Finite Group Databasebeta

← #27286633

Subgroups — #27286633

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810996C211yesmaximal, characteristic, complemented
2C2×#16810903C211yesmaximal, characteristic, complemented
3C2×#16810818C211yesmaximal, characteristic
4C2×#16810370C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272550C211yesmaximal, characteristic, complemented
7C53×#1954C211yesmaximal, characteristic, complemented
8C22×#11719913—22no—
9C22×#11719922—21no—
10C2×#13133217C2211yescharacteristic
11C2×#13133214C2211yescharacteristic
12#16810996—22no—
13#16810996—22no—
14#16810903—22no—
15#16810903—22no—
16#16810818—22no—
17#16810818—22no—
18C2×#13133150—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133138—21no—
22C2×#13133137—21no—
23#16810370—22no—
24#16810366—22no—
25C2×C4×D4×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C22⋊C8×C53—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212—21no—
33C2×#11719913—28no—
34C2×#11719913—44no—
35C2×#11719913—44no—
36C2×#11719922—24no—
37C2×#11719922—42no—
38C2×#11719922—42no—
39C22×#88322—21no—
40C22×#88322—21no—
41#13133217—21no—
42#13133217—21no—
43#13133217—21no—
44#13133217D412yes—
45#13133214—21no—
46#13133214—21no—
47#13133214—21no—
48#13133214D412yes—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51#13133150—24no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322D412yes—
55C4⋊#88322D412yes—
56#13133138—24no—
57#13133137—24no—
58C22×D4×C53D411yescharacteristic
59C23×C4×C53—21no—
60C22×C8×C53—21no—
61C22×C8×C53D412yes—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C22⋊C8×C53—44no—
72C4×C8×C53—21no—
73C4×C8×C53—21no—
74C24×C53—21no—
75C23×D53—41no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53SD1612yescomplemented
82C22×C4×C53—21no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×Dic53—41no—
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—42no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—42no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—82no—
104C2×Dic106—22no—
105C2×Dic106—42no—
106C2×Dic106—42no—
107C2×Dic106—41no—
108C2×C8×C53—22no—
109C2×C8×C53—22no—
110C2×C8×C53—41no—
111C2×C8×C53—42no—
112C2×C8×C53D814yescomplemented
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121#11719922—48no—
122C4⋊Dic53—22no—
123C4⋊Dic53—22no—
124C4⋊Dic53—41no—
125C4⋊Dic53—41no—
126C2×#88322—22no—
127C2×#88322—22no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132#11719913—416no—
133C23×C53—22no—
134C23×C53—41no—
135C23×C53—42no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53#12111yescharacteristic
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—81no—
142C22×D53—81no—
143C22×D53—82no—
144D4×C53—22no—
145D4×C53—22no—
146D4×C53—41no—
147D4×C53—41no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—82no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12011yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12711yescharacteristic
160C2×C4×C53C2×D812yes—
161C2×C4×C53C2×SD1611yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)12yes—
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53#13711yescharacteristic
166C2×Dic53—42no—
167C2×Dic53—82no—
168D212—44no—
169D212—42no—
170D212—42no—
171D212—82no—
172D212—82no—
173D212—81no—
174D212—81no—
175D212—81no—
176D212—81no—
177Dic106—44no—
178Dic106—81no—
179Dic106—81no—
180C8×C53—24no—
181C8×C53—81no—
182#88322—42no—
183#88322—42no—
184#10266—531nomaximal, Sylow
185C22×C53—44no—
186C22×C53—42no—
187C22×C53—82no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53#39211yescharacteristic
191C22×C53#45912yes—
192C22×C53#46511yescharacteristic
193C22×C53#46811yescharacteristic
194C22×C53#47112yes—
195C2×D53—82no—
196C2×D53—82no—
197C2×D53—82no—
198C2×D53—162no—
199C2×D53—162no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—81no—
203C4×C53#44712yes—
204C4×C53#44811yescharacteristic
205C4×C53#44911yescharacteristic
206C4×C53#45011yescharacteristic
207C4×C53#45211yescharacteristic
208C4×C53#49311yescharacteristic
209C4×C53#49611yescharacteristic
210Dic53—82no—
211C2×#492—531no—
212C2×#474—531no—
213C2×#459—531no—
214#1954D5311yescharacteristic, complemented
215C2×#332—531no—
216C2×#331—531no—
217#1277—531no—
218C2×C53—82no—
219C2×C53#166512yes—
220C2×C53#168512yes—
221C2×C53#200011yescharacteristic
222C2×C53#201111yescharacteristic
223C2×C53#201911yescharacteristic
224D53—162no—
225C22×SD16—1062no—
226C22×D8—1061no—
227C2×C4⋊Q8—531no—
228C2×#127—531no—
229C2×C4×D4C2×D5311yescharacteristic
230#492—1062no—
231#492—1062no—
232#474—1062no—
233#474—1062no—
234#459—1062no—
235#459—1062no—
236C2×#107—1061no—
237C2×C4⋊C8—531no—
238C2×C4⋊C8—531no—
239C2×C4⋊C8C2×D5311yescharacteristic
240C2×#102—1061no—
241C2×#102—1061no—
242C2×C22⋊C8—22no—
243C2×C4×C8C2×D5311yescharacteristic
244#332—1062no—
245#331—1062no—
246C53#1026611yescharacteristic, complemented, Sylow
247C22×Q8—1061no—
248C22×D4—1061no—
249C22×D4—1061no—
250C22×D4—1061no—
251C22×D4D21211yescharacteristic
252C23×C4—21no—
253C2×SD16—1068no—
254C2×SD16—2124no—
255C2×SD16—2124no—
256C2×D8—1064no—
257C2×D8—2122no—
258C2×D8—2122no—
259C22×C8—21no—
260C22×C8—1061no—
261C22×C8—1061no—
262C22×C8C22⋊D5312yes—
263C4⋊Q8—532no—
264C4⋊Q8—1061no—
265C4⋊Q8—1061no—
266C4⋊Q8—1061no—
267#127—532no—
268#127—1061no—
269#127—1061no—
270#127—1061no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4D21211yescharacteristic
278C2×C22⋊C4—21no—
279C2×C42C22×D5311yescharacteristic
280#107—1064no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—532no—
284C4⋊C8—532no—
285C4⋊C8—1061no—
286C4⋊C8—1061no—
287#102—1064no—
288#102—1064no—
289C22⋊C8—44no—
290C4×C8—21no—
291C4×C8—21no—
292C24—21no—
293C24—2121no—
294C2×Q8—1062no—
295C2×Q8—2122no—
296C2×Q8—2122no—
297C2×Q8—2121no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—1062no—
304C2×D4—1062no—
305C2×D4—1062no—
306C2×D4—4242no—
307C2×D4—2121no—
308C2×D4—2121no—
309C2×D4—2121no—
310C2×D4—2122no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2122no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4#1171991312yescomplemented
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—2121no—
323C22×C4C2×D21211yescharacteristic
324C22×C4#1171994511yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326SD16—21216no—
327D8—2128no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—1062no—
333C2×C8—1062no—
334C2×C8—2121no—
335C2×C8—2121no—
336C2×C8—2121no—
337C2×C8—2121no—
338C2×C8#1171992214yescomplemented
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—1062no—
342C4⋊C4—1062no—
343C4⋊C4—2121no—
344C4⋊C4—2121no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42D4×D5311yescharacteristic
348C42D4×D5311yescharacteristic
349C42D4×D5311yescharacteristic
350C42D4×D5311yescharacteristic
351C23—22no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—42no—
356C23—4242no—
357C23—4241no—
358C23—4241no—
359C23—4242no—
360C23—2122no—
361C23#1313319111yescharacteristic
362Q8—4241no—
363Q8—4241no—
364Q8—2124no—
365D4—22no—
366D4—22no—
367D4—41no—
368D4—41no—
369D4—4242no—
370D4—4242no—
371D4—4241no—
372D4—4241no—
373D4—4241no—
374D4—4241no—
375D4—2124no—
376D4—2122no—
377D4—2122no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—4242no—
385C2×C4—2122no—
386C2×C4#1313318511yescharacteristic
387C2×C4#1313318711yescharacteristic
388C2×C4#1313318711yescharacteristic
389C2×C4#1313319011yescharacteristic
390C2×C4#1313319011yescharacteristic
391C2×C4#1313321311yescharacteristic
392C2×C4C2×#1171992212yes—
393C2×C4C2×#1171991311yescharacteristic
394C2×C4#1313323911yescharacteristic
395C2×C4#1313324212yes—
396C2×C4#1313324511yescharacteristic
397C2×C4#1313325011yescharacteristic
398C8—24no—
399C8—81no—
400C8—2122no—
401C8—2122no—
402C22—42no—
403C22—44no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—8482no—
408C22—8482no—
409C22—4242no—
410C22—4242no—
411C22—4242no—
412C22#1681053211yescharacteristic
413C22#1681081812yes—
414C22#1681085211yescharacteristic
415C22#1681086911yescharacteristic
416C22#1681088612yes—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—4242no—
421C4#1681075612yes—
422C4#1681076411yescharacteristic
423C4#1681077011yescharacteristic
424C4#1681077611yescharacteristic
425C4#1681078711yescharacteristic
426C4#1681099911yescharacteristic
427C4#1681101311yescharacteristic
428C2—82no—
429C2—8482no—
430C2#2027405612yes—
431C2#2027418012yes—
432C2#2027572011yescharacteristic
433C2#2027578511yescharacteristic
434C2#2027583011yescharacteristic
subgroupG/H
centerC23#13133191
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1954C2
socleC23×C53#121
prime radicalC2×#16810818C2
hypercenter#1954D53
hypocenterC53#10266
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. C2×#16810996C2semidirect
2. C2×#16810903C2semidirect
4. C2×#16810370C2semidirect
5. C2×#16810366C2semidirect
6. #20272550C2semidirect
7. C53×#1954C2semidirect
81. C2×D4×C53SD16semidirect
112. C2×C8×C53D8semidirect
214. #1954D53semidirect
246. C53#10266semidirect
316. C2×D4#11719913semidirect
338. C2×C8#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109962
C2×#168109032
C2×#168108182
C2×#168103702
C2×#168103662
#202725502
C53×#19542
#1026653