Finite Group Databasebeta

← #27284337

Subgroups — #27284337

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810941C211yesmaximal, characteristic
2#20280457C211yesmaximal, characteristic
3#20274805C211yesmaximal, characteristic, complemented
4C2×#16810356C211yesmaximal, characteristic
5C2×#16810349C211yesmaximal, characteristic, complemented
6#20272594C211yesmaximal, characteristic, complemented
7C2×C53×#400C211yesmaximal, characteristic, complemented
8C2×#13133200—21no—
9C2×#13133191C2211yescharacteristic
10#16810941—22no—
11#16810941—22no—
12C2×#13133148—21no—
13C2×#13133137—21no—
14C2×C22⋊#88322C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16#16810553—21no—
17#16810545—22no—
18#16810540—21no—
19#16810502—21no—
20#16810483C2211yescharacteristic
21#16810385—22no—
22#16810356—22no—
23#16810349—22no—
24C22×C4⋊C4×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#400—22no—
27C53×#400—22no—
28C2×#95×C53—21no—
29C2×#95×C53—21no—
30C2×#95×C53C2211yescharacteristic
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133200—42no—
36#13133200—42no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40#13133191—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C4⋊Dic53D412yes—
43C2×#11719919—22no—
44C2×C22⋊Dic53—21no—
45C2×#11719921—22no—
46C2×#11719921—22no—
47C2×#11719921—21no—
48#13133148—24no—
49#13133137—24no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322D412yes—
54#13133124—22no—
55#13133124—21no—
56C23×C4×C53—21no—
57C23×C4×C53—21no—
58C23×C4×C53D411yescharacteristic
59C23×C4×C53D411yescharacteristic
60C23×C4×C53C2311yescharacteristic
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×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×C53—21no—
72C2×C4⋊C4×C53D412yes—
73#95×C53—24no—
74#95×C53—24no—
75#95×C53—21no—
76#95×C53D412yes—
77C24×C53C2×D411yescharacteristic
78C23×D53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—21no—
87C22×C4×C53—22no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—44no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53SD1612yes—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C2×C22⋊D53—41no—
107C2×C22⋊D53—41no—
108C2×C22⋊D53—41no—
109C2×C22⋊D53—41no—
110C2×C22⋊D53—41no—
111C2×C22⋊D53—41no—
112C22×Dic53—22no—
113C22×Dic53—21no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212—42no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C22⋊Dic53—41no—
126C22⋊Dic53—41no—
127#11719921—44no—
128#11719921—41no—
129#11719921—41no—
130#11719921—82no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—42no—
134#11719919—44no—
135C2×#88322—22no—
136C2×#88322—42no—
137C2×#88322—42no—
138C2×#88322—41no—
139C23×C53—22no—
140C23×C53—22no—
141C23×C53#12011yescharacteristic
142C23×C53#12011yescharacteristic
143C23×C53#12111yescharacteristic
144C23×C53C2×SD1611yescharacteristic
145C23×C53#13511yescharacteristic
146C23×C53AGL(1,ℤ/8)11yescharacteristic
147C23×C53#13711yescharacteristic
148C22×D53—41no—
149C22×D53—41no—
150C22×D53—41no—
151C22×D53—42no—
152C22×D53—42no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—44no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—82no—
175C2×C4×C53—82no—
176C2×C4×C53—82no—
177C2×C4×C53—82no—
178C2×C4×C53—81no—
179C2×C4×C53#12011yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12711yescharacteristic
183C22⋊D53—81no—
184C22⋊D53—81no—
185C22⋊D53—81no—
186C22⋊D53—81no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—42no—
190C2×Dic53—41no—
191C2×Dic53—42no—
192C2×Dic53—42no—
193D212—42no—
194D212—81no—
195#88322—42no—
196#88322—81no—
197#9921—531nomaximal, Sylow
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—22no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53#39211yescharacteristic
207C22×C53#44811yescharacteristic
208C22×C53#44911yescharacteristic
209C22×C53#45011yescharacteristic
210C22×C53#45211yescharacteristic
211C22×C53#45412yes—
212C22×C53#45812yes—
213C22×C53#46511yescharacteristic
214C22×C53#46811yescharacteristic
215C2×D53—82no—
216C2×D53—82no—
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—42no—
225C4×C53—82no—
226C4×C53—82no—
227C4×C53—81no—
228Dic53—82no—
229Dic53—81no—
230C2×#481—531no—
231#2618—531no—
232C2×#400D5311yescharacteristic, complemented
233#1824—531no—
234C2×#327—531no—
235C2×#324—531no—
236#1294—531no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53—21no—
241C2×C53#161512yes—
242C2×C53#161612yes—
243C2×C53#200011yescharacteristic
244C2×C53#200311yescharacteristic
245C2×C53#202211yescharacteristic
246D53—161no—
247C2×#123—1061no—
248C2×#121—531no—
249C22×C4⋊C4C2×D5311yescharacteristic
250C22×C4⋊C4C2×D5311yescharacteristic
251#481—1062no—
252#481—1062no—
253C2×#106—1061no—
254C2×#102—1061no—
255C2×C22⋊C8—531no—
256C2×C22⋊C8—531no—
257#400—22no—
258#400—22no—
259#398—1061no—
260#396—1062no—
261#394—1061no—
262#386—1061no—
263#380—531no—
264C2×#95—21no—
265C2×#95—21no—
266C2×#95C2×D5311yescharacteristic
267#340—1062no—
268#327—1062no—
269#324—1062no—
270C53#992111yescharacteristic, complemented, Sylow
271C22×D4—531no—
272C22×D4—1061no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4C22⋊D5311yescharacteristic
276C23×C4C22⋊D5311yescharacteristic
277C23×C4C22×D5311yescharacteristic
278C22×C8—1061no—
279C22×C8—1061no—
280#123—2122no—
281#123—2122no—
282#121—1061no—
283#121—1061no—
284#121—1061no—
285#121—1061no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—531no—
298C2×C4⋊C4—532no—
299C2×C4⋊C4—1062no—
300C2×C4⋊C4C22⋊D5312yes—
301C2×C22⋊C4—1062no—
302C2×C22⋊C4—1062no—
303C2×C22⋊C4—1061no—
304C2×C22⋊C4—1061no—
305#106—1064no—
306#102—1064no—
307C22⋊C8—532no—
308C22⋊C8—1061no—
309C22⋊C8—1061no—
310C22⋊C8—1061no—
311#95—24no—
312#95—24no—
313#95—21no—
314#95—1062no—
315#95—1061no—
316#95C22⋊D5312yes—
317C24—1061no—
318C24C2×C22⋊D5311yescharacteristic
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—2122no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4—2121no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—1062no—
349C22×C4—1061no—
350C22×C4#1171992412yes—
351C22×C4D4×D5311yescharacteristic
352C22×C4D4×D5311yescharacteristic
353C22×C4D4×D5311yescharacteristic
354C22×C4#1171994711yescharacteristic
355C22×C4C2×C22⋊D5311yescharacteristic
356C22×C4C2×C22⋊D5311yescharacteristic
357C2×C8—1062no—
358C2×C8—2122no—
359C2×C8—2121no—
360C2×C8—2122no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—1061no—
370C4⋊C4—1061no—
371C4⋊C4—2122no—
372C4⋊C4—2124no—
373C22⋊C4—4242no—
374C22⋊C4—2124no—
375C22⋊C4—2121no—
376C22⋊C4—2121no—
377C22⋊C4—2121no—
378C22⋊C4—2121no—
379C23—22no—
380C23—22no—
381C23—2121no—
382C23—2121no—
383C23—2121no—
384C23—2122no—
385C23—2122no—
386C23#1313318511yescharacteristic
387C23#1313318611yescharacteristic
388C23#1313318911yescharacteristic
389C23C2×#1171992411yescharacteristic
390C23#1313323711yescharacteristic
391C23#1313324211yescharacteristic
392C23#1313324811yescharacteristic
393D4—4241no—
394D4—4241no—
395D4—4241no—
396D4—4241no—
397D4—4241no—
398D4—2122no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—2122no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4—2122no—
430C2×C4—2122no—
431C2×C4#1313318511yescharacteristic
432C2×C4#1313318811yescharacteristic
433C2×C4#1313318911yescharacteristic
434C2×C4#1313321311yescharacteristic
435C8—4241no—
436C8—2122no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—21no—
444C22—21no—
445C22—4241no—
446C22—4241no—
447C22—4242no—
448C22—4242no—
449C22—4241no—
450C22#1681053111yescharacteristic
451C22#1681076211yescharacteristic
452C22#1681076811yescharacteristic
453C22#1681077311yescharacteristic
454C22#1681078311yescharacteristic
455C22#1681079512yes—
456C22#1681081212yes—
457C22#1681085411yescharacteristic
458C22#1681087111yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—82no—
465C4—82no—
466C4—81no—
467C4—4241no—
468C4—4242no—
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—8481no—
474C2#2027374412yes—
475C2#2027375112yes—
476C2#2027571611yescharacteristic
477C2#2027573711yescharacteristic
478C2#2027584311yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#400C2
socleC23×C53#120
prime radicalC2×#13133191C22
hypercenterC2×#400D53
hypocenterC53#9921
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
3. #20274805C2semidirect
5. C2×#16810349C2semidirect
6. #20272594C2semidirect
7. C2×C53×#400C2semidirect
232. C2×#400D53semidirect
270. C53#9921semidirect

Maximal subgroups

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

subgroupindex
C2×#168109412
#202804572
#202748052
C2×#168103562
C2×#168103492
#202725942
C2×C53×#4002
#992153