Finite Group Databasebeta

← #27284686

Subgroups — #27284686

Order 13568.

632 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 115 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
2C2×#16810951C211yesmaximal, characteristic
3#20280831C211yesmaximal, characteristic
4C2×#16810359C211yesmaximal, characteristic
5#20272984C211yesmaximal, characteristic
6#20272557C211yesmaximal, characteristic
7C2×C53×#328C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10#16810954—22no—
11#16810954—22no—
12#16810951—22no—
13#16810951—22no—
14C2×#13133149—21no—
15C2×#13133148—21no—
16C2×#13133142—21no—
17C2×#13133141—21no—
18C2×C22⋊#88322C2211yescharacteristic
19#16810554—21no—
20#16810553—21no—
21#16810543—21no—
22#16810542—21no—
23#16810481—22no—
24C2×#13133124—22no—
25C2×#13133124C2211yescharacteristic
26#16810359—22no—
27C2×C22⋊Q8×C53C2211yescharacteristic
28C2×C22⋊C8×C53C2211yescharacteristic
29C2×#95×C53C2211yescharacteristic
30C53×#328—22no—
31C22×Dic106D411yescharacteristic
32C23×Dic53—21no—
33C23×Dic53D412yes—
34C22×#88322—21no—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53D411yescharacteristic
45C2×C4⋊Dic53D412yes—
46C2×#11719919—22no—
47C2×#11719919—22no—
48C2×#11719919—21no—
49C2×C22⋊Dic53—21no—
50#13133149—24no—
51#13133148—24no—
52#13133142—24no—
53#13133141—24no—
54C22⋊#88322—21no—
55C22⋊#88322D412yes—
56#13133124—22no—
57#13133124—22no—
58#13133124—21no—
59#13133124—21no—
60#13133124—44no—
61C22×Q8×C53D411yescharacteristic
62C23×C4×C53—21no—
63C23×C4×C53C2311yescharacteristic
64C22×C8×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53D412yes—
74#95×C53—21no—
75#95×C53—21no—
76C24×C53C2×D411yescharacteristic
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—42no—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—42no—
84C22×C4×C53—41no—
85C22×C4×C53—42no—
86C22×C4×C53—41no—
87C22×C4×C53—42no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×Dic53—22no—
95C22×Dic53—22no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—22no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53—42no—
104C22×Dic53—42no—
105C22×Dic53SD1614yes—
106C2×Dic106—21no—
107C2×Dic106—21no—
108C2×Dic106—42no—
109C2×C8×C53—22no—
110C2×C8×C53—42no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊Dic53—41no—
118C22⋊Dic53—41no—
119C4⋊Dic53—22no—
120C4⋊Dic53—22no—
121C4⋊Dic53—21no—
122C4⋊Dic53—21no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125#11719919—41no—
126#11719919—41no—
127#11719919—82no—
128#11719919—82no—
129C2×#88322—22no—
130C2×#88322—42no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—22no—
134C23×C53#12011yescharacteristic
135C23×C53#12111yescharacteristic
136C23×C53#12711yescharacteristic
137C23×C53C2×SD1612yes—
138C23×C53#13712yes—
139Q8×C53—42no—
140Q8×C53—81no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—44no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—82no—
155C2×C4×C53—81no—
156C2×C4×C53—81no—
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12011yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×Dic53—24no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—44no—
165C2×Dic53—41no—
166C2×Dic53—42no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171C2×Dic53—44no—
172C2×Dic53—82no—
173C2×Dic53—82no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176Dic106—42no—
177Dic106—81no—
178C8×C53—42no—
179#88322—42no—
180#9973—531nomaximal, Sylow
181C22×C53—22no—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53#39211yescharacteristic
190C22×C53#44812yes—
191C22×C53#45511yescharacteristic
192C22×C53#45611yescharacteristic
193C22×C53#45812yes—
194C22×C53#46112yes—
195C22×C53#49211yescharacteristic
196C22×C53#49711yescharacteristic
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—82no—
202C4×C53—81no—
203C4×C53—81no—
204Dic53—44no—
205Dic53—81no—
206Dic53—82no—
207Dic53—81no—
208C2×#483—531no—
209C2×#483—531no—
210#2659—531no—
211C2×#328—531no—
212C2×#328D5311yescharacteristic
213#1445—531no—
214#1280—531no—
215C2×C53—21no—
216C2×C53—22no—
217C2×C53—21no—
218C2×C53#160812yes—
219C2×C53#161112yes—
220C2×C53#200711yescharacteristic
221C2×C53#201311yescharacteristic
222C2×C53#202211yescharacteristic
223C2×C22⋊Q8—531no—
224C2×C22⋊Q8C2×D5311yescharacteristic
225C22×C4⋊C4—531no—
226#483—1062no—
227#483—1062no—
228#483—1062no—
229#483—1062no—
230C2×#106—1061no—
231C2×#106—1061no—
232C2×#103—1061no—
233C2×#103—1061no—
234C2×C22⋊C8—531no—
235C2×C22⋊C8C2×D5311yescharacteristic
236#398—1061no—
237#398—1061no—
238#395—1061no—
239#395—1061no—
240#380—1062no—
241C2×#95—531no—
242C2×#95—1062no—
243C2×#95C2×D5311yescharacteristic
244#328—22no—
245#328—1062no—
246C53#997311yescharacteristic, complemented, Sylow
247C22×Q8—531no—
248C22×Q8D21211yescharacteristic
249C23×C4—21no—
250C23×C4—532no—
251C23×C4—1061no—
252C23×C4C22×D5311yescharacteristic
253C22×C8—21no—
254C22×C8—1061no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C22⋊Q8—1061no—
260C22⋊Q8—1061no—
261C22⋊Q8—1061no—
262C22⋊Q8—1061no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—532no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1062no—
271C2×C4⋊C4—1062no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4D21211yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—1061no—
277#106—1064no—
278#106—1064no—
279#103—1064no—
280#103—1064no—
281C22⋊C8—21no—
282C22⋊C8—532no—
283C22⋊C8—1061no—
284C22⋊C8C22⋊D5312yes—
285#95—21no—
286#95—21no—
287#95—1062no—
288#95—1062no—
289#95—1061no—
290#95—1061no—
291#95—2124no—
292C24D4×D5311yescharacteristic
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—42no—
296C2×Q8—1061no—
297C2×Q8—1061no—
298C2×Q8—2122no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—534no—
308C22×C4—1062no—
309C22×C4—1062no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1062no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2122no—
318C22×C4—2122no—
319C22×C4C2×D21211yescharacteristic
320C22×C4#1171994511yescharacteristic
321C22×C4D4×D5311yescharacteristic
322C22×C4D4×D5311yescharacteristic
323C22×C4D4×D5311yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325C2×C8—22no—
326C2×C8—42no—
327C2×C8—1062no—
328C2×C8—2122no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—1062no—
334C4⋊C4—1061no—
335C4⋊C4—1062no—
336C4⋊C4—1061no—
337C4⋊C4—4242no—
338C4⋊C4—4242no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23#1313318711yescharacteristic
351C23#1313319011yescharacteristic
352C23#1313321311yescharacteristic
353C23SD16×D5312yes—
354C23#1313324912yes—
355Q8—42no—
356Q8—81no—
357Q8—4241no—
358Q8—2122no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—1064no—
376C2×C4—4241no—
377C2×C4—4242no—
378C2×C4—4241no—
379C2×C4—4242no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2122no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2122no—
386C2×C4—2124no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2124no—
390C2×C4#1313318511yescharacteristic
391C2×C4#1313318711yescharacteristic
392C2×C4#1313319011yescharacteristic
393C2×C4#1313319111yescharacteristic
394C8—42no—
395C8—2122no—
396C22—22no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—21no—
403C22—21no—
404C22#1681053211yescharacteristic
405C22#1681076312yes—
406C22#1681080211yescharacteristic
407C22#1681080711yescharacteristic
408C22#1681081412yes—
409C22#1681082712yes—
410C22#1681099411yescharacteristic
411C22#1681101611yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—82no—
417C4—81no—
418C4—81no—
419C4—4241no—
420C4—4242no—
421C4—4241no—
422C4—2124no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2#2027370712yes—
427C2#2027372412yes—
428C2#2027576311yescharacteristic
429C2#2027579911yescharacteristic
430C2#2027584511yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#328C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenterC2×#328D53
hypocenterC53#9973
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
246. C53#9973semidirect

Maximal subgroups

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

subgroupindex
C2×#168109542
C2×#168109512
#202808312
C2×#168103592
#202729842
#202725572
C2×C53×#3282
#997353