Finite Group Databasebeta

← #27287976

Subgroups — #27287976

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810904C211yesmaximal, characteristic
2#20275516C211yesmaximal, characteristic, complemented
3#20275508C211yesmaximal, characteristic
4#20274712C211yesmaximal, characteristic
5C2×#16810375C211yesmaximal, characteristic, complemented
6C2×#16810370C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810904—22no—
11#16810904—22no—
12C2×#13133151—21no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133140—21no—
15C2×#13133140—21no—
16C2×#13133139—21no—
17C2×#13133137—21no—
18C2×#13133129—21no—
19C2×#13133129—21no—
20C2×C4×#88322C2211yescharacteristic
21#16810385—21no—
22#16810375—22no—
23#16810370—22no—
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#384C2211yescharacteristic
27C22×Dic106—21no—
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C22×#88322—21no—
31C22×#88322D411yescharacteristic
32C22×#88322D411yescharacteristic
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217D412yes—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×#11719921—21no—
45#13133151—24no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133140—42no—
49#13133140—42no—
50#13133139—24no—
51#13133137—42no—
52#13133129—42no—
53#13133129—42no—
54C4×#88322—21no—
55C4×#88322—21no—
56C22×C8×C53—21no—
57C22×C8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53D412yes—
66C23×D53—21no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4111yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—41no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212Q1612yes—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×Dic106—22no—
86C2×Dic106—41no—
87C2×Dic106—41no—
88C2×Dic106—41no—
89C2×Dic106—41no—
90C2×Dic106—41no—
91C2×C8×C53—22no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—81no—
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C42×C53#4111yescharacteristic
106#11719921—41no—
107#11719921—41no—
108C4⋊Dic53—22no—
109C4⋊Dic53—21no—
110C4⋊Dic53—22no—
111C4⋊Dic53—21no—
112C4⋊Dic53—41no—
113C4⋊Dic53—41no—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322—21no—
118C2×#88322—21no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322D812yes—
123C23×C53#12111yescharacteristic
124C22×D53—22no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12311yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53#12411yescharacteristic
142C2×C4×C53C2×D811yescharacteristic
143C2×C4×C53C2×Q1611yescharacteristic
144C2×C4×C53#13511yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C2×Dic53—42no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—81no—
155C2×Dic53—81no—
156D212—22no—
157D212—22no—
158D212—41no—
159D212—41no—
160C4×D53—81no—
161C4×D53—81no—
162C4×D53—81no—
163C4×D53—81no—
164Dic106—42no—
165Dic106—42no—
166Dic106—81no—
167Dic106—81no—
168C8×C53—42no—
169C8×C53—81no—
170#88322—22no—
171#88322—41no—
172#88322—81no—
173#10485—531nomaximal, Sylow
174C22×C53#39411yescharacteristic
175C22×C53#45911yescharacteristic
176C22×C53#46311yescharacteristic
177C22×C53#46711yescharacteristic
178C22×C53#46811yescharacteristic
179C22×C53#47111yescharacteristic
180C22×C53#47311yescharacteristic
181C2×D53—42no—
182C2×D53—42no—
183C2×D53—42no—
184C2×D53—81no—
185C2×D53—81no—
186C2×D53—81no—
187C2×D53—81no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—161no—
191C4×C53#44911yescharacteristic
192C4×C53#45111yescharacteristic
193C4×C53#45911yescharacteristic
194C4×C53#46311yescharacteristic
195C4×C53#46611yescharacteristic
196C4×C53#46911yescharacteristic
197C4×C53#48411yescharacteristic
198C4×C53#48511yescharacteristic
199Dic53—82no—
200Dic53—81no—
201C2×#474—531no—
202#1956—531no—
203#1954—531no—
204#1802—531no—
205C2×#335—531no—
206C2×#332—531no—
207#1296D5311yescharacteristic, complemented
208C2×C53#166911yescharacteristic
209C2×C53#168311yescharacteristic
210C2×C53#168511yescharacteristic
211C2×C53#169911yescharacteristic
212C2×C53#203611yescharacteristic
213C2×C53#205211yescharacteristic
214C2×C53#205411yescharacteristic
215D53—82no—
216C2×C4⋊Q8—531no—
217C2×C4×D4—531no—
218#474—1062no—
219#474—1062no—
220C2×#107—1061no—
221C2×C4⋊C8—531no—
222C2×C4⋊C8C2×D5311yescharacteristic
223C2×C4⋊C8C2×D5311yescharacteristic
224C2×#103—1061no—
225C2×#103—1061no—
226C2×#102—1061no—
227C2×#102—1061no—
228C2×C22⋊C8—1061no—
229C2×C22⋊C8—1061no—
230C2×C4×C8—531no—
231#384C2×D5311yescharacteristic
232#340—1061no—
233#335—1062no—
234#332—1062no—
235C53#1048511yescharacteristic, complemented, Sylow
236C22×Q8—1061no—
237C22×D4—531no—
238C23×C4—1061no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—531no—
242C22×C8—531no—
243C22×C8—1061no—
244C4⋊Q8—532no—
245C4⋊Q8—1061no—
246C4⋊Q8—1061no—
247C4⋊Q8—1061no—
248C4×D4—1061no—
249C4×D4—1061no—
250C4×D4—1061no—
251C4×D4—1061no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4—1061no—
258C2×C22⋊C4—1061no—
259C2×C42C22×D5311yescharacteristic
260#107—1064no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266C4⋊C8D21212yes—
267#103—2122no—
268#103—2122no—
269#102—1064no—
270#102—2122no—
271C22⋊C8—2122no—
272C22⋊C8—2122no—
273C4×C8—1061no—
274C4×C8—1061no—
275C24—1061no—
276C2×Q8—1062no—
277C2×Q8—2121no—
278C2×Q8—2121no—
279C2×Q8—2121no—
280C2×Q8—2121no—
281C2×Q8—2121no—
282C2×D4—532no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C22×C4—41no—
289C22×C4—1061no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4#1171994511yescharacteristic
298C22×C4D4×D5311yescharacteristic
299C22×C4D4×D5311yescharacteristic
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—532no—
307C2×C8—1061no—
308C2×C8—1061no—
309C2×C8—1061no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—2121no—
313C2×C8—2121no—
314C2×C8—2121no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—81no—
320C4⋊C4—1062no—
321C4⋊C4—1061no—
322C4⋊C4—1061no—
323C4⋊C4—1062no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C42C2×D21211yescharacteristic
329C42#1171994511yescharacteristic
330C42D4×D5311yescharacteristic
331C42#1171994911yescharacteristic
332C23—1062no—
333C23—2121no—
334C23—2121no—
335C23—2121no—
336C23—2121no—
337C23—2121no—
338C23—2121no—
339C23#1313319011yescharacteristic
340Q8—4241no—
341Q8—4241no—
342Q8—2122no—
343Q8—2122no—
344D4—1062no—
345D4—1062no—
346D4—2121no—
347D4—2121no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—4241no—
359C2×C4—2121no—
360C2×C4—2122no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4#1313318711yescharacteristic
364C2×C4#1313319011yescharacteristic
365C2×C4#1313319211yescharacteristic
366C2×C4#1313320211yescharacteristic
367C2×C4#1313320211yescharacteristic
368C2×C4#1313320511yescharacteristic
369C2×C4D8×D5311yescharacteristic
370C2×C4Q16×D5311yescharacteristic
371C2×C4#1313323811yescharacteristic
372C2×C4#1313324011yescharacteristic
373C2×C4#1313324111yescharacteristic
374C2×C4#1313324411yescharacteristic
375C2×C4#1313324411yescharacteristic
376C2×C4#1313324611yescharacteristic
377C8—42no—
378C8—81no—
379C8—1062no—
380C8—4241no—
381C8—2121no—
382C22—4241no—
383C22—4241no—
384C22—4241no—
385C22—4241no—
386C22—2122no—
387C22—2122no—
388C22—2122no—
389C22#1681054111yescharacteristic
390C22#1681081711yescharacteristic
391C22#1681084111yescharacteristic
392C22#1681086311yescharacteristic
393C22#1681086811yescharacteristic
394C22#1681088511yescharacteristic
395C22#1681089711yescharacteristic
396C4—21no—
397C4—21no—
398C4—161no—
399C4—4241no—
400C4—4242no—
401C4#1681076911yescharacteristic
402C4#1681078011yescharacteristic
403C4#1681081711yescharacteristic
404C4#1681084111yescharacteristic
405C4#1681086011yescharacteristic
406C4#1681087711yescharacteristic
407C4#1681096011yescharacteristic
408C4#1681096711yescharacteristic
409C2—4242no—
410C2#2027407911yescharacteristic
411C2#2027416911yescharacteristic
412C2#2027417911yescharacteristic
413C2#2027426511yescharacteristic
414C2#2027592111yescharacteristic
415C2#2027601711yescharacteristic
416C2#2027602711yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenter#1296D53
hypocenterC53#10485
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #20275516C2semidirect
5. C2×#16810375C2semidirect
6. C2×#16810370C2semidirect
7. C53×#1296C2semidirect
207. #1296D53semidirect
235. C53#10485semidirect

Maximal subgroups

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

subgroupindex
C2×#168109042
#202755162
#202755082
#202747122
C2×#168103752
C2×#168103702
C53×#12962
#1048553