Finite Group Databasebeta

← #27285162

Subgroups — #27285162

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810997C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20275642C211yesmaximal, characteristic, complemented
4C2×#16810372C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2563C211yesmaximal, characteristic, complemented
8C22×#11719924—22no—
9C22×#11719922—21no—
10C2×#13133214C2211yescharacteristic
11#16810997—22no—
12#16810997—22no—
13#16810818—22no—
14#16810818—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137—21no—
18C2×#13133137—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810372—22no—
21D4⋊#88322—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#393—22no—
25C53×#390—21no—
26C53×#382—21no—
27C53×#380—22no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719924—28no—
33C2×#11719924—48no—
34C2×#11719922—24no—
35C2×#11719922—42no—
36C2×#11719922—42no—
37C22×#88322—21no—
38C22×#88322—21no—
39C22×#88322—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214D412yes—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D412yes—
48#13133137—24no—
49#13133137—44no—
50C4×#88322—21no—
51C4×#88322D412yes—
52C22×Q8×C53D412yes—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53D412yes—
67C2×C22⋊C4×C53—24no—
68C2×C22⋊C4×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#95×C53—24no—
72#95×C53—21no—
73C24×C53—21no—
74C23×D53—41no—
75C2×Q8×C53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53—44no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53SD1612yescomplemented
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—42no—
90C22×C4×C53—42no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—44no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—42no—
101C2×D212—42no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—84no—
118C42×C53—41no—
119C42×C53—41no—
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53#4111yescharacteristic
124#11719924—416no—
125#11719922—48no—
126C2×#88322—24no—
127C2×#88322—22no—
128C2×#88322—22no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—42no—
133C23×C53—22no—
134C23×C53—41no—
135C23×C53—42no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53#12011yescharacteristic
139C22×D53—42no—
140C22×D53—81no—
141C22×D53—81no—
142C22×D53—82no—
143C22×D53—82no—
144Q8×C53—44no—
145Q8×C53—82no—
146D4×C53—22no—
147D4×C53—22no—
148D4×C53—41no—
149D4×C53—41no—
150C2×C4×C53—22no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—44no—
157C2×C4×C53—42no—
158C2×C4×C53—82no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12711yescharacteristic
167C2×C4×C53C2×SD1611yescharacteristic
168C2×C4×C53#13511yescharacteristic
169C2×C4×C53#13512yes—
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)12yes—
172C2×C4×C53#13711yescharacteristic
173D212—44no—
174D212—42no—
175D212—42no—
176D212—44no—
177D212—82no—
178D212—81no—
179D212—81no—
180D212—81no—
181D212—81no—
182#88322—24no—
183#88322—42no—
184#88322—81no—
185#10048—531nomaximal, Sylow
186C22×C53—44no—
187C22×C53—42no—
188C22×C53—82no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53#39211yescharacteristic
192C22×C53#44811yescharacteristic
193C22×C53#44912yes—
194C22×C53#44911yescharacteristic
195C22×C53#45312yes—
196C2×D53—82no—
197C2×D53—82no—
198C2×D53—82no—
199C2×D53—162no—
200C2×D53—162no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—82no—
204C4×C53—82no—
205C4×C53—81no—
206C4×C53#45011yescharacteristic
207C4×C53#45211yescharacteristic
208C4×C53#46312yes—
209C4×C53#46511yescharacteristic
210C4×C53#46811yescharacteristic
211C4×C53#49511yescharacteristic
212C4×C53#49611yescharacteristic
213C2×#492—531no—
214C2×#459—531no—
215#2563D5311yescharacteristic, complemented
216#1982—531no—
217C2×#333—531no—
218C2×D4⋊C8—531no—
219#1279—531no—
220C2×C53—82no—
221C2×C53#162212yes—
222C2×C53#162412yes—
223C2×C53#200011yescharacteristic
224C2×C53#200211yescharacteristic
225C2×C53#201211yescharacteristic
226D53—162no—
227C22×SD16—1062no—
228C22×D8—1061no—
229C2×C4⋊Q8C2×D5311yescharacteristic
230C2×#127—531no—
231C2×C4×D4C2×D5311yescharacteristic
232#492—1062no—
233#492—1062no—
234#459—1062no—
235#459—1062no—
236C2×C4⋊C8—531no—
237C2×C4⋊C8—531no—
238C2×#102—1061no—
239C2×#102—1062no—
240C2×C4×C8—531no—
241#393—22no—
242#390—21no—
243#382—21no—
244#380—22no—
245C4×C4⋊C4C2×D5311yescharacteristic
246#333—1062no—
247D4⋊C8—1062no—
248C53#1004811yescharacteristic, complemented, Sylow
249C22×Q8C22⋊D5312yes—
250C22×D4—1061no—
251C22×D4—1061no—
252C22×D4—1061no—
253C22×D4C22⋊D5311yescharacteristic
254C23×C4—21no—
255C2×SD16—1068no—
256C2×SD16—2128no—
257C2×D8—1064no—
258C2×D8—2122no—
259C2×D8—2122no—
260C22×C8—1061no—
261C22×C8—1061no—
262C22×C8—1061no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
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—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22⋊D5311yescharacteristic
278C2×C4⋊C4C22⋊D5312yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—24no—
281C2×C42—21no—
282C2×C42C22×D5311yescharacteristic
283C4⋊C8—532no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8—1061no—
287#102—1064no—
288#102—2124no—
289C4×C8—532no—
290C4×C8—1061no—
291#95—24no—
292#95—21no—
293C24—21no—
294C24—2121no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—44no—
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—2124no—
307C2×D4—2121no—
308C2×D4—2121no—
309C2×D4—2121no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2122no—
313C2×D4—2122no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4#1171992412yescomplemented
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C2×C22⋊D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326C22×C4C2×C22⋊D5311yescharacteristic
327SD16—21216no—
328D8—2128no—
329C2×C8—1064no—
330C2×C8—1062no—
331C2×C8—1062no—
332C2×C8—2121no—
333C2×C8—2122no—
334C2×C8—2121no—
335C2×C8—2121no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—84no—
347C42—41no—
348C42—41no—
349C42D4×D5311yescharacteristic
350C42D4×D5311yescharacteristic
351C42D4×D5311yescharacteristic
352C42#1171994711yescharacteristic
353C23—22no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—4241no—
359C23—4242no—
360C23—4241no—
361C23—4242no—
362C23—2122no—
363C23#1313318611yescharacteristic
364Q8—44no—
365Q8—82no—
366D4—22no—
367D4—22no—
368D4—41no—
369D4—41no—
370D4—4242no—
371D4—4241no—
372D4—4241no—
373D4—4241no—
374D4—4241no—
375D4—2124no—
376D4—2122no—
377D4—2122no—
378D4—2124no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4#1313318511yescharacteristic
391C2×C4#1313318511yescharacteristic
392C2×C4#1313318811yescharacteristic
393C2×C4#1313318911yescharacteristic
394C2×C4#1313318911yescharacteristic
395C2×C4#1313321311yescharacteristic
396C2×C4C2×#1171992411yescharacteristic
397C2×C4#1313323711yescharacteristic
398C2×C4#1313323712yes—
399C2×C4#1313324212yes—
400C2×C4#1313324211yescharacteristic
401C2×C4#1313324811yescharacteristic
402C8—1064no—
403C8—4241no—
404C8—2122no—
405C22—42no—
406C22—44no—
407C22—81no—
408C22—81no—
409C22—82no—
410C22—8482no—
411C22—8482no—
412C22—4242no—
413C22—4242no—
414C22—4242no—
415C22#1681053111yescharacteristic
416C22#1681076211yescharacteristic
417C22#1681076811yescharacteristic
418C22#1681076812yes—
419C22#1681079112yes—
420C4—21no—
421C4—21no—
422C4—82no—
423C4—81no—
424C4—82no—
425C4#1681077311yescharacteristic
426C4#1681078311yescharacteristic
427C4#1681083512yes—
428C4#1681085411yescharacteristic
429C4#1681087111yescharacteristic
430C4#1681100511yescharacteristic
431C4#1681101311yescharacteristic
432C2—82no—
433C2—8482no—
434C2#2027378712yes—
435C2#2027379812yes—
436C2#2027571611yescharacteristic
437C2#2027572811yescharacteristic
438C2#2027579211yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2563C2
socleC23×C53#120
prime radicalC2×#16810818C2
hypercenter#2563D53
hypocenterC53#10048
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C2×#16810997C2semidirect
3. #20275642C2semidirect
4. C2×#16810372C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272556C2semidirect
7. C53×#2563C2semidirect
83. C2×D4×C53SD16semidirect
215. #2563D53semidirect
248. C53#10048semidirect
316. C2×D4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109972
C2×#168108182
#202756422
C2×#168103722
C2×D4⋊#883222
#202725562
C53×#25632
#1004853