Finite Group Databasebeta

← #27272291

Subgroups — #27272291

Order 13568.

590 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 111 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×#13133191C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic, complemented
3#20272539C212yesmaximal, complemented
4#20272535C212yesmaximal, complemented
5C53×#1817C211yesmaximal, characteristic, complemented
6C2×#13133191C411yescharacteristic
7C2×C4×D212C411yescharacteristic
8C2×C4×C22⋊D53—21no—
9C2×C22⋊#88322—21no—
10C2×#13133128—21no—
11#16810525C2211yescharacteristic
12#16810521C2211yescharacteristic
13#16810514—21no—
14#16810502—21no—
15#16810483C411yescharacteristic
16C4×C4⋊Dic53C411yescharacteristic
17C4×#11719921—21no—
18#16810378C2212yes—
19C22×C42×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#336—21no—
22C53×#336C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C4×D53—21no—
26C22×#88322—22no—
27#13133191—22no—
28#13133191—22no—
29C4×D212—42no—
30C4×C22⋊D53—44no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53—21no—
41C22⋊#88322—24no—
42#13133128—42no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C22×C8×C53—22no—
50C2×C42×C53—42no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53C2311yescharacteristic
55C22⋊C8×C53—42no—
56C24×C53C22⋊C411yescharacteristic
57C23×D53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C2×C22⋊D53—24no—
75C2×C22⋊D53—42no—
76C2×C22⋊D53—42no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×D212—21no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212M1612yes—
84C2×C4×D53—42no—
85C2×C4×D53—42no—
86C2×C4×D53—42no—
87C2×C8×C53—42no—
88C2×C8×C53—42no—
89C2×C8×C53—42no—
90C42×C53—22no—
91C42×C53—22no—
92C42×C53—41no—
93C42×C53—41no—
94C22⋊Dic53—42no—
95#11719921—24no—
96#11719921—42no—
97#11719921—42no—
98#11719921—42no—
99C4⋊Dic53—21no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C4⋊Dic53M1612yes—
103#11719919—42no—
104C4×Dic53—42no—
105C2×#88322—24no—
106C2×#88322—42no—
107C2×#88322—42no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53#10412yes—
115C23×C53C2×C22⋊C411yescharacteristic
116C22×D53—22no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—44no—
138C2×C4×C53—44no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53#9912yes—
142C2×C4×C53C2×M1611yescharacteristic
143C2×C4×C53C2×M1611yescharacteristic
144C2×C4×C53#13512yes—
145C2×C4×C53AGL(1,ℤ/8)12yes—
146C22⋊D53—48no—
147C2×Dic53—22no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156D212—42no—
157D212—42no—
158C4×D53—84no—
159C8×C53—82no—
160#88322—44no—
161#7620—531nomaximal, Sylow
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53#40711yescharacteristic
170C22×C53C2×#9911yescharacteristic
171C22×C53#41311yescharacteristic
172C22×C53#41611yescharacteristic
173C22×C53#41811yescharacteristic
174C22×C53C2×#10411yescharacteristic
175C22×C53#42111yescharacteristic
176C2×D53—42no—
177C2×D53—42no—
178C2×D53—42no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—22no—
186C4×C53—22no—
187C4×C53—22no—
188C4×C53—41no—
189C4×C53—41no—
190Dic53—42no—
191Dic53—81no—
192Dic53—81no—
193C4×#121—531no—
194#1817D5311yescharacteristic, complemented
195#1749—531no—
196#1271—532no—
197#1270—532no—
198C2×C53—42no—
199C2×C53#146511yescharacteristic
200C2×C53#148811yescharacteristic
201C2×C53#149011yescharacteristic
202C2×C53#150111yescharacteristic
203C2×C53#151011yescharacteristic
204C2×C53#153211yescharacteristic
205C2×C53#155311yescharacteristic
206D53—82no—
207C2×#121—531no—
208C2×C4×D4—531no—
209C2×C4×D4—1061no—
210C22×C42C2×D5311yescharacteristic
211C2×C22⋊C8—21no—
212C2×C22⋊C8—1061no—
213C2×#97—1061no—
214#390—531no—
215#389—531no—
216#388—1061no—
217#386—1061no—
218#380—531no—
219C4×C4⋊C4—531no—
220C4×C22⋊C4—1061no—
221#336—21no—
222#336—532no—
223#336C2×D5312yes—
224C53#762011yescharacteristic, complemented, Sylow
225C22×D4—531no—
226C22×D4—1061no—
227C23×C4—1061no—
228C23×C4C4×D5311yescharacteristic
229C23×C4D21211yescharacteristic
230C23×C4C22⋊D5311yescharacteristic
231C22×C8—22no—
232C22×C8—1062no—
233#121—1062no—
234#121—1062no—
235C4×D4—2122no—
236C4×D4—2124no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C42—42no—
247C2×C42—1061no—
248C2×C42C4×D5311yescharacteristic
249C2×C42D21211yescharacteristic
250C2×C42C22⋊D5311yescharacteristic
251C2×C42C22×D5311yescharacteristic
252C22⋊C8—42no—
253C22⋊C8—1064no—
254#97—2122no—
255#95—1061no—
256#95—1061no—
257#95—1061no—
258C24—1061no—
259C24#1171992111yescharacteristic
260C2×D4—532no—
261C2×D4—1064no—
262C2×D4—1061no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2122no—
266C2×D4—2122no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—2122no—
281C22×C4—2122no—
282C22×C4—2122no—
283C22×C4#1171992111yescharacteristic
284C22×C4#1171992111yescharacteristic
285C22×C4#1171992111yescharacteristic
286C22×C4C2×C4×D5311yescharacteristic
287C22×C4C2×D21211yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—1064no—
293C2×C8—2122no—
294C2×C8—2122no—
295C4⋊C4—532no—
296C4⋊C4—1061no—
297C4⋊C4—2121no—
298C4⋊C4—2121no—
299C4⋊C4—2122no—
300C22⋊C4—1064no—
301C22⋊C4—2122no—
302C22⋊C4—2122no—
303C22⋊C4—2122no—
304C22⋊C4—2122no—
305C42—22no—
306C42—22no—
307C42—41no—
308C42—41no—
309C42—2122no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—1062no—
317C23—2121no—
318C23—2121no—
319C23—2121no—
320C23—2121no—
321C23—2121no—
322C23—2121no—
323C23#1313314312yes—
324C23C2×#1171992111yescharacteristic
325D4—2122no—
326D4—2122no—
327D4—2128no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—1062no—
347C2×C4—4244no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4#1313313112yes—
357C2×C4M16×D5311yescharacteristic
358C2×C4M16×D5311yescharacteristic
359C2×C4#1313323912yes—
360C2×C4#1313324212yes—
361C8—82no—
362C8—2124no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—4241no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22—2122no—
375C22—2122no—
376C22—2122no—
377C22#1681058611yescharacteristic
378C22C2×#1313313111yescharacteristic
379C22#1681060511yescharacteristic
380C22#1681062011yescharacteristic
381C22#1681062711yescharacteristic
382C22C2×#1313314311yescharacteristic
383C22#1681064111yescharacteristic
384C4—22no—
385C4—22no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—4241no—
392C4—4241no—
393C4—2122no—
394C2—42no—
395C2—4242no—
396C2#2027305111yescharacteristic
397C2#2027315011yescharacteristic
398C2#2027315711yescharacteristic
399C2#2027320711yescharacteristic
400C2#2027324311yescharacteristic
401C2#2027333611yescharacteristic
402C2#2027341811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1817C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C4
cosocleC2×C42×C53C23
hypercenter#1817D53
hypocenterC53#7620
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #20274477C2semidirect
3. #20272539C2semidirect
4. #20272535C2semidirect
5. C53×#1817C2semidirect
194. #1817D53semidirect
224. C53#7620semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
#202744772
#202725392
#202725352
C53×#18172
#762053