Finite Group Databasebeta

← #27272347

Subgroups — #27272347

Order 13568.

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

normal subgroups only

This group has 556 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#20277253C211yesmaximal, characteristic
2#20274476C211yesmaximal, characteristic
3#20274407C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic, complemented
5#20272539C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1272C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C2×#13133193C411yescharacteristic
10C2×Q8×Dic53C411yescharacteristic
11C2×C42×D53C2211yescharacteristic
12C2×#13133129—21no—
13C2×#13133127—21no—
14#16810521C2211yescharacteristic
15#16810510—21no—
16#16810497—21no—
17#16810480C411yescharacteristic
18C4⋊C4×Dic53C411yescharacteristic
19#16810474—21no—
20#16810470—21no—
21#16810378C2211yescharacteristic
22#16810378C2211yescharacteristic
23#16810377—21no—
24C53×#391C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C2×#11719912—24no—
31C2×#11719912—42no—
32C2×#11719912—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133193—22no—
36#13133193—22no—
37Q8×Dic53—42no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53C2×C411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C42×D53—42no—
51#13133129—24no—
52#13133127—42no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56C22×Q8×C53C2×C411yescharacteristic
57C22×C8×C53—21no—
58C22×C8×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61#95×C53—21no—
62#95×C53—21no—
63C23×D53C22⋊C411yescharacteristic
64C2×Q8×C53—21no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53M1612yescomplemented
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53C22⋊C411yescharacteristic
78C2×C4×D53—22no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—22no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—22no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C8×C53—22no—
91C2×C8×C53—22no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C4⋊C4×C53—41no—
97C42×C53—41no—
98#11719921—24no—
99#11719921—42no—
100C4⋊Dic53—21no—
101C4⋊Dic53—42no—
102C4⋊Dic53—41no—
103C4⋊Dic53M1612yescomplemented
104#11719919—24no—
105#11719919—41no—
106#11719919—41no—
107C4×Dic53—22no—
108C4×Dic53—22no—
109C4×Dic53—41no—
110C4×Dic53—42no—
111C4×Dic53—42no—
112C2×#88322—22no—
113C2×#88322—22no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118#11719912—48no—
119C23×C53C2×C22⋊C411yescharacteristic
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53#10412yes—
127Q8×C53—42no—
128Q8×C53—42no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53C2×M1611yescharacteristic
142C2×C4×C53C2×M1611yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53#13711yescharacteristic
145C2×C4×C53#13711yescharacteristic
146C2×C4×C53#13711yescharacteristic
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—22no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53#9912yes—
161C2×Dic53#10412yes—
162C4×D53—24no—
163C4×D53—24no—
164C4×D53—42no—
165C4×D53—42no—
166C4×D53—42no—
167C4×D53—42no—
168C8×C53—42no—
169C8×C53—42no—
170#88322—42no—
171#88322—42no—
172#7634—531nomaximal, Sylow
173C22×C53#40711yescharacteristic
174C22×C53C2×#9911yescharacteristic
175C22×C53#41311yescharacteristic
176C22×C53#41711yescharacteristic
177C22×C53#41911yescharacteristic
178C22×C53C2×#10411yescharacteristic
179C22×C53C2×#10411yescharacteristic
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—42no—
194C4×C53—81no—
195Dic53—22no—
196Dic53—22no—
197Dic53—41no—
198Dic53—42no—
199Dic53—81no—
200C4×C22⋊Q8—531no—
201#1749—531no—
202#1728—531no—
203#1272—531no—
204#1272D5311yescharacteristic, complemented
205#1271—531no—
206#1271—531no—
207C2×C53#146511yescharacteristic
208C2×C53#148911yescharacteristic
209C2×C53#149111yescharacteristic
210C2×C53#150411yescharacteristic
211C2×C53#150811yescharacteristic
212C2×C53#152411yescharacteristic
213C2×C53#155911yescharacteristic
214D53—42no—
215C22×M16—1061no—
216C2×C22⋊Q8—531no—
217C2×C4×Q8—531no—
218C22×C42—531no—
219C2×C22⋊C8—1061no—
220C2×#97—1061no—
221#391C2×D5311yescharacteristic
222#389—531no—
223#387—1061no—
224#385—1061no—
225#380—531no—
226C4×C4⋊C4—531no—
227C4×C4⋊C4—1061no—
228C4×C22⋊C4—1061no—
229#336—531no—
230#336—531no—
231#336—1061no—
232#336C2×D5311yescharacteristic
233#336C2×D5311yescharacteristic
234C53#763411yescharacteristic, complemented, Sylow
235C22×Q8C4×D5311yescharacteristic
236C23×C4—531no—
237C23×C4—531no—
238C23×C4—531no—
239C2×M16—1064no—
240C2×M16—2122no—
241C2×M16—2122no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—1061no—
245C22×C8—1061no—
246C22⋊Q8—1062no—
247C22⋊Q8—1062no—
248C4×Q8—2122no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4C4×D5311yescharacteristic
255C2×C22⋊C4—1061no—
256C2×C22⋊C4—1061no—
257C2×C42—531no—
258C2×C42—531no—
259C2×C42—531no—
260C2×C42—1061no—
261C2×C42—1061no—
262C2×C42—2122no—
263C2×C42C22×D5311yescharacteristic
264C22⋊C8—1064no—
265#97—2122no—
266#95—21no—
267#95—21no—
268#95—1061no—
269#95—1061no—
270#95—1061no—
271C24—531no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8#1171991212yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—1062no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4C2×C4×D5311yescharacteristic
296C22×C4D4×D5311yescharacteristic
297C22×C4D4×D5311yescharacteristic
298M16—2128no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—1062no—
306C2×C8—1062no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8—2121no—
310C2×C8—2121no—
311C4⋊C4—41no—
312C4⋊C4—532no—
313C4⋊C4—1064no—
314C4⋊C4—1061no—
315C4⋊C4—2121no—
316C4⋊C4—2122no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C22⋊C4—1064no—
320C22⋊C4—2122no—
321C42—41no—
322C42—1062no—
323C42—1062no—
324C42—2121no—
325C42—2122no—
326C42—2122no—
327C23—532no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23C22⋊C4×D5311yescharacteristic
335Q8—42no—
336Q8—42no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—532no—
350C2×C4—532no—
351C2×C4—1064no—
352C2×C4—1064no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1062no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4M16×D5311yescharacteristic
371C2×C4C2×#1171991211yescharacteristic
372C2×C4#1313324411yescharacteristic
373C2×C4#1313324611yescharacteristic
374C2×C4#1313324911yescharacteristic
375C2×C4#1313324911yescharacteristic
376C8—42no—
377C8—42no—
378C8—2122no—
379C8—2122no—
380C22—1062no—
381C22—1062no—
382C22—1062no—
383C22—2121no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22#1681058511yescharacteristic
388C22#99×D5311yescharacteristic
389C22#1681060411yescharacteristic
390C22#1681062311yescharacteristic
391C22#1681063011yescharacteristic
392C22#104×D5311yescharacteristic
393C22#104×D5311yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—1062no—
403C4—1062no—
404C4—4241no—
405C4—2122no—
406C4—2121no—
407C2—2122no—
408C2#2027305011yescharacteristic
409C2#2027315311yescharacteristic
410C2#2027316011yescharacteristic
411C2#2027321811yescharacteristic
412C2#2027323311yescharacteristic
413C2#2027330011yescharacteristic
414C2#2027344211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7634
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #20272541C2semidirect
5. #20272539C2semidirect
6. #20272537C2semidirect
7. C53×#1272C2semidirect
67. C2×Q8×C53M16semidirect
103. C4⋊Dic53M16semidirect
204. #1272D53semidirect
234. C53#7634semidirect
275. C2×Q8#11719912semidirect

Maximal subgroups

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

subgroupindex
#202772532
#202744762
#202744072
#202725412
#202725392
#202725372
C53×#12722
#763453