Finite Group Databasebeta

← #27284322

Subgroups — #27284322

Order 13568.

716 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16810934C211yesmaximal, characteristic
2#20280459C211yesmaximal, characteristic
3C2×#16810555C211yesmaximal, characteristic, complemented
4#20274802C211yesmaximal, characteristic, complemented
5C2×#16810348C211yesmaximal, characteristic, complemented
6#20272594C211yesmaximal, characteristic, complemented
7C2×C53×#327C211yesmaximal, characteristic
8C2×#13133198—21no—
9C22×C4⋊Dic53—21no—
10C22×C4⋊Dic53C2211yescharacteristic
11C22×#11719919C2211yescharacteristic
12#16810934—22no—
13#16810934—22no—
14C2×#13133150—21no—
15C2×#13133138—21no—
16C2×C22⋊#88322C2211yescharacteristic
17#16810555—22no—
18#16810555—22no—
19#16810554—21no—
20#16810546—22no—
21#16810537—21no—
22#16810503—21no—
23#16810483C2211yescharacteristic
24C2×#13133124—21no—
25#16810386—22no—
26#16810348—22no—
27C2×#121×C53C2211yescharacteristic
28C2×C22⋊C8×C53C2211yescharacteristic
29C2×#95×C53C2211yescharacteristic
30C53×#327—22no—
31C23×Dic53—21no—
32C23×Dic53D411yescharacteristic
33C23×Dic53D411yescharacteristic
34C22×#88322—21no—
35#13133198—42no—
36#13133198—42no—
37C2×C4⋊Dic53—24no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—42no—
44C2×C4⋊Dic53—42no—
45C2×C4⋊Dic53D412yes—
46C2×C4⋊Dic53D412yes—
47C2×#11719919—22no—
48C2×#11719919—21no—
49C2×#11719919—21no—
50C2×#11719919—21no—
51C2×#11719919—21no—
52C2×#11719919—21no—
53C2×#11719919—21no—
54C2×C22⋊Dic53—22no—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—21no—
57#13133150—24no—
58#13133138—24no—
59C22⋊#88322—21no—
60C22⋊#88322D412yes—
61#13133124—22no—
62#13133124—24no—
63#13133124—21no—
64C22×D4×C53—21no—
65C22×D4×C53D411yescharacteristic
66C23×C4×C53—21no—
67C23×C4×C53C2311yescharacteristic
68C22×C8×C53—21no—
69#121×C53—21no—
70#121×C53—21no—
71#121×C53—21no—
72#121×C53—21no—
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C22⋊C4×C53—21no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53—21no—
77#95×C53—21no—
78#95×C53D412yes—
79C24×C53—21no—
80C24×C53C2×D411yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C22×C4×C53—24no—
91C22×C4×C53—22no—
92C22×C4×C53—21no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—21no—
108C22×Dic53—21no—
109C22×Dic53—21no—
110C22×Dic53—22no—
111C22×Dic53—21no—
112C22×Dic53—22no—
113C22×Dic53—21no—
114C22×Dic53—42no—
115C22×Dic53—41no—
116C22×Dic53—41no—
117C22×Dic53—41no—
118C22×Dic53—41no—
119C22×Dic53—41no—
120C22×Dic53—41no—
121C22×Dic53D812yes—
122C2×C8×C53—42no—
123C2×C8×C53—41no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C22⋊Dic53—44no—
129C22⋊Dic53—41no—
130C22⋊Dic53—41no—
131C22⋊Dic53—82no—
132C4⋊Dic53—22no—
133C4⋊Dic53—22no—
134C4⋊Dic53—48no—
135C4⋊Dic53—41no—
136C4⋊Dic53—41no—
137C4⋊Dic53—42no—
138#11719919—44no—
139#11719919—41no—
140#11719919—41no—
141#11719919—41no—
142#11719919—41no—
143C2×#88322—22no—
144C2×#88322—42no—
145C23×C53—22no—
146C23×C53—22no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53#12011yescharacteristic
153C23×C53#12011yescharacteristic
154C23×C53#12111yescharacteristic
155C23×C53C2×D811yescharacteristic
156C23×C53#13511yescharacteristic
157C23×C53AGL(1,ℤ/8)11yescharacteristic
158C23×C53AGL(1,ℤ/8)11yescharacteristic
159D4×C53—42no—
160D4×C53—81no—
161D4×C53—81no—
162D4×C53—81no—
163D4×C53—81no—
164D4×C53—81no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—24no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—44no—
173C2×C4×C53—44no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53#12011yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12711yescharacteristic
183C2×Dic53—22no—
184C2×Dic53—22no—
185C2×Dic53—22no—
186C2×Dic53—41no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—42no—
191C2×Dic53—41no—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—81no—
200C2×Dic53—81no—
201C2×Dic53—81no—
202C2×Dic53—81no—
203C2×Dic53—81no—
204C8×C53—81no—
205#88322—42no—
206#9919—531nomaximal, Sylow
207C22×C53—22no—
208C22×C53—22no—
209C22×C53—22no—
210C22×C53—22no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—82no—
216C22×C53—82no—
217C22×C53—81no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53#39211yescharacteristic
221C22×C53#44711yescharacteristic
222C22×C53#44711yescharacteristic
223C22×C53#44911yescharacteristic
224C22×C53#44911yescharacteristic
225C22×C53#45612yes—
226C22×C53#45812yes—
227C22×C53#46611yescharacteristic
228C22×C53#46911yescharacteristic
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—44no—
234C4×C53—81no—
235Dic53—42no—
236Dic53—82no—
237Dic53—82no—
238Dic53—81no—
239C2×#480—531no—
240#2618—531no—
241C2×#398—531no—
242#1823—531no—
243C2×#327D5311yescharacteristic
244C2×#324—531no—
245#1294—531no—
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—161no—
251C2×C53#160912yes—
252C2×C53#161012yes—
253C2×C53#199911yescharacteristic
254C2×C53#200511yescharacteristic
255C2×C53#202211yescharacteristic
256C2×#123—1061no—
257C2×#121C2×D5311yescharacteristic
258C22×C4⋊C4—531no—
259C22×C4⋊C4—531no—
260C22×C4⋊C4—1061no—
261#480—1062no—
262#480—1062no—
263C2×#107—1061no—
264C2×#102—1061no—
265C2×C22⋊C8—531no—
266C2×C22⋊C8C2×D5311yescharacteristic
267#398—1062no—
268#398—1061no—
269#398—1062no—
270#396—1062no—
271#394—1061no—
272#386—1061no—
273#380—531no—
274C2×#95—1061no—
275C2×#95C2×D5311yescharacteristic
276#340—1062no—
277#327—22no—
278#324—1062no—
279C53#991911yescharacteristic, complemented, Sylow
280C22×D4—21no—
281C22×D4D21211yescharacteristic
282C23×C4—21no—
283C23×C4—531no—
284C23×C4—531no—
285C23×C4—1061no—
286C23×C4C22×D5311yescharacteristic
287C22×C8—21no—
288C22×C8—1061no—
289#123—2122no—
290#123—2122no—
291#121—21no—
292#121—21no—
293#121—21no—
294#121—21no—
295C2×C4⋊C4—532no—
296C2×C4⋊C4—532no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4—1061no—
299C2×C4⋊C4—1061no—
300C2×C4⋊C4—1061no—
301C2×C4⋊C4—1064no—
302C2×C4⋊C4—1062no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—1061no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—2122no—
311C2×C4⋊C4—2122no—
312C2×C4⋊C4D21211yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—1062no—
315C2×C22⋊C4—1062no—
316C2×C22⋊C4—1061no—
317#107—1064no—
318#102—1064no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8—532no—
322C22⋊C8—1061no—
323#95—21no—
324#95—1064no—
325#95—1062no—
326#95—1061no—
327#95C22⋊D5312yes—
328C24—21no—
329C24D4×D5311yescharacteristic
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—532no—
347C22×C4—1061no—
348C22×C4—1061no—
349C22×C4—1061no—
350C22×C4—1061no—
351C22×C4—1061no—
352C22×C4—1061no—
353C22×C4—1061no—
354C22×C4—1061no—
355C22×C4—1062no—
356C22×C4—1062no—
357C22×C4—1061no—
358C22×C4—2121no—
359C22×C4—2121no—
360C22×C4—2121no—
361C22×C4—2121no—
362C22×C4—2121no—
363C22×C4—2121no—
364C22×C4—2122no—
365C22×C4C2×D21211yescharacteristic
366C22×C4#1171994511yescharacteristic
367C22×C4D4×D5311yescharacteristic
368C22×C4D4×D5311yescharacteristic
369C22×C4D4×D5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—1062no—
374C2×C8—2122no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—1062no—
378C4⋊C4—1062no—
379C4⋊C4—2122no—
380C4⋊C4—2128no—
381C4⋊C4—2124no—
382C4⋊C4—2121no—
383C4⋊C4—2121no—
384C4⋊C4—2121no—
385C4⋊C4—2121no—
386C4⋊C4—2121no—
387C4⋊C4—2121no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—4242no—
391C22⋊C4—2124no—
392C22⋊C4—2121no—
393C22⋊C4—2121no—
394C23—22no—
395C23—22no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23#1313318511yescharacteristic
402C23#1313318711yescharacteristic
403C23#1313319011yescharacteristic
404C23D8×D5311yescharacteristic
405C23#1313323811yescharacteristic
406C23#1313324111yescharacteristic
407C23#1313324411yescharacteristic
408D4—42no—
409D4—81no—
410D4—81no—
411D4—81no—
412D4—81no—
413D4—81no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—1062no—
429C2×C4—1062no—
430C2×C4—1062no—
431C2×C4—4241no—
432C2×C4—4241no—
433C2×C4—4241no—
434C2×C4—4241no—
435C2×C4—4241no—
436C2×C4—2122no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4—2122no—
440C2×C4—2122no—
441C2×C4—2121no—
442C2×C4—2122no—
443C2×C4—2122no—
444C2×C4—2122no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4#1313318711yescharacteristic
450C2×C4#1313319011yescharacteristic
451C2×C4#1313319111yescharacteristic
452C2×C4#1313321311yescharacteristic
453C8—81no—
454C8—2122no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—21no—
462C22—21no—
463C22—81no—
464C22—81no—
465C22—82no—
466C22—82no—
467C22—81no—
468C22#1681053211yescharacteristic
469C22#1681075511yescharacteristic
470C22#1681075811yescharacteristic
471C22#1681076711yescharacteristic
472C22#1681076911yescharacteristic
473C22#1681080712yes—
474C22#1681081412yes—
475C22#1681085811yescharacteristic
476C22#1681087511yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—44no—
482C4—81no—
483C4—4242no—
484C4—4242no—
485C4—4241no—
486C4—2122no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—161no—
492C2#2027371312yes—
493C2#2027371912yes—
494C2#2027571211yescharacteristic
495C2#2027575011yescharacteristic
496C2#2027584511yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#327C2
socleC23×C53#120
prime radicalC2×#121×C53C22
hypercenterC2×#327D53
hypocenterC53#9919
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
3. C2×#16810555C2semidirect
4. #20274802C2semidirect
5. C2×#16810348C2semidirect
6. #20272594C2semidirect
279. C53#9919semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
#202804592
C2×#168105552
#202748022
C2×#168103482
#202725942
C2×C53×#3272
#991953