Finite Group Databasebeta

← #27285148

Subgroups — #27285148

Order 13568.

938 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810999C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic, complemented
3#20275642C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2541C211yesmaximal, characteristic, complemented
8C22×#11719922—22no—
9C22×#11719922—21no—
10C2×#13133214C2211yescharacteristic
11#16810999—22no—
12#16810999—22no—
13#16810818—22no—
14#16810818—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137—21no—
18C2×#13133137—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810367—22no—
21D4⋊#88322—22no—
22C2×#127×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C2×#121×C53—22no—
25C2×C4×D4×C53C2211yescharacteristic
26C53×#392—22no—
27C53×#390—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719922—24no—
33C2×#11719922—28no—
34C2×#11719922—42no—
35C2×#11719922—48no—
36C2×#11719922—42no—
37C22×#88322—21no—
38C22×#88322—21no—
39C22×#88322—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214D412yescomplemented
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D412yescomplemented
48#13133137—24no—
49#13133137—44no—
50C4×#88322—21no—
51C4×#88322D412yescomplemented
52C22×D4×C53—24no—
53C22×D4×C53—21no—
54C22×D4×C53—21no—
55C22×D4×C53D411yescharacteristic
56C22×D4×C53D412yes—
57C23×C4×C53—21no—
58#127×C53—21no—
59#127×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53—42no—
63#127×C53—42no—
64#121×C53—44no—
65#121×C53—44no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C4⋊C4×C53D412yes—
72C2×C22⋊C4×C53—24no—
73C2×C22⋊C4×C53—21no—
74C2×C42×C53—21no—
75C2×C42×C53C2311yescharacteristic
76#95×C53—21no—
77C24×C53—21no—
78C24×C53—22no—
79C23×D53—41no—
80C2×D4×C53—21no—
81C2×D4×C53—22no—
82C2×D4×C53—21no—
83C2×D4×C53—22no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—44no—
88C2×D4×C53—48no—
89C2×D4×C53—48no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—42no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—42no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—84no—
101C2×D4×C53D812yescomplemented
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C2×D212—22no—
112C2×D212—22no—
113C2×D212—22no—
114C2×D212—44no—
115C2×D212—42no—
116C2×D212—41no—
117C2×D212—41no—
118C2×D212—41no—
119C2×D212—41no—
120C2×D212—42no—
121C2×D212—41no—
122C2×D212—41no—
123C2×D212—41no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53D814yescomplemented
128C22⋊C4×C53—48no—
129C22⋊C4×C53—41no—
130C22⋊C4×C53—41no—
131C42×C53—41no—
132C42×C53—41no—
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53C2×D411yescharacteristic
136C42×C53#4111yescharacteristic
137#11719922—416no—
138#11719922—48no—
139C2×#88322—24no—
140C2×#88322—22no—
141C2×#88322—22no—
142C2×#88322—41no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—42no—
146C23×C53—22no—
147C23×C53—41no—
148C23×C53—42no—
149C23×C53—41no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—42no—
153C23×C53—44no—
154C23×C53—42no—
155C23×C53—44no—
156C23×C53#12011yescharacteristic
157C22×D53—42no—
158C22×D53—81no—
159C22×D53—82no—
160C22×D53—81no—
161C22×D53—82no—
162D4×C53—22no—
163D4×C53—22no—
164D4×C53—44no—
165D4×C53—41no—
166D4×C53—41no—
167D4×C53—82no—
168D4×C53—816no—
169D4×C53—81no—
170D4×C53—81no—
171D4×C53—82no—
172D4×C53—81no—
173D4×C53—81no—
174D4×C53—81no—
175D4×C53—81no—
176C2×C4×C53—22no—
177C2×C4×C53—24no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—82no—
183C2×C4×C53—81no—
184C2×C4×C53—81no—
185C2×C4×C53#12011yescharacteristic
186C2×C4×C53#12011yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12711yescharacteristic
191C2×C4×C53C2×D811yescharacteristic
192C2×C4×C53C2×D812yes—
193C2×C4×C53#13511yescharacteristic
194C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
195C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C53AGL(1,ℤ/8)12yes—
197D212—44no—
198D212—42no—
199D212—42no—
200D212—44no—
201D212—82no—
202D212—81no—
203D212—81no—
204D212—81no—
205D212—81no—
206#88322—24no—
207#88322—42no—
208#88322—81no—
209#10046—531nomaximal, Sylow
210C22×C53—44no—
211C22×C53—42no—
212C22×C53—82no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53—84no—
216C22×C53—84no—
217C22×C53—82no—
218C22×C53—82no—
219C22×C53—82no—
220C22×C53#39211yescharacteristic
221C22×C53#44711yescharacteristic
222C22×C53#44712yes—
223C22×C53#44911yescharacteristic
224C22×C53#45312yes—
225C2×D53—82no—
226C2×D53—82no—
227C2×D53—82no—
228C2×D53—162no—
229C2×D53—162no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—44no—
233C4×C53—81no—
234C4×C53#44711yescharacteristic
235C4×C53#44911yescharacteristic
236C4×C53#45912yes—
237C4×C53#46611yescharacteristic
238C4×C53#46911yescharacteristic
239C4×C53#49311yescharacteristic
240C4×C53#49611yescharacteristic
241C2×#493—531no—
242C2×#459—531no—
243#2541D5311yescharacteristic, complemented
244#1982—531no—
245C2×#331—531no—
246C2×D4⋊C8—531no—
247#1279—531no—
248C2×C53—82no—
249C2×C53—162no—
250C2×C53#161912yes—
251C2×C53#162112yes—
252C2×C53#199911yescharacteristic
253C2×C53#200211yescharacteristic
254C2×C53#201111yescharacteristic
255D53—162no—
256C22×D8—1062no—
257C22×D8—1061no—
258C2×#127—21no—
259C2×#127—531no—
260C2×#127C2×D5311yescharacteristic
261C2×#121—22no—
262C2×C4×D4C2×D5311yescharacteristic
263#493—1062no—
264#493—1062no—
265#459—1062no—
266#459—1062no—
267C2×C4⋊C8—531no—
268C2×C4⋊C8—531no—
269C2×#102—1061no—
270C2×#102—1062no—
271C2×C4×C8—531no—
272#392—22no—
273#390—21no—
274C4×C4⋊C4C2×D5311yescharacteristic
275#331—1062no—
276D4⋊C8—1062no—
277C53#1004611yescharacteristic, complemented, Sylow
278C22×D4—24no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—1061no—
282C22×D4—1061no—
283C22×D4—1061no—
284C22×D4C22⋊D5311yescharacteristic
285C22×D4C22⋊D5312yes—
286C23×C4—21no—
287C2×D8—1064no—
288C2×D8—1068no—
289C2×D8—2128no—
290C2×D8—2122no—
291C2×D8—2122no—
292C22×C8—1061no—
293C22×C8—1061no—
294C22×C8—1061no—
295#127—21no—
296#127—21no—
297#127—21no—
298#127—21no—
299#127—42no—
300#127—42no—
301#127—532no—
302#127—1061no—
303#127—1061no—
304#127—1061no—
305#121—44no—
306#121—44no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—21no—
311C2×C4⋊C4C22⋊D5311yescharacteristic
312C2×C4⋊C4C22⋊D5312yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—21no—
315C2×C42—21no—
316C2×C42C22×D5311yescharacteristic
317C4⋊C8—532no—
318C4⋊C8—1061no—
319C4⋊C8—1061no—
320C4⋊C8—1061no—
321#102—1064no—
322#102—2124no—
323C4×C8—532no—
324C4×C8—1061no—
325#95—21no—
326C24—21no—
327C24—22no—
328C24—2121no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—44no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—42no—
348C2×D4—41no—
349C2×D4—84no—
350C2×D4—1062no—
351C2×D4—1062no—
352C2×D4—1062no—
353C2×D4—2124no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4—2122no—
358C2×D4—2121no—
359C2×D4—2121no—
360C2×D4—2121no—
361C2×D4—2122no—
362C2×D4—2121no—
363C2×D4#1171992212yescomplemented
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4C2×C22⋊D5311yescharacteristic
371C22×C4C2×C22⋊D5311yescharacteristic
372C22×C4C2×C22⋊D5311yescharacteristic
373D8—21216no—
374D8—2128no—
375C2×C8—1064no—
376C2×C8—1062no—
377C2×C8—1062no—
378C2×C8—2121no—
379C2×C8—2122no—
380C2×C8—2121no—
381C2×C8—2121no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4#1171992214yescomplemented
386C22⋊C4—48no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C42—41no—
390C42—41no—
391C42D4×D5311yescharacteristic
392C42D4×D5311yescharacteristic
393C42D4×D5311yescharacteristic
394C42#1171994711yescharacteristic
395C23—22no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—4241no—
406C23—4242no—
407C23—4241no—
408C23—4242no—
409C23—2122no—
410C23#1313318611yescharacteristic
411D4—22no—
412D4—22no—
413D4—44no—
414D4—41no—
415D4—41no—
416D4—82no—
417D4—816no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—82no—
422D4—81no—
423D4—81no—
424D4—81no—
425D4—4242no—
426D4—4241no—
427D4—4241no—
428D4—4241no—
429D4—4241no—
430D4—2124no—
431D4—2122no—
432D4—2122no—
433D4—2124no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4#1313318511yescharacteristic
444C2×C4#1313318511yescharacteristic
445C2×C4#1313318811yescharacteristic
446C2×C4#1313318911yescharacteristic
447C2×C4#1313318911yescharacteristic
448C2×C4#1313321311yescharacteristic
449C2×C4C2×#1171992211yescharacteristic
450C2×C4C2×#1171992212yes—
451C2×C4#1313323711yescharacteristic
452C2×C4#1313324211yescharacteristic
453C2×C4#1313324312yes—
454C2×C4#1313324311yescharacteristic
455C8—1064no—
456C8—4241no—
457C8—2122no—
458C22—44no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—84no—
466C22—84no—
467C22—82no—
468C22—8482no—
469C22—8482no—
470C22—4242no—
471C22—4242no—
472C22—4242no—
473C22#1681053111yescharacteristic
474C22#1681075712yes—
475C22#1681075711yescharacteristic
476C22#1681076811yescharacteristic
477C22#1681079112yes—
478C4—21no—
479C4—21no—
480C4—44no—
481C4—81no—
482C4#1681075611yescharacteristic
483C4#1681076611yescharacteristic
484C4#1681081512yes—
485C4#1681085711yescharacteristic
486C4#1681087411yescharacteristic
487C4#1681099911yescharacteristic
488C4#1681101011yescharacteristic
489C2—82no—
490C2—162no—
491C2—8482no—
492C2#2027376912yes—
493C2#2027378012yes—
494C2#2027571111yescharacteristic
495C2#2027572811yescharacteristic
496C2#2027578611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2541C2
socleC23×C53#120
hypercenter#2541D53
hypocenterC53#10046
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C2×#16810999C2semidirect
2. C2×#16810818C2semidirect
3. #20275642C2semidirect
4. C2×#16810367C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272556C2semidirect
7. C53×#2541C2semidirect
43. #13133214D4semidirect
47. C4⋊#88322D4semidirect
51. C4×#88322D4semidirect
101. C2×D4×C53D8semidirect
127. C4⋊C4×C53D8semidirect
243. #2541D53semidirect
277. C53#10046semidirect
363. C2×D4#11719922semidirect
385. C4⋊C4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109992
C2×#168108182
#202756422
C2×#168103672
C2×D4⋊#883222
#202725562
C53×#25412
#1004653