Finite Group Databasebeta

← #27280519

Subgroups — #27280519

Order 13568.

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

normal subgroups only

This group has 552 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×#16810954C211yesmaximal, characteristic
2#20277586C211yesmaximal, characteristic
3#20275699C211yesmaximal, characteristic
4#20274799C211yesmaximal, characteristic
5#20273286C211yesmaximal, characteristic
6#20272562C211yesmaximal, characteristic
7C53×#1289C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9C2×#13133173—21no—
10C22×C4⋊Dic53C2211yescharacteristic
11#16810954—21no—
12#16810954—22no—
13#16810954—21no—
14C2×#13133149C412yescomplemented
15C2×#13133142C412yescomplemented
16C2×C22⋊#88322C2211yescharacteristic
17#16810519—21no—
18#16810506—22no—
19#16810484—21no—
20C4⋊C4×Dic53—21no—
21C4×C22⋊Dic53C2211yescharacteristic
22#16810389—21no—
23#16810389—21no—
24#16810385—22no—
25#16810359—22no—
26C2×C22⋊C8×C53C2211yescharacteristic
27C53×#380C2211yescharacteristic
28C53×#380C2211yescharacteristic
29C22×Dic106C2×C411yescharacteristic
30C23×Dic53—21no—
31C22×#88322—21no—
32#13133196—21no—
33#13133196—21no—
34#13133196—21no—
35#13133196—21no—
36#13133173—42no—
37#13133173—42no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C4⋊Dic53C2×C412yes—
43C2×C4⋊Dic53D411yescharacteristic
44C2×C4⋊Dic53D412yes—
45C2×C4⋊Dic53D411yescharacteristic
46C2×#11719919—22no—
47C2×#11719919—21no—
48C2×C22⋊Dic53—21no—
49C2×C22⋊Dic53D411yescharacteristic
50C2×C22⋊Dic53D411yescharacteristic
51C2×C4×Dic53—22no—
52C2×C4×Dic53—21no—
53#13133149—22no—
54#13133149—22no—
55#13133142—22no—
56#13133142—22no—
57C22⋊#88322—21no—
58C22⋊#88322—21no—
59#13133124—22no—
60#13133124—22no—
61#13133124—21no—
62C23×C4×C53C2311yescharacteristic
63C22×C8×C53C2×C412yes—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C22⋊C8×C53—21no—
69C22⋊C8×C53—21no—
70#95×C53—21no—
71#95×C53D412yes—
72C24×C53C2×D411yescharacteristic
73C22×C4×C53—22no—
74C22×C4×C53—42no—
75C22×C4×C53—41no—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×Dic53—22no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—22no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C2×Dic106—21no—
95C2×Dic106—21no—
96C2×Dic106—42no—
97C2×C8×C53—22no—
98C2×C8×C53—42no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—42no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊Dic53—21no—
106C22⋊Dic53—21no—
107C22⋊Dic53—21no—
108C22⋊Dic53—41no—
109C22⋊Dic53—41no—
110C22⋊Dic53SD1612yes—
111C4⋊Dic53—22no—
112C4⋊Dic53—22no—
113C4⋊Dic53—21no—
114C4⋊Dic53—21no—
115C4⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4⋊Dic53C22⋊C412yes—
118C4⋊Dic53C22⋊C412yes—
119#11719919—44no—
120#11719919—41no—
121#11719919—41no—
122C4×Dic53—44no—
123C4×Dic53—42no—
124C2×#88322—42no—
125C2×#88322—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53C4×D411yescharacteristic
129C23×C53#12011yescharacteristic
130C23×C53#12111yescharacteristic
131C23×C53C2×SD1611yescharacteristic
132C23×C53#13511yescharacteristic
133C23×C53AGL(1,ℤ/8)11yescharacteristic
134C23×C53#13711yescharacteristic
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C2×C4×C53C2×C22⋊C411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12311yescharacteristic
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171Dic106—42no—
172Dic106—81no—
173C8×C53—42no—
174#88322—81no—
175#9256—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53#38611yescharacteristic
187C22×C53C4×SD1611yescharacteristic
188C22×C53#44011yescharacteristic
189C22×C53#45011yescharacteristic
190C22×C53#45211yescharacteristic
191C22×C53#46111yescharacteristic
192C22×C53#46311yescharacteristic
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—82no—
197C4×C53—161no—
198C4×C53#41012yes—
199Dic53—42no—
200Dic53—42no—
201Dic53—42no—
202Dic53—81no—
203Dic53—81no—
204Dic53—81no—
205C2×#483—531no—
206#2344—531no—
207#1996—531no—
208#1823—531no—
209#1520—531no—
210#1289D5311yescharacteristic
211#1282—531no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—21no—
215C2×C53—21no—
216C2×C53—21no—
217C2×C53—21no—
218C2×C53#187711yescharacteristic
219C2×C53#189911yescharacteristic
220C2×C53#190511yescharacteristic
221C2×C22⋊Q8—531no—
222C2×#117—1061no—
223C22×C4⋊C4—531no—
224#483—1062no—
225#483—1061no—
226#483—1061no—
227C2×#106—532no—
228C2×#103—532no—
229C2×C22⋊C8—531no—
230C2×C22⋊C8C2×D5311yescharacteristic
231#389—1061no—
232#387—1062no—
233#381—1061no—
234#380C2×D5311yescharacteristic
235#380C2×D5311yescharacteristic
236C4×C4⋊C4—1061no—
237C4×C22⋊C4—531no—
238#342—1061no—
239#342—1061no—
240#340—1062no—
241#328—1062no—
242C53#925611yescharacteristic, complemented, Sylow
243C22×Q8—531no—
244C23×C4—1061no—
245C23×C4C22×D5311yescharacteristic
246C22×C8—1061no—
247C22×C8C4×D5312yes—
248C22⋊Q8—1061no—
249C22⋊Q8—1061no—
250C22⋊Q8—1061no—
251C22⋊Q8—1061no—
252#117—2122no—
253#117—2122no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—532no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4—531no—
259C2×C4⋊C4—531no—
260C2×C4⋊C4—531no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—1062no—
265C2×C4⋊C4—1061no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—531no—
269C2×C22⋊C4—531no—
270C2×C22⋊C4—1061no—
271C2×C42—1062no—
272C2×C42—1061no—
273#106—1062no—
274#106—1062no—
275#103—1062no—
276#103—1062no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—1061no—
280C22⋊C8—1061no—
281#95—21no—
282#95—1062no—
283#95—1062no—
284#95—1061no—
285#95D21212yes—
286C24D4×D5311yescharacteristic
287C2×Q8—1061no—
288C2×Q8—1061no—
289C2×Q8—2122no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—1062no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1062no—
298C22×C4—1061no—
299C22×C4—1061no—
300C22×C4—1061no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4—2121no—
305C22×C4C2×C4×D5311yescharacteristic
306C22×C4C2×D21211yescharacteristic
307C22×C4#1171994511yescharacteristic
308C22×C4D4×D5311yescharacteristic
309C22×C4D4×D5311yescharacteristic
310C22×C4#1171994911yescharacteristic
311C2×C8—22no—
312C2×C8—42no—
313C2×C8—2122no—
314C2×C8—2121no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—532no—
318C4⋊C4—532no—
319C4⋊C4—1062no—
320C4⋊C4—1062no—
321C4⋊C4—1061no—
322C4⋊C4—1061no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4—2124no—
326C4⋊C4—2121no—
327C4⋊C4—2121no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—532no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C22⋊C4—2121no—
337C22⋊C4—2121no—
338C42—2122no—
339C42—2124no—
340C23—41no—
341C23—41no—
342C23#1313317911yescharacteristic
343C23#1313318711yescharacteristic
344C23#1313319011yescharacteristic
345C23SD16×D5311yescharacteristic
346C23#1313323811yescharacteristic
347C23#1313324111yescharacteristic
348C23#1313324911yescharacteristic
349Q8—4241no—
350Q8—2122no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—1062no—
364C2×C4—1062no—
365C2×C4—1062no—
366C2×C4—4241no—
367C2×C4—4241no—
368C2×C4—2121no—
369C2×C4—2121no—
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—2122no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4C22⋊C4×D5311yescharacteristic
384C2×C4C4×D21211yescharacteristic
385C2×C4#1313319211yescharacteristic
386C2×C4#1313320211yescharacteristic
387C8—42no—
388C8—4241no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22#1681050111yescharacteristic
400C22#1681071011yescharacteristic
401C22#1681072211yescharacteristic
402C22#1681077511yescharacteristic
403C22#1681078811yescharacteristic
404C22#1681082711yescharacteristic
405C22#1681083711yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—82no—
410C4—161no—
411C4—4241no—
412C4—4241no—
413C4—4241no—
414C4—2122no—
415C4—2122no—
416C4—2122no—
417C4#1681059512yes—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2#2027505011yescharacteristic
425C2#2027517211yescharacteristic
426C2#2027521011yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1289C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1289D53
hypocenterC53#9256
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
14. C2×#13133149C4semidirect
15. C2×#13133142C4semidirect
242. C53#9256semidirect

Maximal subgroups

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

subgroupindex
C2×#168109542
#202775862
#202756992
#202747992
#202732862
#202725622
C53×#12892
#925653