Finite Group Databasebeta

← #27287899

Subgroups — #27287899

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810824C211yesmaximal, characteristic
2#20275650C211yesmaximal, characteristic, complemented
3#20275520C211yesmaximal, characteristic
4#20274717C211yesmaximal, characteristic, complemented
5C2×#16810375C211yesmaximal, characteristic, complemented
6C2×#16810366C211yesmaximal, characteristic
7C53×#1296C211yesmaximal, characteristic, complemented
8C22×#11719913—21no—
9C2×#13133214C2211yescharacteristic
10C2×C4×Dic106C2211yescharacteristic
11#16810824—22no—
12#16810824—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133140—21no—
15C2×#13133139—21no—
16C2×#13133137—21no—
17C2×#13133137—21no—
18C2×C4×#88322C2211yescharacteristic
19#16810385—21no—
20#16810377—21no—
21#16810377—21no—
22#16810375—22no—
23#16810366—22no—
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#384C2211yescharacteristic
27C22×Dic106D411yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719913—24no—
32C2×#11719913—42no—
33C2×#11719913—42no—
34C22×#88322—21no—
35C22×#88322D411yescharacteristic
36C22×#88322D411yescharacteristic
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214D412yes—
41C4×Dic106—21no—
42C4×Dic106—21no—
43C4×Dic106—21no—
44C4×Dic106—21no—
45C2×C4⋊Dic53D411yescharacteristic
46C2×#11719919—21no—
47C2×C4×Dic53—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50#13133140—42no—
51#13133139—24no—
52#13133137—42no—
53#13133137—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—41no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4111yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—81no—
88C2×D212—81no—
89C2×Dic106—21no—
90C2×Dic106—21no—
91C2×Dic106—21no—
92C2×Dic106—21no—
93C2×Dic106—21no—
94C2×Dic106D812yes—
95C2×C8×C53—22no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—81no—
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112#11719919—41no—
113#11719919—41no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116C2×#88322—21no—
117C2×#88322—21no—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322SD1612yescomplemented
125#11719913—48no—
126C23×C53#12111yescharacteristic
127C22×D53—42no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—81no—
131C22×D53—81no—
132C22×D53—81no—
133C22×D53—81no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12311yescharacteristic
143C2×C4×C53#12311yescharacteristic
144C2×C4×C53#12411yescharacteristic
145C2×C4×C53C2×D811yescharacteristic
146C2×C4×C53C2×SD1611yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C2×C4×C53#13711yescharacteristic
153C2×Dic53—22no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159D212—42no—
160D212—42no—
161D212—42no—
162D212—42no—
163D212—82no—
164D212—82no—
165D212—81no—
166D212—81no—
167D212—81no—
168D212—81no—
169Dic106—22no—
170Dic106—22no—
171Dic106—41no—
172Dic106—41no—
173C8×C53—42no—
174C8×C53—81no—
175#88322—22no—
176#88322—41no—
177#88322—81no—
178#10474—531nomaximal, Sylow
179C22×C53#39411yescharacteristic
180C22×C53#46011yescharacteristic
181C22×C53#46411yescharacteristic
182C22×C53#46611yescharacteristic
183C22×C53#46811yescharacteristic
184C22×C53#47111yescharacteristic
185C22×C53#47211yescharacteristic
186C2×D53—82no—
187C2×D53—82no—
188C2×D53—82no—
189C2×D53—161no—
190C2×D53—161no—
191C2×D53—161no—
192C2×D53—161no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—161no—
196C4×C53#44711yescharacteristic
197C4×C53#45211yescharacteristic
198C4×C53#46111yescharacteristic
199C4×C53#46311yescharacteristic
200C4×C53#46511yescharacteristic
201C4×C53#46811yescharacteristic
202C4×C53#48011yescharacteristic
203C4×C53#48211yescharacteristic
204Dic53—42no—
205Dic53—81no—
206C2×#460—531no—
207#1984—531no—
208#1957—531no—
209#1803—531no—
210C2×#335—531no—
211C2×#331—531no—
212#1296D5311yescharacteristic, complemented
213C2×C53#166811yescharacteristic
214C2×C53#168211yescharacteristic
215C2×C53#168611yescharacteristic
216C2×C53#169811yescharacteristic
217C2×C53#203311yescharacteristic
218C2×C53#205111yescharacteristic
219C2×C53#205511yescharacteristic
220D53—162no—
221C22×SD16—1061no—
222C2×#127—531no—
223C2×C4×Q8—531no—
224#460—1062no—
225#460—1062no—
226C2×C4⋊C8—531no—
227C2×C4⋊C8C2×D5311yescharacteristic
228C2×C4⋊C8C2×D5311yescharacteristic
229C2×#103—1061no—
230C2×#102—1061no—
231C2×#102—1061no—
232C2×#102—1061no—
233C2×C4×C8—531no—
234#384C2×D5311yescharacteristic
235#340—1061no—
236#336—1061no—
237#336—1061no—
238#335—1062no—
239#331—1062no—
240C53#1047411yescharacteristic, complemented, Sylow
241C22×Q8—531no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4—1061no—
245C2×SD16—1064no—
246C2×SD16—2122no—
247C2×SD16—2122no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—531no—
251C22×C8—531no—
252C22×C8—1061no—
253#127—532no—
254#127—1061no—
255#127—1061no—
256#127—1061no—
257C4×Q8—1061no—
258C4×Q8—1061no—
259C4×Q8—1061no—
260C4×Q8—1061no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—1061no—
266C2×C42—1061no—
267C2×C42C22×D5311yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1061no—
272C4⋊C8—1061no—
273C4⋊C8D21212yes—
274#103—2122no—
275#102—1064no—
276#102—2122no—
277#102—2122no—
278C4×C8—1061no—
279C4×C8—1061no—
280C24—2121no—
281C2×Q8—532no—
282C2×Q8—1061no—
283C2×Q8—1061no—
284C2×Q8—1061no—
285C2×Q8—1061no—
286C2×Q8—1061no—
287C2×D4—1062no—
288C2×D4—1062no—
289C2×D4—1062no—
290C2×D4—4241no—
291C2×D4—4241no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C22×C4—41no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4#1171994511yescharacteristic
307C22×C4D4×D5311yescharacteristic
308C22×C4D4×D5311yescharacteristic
309SD16—2128no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—532no—
317C2×C8—1061no—
318C2×C8—1061no—
319C2×C8—1061no—
320C2×C8—1061no—
321C2×C8—1061no—
322C2×C8—2121no—
323C2×C8—2121no—
324C2×C8—2121no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—81no—
330C4⋊C4—1061no—
331C4⋊C4—1061no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C42—2121no—
335C42—2121no—
336C42C2×D21211yescharacteristic
337C42#1171994511yescharacteristic
338C42D4×D5311yescharacteristic
339C42#1171994911yescharacteristic
340C23—4241no—
341C23—4241no—
342C23—4241no—
343C23—4241no—
344C23—4241no—
345C23—4241no—
346C23—2122no—
347C23#1313319011yescharacteristic
348Q8—1062no—
349Q8—1062no—
350Q8—2121no—
351Q8—2121no—
352D4—4242no—
353D4—4242no—
354D4—4241no—
355D4—4241no—
356D4—4241no—
357D4—4241no—
358D4—2122no—
359D4—2122no—
360D4—2122no—
361D4—2122no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—1062no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4#1313318711yescharacteristic
374C2×C4#1313319011yescharacteristic
375C2×C4#1313319211yescharacteristic
376C2×C4#1313320211yescharacteristic
377C2×C4#1313320211yescharacteristic
378C2×C4#1313320511yescharacteristic
379C2×C4D8×D5311yescharacteristic
380C2×C4SD16×D5311yescharacteristic
381C2×C4#1313323811yescharacteristic
382C2×C4#1313323811yescharacteristic
383C2×C4#1313324111yescharacteristic
384C2×C4#1313324411yescharacteristic
385C2×C4#1313324911yescharacteristic
386C2×C4#1313324911yescharacteristic
387C8—42no—
388C8—81no—
389C8—1062no—
390C8—4241no—
391C8—2121no—
392C22—8481no—
393C22—8481no—
394C22—8481no—
395C22—8481no—
396C22—4242no—
397C22—4242no—
398C22—4242no—
399C22#1681054111yescharacteristic
400C22#1681082211yescharacteristic
401C22#1681084811yescharacteristic
402C22#1681085811yescharacteristic
403C22#1681087211yescharacteristic
404C22#1681088911yescharacteristic
405C22#1681089211yescharacteristic
406C4—21no—
407C4—21no—
408C4—161no—
409C4—4241no—
410C4—2122no—
411C4#1681075811yescharacteristic
412C4#1681078811yescharacteristic
413C4#1681082711yescharacteristic
414C4#1681083711yescharacteristic
415C4#1681085311yescharacteristic
416C4#1681087011yescharacteristic
417C4#1681093811yescharacteristic
418C4#1681094911yescharacteristic
419C2—8482no—
420C2#2027407411yescharacteristic
421C2#2027416211yescharacteristic
422C2#2027419011yescharacteristic
423C2#2027425811yescharacteristic
424C2#2027590011yescharacteristic
425C2#2027601011yescharacteristic
426C2#2027603411yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#121
prime radicalC2×#13133214C22
hypercenter#1296D53
hypocenterC53#10474
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. #20275650C2semidirect
4. #20274717C2semidirect
5. C2×#16810375C2semidirect
7. C53×#1296C2semidirect
124. C2×#88322SD16semidirect
212. #1296D53semidirect
240. C53#10474semidirect

Maximal subgroups

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

subgroupindex
C2×#168108242
#202756502
#202755202
#202747172
C2×#168103752
C2×#168103662
C53×#12962
#1047453