Finite Group Databasebeta

← #27286864

Subgroups — #27286864

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811000C211yesmaximal, characteristic
2C2×#16810997C211yesmaximal, characteristic, complemented
3C2×#16810971C211yesmaximal, characteristic, complemented
4#20275520C211yesmaximal, characteristic, complemented
5#20275520C211yesmaximal, characteristic, complemented
6#20272542C211yesmaximal, characteristic, complemented
7C53×#1990C211yesmaximal, characteristic, complemented
8C22×#11719924—22no—
9C22×D424—22no—
10C2×#13133214C2211yescharacteristic
11C2×#13133214C2211yescharacteristic
12#16811000—22no—
13#16811000—22no—
14#16810997—22no—
15#16810997—22no—
16#16810971—22no—
17#16810971—22no—
18C2×#13133137—22no—
19C2×#13133137—22no—
20C2×#13133137—22no—
21C2×C4×#88322C2211yescharacteristic
22C2×C4×#88322C2211yescharacteristic
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4×C8×C53C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C53×#340—22no—
27C22×D212—21no—
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212—21no—
33C2×#11719924—28no—
34C2×#11719924—44no—
35C2×#11719924—44no—
36C2×D424—28no—
37C2×D424—44no—
38C2×D424—44no—
39C22×#88322D412yes—
40C22×#88322D412yes—
41#13133214—21no—
42#13133214—21no—
43#13133214—21no—
44#13133214—21no—
45#13133214—21no—
46#13133214—21no—
47#13133214D412yes—
48#13133214D412yes—
49#13133137—28no—
50#13133137—44no—
51#13133137—44no—
52C4×#88322—21no—
53C4×#88322—21no—
54C4×#88322—21no—
55C4×#88322—21no—
56C22×Q8×C53—21no—
57C22×C8×C53D412yes—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53D412yes—
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×C42×C53C2311yescharacteristic
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70C23×D53—41no—
71C23×D53—41no—
72C2×Q8×C53—22no—
73C2×Q8×C53—41no—
74C2×Q8×C53—42no—
75C2×Q8×C53—42no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C2×D212—22no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—42no—
88C2×D212—42no—
89C2×D212—42no—
90C2×D212—42no—
91C2×D212—42no—
92C2×D212—42no—
93C2×D212—42no—
94C2×D212—42no—
95C2×D212—42no—
96C2×D212—42no—
97C2×D212—42no—
98C2×D212—42no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—41no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53SD1614yescomplemented
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—42no—
113C4⋊C4×C53—42no—
114C4⋊C4×C53—42no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119#11719924—416no—
120C2×#88322—22no—
121C2×#88322—22no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322D814yescomplemented
126D424—416no—
127C23×C53#12711yescharacteristic
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—81no—
131C22×D53—81no—
132C22×D53—81no—
133C22×D53—81no—
134C22×D53—81no—
135C22×D53—81no—
136C22×D53—81no—
137C22×D53—81no—
138C22×D53—81no—
139C22×D53—81no—
140C22×D53—81no—
141C22×D53—81no—
142Q8×C53—44no—
143Q8×C53—82no—
144C2×C4×C53—42no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53C2×D812yes—
157C2×C4×C53C2×SD1612yes—
158C2×C4×C53#13512yes—
159C2×C4×C53AGL(1,ℤ/8)12yes—
160C2×C4×C53AGL(1,ℤ/8)12yes—
161C2×C4×C53AGL(1,ℤ/8)12yes—
162D212—44no—
163D212—44no—
164D212—44no—
165D212—44no—
166D212—44no—
167D212—44no—
168D212—82no—
169D212—82no—
170D212—82no—
171D212—82no—
172D212—82no—
173D212—82no—
174C8×C53—24no—
175#88322—24no—
176#88322—42no—
177#10305—531nomaximal, Sylow
178C22×C53#39211yescharacteristic
179C22×C53#49211yescharacteristic
180C22×C53#49311yescharacteristic
181C22×C53#49511yescharacteristic
182C22×C53#49611yescharacteristic
183C22×C53#49611yescharacteristic
184C22×C53#49611yescharacteristic
185C2×D53—82no—
186C2×D53—82no—
187C2×D53—82no—
188C2×D53—82no—
189C2×D53—82no—
190C2×D53—82no—
191C2×D53—161no—
192C2×D53—161no—
193C2×D53—161no—
194C2×D53—161no—
195C2×D53—161no—
196C2×D53—161no—
197C2×D53—161no—
198C2×D53—161no—
199C4×C53—82no—
200C4×C53—161no—
201C4×C53#44712yes—
202C4×C53#44912yes—
203C4×C53#45012yes—
204C4×C53#45912yes—
205C4×C53#46012yes—
206C4×C53#46312yes—
207C2×#493—531no—
208C2×#492—531no—
209C2×#486—531no—
210#1990D5311yescharacteristic, complemented
211#1957—531no—
212#1957—531no—
213#1273—531no—
214C2×C53#170611yescharacteristic
215C2×C53#171511yescharacteristic
216C2×C53#171711yescharacteristic
217C2×C53#172111yescharacteristic
218C2×C53#201111yescharacteristic
219C2×C53#201211yescharacteristic
220C2×C53#201411yescharacteristic
221D53—162no—
222D53—162no—
223C22×SD16—1062no—
224C22×D8—1062no—
225C2×C4⋊Q8C2×D5311yescharacteristic
226C2×#127—531no—
227C2×#127—531no—
228#493—1062no—
229#493—1062no—
230#492—1062no—
231#492—1062no—
232#486—1062no—
233#486—1062no—
234C2×#102—1062no—
235C2×#102—1062no—
236C2×#102—1062no—
237C2×C4×C8—531no—
238C2×C4×C8—531no—
239C2×C4×C8C2×D5311yescharacteristic
240#384C2×D5311yescharacteristic
241#340—22no—
242C53#1030511yescharacteristic, complemented, Sylow
243C22×Q8—21no—
244C22×D4—1061no—
245C22×D4—1061no—
246C22×D4—1061no—
247C22×D4—1061no—
248C22×D4—1061no—
249C22×D4—1061no—
250C2×SD16—1068no—
251C2×SD16—2124no—
252C2×SD16—2124no—
253C2×D8—1068no—
254C2×D8—2124no—
255C2×D8—2124no—
256C22×C8—532no—
257C22×C8—532no—
258C22×C8C22⋊D5312yes—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8D21212yes—
263#127—532no—
264#127—532no—
265#127—1061no—
266#127—1061no—
267#127—1061no—
268#127—1061no—
269#127—1061no—
270#127—1061no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C42C22×D5311yescharacteristic
277#102—1068no—
278#102—2124no—
279#102—2124no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—1061no—
283C4×C8—1061no—
284C4×C8—1061no—
285C4×C8—1061no—
286C24—2121no—
287C24—2121no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×D4—1062no—
293C2×D4—1062no—
294C2×D4—1062no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2122no—
305C2×D4—2122no—
306C2×D4—2122no—
307C2×D4—2122no—
308C2×D4—2122no—
309C2×D4—2122no—
310C2×D4—2122no—
311C2×D4—2122no—
312C2×D4—2122no—
313C2×D4—2122no—
314C2×D4—2122no—
315C2×D4—2122no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4D4×D5311yescharacteristic
319C22×C4D4×D5311yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—21216no—
322D8—21216no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—534no—
326C2×C8—1062no—
327C2×C8—1062no—
328C2×C8—1062no—
329C2×C8—1062no—
330C2×C8—1062no—
331C2×C8#1171992414yescomplemented
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C42C2×D21211yescharacteristic
340C42#1171994511yescharacteristic
341C42D4×D5311yescharacteristic
342C42D4×D5311yescharacteristic
343C23—4241no—
344C23—4241no—
345C23—4241no—
346C23—4241no—
347C23—4241no—
348C23—4241no—
349C23—4241no—
350C23—4241no—
351C23—4241no—
352C23—4241no—
353C23—4241no—
354C23—4241no—
355C23—2122no—
356C23—2122no—
357C23#1313321311yescharacteristic
358Q8—44no—
359Q8—82no—
360D4—4242no—
361D4—4242no—
362D4—4242no—
363D4—4242no—
364D4—4242no—
365D4—4242no—
366D4—2124no—
367D4—2124no—
368D4—2124no—
369D4—2124no—
370D4—2124no—
371D4—2124no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4#1313318511yescharacteristic
379C2×C4#1313318711yescharacteristic
380C2×C4#1313318711yescharacteristic
381C2×C4#1313319011yescharacteristic
382C2×C4#1313319011yescharacteristic
383C2×C4#1313319111yescharacteristic
384C2×C4D8×D5312yes—
385C2×C4C2×#1171992412yes—
386C2×C4#1313324012yes—
387C2×C4#1313324212yes—
388C2×C4#1313324412yes—
389C2×C4#1313324412yes—
390C8—24no—
391C8—1064no—
392C8—2122no—
393C22—8481no—
394C22—8481no—
395C22—8481no—
396C22—8481no—
397C22—8481no—
398C22—8481no—
399C22—8481no—
400C22—8481no—
401C22—4242no—
402C22—4242no—
403C22—4242no—
404C22—4242no—
405C22—4242no—
406C22—4242no—
407C22#1681053211yescharacteristic
408C22#1681099711yescharacteristic
409C22#1681099811yescharacteristic
410C22#1681100811yescharacteristic
411C22#1681100911yescharacteristic
412C22#1681100911yescharacteristic
413C22#1681101311yescharacteristic
414C4—82no—
415C4—161no—
416C4#1681075812yes—
417C4#1681077112yes—
418C4#1681077312yes—
419C4#1681081712yes—
420C4#1681082312yes—
421C4#1681084112yes—
422C2—8482no—
423C2—8482no—
424C2#2027430811yescharacteristic
425C2#2027434411yescharacteristic
426C2#2027435711yescharacteristic
427C2#2027438311yescharacteristic
428C2#2027578711yescharacteristic
429C2#2027579311yescharacteristic
430C2#2027580211yescharacteristic
subgroupG/H
centerC23#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1990C2
socleC23×C53#127
prime radicalC2×#16811000C2
hypercenter#1990D53
hypocenterC53#10305
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#16810997C2semidirect
3. C2×#16810971C2semidirect
4. #20275520C2semidirect
5. #20275520C2semidirect
6. #20272542C2semidirect
7. C53×#1990C2semidirect
107. C2×C8×C53SD16semidirect
125. C2×#88322D8semidirect
210. #1990D53semidirect
242. C53#10305semidirect
331. C2×C8#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168110002
C2×#168109972
C2×#168109712
#202755202
#202725422
C53×#19902
#1030553