Finite Group Databasebeta

← #27277574

Subgroups — #27277574

Order 13568.

418 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 83 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#20277211C211yesmaximal, characteristic
2#20274796C211yesmaximal, characteristic
3#20274790C211yesmaximal, characteristic
4#20272569C211yesmaximal, characteristic
5#20272568C211yesmaximal, characteristic
6#20272567C211yesmaximal, characteristic
7C53×#1284C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810517—21no—
11#16810511—21no—
12#16810508—21no—
13#16810506—21no—
14#16810486C2211yescharacteristic
15C4×#11719919C411yescharacteristic, complemented
16C4×#11719919C411yescharacteristic, complemented
17C4×C22⋊Dic53C2211yescharacteristic
18#16810469C411yescharacteristic, complemented
19C42×Dic53C411yescharacteristic, complemented
20#16810391—21no—
21#16810391—21no—
22#16810389—21no—
23#16810389—21no—
24C53×#407C2211yescharacteristic
25C53×#381C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C2×#11719917—21no—
28#13133176—21no—
29#13133176—21no—
30#13133176—21no—
31#13133176—21no—
32C2×C4⋊Dic53D411yescharacteristic
33C2×C4⋊Dic53D411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53D411yescharacteristic
39C2×C22⋊Dic53D411yescharacteristic
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53C2×C411yescharacteristic
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55C23×C4×C53C2311yescharacteristic
56C2×M16×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C42×C53C2×C411yescharacteristic
60C2×C42×C53C2×C411yescharacteristic
61C22⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65#95×C53—21no—
66C24×C53C2×D411yescharacteristic
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4111yescharacteristic
75C22×C4×C53#4111yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—41no—
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C22⋊C411yescharacteristic
85M16×C53—41no—
86M16×C53—41no—
87C2×C8×C53—41no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C42×C53—41no—
92C42×C53—41no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53—42no—
98C22⋊Dic53—41no—
99C22⋊Dic53—41no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53—21no—
104#11719919—42no—
105#11719919—42no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53—21no—
109C4×Dic53—21no—
110C4×Dic53—42no—
111C4×Dic53—42no—
112C4×Dic53—41no—
113C4×Dic53—41no—
114#11719917—41no—
115#11719917—41no—
116C2×#88322—41no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53#12311yescharacteristic
123C23×C53#12311yescharacteristic
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—22no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53#11711yescharacteristic
142C2×C4×C53#11711yescharacteristic
143C2×C4×C53#12311yescharacteristic
144C2×C4×C53#12311yescharacteristic
145C2×Dic53—22no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—81no—
160C2×Dic53—81no—
161C2×Dic53—81no—
162C8×C53—81no—
163#88322—81no—
164#8741—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53#38111yescharacteristic
176C22×C53#41011yescharacteristic
177C22×C53#41011yescharacteristic
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—42no—
183C4×C53—81no—
184C4×C53—161no—
185Dic53—42no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190Dic53—81no—
191Dic53—161no—
192#2298—531no—
193#1822—531no—
194#1820—531no—
195#1284—531no—
196#1284—531no—
197#1284—531no—
198#1284D5311yescharacteristic
199C2×C53—21no—
200C2×C53—21no—
201C2×C53—41no—
202C2×C53—41no—
203C2×C53#182011yescharacteristic
204C2×C53#182711yescharacteristic
205C2×C53#182711yescharacteristic
206C2×#117—531no—
207#407—531no—
208#407C2×D5311yescharacteristic
209#388—1061no—
210#388—1061no—
211#387—1061no—
212#387—1061no—
213#381—531no—
214#381C2×D5311yescharacteristic
215C4×C4⋊C4—531no—
216C4×C4⋊C4—531no—
217C4×C22⋊C4—531no—
218C4×C22⋊C4C2×D5311yescharacteristic
219#376—531no—
220C43—531no—
221#343—1061no—
222#343—1061no—
223#342—1061no—
224#342—1061no—
225C53#874111yescharacteristic, complemented, Sylow
226C23×C4C22×D5311yescharacteristic
227C2×M16—21no—
228C2×M16—1061no—
229#117—1061no—
230#117—1061no—
231#117—1061no—
232#117—1061no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C42—531no—
244C2×C42—531no—
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—1061no—
248C2×C42—1061no—
249C2×C42C4×D5311yescharacteristic
250C2×C42C4×D5311yescharacteristic
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—1061no—
254C22⋊C8—1061no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262#95—1061no—
263#95—1061no—
264#95—1061no—
265C24D4×D5311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—2121no—
278C22×C4C2×C4×D5311yescharacteristic
279C22×C4#1171994511yescharacteristic
280C22×C4#1171994511yescharacteristic
281C22×C4D4×D5311yescharacteristic
282C22×C4#1171994711yescharacteristic
283C22×C4#1171994911yescharacteristic
284M16—41no—
285M16—41no—
286M16—2121no—
287M16—2121no—
288C2×C8—41no—
289C2×C8—2121no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—2122no—
295C4⋊C4—2122no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—2122no—
304C22⋊C4—2121no—
305C22⋊C4—2121no—
306C42—41no—
307C42—41no—
308C42—1061no—
309C42—1061no—
310C42—1061no—
311C42—1061no—
312C42—2121no—
313C42—2121no—
314C42—2122no—
315C42—2122no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23C22⋊C4×D5311yescharacteristic
321C23#1313319911yescharacteristic
322C23#1313320211yescharacteristic
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—1062no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—4241no—
346C2×C4—4241no—
347C2×C4—4241no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4#1313317411yescharacteristic
358C2×C4#1313317511yescharacteristic
359C2×C4#1313319911yescharacteristic
360C2×C4#1313320211yescharacteristic
361C8—81no—
362C8—4241no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—81no—
372C22—81no—
373C22#1681048511yescharacteristic
374C22#1681059511yescharacteristic
375C22#1681059511yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—81no—
382C4—161no—
383C4—8481no—
384C4—4241no—
385C4—2122no—
386C4—2121no—
387C4—2121no—
388C4—2121no—
389C4—2121no—
390C2—21no—
391C2—21no—
392C2—41no—
393C2—41no—
394C2#2027479111yescharacteristic
395C2#2027481711yescharacteristic
396C2#2027481811yescharacteristic
subgroupG/H
centerC22#16810485
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1284C2
socleC22×C53#381
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1284D53
hypocenterC53#8741
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
15. C4×#11719919C4semidirect
16. C4×#11719919C4semidirect
18. #16810469C4semidirect
19. C42×Dic53C4semidirect
225. C53#8741semidirect

Maximal subgroups

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

subgroupindex
#202772112
#202747962
#202747902
#202725692
#202725682
#202725672
C53×#12842
#874153