Finite Group Databasebeta

← #27358008

Subgroups — #27358008

Order 13568.

1002 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285468C212yesmaximal
2#20280457C211yesmaximal, characteristic
3#20273723C214yesmaximal, complemented
4#20273655C214yesmaximal, complemented
5#20273287C212yesmaximal, complemented
6C2×#16810349C211yesmaximal, characteristic, complemented
7C2×C53×#544C211yesmaximal, characteristic, complemented
8C2×#13133200—21no—
9C2×#13133191C2211yescharacteristic
10#16810951C2214yes—
11#16810941—22no—
12#16810854—24no—
13#16810773C2214yes—
14#16810762—22no—
15C2×C22⋊#88322C2212yes—
16#16810553C2212yes—
17#16810545—22no—
18#16810540C2212yes—
19#16810502—21no—
20#16810483C2211yescharacteristic
21#16810360C2218yes—
22#16810349C2214yes—
23C53×#544—24no—
24C53×#544—22no—
25C53×#544C2214yes—
26C2×C4⋊Q8×C53—22no—
27C2×#125×C53—22no—
28C2×C22⋊Q8×C53—21no—
29C2×C22⋊Q8×C53C2212yes—
30C2×#117×C53—21no—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×D212C2311yescharacteristic
33C22×C22⋊D53—21no—
34C2×#11719924—24no—
35C22×#88322—22no—
36#13133200—44no—
37#13133191—22no—
38#13133191—22no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2311yescharacteristic
41C2×#11719919—22no—
42C2×C22⋊Dic53—21no—
43C2×#11719921—22no—
44C2×#11719921—21no—
45C2×#11719921—22no—
46#13133148—24no—
47#13133141—24no—
48#13133137—24no—
49C22⋊#88322—22no—
50C22⋊#88322C2314yes—
51#13133124—22no—
52#13133124C2312yes—
53C22×Q8×C53—21no—
54C22×Q8×C53D412yes—
55C23×C4×C53D411yescharacteristic
56C23×C4×C53D411yescharacteristic
57C23×C4×C53C2311yescharacteristic
58C4⋊Q8×C53—28no—
59C4⋊Q8×C53—28no—
60#125×C53—28no—
61#125×C53—28no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—24no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53D414yes—
67C22⋊Q8×C53C2314yes—
68#117×C53—24no—
69#117×C53—24no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—22no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53D412yes—
78C2×C4⋊C4×C53D412yes—
79C2×C4⋊C4×C53C2312yes—
80C2×C22⋊C4×C53—22no—
81C2×C42×C53—22no—
82C24×C53C2×D411yescharacteristic
83C23×D53—21no—
84C2×Q8×C53—24no—
85C2×Q8×C53—22no—
86C2×Q8×C53—82no—
87C2×Q8×C53—82no—
88C2×Q8×C53C2×D414yes—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—81no—
97C22×C4×C53—81no—
98C22×C4×C53SD1614yescomplemented
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D412yes—
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2411yescharacteristic
105C2×C22⋊D53—41no—
106C2×C22⋊D53—41no—
107C2×C22⋊D53—42no—
108C2×C22⋊D53—42no—
109C22×Dic53—21no—
110C22×Dic53—22no—
111C2×D212—22no—
112C2×D212—81no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—28no—
117C4⋊C4×C53—24no—
118C4⋊C4×C53—24no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—24no—
121C4⋊C4×C53—24no—
122C4⋊C4×C53C2×D414yes—
123C22⋊C4×C53—24no—
124C22⋊C4×C53—24no—
125C42×C53—28no—
126C22⋊Dic53—42no—
127#11719924—84no—
128#11719921—42no—
129#11719921—44no—
130#11719921—82no—
131C4⋊Dic53—22no—
132C4⋊Dic53—81no—
133#11719919—44no—
134C2×#88322—24no—
135C2×#88322—82no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53#12011yescharacteristic
139C23×C53#12011yescharacteristic
140C23×C53C2×SD1612yes—
141C23×C53#13712yes—
142C23×C53C22×D411yescharacteristic
143C22×D53—42no—
144C22×D53—42no—
145C22×D53—41no—
146C22×D53—41no—
147C22×D53—41no—
148Q8×C53—44no—
149Q8×C53—82no—
150Q8×C53—84no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—24no—
154C2×C4×C53—24no—
155C2×C4×C53—22no—
156C2×C4×C53—24no—
157C2×C4×C53—24no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—44no—
163C2×C4×C53—42no—
164C2×C4×C53—44no—
165C2×C4×C53—42no—
166C2×C4×C53—81no—
167C2×C4×C53#12012yes—
168C2×C4×C53C2×SD1614yes—
169C2×C4×C53C22×D412yes—
170C22⋊D53—82no—
171C22⋊D53—82no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—42no—
177C2×Dic53—41no—
178D212—44no—
179#88322—44no—
180#18294—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53#44812yes—
190C22×C53#44812yes—
191C22×C53C2×#12011yescharacteristic
192C22×C53C22×SD1611yescharacteristic
193C22×C53C2×#13711yescharacteristic
194C2×D53—82no—
195C2×D53—82no—
196C2×D53—81no—
197C2×D53—81no—
198C2×D53—81no—
199C4×C53—24no—
200C4×C53—22no—
201C4×C53—22no—
202C4×C53—44no—
203C4×C53—42no—
204C4×C53—44no—
205Dic53—82no—
206Dic53—81no—
207C2×#544D5311yescharacteristic, complemented
208#3187—532no—
209#2618—531no—
210#1611—534no—
211#1599—534no—
212#1520—532no—
213C2×#324—531no—
214C2×C53—21no—
215C2×C53—21no—
216C2×C53—22no—
217C2×C53—22no—
218C2×C53C2×#44811yescharacteristic
219C2×C53#302011yescharacteristic
220C2×C53#302611yescharacteristic
221D53—161no—
222#544—24no—
223#544—22no—
224#544C2×D5314yes—
225C2×C4⋊Q8—22no—
226C2×#125—22no—
227C2×#123—1061no—
228C2×C22⋊Q8—21no—
229C2×C22⋊Q8C2×D5312yes—
230C2×#121—531no—
231C2×#117—21no—
232C22×C4⋊C4C2×D5311yescharacteristic
233#483—534no—
234#481—1062no—
235#465—1064no—
236#450—534no—
237#448—1062no—
238C2×C22⋊C8—532no—
239#398—532no—
240#396—1062no—
241#394—532no—
242#386—1061no—
243#380—531no—
244#328—538no—
245#324—534no—
246C53#1829411yescharacteristic, complemented, Sylow
247C22×Q8—21no—
248C22×Q8C22⋊D5312yes—
249C22×D4—531no—
250C22×D4—1061no—
251C23×C4C22⋊D5311yescharacteristic
252C23×C4C22⋊D5311yescharacteristic
253C23×C4C22×D5311yescharacteristic
254C2×SD16—1064no—
255C22×C8—1062no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258#125—28no—
259#125—28no—
260#123—2124no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8C22⋊D5314yes—
266C22⋊Q8C22×D5314yes—
267#121—1062no—
268#121—1062no—
269#117—24no—
270#117—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4—1062no—
281C2×C4⋊C4C22⋊D5312yes—
282C2×C4⋊C4C22⋊D5312yes—
283C2×C4⋊C4C22×D5312yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—1062no—
286C2×C22⋊C4—1061no—
287C2×C22⋊C4—1061no—
288C2×C22⋊C4—1062no—
289C2×C42—22no—
290#106—1064no—
291#103—1064no—
292#102—1064no—
293C22⋊C8—534no—
294C22⋊C8—1062no—
295#95—532no—
296#95—1062no—
297C24—1061no—
298C24C2×C22⋊D5311yescharacteristic
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—82no—
302C2×Q8—82no—
303C2×Q8C2×C22⋊D5314yes—
304C2×D4—1062no—
305C2×D4—4241no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4—2122no—
309C2×D4—2122no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—1062no—
320C22×C4—1061no—
321C22×C4#1171992414yescomplemented
322C22×C4C2×C22⋊D5312yes—
323C22×C4C2×C22⋊D5312yes—
324C22×C4C2×C22⋊D5311yescharacteristic
325C22×C4C2×C22⋊D5312yes—
326C22×C4C2×C22⋊D5312yes—
327C22×C4C23×D5311yescharacteristic
328SD16—4244no—
329C2×C8—1064no—
330C2×C8—4242no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—1062no—
341C4⋊C4—4241no—
342C4⋊C4—2124no—
343C4⋊C4C2×C22⋊D5314yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—4242no—
347C22⋊C4—2122no—
348C22⋊C4—2122no—
349C22⋊C4—2124no—
350C42—28no—
351C23—41no—
352C23—41no—
353C23—2122no—
354C23—2122no—
355C23—2121no—
356C23—2121no—
357C23—2121no—
358C23#1313318611yescharacteristic
359C23#1313318611yescharacteristic
360C23C2×#1171992412yes—
361C23#1313324812yes—
362C23C22×C22⋊D5311yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366D4—4242no—
367D4—4242no—
368D4—2124no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—81no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2121no—
388C2×C4—2122no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4#1313318612yes—
392C2×C4C2×#1171992414yes—
393C2×C4C22×C22⋊D5312yes—
394C8—2124no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—4242no—
404C22—4242no—
405C22—4241no—
406C22—4241no—
407C22—4241no—
408C22#1681076212yes—
409C22#1681076212yes—
410C22C2×#1313318611yescharacteristic
411C22C22×#1171992411yescharacteristic
412C22C2×#1313324811yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—44no—
417C4—42no—
418C4—44no—
419C4—4242no—
420C4—4241no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—8481no—
426C2C2×#1681076211yescharacteristic
427C2#2028407911yescharacteristic
428C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#544C2
socleC22×C53C2×#120
prime radicalC2×#13133191C22
hypercenterC2×#544D53
hypocenterC53#18294
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20273723C2semidirect
4. #20273655C2semidirect
5. #20273287C2semidirect
6. C2×#16810349C2semidirect
7. C2×C53×#544C2semidirect
98. C22×C4×C53SD16semidirect
207. C2×#544D53semidirect
246. C53#18294semidirect
321. C22×C4#11719924semidirect

Maximal subgroups

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

subgroupindex
#202854682
#202804572
#202737232
#202736552
#202732872
C2×#168103492
C2×C53×#5442
#1829453