Finite Group Databasebeta

← #27284655

Subgroups — #27284655

Order 13568.

772 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16810941C211yesmaximal, characteristic, complemented
2#20280819C211yesmaximal, characteristic
3C2×#16810358C211yesmaximal, characteristic, complemented
4C2×#16810357C211yesmaximal, characteristic, complemented
5#20272983C211yesmaximal, characteristic, complemented
6#20272557C211yesmaximal, characteristic, complemented
7C2×C53×#481C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9C2×#13133189—22no—
10#16810941—22no—
11#16810941—22no—
12C2×#13133148—21no—
13C2×#13133137—21no—
14C2×C22⋊#88322C2211yescharacteristic
15#16810559—21no—
16#16810558—21no—
17#16810532—21no—
18#16810531—21no—
19C2×#13133124—22no—
20C2×#13133124C2211yescharacteristic
21C2×#13133124C2211yescharacteristic
22#16810358—22no—
23#16810357—22no—
24C2×#121×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#481—22no—
27C53×#481—22no—
28C2×#106×C53—21no—
29C2×#102×C53—21no—
30C2×C22⋊C8×C53C2211yescharacteristic
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C22×C22⋊D53—22no—
34C23×Dic53—21no—
35C23×Dic53D412yes—
36C22×#88322—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40#13133191—21no—
41#13133189—44no—
42#13133189—44no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×#11719919—22no—
45C2×C22⋊Dic53—22no—
46C2×#11719921—22no—
47C2×#11719921—21no—
48#13133148—24no—
49#13133137—24no—
50C22⋊#88322—21no—
51C22⋊#88322D412yescomplemented
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56#13133124—44no—
57C22×D4×C53—21no—
58C22×D4×C53D411yescharacteristic
59C23×C4×C53—21no—
60C23×C4×C53C2311yescharacteristic
61C22×C8×C53—21no—
62#121×C53—21no—
63#121×C53—21no—
64#121×C53—21no—
65#121×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C4⋊C4×C53D412yescomplemented
73C2×C22⋊C4×C53—21no—
74#106×C53—24no—
75#102×C53—24no—
76C22⋊C8×C53—21no—
77C22⋊C8×C53D412yescomplemented
78C24×C53—21no—
79C24×C53C2×D411yescharacteristic
80C23×D53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53—42no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53#4111yescharacteristic
101C2×C22⋊D53—44no—
102C2×C22⋊D53—41no—
103C2×C22⋊D53—41no—
104C2×C22⋊D53—42no—
105C2×C22⋊D53—42no—
106C2×C22⋊D53—82no—
107C2×C22⋊D53—82no—
108C22×Dic53—22no—
109C22×Dic53—22no—
110C22×Dic53—22no—
111C22×Dic53—21no—
112C22×Dic53—42no—
113C22×Dic53—42no—
114C22×Dic53—41no—
115C22×Dic53—42no—
116C22×Dic53—41no—
117C22×Dic53—42no—
118C22×Dic53SD1614yes—
119C2×D212—21no—
120C2×D212—21no—
121C2×D212—42no—
122C2×C8×C53—22no—
123C2×C8×C53—42no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—41no—
129C4⋊C4×C53—41no—
130C22⋊C4×C53—41no—
131C22⋊C4×C53—41no—
132C22⋊Dic53—44no—
133#11719921—44no—
134#11719921—41no—
135#11719921—41no—
136C4⋊Dic53—21no—
137C4⋊Dic53—21no—
138#11719919—44no—
139C2×#88322—22no—
140C2×#88322—42no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—22no—
144C23×C53—41no—
145C23×C53—42no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—42no—
149C23×C53#12011yescharacteristic
150C23×C53#12111yescharacteristic
151C23×C53#12711yescharacteristic
152C23×C53C2×SD1612yes—
153C23×C53AGL(1,ℤ/8)12yes—
154C22×D53—41no—
155C22×D53—42no—
156C22×D53—41no—
157C22×D53—41no—
158C22×D53—42no—
159D4×C53—42no—
160D4×C53—81no—
161D4×C53—82no—
162D4×C53—81no—
163D4×C53—81no—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—44no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—82no—
175C2×C4×C53—81no—
176C2×C4×C53—81no—
177C2×C4×C53#12011yescharacteristic
178C2×C4×C53#12011yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C22⋊D53—88no—
182C22⋊D53—82no—
183C22⋊D53—81no—
184C22⋊D53—81no—
185C2×Dic53—24no—
186C2×Dic53—42no—
187C2×Dic53—42no—
188C2×Dic53—44no—
189C2×Dic53—44no—
190C2×Dic53—42no—
191C2×Dic53—41no—
192C2×Dic53—42no—
193C2×Dic53—82no—
194C2×Dic53—82no—
195C2×Dic53—81no—
196C2×Dic53—81no—
197D212—42no—
198D212—81no—
199C8×C53—42no—
200#88322—42no—
201#9968—531nomaximal, Sylow
202C22×C53—22no—
203C22×C53—22no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—82no—
211C22×C53—82no—
212C22×C53—81no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53#39211yescharacteristic
216C22×C53#45012yes—
217C22×C53#45311yescharacteristic
218C22×C53#45411yescharacteristic
219C22×C53#45812yes—
220C22×C53#46012yes—
221C22×C53#49211yescharacteristic
222C22×C53#49611yescharacteristic
223C2×D53—82no—
224C2×D53—82no—
225C2×D53—81no—
226C2×D53—81no—
227C2×D53—81no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—82no—
233C4×C53—81no—
234Dic53—44no—
235Dic53—82no—
236Dic53—81no—
237C2×#481—531no—
238C2×#481D5311yescharacteristic, complemented
239#2656—531no—
240C2×#327—531no—
241C2×#327—531no—
242#1445—531no—
243#1280—531no—
244C2×C53—21no—
245C2×C53—22no—
246C2×C53—21no—
247C2×C53—161no—
248C2×C53#161312yes—
249C2×C53#161612yes—
250C2×C53#200611yescharacteristic
251C2×C53#201411yescharacteristic
252C2×C53#202211yescharacteristic
253D53—161no—
254C2×#121—531no—
255C2×#121—1062no—
256C2×#121C2×D5311yescharacteristic
257C22×C4⋊C4C2×D5311yescharacteristic
258#481—22no—
259#481—22no—
260#481—1062no—
261#481—1062no—
262C2×#106—21no—
263C2×#106—1061no—
264C2×#102—21no—
265C2×#102—1061no—
266C2×C22⋊C8—531no—
267C2×C22⋊C8C2×D5311yescharacteristic
268#399—1061no—
269#399—1061no—
270#392—1061no—
271#392—1061no—
272C2×#95—531no—
273C2×#95—531no—
274C2×#95—1062no—
275#327—1062no—
276#327—1062no—
277C53#996811yescharacteristic, complemented, Sylow
278C22×D4—21no—
279C22×D4—531no—
280C22×D4—1062no—
281C22×D4—1061no—
282C22×D4C22⋊D5311yescharacteristic
283C23×C4—21no—
284C23×C4—532no—
285C23×C4—1061no—
286C23×C4C22×D5311yescharacteristic
287C22×C8—21no—
288C22×C8—1061no—
289#121—21no—
290#121—21no—
291#121—21no—
292#121—21no—
293#121—1061no—
294#121—1061no—
295#121—1061no—
296#121—1061no—
297#121—2124no—
298#121—2124no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—531no—
305C2×C4⋊C4—1062no—
306C2×C4⋊C4C22⋊D5311yescharacteristic
307C2×C4⋊C4C22⋊D5312yescomplemented
308C2×C22⋊C4—21no—
309C2×C22⋊C4—1062no—
310C2×C22⋊C4—1062no—
311C2×C22⋊C4—1061no—
312#106—24no—
313#106—1064no—
314#102—24no—
315#102—1064no—
316C22⋊C8—21no—
317C22⋊C8—532no—
318C22⋊C8—1061no—
319C22⋊C8C22⋊D5312yescomplemented
320#95—1061no—
321#95—1061no—
322#95—1061no—
323#95—1061no—
324#95—2124no—
325C24—21no—
326C24—1061no—
327C24D4×D5311yescharacteristic
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—1061no—
336C2×D4—1061no—
337C2×D4—4242no—
338C2×D4—4242no—
339C2×D4—2122no—
340C2×D4—2124no—
341C2×D4—2121no—
342C2×D4—2121no—
343C2×D4—2122no—
344C2×D4—2122no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—534no—
353C22×C4—1062no—
354C22×C4—1062no—
355C22×C4—1062no—
356C22×C4—1061no—
357C22×C4—2121no—
358C22×C4—2122no—
359C22×C4—2121no—
360C22×C4—2122no—
361C22×C4—2122no—
362C22×C4—2122no—
363C22×C4D4×D5311yescharacteristic
364C22×C4D4×D5311yescharacteristic
365C22×C4#1171994711yescharacteristic
366C22×C4C2×C22⋊D5311yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368C22×C4C2×C22⋊D5311yescharacteristic
369C2×C8—22no—
370C2×C8—42no—
371C2×C8—1062no—
372C2×C8—2122no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—1061no—
380C4⋊C4—1061no—
381C4⋊C4—2124no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—2124no—
385C22⋊C4—2124no—
386C22⋊C4—2121no—
387C22⋊C4—2121no—
388C23—21no—
389C23—21no—
390C23—22no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—2122no—
397C23—2121no—
398C23—2121no—
399C23—2121no—
400C23—2122no—
401C23#1313318511yescharacteristic
402C23#1313318911yescharacteristic
403C23#1313321311yescharacteristic
404C23SD16×D5312yes—
405C23#1313324412yes—
406D4—42no—
407D4—81no—
408D4—81no—
409D4—82no—
410D4—81no—
411D4—4241no—
412D4—4241no—
413D4—4242no—
414D4—4241no—
415D4—4248no—
416D4—2122no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—1064no—
431C2×C4—4242no—
432C2×C4—4241no—
433C2×C4—4241no—
434C2×C4—4242no—
435C2×C4—2122no—
436C2×C4—2124no—
437C2×C4—2122no—
438C2×C4—2121no—
439C2×C4—2122no—
440C2×C4—2122no—
441C2×C4—2124no—
442C2×C4#1313318511yescharacteristic
443C2×C4#1313318611yescharacteristic
444C2×C4#1313318811yescharacteristic
445C2×C4#1313318911yescharacteristic
446C8—42no—
447C8—2122no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—81no—
457C22—82no—
458C22—82no—
459C22—81no—
460C22—81no—
461C22—4241no—
462C22—4242no—
463C22—4242no—
464C22—4241no—
465C22—4241no—
466C22#1681053111yescharacteristic
467C22#1681077212yes—
468C22#1681078911yescharacteristic
469C22#1681079411yescharacteristic
470C22#1681081212yes—
471C22#1681082112yes—
472C22#1681099411yescharacteristic
473C22#1681100911yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—82no—
479C4—81no—
480C4—4242no—
481C4—4241no—
482C4—2124no—
483C2—21no—
484C2—22no—
485C2—21no—
486C2—161no—
487C2—8481no—
488C2#2027373312yes—
489C2#2027375012yes—
490C2#2027575511yescharacteristic
491C2#2027580111yescharacteristic
492C2#2027584311yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#481C2
socleC23×C53#120
prime radical#20280819C2
hypercenterC2×#481D53
hypocenterC53#9968
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
1. C2×#16810941C2semidirect
3. C2×#16810358C2semidirect
4. C2×#16810357C2semidirect
5. #20272983C2semidirect
6. #20272557C2semidirect
7. C2×C53×#481C2semidirect
51. C22⋊#88322D4semidirect
72. C2×C4⋊C4×C53D4semidirect
77. C22⋊C8×C53D4semidirect
238. C2×#481D53semidirect
277. C53#9968semidirect
307. C2×C4⋊C4C22⋊D53semidirect
319. C22⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109412
#202808192
C2×#168103582
C2×#168103572
#202729832
#202725572
C2×C53×#4812
#996853