Finite Group Databasebeta

← #27284396

Subgroups — #27284396

Order 13568.

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

normal subgroups only

This group has 610 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×#16810959C211yesmaximal, characteristic
2#20280531C211yesmaximal, characteristic
3C2×#16810563C211yesmaximal, characteristic
4#20274803C211yesmaximal, characteristic
5C2×#16810359C211yesmaximal, characteristic
6C2×#16810348C211yesmaximal, characteristic
7C53×#1294C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10C22×#11719919C2211yescharacteristic
11#16810959—22no—
12#16810959—22no—
13C2×#13133151—21no—
14C2×#13133142—21no—
15C2×C22⋊#88322C2211yescharacteristic
16#16810563—22no—
17#16810563—21no—
18#16810563—22no—
19#16810553—22no—
20#16810544—21no—
21#16810498—21no—
22#16810484—21no—
23C2×#13133124—21no—
24C2×#13133124—21no—
25#16810385—22no—
26#16810359—22no—
27#16810348—22no—
28C2×C22⋊C8×C53C2211yescharacteristic
29C53×#380C2211yescharacteristic
30C2×#95×C53C2211yescharacteristic
31C22×Dic106D411yescharacteristic
32C23×Dic53—21no—
33C23×Dic53D411yescharacteristic
34C23×Dic53D411yescharacteristic
35C22×#88322—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39#13133196—21no—
40C2×C4⋊Dic53—22no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D412yes—
47C2×C4⋊Dic53D411yescharacteristic
48C2×#11719919—21no—
49C2×#11719919—21no—
50C2×#11719919—21no—
51C2×#11719919—21no—
52C2×#11719919—21no—
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×C22⋊Dic53—21no—
56#13133151—24no—
57#13133142—24no—
58C22⋊#88322—21no—
59C22⋊#88322—21no—
60#13133124—22no—
61#13133124—22no—
62#13133124—22no—
63#13133124—24no—
64#13133124—24no—
65C23×C4×C53—21no—
66C23×C4×C53C2311yescharacteristic
67C22×C8×C53—21no—
68C2×C4⋊C4×C53D412yes—
69C2×C22⋊C4×C53—21no—
70C22⋊C8×C53—21no—
71C22⋊C8×C53D412yes—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53D412yes—
75C24×C53C2×D411yescharacteristic
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—22no—
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—21no—
101C22×Dic53—42no—
102C22×Dic53—44no—
103C22×Dic53—41no—
104C22×Dic53—41no—
105C22×Dic53Q1612yes—
106C2×Dic106—21no—
107C2×Dic106—21no—
108C2×Dic106—42no—
109C2×C8×C53—22no—
110C2×C8×C53—42no—
111C4⋊C4×C53—42no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—41no—
116C4⋊Dic53—22no—
117C4⋊Dic53—21no—
118C4⋊Dic53—22no—
119C4⋊Dic53—21no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C4⋊Dic53—82no—
123#11719919—41no—
124#11719919—41no—
125#11719919—41no—
126#11719919—41no—
127#11719919—41no—
128#11719919—41no—
129C2×#88322—42no—
130C2×#88322—41no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53#12011yescharacteristic
134C23×C53#12011yescharacteristic
135C23×C53#12111yescharacteristic
136C23×C53C2×Q1611yescharacteristic
137C23×C53#13511yescharacteristic
138C23×C53#13711yescharacteristic
139C23×C53#13711yescharacteristic
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—82no—
153C2×C4×C53—82no—
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12711yescharacteristic
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—44no—
166C2×Dic53—42no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—42no—
170C2×Dic53—41no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—82no—
175C2×Dic53—82no—
176C2×Dic53—81no—
177Dic106—42no—
178Dic106—81no—
179C8×C53—42no—
180#88322—81no—
181#9929—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53#39211yescharacteristic
191C22×C53#45111yescharacteristic
192C22×C53#45111yescharacteristic
193C22×C53#45211yescharacteristic
194C22×C53#45211yescharacteristic
195C22×C53#45412yes—
196C22×C53#45812yes—
197C22×C53#46711yescharacteristic
198C22×C53#47011yescharacteristic
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—82no—
204C4×C53—82no—
205Dic53—42no—
206Dic53—81no—
207Dic53—82no—
208Dic53—81no—
209Dic53—81no—
210C2×#484—531no—
211#2627—531no—
212C2×#400—531no—
213#1824—531no—
214C2×#328—531no—
215C2×#324—531no—
216#1294D5311yescharacteristic
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—21no—
220C2×C53—21no—
221C2×C53#161512yes—
222C2×C53#161712yes—
223C2×C53#200111yescharacteristic
224C2×C53#200311yescharacteristic
225C2×C53#202211yescharacteristic
226C2×C22⋊Q8—531no—
227C22×C4⋊C4—531no—
228C22×C4⋊C4—531no—
229#484—1062no—
230#484—1062no—
231C2×#107—1061no—
232C2×#103—1061no—
233C2×C22⋊C8—531no—
234C2×C22⋊C8C2×D5311yescharacteristic
235#400—1061no—
236#400—1062no—
237#400—1062no—
238#398—1062no—
239#395—1061no—
240#385—1061no—
241#381—1061no—
242#380C2×D5311yescharacteristic
243C2×#95—1061no—
244C2×#95—1061no—
245C2×#95C2×D5311yescharacteristic
246#340—1062no—
247#328—1062no—
248#324—1062no—
249C53#992911yescharacteristic, complemented, Sylow
250C22×Q8—531no—
251C23×C4—21no—
252C23×C4—531no—
253C23×C4—531no—
254C23×C4—1061no—
255C23×C4C22×D5311yescharacteristic
256C22×C8—21no—
257C22×C8—1061no—
258C22⋊Q8—1061no—
259C22⋊Q8—1061no—
260C22⋊Q8—1061no—
261C22⋊Q8—1061no—
262C2×C4⋊C4—532no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1062no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4D21212yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—1061no—
280#107—1064no—
281#103—1064no—
282C22⋊C8—21no—
283C22⋊C8—1061no—
284C22⋊C8—1061no—
285C22⋊C8D21212yes—
286#95—21no—
287#95—21no—
288#95—1064no—
289#95—1064no—
290#95—1062no—
291#95—1062no—
292#95—1062no—
293#95D21212yes—
294C24D4×D5311yescharacteristic
295C2×Q8—1061no—
296C2×Q8—1061no—
297C2×Q8—2122no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—532no—
306C22×C4—1061no—
307C22×C4—1061no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—1062no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—2124no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4—2122no—
322C22×C4C2×D21211yescharacteristic
323C22×C4C2×D21211yescharacteristic
324C22×C4C2×D21211yescharacteristic
325C22×C4D4×D5311yescharacteristic
326C22×C4D4×D5311yescharacteristic
327C22×C4#1171994911yescharacteristic
328C2×C8—22no—
329C2×C8—42no—
330C2×C8—2122no—
331C2×C8—2121no—
332C4⋊C4—42no—
333C4⋊C4—1062no—
334C4⋊C4—1061no—
335C4⋊C4—1062no—
336C4⋊C4—1061no—
337C4⋊C4—4242no—
338C4⋊C4—2121no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C4⋊C4—2121no—
345C4⋊C4—2121no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C23—22no—
351C23—22no—
352C23#1313318711yescharacteristic
353C23#1313318711yescharacteristic
354C23#1313319211yescharacteristic
355C23Q16×D5311yescharacteristic
356C23#1313323611yescharacteristic
357C23#1313324611yescharacteristic
358C23#1313324611yescharacteristic
359Q8—4241no—
360Q8—2122no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—1062no—
376C2×C4—1062no—
377C2×C4—1062no—
378C2×C4—4242no—
379C2×C4—4242no—
380C2×C4—4241no—
381C2×C4—2124no—
382C2×C4—2122no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2122no—
386C2×C4—2121no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4#1313318711yescharacteristic
395C2×C4#1313319211yescharacteristic
396C2×C4#1313319211yescharacteristic
397C2×C4#1313321411yescharacteristic
398C8—42no—
399C8—4241no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—21no—
407C22—21no—
408C22#1681053311yescharacteristic
409C22#1681078011yescharacteristic
410C22#1681078011yescharacteristic
411C22#1681078611yescharacteristic
412C22#1681078611yescharacteristic
413C22#1681079712yes—
414C22#1681081412yes—
415C22#1681086511yescharacteristic
416C22#1681088211yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—82no—
422C4—82no—
423C4—4242no—
424C4—4241no—
425C4—4241no—
426C4—4241no—
427C4—2122no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2#2027374712yes—
433C2#2027375812yes—
434C2#2027572611yescharacteristic
435C2#2027574011yescharacteristic
436C2#2027584711yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1294C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1294D53
hypocenterC53#9929
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
249. C53#9929semidirect

Maximal subgroups

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

subgroupindex
C2×#168109592
#202805312
C2×#168105632
#202748032
C2×#168103592
C2×#168103482
C53×#12942
#992953