Finite Group Databasebeta

← #27280522

Subgroups — #27280522

Order 13568.

524 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16810956C211yesmaximal, characteristic
2#20277594C211yesmaximal, characteristic
3#20275701C211yesmaximal, characteristic
4#20274806C211yesmaximal, characteristic
5#20272582C211yesmaximal, characteristic
6#20272561C211yesmaximal, characteristic
7C53×#1520C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810956—21no—
10#16810956—22no—
11#16810956—21no—
12C2×#13133150C412yescomplemented
13C2×#13133141C412yescomplemented
14C2×C22⋊#88322C2211yescharacteristic
15#16810519—21no—
16#16810508—22no—
17#16810483C2211yescharacteristic
18#16810481—21no—
19C4⋊C4×Dic53—21no—
20C22⋊C4×Dic53—21no—
21C4×C22⋊Dic53C2211yescharacteristic
22#16810390—21no—
23#16810390—21no—
24#16810386—22no—
25C2×C22⋊Q8×C53C2211yescharacteristic
26C2×C22⋊C8×C53C2211yescharacteristic
27C53×#380C2211yescharacteristic
28C53×#328—22no—
29C23×Dic53—21no—
30C22×#88322C2×C412yes—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C412yes—
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D412yes—
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719919—22no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53D411yescharacteristic
42C2×C22⋊Dic53D411yescharacteristic
43C2×C4×Dic53—22no—
44C2×C4×Dic53—21no—
45#13133150—22no—
46#13133150—22no—
47#13133141—22no—
48#13133141—22no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51#13133124—22no—
52#13133124—22no—
53#13133124—21no—
54#13133124—21no—
55C22×Q8×C53C2×C411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C22×C8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53C2×C411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C22⋊C8×C53—21no—
68C22⋊C8×C53—21no—
69#95×C53D412yes—
70C24×C53C2×D411yescharacteristic
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—42no—
74C22×C4×C53—22no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—42no—
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—22no—
88C22×Dic53—22no—
89C22×Dic53—21no—
90C22×Dic53—41no—
91C22×Dic53—82no—
92C2×C8×C53—42no—
93C2×C8×C53—41no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—42no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊Dic53—21no—
104C22⋊Dic53—21no—
105C22⋊Dic53—21no—
106C22⋊Dic53—41no—
107C22⋊Dic53—41no—
108C22⋊Dic53Q1612yes—
109C4⋊Dic53—22no—
110C4⋊Dic53—22no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53—42no—
114C4⋊Dic53C22⋊C412yes—
115C4⋊Dic53C22⋊C412yes—
116#11719919—82no—
117C4×Dic53—42no—
118C4×Dic53—82no—
119C2×#88322—22no—
120C2×#88322—42no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53C4×D411yescharacteristic
124C23×C53#12011yescharacteristic
125C23×C53#12111yescharacteristic
126C23×C53C2×Q1611yescharacteristic
127C23×C53#13511yescharacteristic
128C23×C53#13711yescharacteristic
129C23×C53#13711yescharacteristic
130Q8×C53—42no—
131Q8×C53—81no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12311yescharacteristic
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—82no—
160C2×Dic53—82no—
161C2×Dic53—81no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164C8×C53—81no—
165#88322—42no—
166#9257—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53#38611yescharacteristic
178C22×C53C4×Q1611yescharacteristic
179C22×C53#44111yescharacteristic
180C22×C53#45111yescharacteristic
181C22×C53#45211yescharacteristic
182C22×C53#46211yescharacteristic
183C22×C53#46411yescharacteristic
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53#41012yes—
191Dic53—42no—
192Dic53—42no—
193Dic53—42no—
194Dic53—81no—
195Dic53—161no—
196C2×#484—531no—
197#2345—531no—
198#1996—531no—
199#1824—531no—
200#1520D5311yescharacteristic
201#1289—531no—
202#1282—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—21no—
209C2×C53#187811yescharacteristic
210C2×C53#189911yescharacteristic
211C2×C53#190411yescharacteristic
212C2×C22⋊Q8C2×D5311yescharacteristic
213C22×C4⋊C4—531no—
214#484—1062no—
215#484—1061no—
216#484—1061no—
217C2×#107—532no—
218C2×#103—532no—
219C2×C22⋊C8—531no—
220C2×C22⋊C8C2×D5311yescharacteristic
221#389—1061no—
222#387—1062no—
223#380—531no—
224#380—1061no—
225#380C2×D5311yescharacteristic
226C4×C4⋊C4—1061no—
227C4×C22⋊C4—531no—
228C4×C22⋊C4—1061no—
229#342—1061no—
230#342—1061no—
231#340—1062no—
232#328—22no—
233C53#925711yescharacteristic, complemented, Sylow
234C22×Q8C4×D5311yescharacteristic
235C23×C4—1061no—
236C23×C4C22×D5311yescharacteristic
237C22×C8—21no—
238C22×C8—532no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—532no—
246C2×C4⋊C4—532no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1062no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4C4×D5311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—531no—
258C2×C22⋊C4—531no—
259C2×C22⋊C4—1061no—
260C2×C42—1062no—
261C2×C42—1061no—
262#107—1062no—
263#107—1062no—
264#103—1062no—
265#103—1062no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—1061no—
269C22⋊C8—1061no—
270#95—1062no—
271#95—1062no—
272#95—1061no—
273#95—1061no—
274#95C22⋊D5312yes—
275C24D4×D5311yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—4242no—
290C22×C4—2121no—
291C22×C4C2×C4×D5311yescharacteristic
292C22×C4#1171994511yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4D4×D5311yescharacteristic
295C22×C4#1171994711yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—1062no—
300C2×C8—2122no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—532no—
307C4⋊C4—532no—
308C4⋊C4—1062no—
309C4⋊C4—1062no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—4242no—
313C4⋊C4—2122no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—532no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C42—4242no—
325C42—2122no—
326C23—41no—
327C23—41no—
328C23#1313317711yescharacteristic
329C23#1313318511yescharacteristic
330C23#1313319011yescharacteristic
331C23Q16×D5311yescharacteristic
332C23#1313323811yescharacteristic
333C23#1313324611yescharacteristic
334C23#1313324911yescharacteristic
335Q8—42no—
336Q8—81no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—1062no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—4241no—
356C2×C4—4242no—
357C2×C4—4242no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4—2121no—
364C2×C4—2122no—
365C2×C4C22⋊C4×D5311yescharacteristic
366C2×C4C4×C22⋊D5311yescharacteristic
367C2×C4#1313318811yescharacteristic
368C2×C4#1313319911yescharacteristic
369C8—81no—
370C8—2122no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22#1681049911yescharacteristic
382C22#1681071511yescharacteristic
383C22#1681072711yescharacteristic
384C22#1681077711yescharacteristic
385C22#1681078211yescharacteristic
386C22#1681083211yescharacteristic
387C22#1681084811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—81no—
393C4—81no—
394C4—8481no—
395C4—4241no—
396C4—2122no—
397C4—2122no—
398C4—2122no—
399C4#1681059512yes—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2#2027505311yescharacteristic
407C2#2027517011yescharacteristic
408C2#2027519911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1520C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1520D53
hypocenterC53#9257
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
12. C2×#13133150C4semidirect
13. C2×#13133141C4semidirect
233. C53#9257semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
#202775942
#202757012
#202748062
#202725822
#202725612
C53×#15202
#925753