Finite Group Databasebeta

← #27358280

Subgroups — #27358280

Order 13568.

976 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 183 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#20285517C212yesmaximal
2#20280195C211yesmaximal, characteristic
3#20273757C214yesmaximal, complemented
4#20273655C214yesmaximal, complemented
5#20273235C212yesmaximal, complemented
6#20273045C211yesmaximal, characteristic, complemented
7C2×C53×#544C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810956C2214yes—
10#16810936—22no—
11#16810871—24no—
12#16810773C2214yes—
13#16810762—22no—
14#16810587C2212yes—
15#16810562C2212yes—
16#16810540C2212yes—
17#16810513—22no—
18#16810502—21no—
19#16810502—21no—
20C4×C22⋊Dic53C2211yescharacteristic
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—
35C2×#11719917—22no—
36#13133191—22no—
37#13133191—22no—
38C2×C4⋊Dic53C2311yescharacteristic
39C2×C22⋊Dic53—21no—
40C2×#11719921—22no—
41C2×#11719921—21no—
42C2×#11719921—22no—
43C2×C4×Dic53—21no—
44#13133150—24no—
45#13133141—24no—
46#13133137—24no—
47C22⋊#88322—22no—
48C22⋊#88322C2314yes—
49#13133124—22no—
50#13133124—22no—
51#13133124C2312yes—
52C22×Q8×C53—21no—
53C22×Q8×C53D412yes—
54C23×C4×C53D411yescharacteristic
55C23×C4×C53D411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C4⋊Q8×C53—28no—
58C4⋊Q8×C53—28no—
59#125×C53—28no—
60#125×C53—28no—
61C22⋊Q8×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53D414yes—
66C22⋊Q8×C53C2314yes—
67#117×C53—24no—
68#117×C53—24no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—22no—
73C2×C4⋊C4×C53—22no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53D412yes—
77C2×C4⋊C4×C53D412yes—
78C2×C4⋊C4×C53C2312yes—
79C2×C22⋊C4×C53—22no—
80C2×C42×C53—22no—
81C24×C53C2×D411yescharacteristic
82C23×D53—21no—
83C2×Q8×C53—24no—
84C2×Q8×C53—22no—
85C2×Q8×C53—82no—
86C2×Q8×C53—82no—
87C2×Q8×C53C2×D414yes—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—24no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—81no—
96C22×C4×C53—81no—
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2411yescharacteristic
103C2×C22⋊D53—44no—
104C2×C22⋊D53—41no—
105C2×C22⋊D53—41no—
106C22×Dic53—21no—
107C22×Dic53—22no—
108C2×D212—22no—
109C2×D212—81no—
110C4⋊C4×C53—24no—
111C4⋊C4×C53—24no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—28no—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—24no—
118C4⋊C4×C53—24no—
119C4⋊C4×C53C2×D414yes—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—24no—
122C42×C53—28no—
123C22⋊Dic53—42no—
124#11719924—84no—
125#11719921—42no—
126#11719921—44no—
127#11719921—82no—
128C4⋊Dic53—22no—
129C4×Dic53—81no—
130#11719917—82no—
131C2×#88322—24no—
132C23×C53—22no—
133C23×C53—22no—
134C23×C53—41no—
135C23×C53#12011yescharacteristic
136C23×C53#12011yescharacteristic
137C23×C53C22×D411yescharacteristic
138C22×D53—42no—
139C22×D53—42no—
140C22×D53—41no—
141C22×D53—42no—
142Q8×C53—44no—
143Q8×C53—82no—
144Q8×C53—84no—
145C2×C4×C53—24no—
146C2×C4×C53—24no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—24no—
151C2×C4×C53—24no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—44no—
158C2×C4×C53—44no—
159C2×C4×C53—42no—
160C2×C4×C53—81no—
161C2×C4×C53#12012yes—
162C2×C4×C53#13714yes—
163C2×C4×C53C22×D412yes—
164C22⋊D53—82no—
165C22⋊D53—82no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171D212—44no—
172#88322—44no—
173#18326—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—44no—
182C22×C53#45412yes—
183C22×C53C2×#12011yescharacteristic
184C22×C53C2×#13712yes—
185C2×D53—84no—
186C2×D53—81no—
187C2×D53—81no—
188C2×D53—81no—
189C4×C53—24no—
190C4×C53—22no—
191C4×C53—22no—
192C4×C53—44no—
193C4×C53—42no—
194C4×C53—44no—
195Dic53—82no—
196Dic53—81no—
197C2×#544D5311yescharacteristic, complemented
198#3193—532no—
199#2591—531no—
200#1617—534no—
201#1599—534no—
202#1508—532no—
203#1463—531no—
204C2×C53—22no—
205C2×C53—22no—
206C2×C53—41no—
207C2×C53#300411yescharacteristic
208C2×C53C2×#45411yescharacteristic
209C2×C53#302611yescharacteristic
210D53—161no—
211#544—24no—
212#544—22no—
213#544C2×D5314yes—
214C2×C4⋊Q8—22no—
215C2×#125—22no—
216C2×C22⋊Q8—21no—
217C2×C22⋊Q8C2×D5312yes—
218C2×#121—531no—
219C2×#117—21no—
220C22×C4⋊C4C2×D5311yescharacteristic
221#484—534no—
222#480—1062no—
223#468—1064no—
224#450—534no—
225#448—1062no—
226#407—532no—
227#400—532no—
228#394—532no—
229#388—1062no—
230#386—1061no—
231#386—1061no—
232C4×C22⋊C4—531no—
233#328—538no—
234#324—534no—
235C53#1832611yescharacteristic, complemented, Sylow
236C22×Q8—21no—
237C22×Q8C22⋊D5312yes—
238C22×D4—531no—
239C22×D4—1061no—
240C23×C4C22⋊D5311yescharacteristic
241C23×C4C22⋊D5311yescharacteristic
242C23×C4C22×D5311yescharacteristic
243C2×SD16—1064no—
244C2×M16—1062no—
245C4⋊Q8—28no—
246C4⋊Q8—28no—
247#125—28no—
248#125—28no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8C22⋊D5314yes—
254C22⋊Q8C22×D5314yes—
255#121—1062no—
256#121—1062no—
257#117—24no—
258#117—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4C22⋊D5312yes—
268C2×C4⋊C4C22⋊D5312yes—
269C2×C4⋊C4C22×D5312yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—1062no—
272C2×C22⋊C4—1061no—
273C2×C22⋊C4—1061no—
274C2×C22⋊C4—1062no—
275C2×C42—22no—
276C2×C42—1061no—
277#107—1064no—
278#103—1064no—
279#102—1064no—
280C22⋊C8—534no—
281C22⋊C8—1062no—
282#95—532no—
283#95—1062no—
284#95—1062no—
285C24—1061no—
286C24C2×C22⋊D5311yescharacteristic
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8C2×C22⋊D5314yes—
292C2×D4—1062no—
293C2×D4—4241no—
294C2×D4—2124no—
295C2×D4—2121no—
296C2×D4—2121no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—1062no—
307C22×C4—1061no—
308C22×C4C2×C22⋊D5312yes—
309C22×C4C2×C22⋊D5312yes—
310C22×C4C2×C22⋊D5311yescharacteristic
311C22×C4C2×C22⋊D5312yes—
312C22×C4C2×C22⋊D5312yes—
313C22×C4C23×D5311yescharacteristic
314SD16—4244no—
315M16—4242no—
316C2×C8—1064no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—1062no—
327C4⋊C4C2×C22⋊D5314yes—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—4242no—
331C22⋊C4—2122no—
332C22⋊C4—2122no—
333C22⋊C4—2124no—
334C42—28no—
335C42—4241no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—2122no—
340C23—2122no—
341C23—2121no—
342C23—2122no—
343C23#1313318611yescharacteristic
344C23#1313318611yescharacteristic
345C23C22×C22⋊D5311yescharacteristic
346Q8—44no—
347Q8—82no—
348Q8—84no—
349D4—4242no—
350D4—4242no—
351D4—2124no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2121no—
372C2×C4—2122no—
373C2×C4#1313318612yes—
374C2×C4#1313324814yes—
375C2×C4C22×C22⋊D5312yes—
376C8—2124no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—44no—
385C22—4244no—
386C22—4241no—
387C22—4241no—
388C22—4241no—
389C22#1681079612yes—
390C22C2×#1313318611yescharacteristic
391C22C2×#1313324812yes—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—44no—
396C4—42no—
397C4—44no—
398C4—4242no—
399C4—4241no—
400C2—22no—
401C2—22no—
402C2—41no—
403C2—8481no—
404C2#2028394911yescharacteristic
405C2C2×#1681079611yescharacteristic
406C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#544C2
socleC22×C53C2×#120
prime radicalC2×#13133191C22
hypercenterC2×#544D53
hypocenterC53#18326
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. #20273757C2semidirect
4. #20273655C2semidirect
5. #20273235C2semidirect
6. #20273045C2semidirect
7. C2×C53×#544C2semidirect
197. C2×#544D53semidirect
235. C53#18326semidirect

Maximal subgroups

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

subgroupindex
#202855172
#202801952
#202737572
#202736552
#202732352
#202730452
C2×C53×#5442
#1832653