Finite Group Databasebeta

← #27281752

Subgroups — #27281752

Order 13568.

568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 107 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#20283744C211yesmaximal, characteristic, complemented
2#20273339C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic, complemented
4#20272571C212yesmaximal, complemented
5C53×#2544C211yesmaximal, characteristic
6#16810695—24no—
7#16810680C2212yes—
8#16810680C2212yes—
9#16810587—21no—
10C2×C22⋊#88322—21no—
11#16810363—24no—
12#16810347—21no—
13#16810347—21no—
14C2×#127×C53C411yescharacteristic
15C2×#124×C53C411yescharacteristic
16C2×#124×C53C2211yescharacteristic
17C2×C4×D4×C53C2211yescharacteristic
18C53×#394—21no—
19C53×#394—21no—
20C53×#391C411yescharacteristic
21C53×#390C411yescharacteristic
22C53×#386—21no—
23C53×#386—21no—
24C4×C4⋊C4×C53C2211yescharacteristic
25C2×#11719917—22no—
26C22×#88322—22no—
27C4⋊#88322—22no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133128—22no—
34#13133128D414yescomplemented
35C22×Q8×C53C2×C411yescharacteristic
36C22×D4×C53—21no—
37C22×D4×C53—21no—
38C22×D4×C53C2×C411yescharacteristic
39C22×D4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42#127×C53—42no—
43#124×C53—22no—
44#124×C53—22no—
45#124×C53—42no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53C2×C411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64C24×C53—21no—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C22⋊C411yescharacteristic
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—41no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—42no—
79C2×D4×C53—42no—
80C2×D4×C53—81no—
81C2×D4×C53—81no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—81no—
90C22×C4×C53—81no—
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22×C411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C4⋊C4×C53—22no—
97C4⋊C4×C53—42no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—81no—
106C22⋊C4×C53—81no—
107C42×C53—41no—
108C42×C53—41no—
109C42×C53C2×D412yes—
110C42×C53#4112yes—
111#11719917—82no—
112C2×#88322—24no—
113C2×#88322—42no—
114C2×#88322—82no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53#10012yes—
132C23×C53C2×C22⋊C411yescharacteristic
133Q8×C53—82no—
134D4×C53—44no—
135D4×C53—82no—
136D4×C53—84no—
137D4×C53—84no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—22no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53C4×D412yes—
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12112yes—
162C2×C4×C53#12311yescharacteristic
163#88322—44no—
164#88322—82no—
165#9447—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—22no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53#38611yescharacteristic
186C22×C53C2×#10011yescharacteristic
187C22×C53#41311yescharacteristic
188C4×C53—22no—
189C4×C53—22no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—42no—
193C4×C53—41no—
194C4×C53—41no—
195C4×C53—81no—
196C4×C53—81no—
197#2973—531no—
198#2544D5311yescharacteristic
199#1533—532no—
200#1466—531no—
201#1285—532no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—42no—
205C2×C53—81no—
206C2×C53—161no—
207C2×C53#188411yescharacteristic
208C2×C53#190011yescharacteristic
209C2×C53#191111yescharacteristic
210C2×#127Dic5311yescharacteristic
211C2×#124Dic5311yescharacteristic
212C2×#124C2×D5311yescharacteristic
213C2×C4×D4C2×D5311yescharacteristic
214#435—1064no—
215#432—532no—
216#432—532no—
217#407—1061no—
218C2×C22⋊C8—1061no—
219#394—21no—
220#394—21no—
221#391Dic5311yescharacteristic
222#390Dic5311yescharacteristic
223#386—21no—
224#386—21no—
225C4×C4⋊C4C2×D5311yescharacteristic
226#329—1064no—
227#323—1061no—
228#323—1061no—
229C53#944711yescharacteristic, complemented, Sylow
230C22×Q8C2×Dic5311yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C2×Dic5311yescharacteristic
234C22×D4C2×Dic5311yescharacteristic
235C23×C4C22⋊D5311yescharacteristic
236C23×C4C22⋊D5311yescharacteristic
237C2×M16—1062no—
238C22×C8—1062no—
239#127—42no—
240#124—22no—
241#124—22no—
242#124—42no—
243C4×D4—22no—
244C4×D4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C2×Dic5311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4C22⋊D5311yescharacteristic
254C2×C22⋊C4C22⋊D5311yescharacteristic
255C2×C42C2×Dic5311yescharacteristic
256C2×C42C2×Dic5311yescharacteristic
257C2×C42C22×D5311yescharacteristic
258C4⋊C8—1062no—
259C4⋊C8—1062no—
260C22⋊C8—1062no—
261C22⋊C8—1062no—
262C22⋊C8—1062no—
263C22⋊C8—1062no—
264#97—534no—
265#97—1062no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24—21no—
270C24C22⋊Dic5311yescharacteristic
271C24C22⋊Dic5311yescharacteristic
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—81no—
286C2×D4—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4C22⋊Dic5311yescharacteristic
297C22×C4C22⋊Dic5311yescharacteristic
298C22×C4C22×Dic5311yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300C22×C4C2×C22⋊D5311yescharacteristic
301M16—4242no—
302C2×C8—1064no—
303C2×C8—4242no—
304C2×C8—2122no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C42—41no—
317C42—41no—
318C42D4×D5312yes—
319C42#1171994712yes—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1313313412yes—
337C23C2×C22⋊Dic5311yescharacteristic
338Q8—82no—
339D4—44no—
340D4—82no—
341D4—84no—
342D4—84no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4D4×Dic5312yes—
365C2×C4#1313318511yescharacteristic
366C2×C4#1313318912yes—
367C2×C4#1313319811yescharacteristic
368C8—4242no—
369C8—2124no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#1681050311yescharacteristic
390C22C2×#1313313411yescharacteristic
391C22#1681060711yescharacteristic
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—42no—
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—81no—
405C2—161no—
406C2#2027509911yescharacteristic
407C2#2027517911yescharacteristic
408C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2544C2
socleC22×C53#386
prime radicalC2×#127×C53C4
cosocleC2×C42×C53C23
hypercenter#2544D53
hypocenterC53#9447
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2544

Complements

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

NH ≅ G/Ntype
1. #20283744C2semidirect
2. #20273339C2semidirect
3. #20273056C2semidirect
4. #20272571C2semidirect
34. #13133128D4semidirect
229. C53#9447semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202733392
#202730562
#202725712
C53×#25442
#944753