4⋊C4—2121no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—4241no—
269C22⋊C4—4241no—
270C22⋊C4—2122no—
271C22⋊C4—2121no—
272C22⋊C4—2121no—
273C22⋊C4—2122no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—1062no—
284C23—2121no—
285C23—2121no—
286C23—2121no—
287C23—2121no—
288C23—2121no—
289C23—2121no—
290C23C2×#1171992111yescharacteristic
291D4—4244no—
292D4—2122no—
293D4—2122no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—4244no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4#1313313111yescharacteristic
319C2×C4#1313313111yescharacteristic
320C2×C4#1313323912yes—
321C8—82no—
322C8—4242no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—4241no—
331C22—4241no—
332C22—4241no—
333C22—4241no—
334C22—2122no—
335C22—2122no—
336C22—2122no—
337C22C2×#1313313111yescharacteristic
338C22#1681062011yescharacteristic
339C22#1681063911yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—4241no—
348C4—4241no—
349C4—4241no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—4242no—
354C2#2027327211yescharacteristic
355C2#2027635411yescharacteristic
356C2#2027640611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1458C2
socleC22×C53C2×#99
prime radical#16810532C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1458D53
hypocenterC53#11847
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #20274798C2semidirect
3. #20272743C2semidirect
4. #20272719C2semidirect
5. C53×#1458C2semidirect
174. #1458D53semidirect
201. C53#11847semidirect

Maximal subgroups

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

subgroupindex
#202802812
#202747982
#202727432
#202727192
C53×#14582
#1184753