Finite Group Databasebeta

← #27281122

Subgroups — #27281122

Order 13568.

478 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277881C211yesmaximal, characteristic
2#20275701C211yesmaximal, characteristic
3#20273248C211yesmaximal, characteristic
4#20273235C211yesmaximal, characteristic
5#20272622C211yesmaximal, characteristic
6#20272569C211yesmaximal, characteristic
7C53×#3195C211yesmaximal, characteristic
8#16810587C2211yescharacteristic
9#16810530—21no—
10#16810522—21no—
11#16810516—21no—
12#16810516—21no—
13#16810509—21no—
14#16810509—21no—
15#16810486C2211yescharacteristic
16#16810483C2211yescharacteristic
17C4×C22⋊Dic53C2211yescharacteristic
18#16810390—21no—
19#16810390—21no—
20#16810360—21no—
21#16810360—21no—
22#16810360—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×C22⋊Q8×C53C2211yescharacteristic
25C53×#483—21no—
26C53×#483C412yescomplemented
27C53×#470—22no—
28C53×#451—21no—
29C53×#451C412yescomplemented
30C53×#407C2211yescharacteristic
31C2×#11719917—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53D411yescharacteristic
38C2×C22⋊Dic53D411yescharacteristic
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124D411yescharacteristic
52#13133124D411yescharacteristic
53C22×Q8×C53C2×C411yescharacteristic
54C22×Q8×C53D411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C2×Q16×C53—22no—
57C2×M16×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53—21no—
63C22⋊Q8×C53—21no—
64C22⋊Q8×C53C2×C412yes—
65C22⋊Q8×C53D412yes—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53C2×C411yescharacteristic
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72#106×C53—22no—
73#103×C53—22no—
74#103×C53—22no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53C2×C412yes—
77C24×C53C2×D411yescharacteristic
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—81no—
82C2×Q8×C53—81no—
83C2×Q8×C53C22⋊C412yes—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—81no—
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—21no—
99C22×Dic53—41no—
100Q16×C53—82no—
101M16×C53—81no—
102C2×C8×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53C22⋊C412yes—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊Dic53—22no—
116C22⋊Dic53—41no—
117C22⋊Dic53—41no—
118C22⋊Dic53—41no—
119C22⋊Dic53—81no—
120C4⋊Dic53—81no—
121C4⋊Dic53—81no—
122C4⋊Dic53—81no—
123C4×Dic53—81no—
124C4×Dic53—81no—
125C4×Dic53—81no—
126#11719917—81no—
127C2×#88322—41no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53C4×D411yescharacteristic
133C23×C53#12011yescharacteristic
134C23×C53#12111yescharacteristic
135Q8×C53—42no—
136Q8×C53—42no—
137Q8×C53—81no—
138Q8×C53—81no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
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—81no—
154C2×C4×C53—81no—
155C2×C4×C53C2×C22⋊C411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—42no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C8×C53—42no—
175#88322—81no—
176#9342—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#38611yescharacteristic
188C22×C53#45511yescharacteristic
189C22×C53#45611yescharacteristic
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—42no—
195C4×C53—42no—
196C4×C53—81no—
197C4×C53—81no—
198Dic53—81no—
199Dic53—81no—
200Dic53—81no—
201Dic53—81no—
202Dic53—82no—
203#3195D5311yescharacteristic
204#2378—531no—
205#1996—531no—
206#1511—531no—
207#1508—531no—
208#1305—531no—
209#1284—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—41no—
213C2×C53—41no—
214C2×C53#188211yescharacteristic
215C2×C53#190511yescharacteristic
216C2×C53#191311yescharacteristic
217C2×C22⋊Q8C2×D5311yescharacteristic
218C2×C22⋊Q8C2×D5311yescharacteristic
219#483—21no—
220#483Dic5312yescomplemented
221#470—22no—
222#451—21no—
223#451Dic5312yescomplemented
224#407—531no—
225#407C2×D5311yescharacteristic
226#391—1061no—
227#389—1061no—
228#388—1061no—
229#388—1061no—
230#387—1061no—
231#387—1061no—
232#381—531no—
233#380—531no—
234C4×C22⋊C4—531no—
235#342—1061no—
236#342—1061no—
237#328—1061no—
238#328—1061no—
239#328—1062no—
240C53#934211yescharacteristic, complemented, Sylow
241C22×Q8C2×Dic5311yescharacteristic
242C22×Q8C22⋊D5311yescharacteristic
243C23×C4C22×D5311yescharacteristic
244C2×Q16—22no—
245C2×M16—21no—
246C2×M16—1061no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8C2×Dic5312yes—
254C22⋊Q8C22⋊D5312yes—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4C2×Dic5311yescharacteristic
261C2×C4⋊C4C22⋊D5311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—531no—
265C2×C22⋊C4—531no—
266C2×C22⋊C4—1061no—
267C2×C22⋊C4—1061no—
268C2×C42—1061no—
269C2×C42—1061no—
270C2×C42—1061no—
271#106—22no—
272#103—22no—
273#103—22no—
274C22⋊C8—21no—
275C22⋊C8—1061no—
276C22⋊C8—1061no—
277C22⋊C8C2×Dic5312yes—
278#95—531no—
279#95—531no—
280#95—1061no—
281#95—1061no—
282#95—1061no—
283#95—1061no—
284#95—1061no—
285#95—1061no—
286#95—1061no—
287C24D4×D5311yescharacteristic
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—81no—
292C2×Q8—81no—
293C2×Q8C22⋊Dic5312yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—81no—
299C22×C4—1061no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—2121no—
304C22×C4D4×D5311yescharacteristic
305C22×C4#1171994711yescharacteristic
306C22×C4#1171994711yescharacteristic
307C22×C4C22×Dic5311yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310Q16—82no—
311M16—81no—
312M16—4241no—
313C2×C8—22no—
314C2×C8—2121no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—4241no—
323C4⋊C4—4241no—
324C4⋊C4—4241no—
325C4⋊C4C22⋊Dic5312yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—1062no—
331C22⋊C4—4241no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C22⋊C4—2121no—
335C42—4241no—
336C42—4241no—
337C42—4241no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23D4×Dic5311yescharacteristic
343C23#1313318511yescharacteristic
344C23#1313318911yescharacteristic
345Q8—42no—
346Q8—42no—
347Q8—81no—
348Q8—81no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
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—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—4241no—
366C2×C4—4241no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2122no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4C2×C22⋊Dic5311yescharacteristic
381C2×C4D4×Dic5311yescharacteristic
382C2×C4#1313318911yescharacteristic
383C2×C4#1313319811yescharacteristic
384C8—42no—
385C8—4241no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—81no—
395C22—81no—
396C22#1681050311yescharacteristic
397C22#1681079911yescharacteristic
398C22#1681080411yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—4241no—
408C4—4241no—
409C4—4241no—
410C4—4241no—
411C4—4242no—
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#2027508611yescharacteristic
417C2#2027521111yescharacteristic
418C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3195C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#3195D53
hypocenterC53#9342
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
26. C53×#483C4semidirect
29. C53×#451C4semidirect
220. #483Dic53semidirect
223. #451Dic53semidirect
240. C53#9342semidirect

Maximal subgroups

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

subgroupindex
#202778812
#202757012
#202732482
#202732352
#202726222
#202725692
C53×#31952
#934253