Finite Group Databasebeta

← #27295114

Subgroups — #27295114

Order 13568.

536 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 95 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#20279612C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic, complemented
3#20272714C212yesmaximal, complemented
4#20272711C212yesmaximal, complemented
5C53×#1799C211yesmaximal, characteristic, complemented
6C2×#13133188—21no—
7C2×C4×D212C411yescharacteristic, complemented
8C2×C22⋊#88322—21no—
9#16810548C411yescharacteristic, complemented
10#16810548C2211yescharacteristic
11#16810532C411yescharacteristic, complemented
12#16810532C2211yescharacteristic
13#16810502—21no—
14#16810502—21no—
15#16810489C411yescharacteristic, complemented
16#16810480—21no—
17#16810349—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C53×#380—21no—
22C53×#342C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C22⋊D53—21no—
26C22×C4×D53—21no—
27C22×#88322—41no—
28#13133188—42no—
29#13133188—42no—
30C4×D212—41no—
31C4×D212—41no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C22⋊#88322—22no—
40C22⋊#88322—41no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C22×D4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51C2×C4⋊C4×C53—41no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53—22no—
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53C2311yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57#99×C53—42no—
58#95×C53—41no—
59C24×C53C22⋊C411yescharacteristic
60C24×C53C2×D411yescharacteristic
61C23×D53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53D812yes—
65C2×D4×C53SD1612yes—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—42no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C2×C22⋊D53—42no—
79C2×C22⋊D53—42no—
80C2×C22⋊D53—42no—
81C2×C22⋊D53—41no—
82C2×C22⋊D53—41no—
83C2×C22⋊D53—81no—
84C2×C22⋊D53—81no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C2×D212—41no—
89C2×D212—42no—
90C2×C4×D53—42no—
91C2×C4×D53—42no—
92C2×C4×D53—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—82no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—42no—
99C42×C53—21no—
100C42×C53—21no—
101C22⋊Dic53—42no—
102C22⋊Dic53—41no—
103C22⋊Dic53—41no—
104#11719921—41no—
105#11719921—41no—
106#11719921—81no—
107#11719921—81no—
108C4⋊Dic53—42no—
109C4⋊Dic53—41no—
110C2×#88322—42no—
111C2×#88322—82no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53#10212yes—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53C2×D811yescharacteristic
123C23×C53C2×SD1611yescharacteristic
124C22×D53—22no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131D4×C53—42no—
132D4×C53—42no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53—82no—
147C2×C4×C53—82no—
148C2×C4×C53—82no—
149C2×C4×C53#9911yescharacteristic
150C2×C4×C53#9911yescharacteristic
151C2×C4×C53#10212yes—
152C22⋊D53—84no—
153C22⋊D53—82no—
154C22⋊D53—82no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164D212—42no—
165D212—42no—
166C4×D53—84no—
167#88322—82no—
168#11875—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—22no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53C2×#9911yescharacteristic
184C22×C53C2×#10211yescharacteristic
185C22×C53#42111yescharacteristic
186C2×D53—42no—
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—81no—
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—81no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—22no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—162no—
200Dic53—81no—
201Dic53—81no—
202Dic53—81no—
203#2535—531no—
204#1799D5311yescharacteristic, complemented
205#1459—531no—
206#1344—532no—
207#1343—532no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—41no—
211C2×C53—42no—
212C2×C53—41no—
213C2×C53#151911yescharacteristic
214C2×C53#212011yescharacteristic
215C2×C53#212711yescharacteristic
216D53—82no—
217C2×#121—1061no—
218C2×C4×D4—531no—
219C2×C4×D4C2×D5311yescharacteristic
220C2×#99—21no—
221C2×C22⋊C8—1061no—
222#397—531no—
223#397—531no—
224#392—531no—
225#392—531no—
226#386—1061no—
227#386—1061no—
228#382—531no—
229#380—21no—
230#380—1061no—
231#342C2×D5312yes—
232#324—1061no—
233#323—532no—
234C53#1187511yescharacteristic, complemented, Sylow
235C22×D4—531no—
236C22×D4—1061no—
237C22×D4—1061no—
238C22×D4C22⋊D5311yescharacteristic
239C23×C4—1061no—
240C23×C4D21211yescharacteristic
241C23×C4C22⋊D5311yescharacteristic
242C22×C8—2121no—
243#121—2122no—
244#121—2122no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—2121no—
248C4×D4—2121no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4D21211yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4—1061no—
257C2×C22⋊C4—1061no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4C4×D5311yescharacteristic
260C2×C22⋊C4C22×D5311yescharacteristic
261C2×C42C4×D5311yescharacteristic
262#99—42no—
263C22⋊C8—1062no—
264C22⋊C8—2121no—
265#95—41no—
266#95—531no—
267#95—531no—
268#95—1061no—
269#95—1061no—
270#95—1061no—
271C24—1061no—
272C24#1171992111yescharacteristic
273C24C2×C22⋊D5311yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—4241no—
277C2×D4—4241no—
278C2×D4—2121no—
279C2×D4—2122no—
280C2×D4—2122no—
281C2×D4—2122no—
282C2×D4—2122no—
283C2×D4—2121no—
284C2×D4—2121no—
285C2×D4#1171992212yes—
286C2×D4#1171992412yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—2122no—
298C22×C4—2122no—
299C22×C4—2122no—
300C22×C4#1171992111yescharacteristic
301C22×C4#1171992111yescharacteristic
302C22×C4#1171992111yescharacteristic
303C22×C4C2×C4×D5311yescharacteristic
304C22×C4C2×D21211yescharacteristic
305C2×C8—4242no—
306C2×C8—2122no—
307C4⋊C4—41no—
308C4⋊C4—82no—
309C4⋊C4—2121no—
310C4⋊C4—2122no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—4241no—
316C22⋊C4—4241no—
317C22⋊C4—2122no—
318C22⋊C4—2121no—
319C22⋊C4—2121no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—1062no—
333C23—2121no—
334C23—2121no—
335C23—2121no—
336C23—2121no—
337C23—2121no—
338C23—2121no—
339C23#1313313712yes—
340C23C2×#1171992111yescharacteristic
341C23C2×#1171992211yescharacteristic
342C23C2×#1171992411yescharacteristic
343D4—42no—
344D4—42no—
345D4—4244no—
346D4—4242no—
347D4—4242no—
348D4—2122no—
349D4—2122no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—4244no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4#1313313111yescharacteristic
377C2×C4#1313313111yescharacteristic
378C2×C4#1313313712yes—
379C8—4242no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—81no—
394C22—4241no—
395C22—4241no—
396C22—4241no—
397C22—4241no—
398C22—2122no—
399C22—2122no—
400C22—2122no—
401C22C2×#1313313111yescharacteristic
402C22C2×#1313313711yescharacteristic
403C22#1681064111yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—162no—
411C4—4241no—
412C4—4241no—
413C4—4241no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—42no—
418C2—41no—
419C2—4242no—
420C2#2027328211yescharacteristic
421C2#2027635411yescharacteristic
422C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1799C2
socleC22×C53C2×#99
prime radical#20279612C2
cosocleC2×C22⋊C4×C53C23
hypercenter#1799D53
hypocenterC53#11875
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #20273033C2semidirect
3. #20272714C2semidirect
4. #20272711C2semidirect
5. C53×#1799C2semidirect
7. C2×C4×D212C4semidirect
9. #16810548C4semidirect
11. #16810532C4semidirect
15. #16810489C4semidirect
204. #1799D53semidirect
234. C53#11875semidirect

Maximal subgroups

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

subgroupindex
#202796122
#202730332
#202727142
#202727112
C53×#17992
#1187553