Finite Group Databasebeta

← #27285185

Subgroups — #27285185

Order 13568.

670 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×#16810970C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20279700C211yesmaximal, characteristic, complemented
4#20275641C211yesmaximal, characteristic
5C2×#16810368C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133217C2211yescharacteristic
10C2×#13133215—21no—
11C2×#13133214C2211yescharacteristic
12#16810970—22no—
13#16810970—22no—
14#16810823—22no—
15#16810823—22no—
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133139—22no—
18C2×#13133139—22no—
19C2×#13133137—21no—
20#16810553—22no—
21#16810528—21no—
22#16810521—22no—
23C4×C4⋊Dic53C2211yescharacteristic
24#16810368—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C53×Q8⋊C8—22no—
29C22×Dic106—21no—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212—21no—
33C2×#11719924—24no—
34C2×#11719924—42no—
35C2×#11719924—42no—
36C22×#88322—21no—
37#13133217—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217—21no—
41#13133215—42no—
42#13133215—42no—
43#13133214—21no—
44#13133214—21no—
45#13133214—21no—
46#13133214D412yes—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53D412yes—
49C2×C4⋊Dic53D412yes—
50C2×#11719919—24no—
51C2×C4×Dic53—21no—
52C4⋊#88322—21no—
53C4⋊#88322D412yes—
54#13133139—28no—
55#13133139—44no—
56#13133137—24no—
57#13133124—24no—
58#13133124—21no—
59C22×Q8×C53D411yescharacteristic
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×Q8×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C42×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4×C8×C53—21no—
73C4×C8×C53D412yes—
74C23×D53—41no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53D812yes—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×Dic53—22no—
87C22×Dic53—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—44no—
92C2×D212—41no—
93C2×D212—42no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—42no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×Dic106—41no—
102C2×Dic106—41no—
103C2×Dic106—41no—
104C2×Dic106—41no—
105C2×Dic106—41no—
106C2×Dic106—41no—
107C2×C8×C53—22no—
108C2×C8×C53—42no—
109C2×C8×C53—41no—
110C2×C8×C53—42no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C42×C53—41no—
116C42×C53—41no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121#11719924—48no—
122C4⋊Dic53—22no—
123C4⋊Dic53—22no—
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53SD1614yescomplemented
128#11719919—48no—
129C4×Dic53—41no—
130C4×Dic53—41no—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—41no—
134C23×C53#12011yescharacteristic
135C22×D53—42no—
136C22×D53—81no—
137C22×D53—82no—
138C22×D53—81no—
139C22×D53—82no—
140Q8×C53—22no—
141Q8×C53—22no—
142Q8×C53—41no—
143Q8×C53—41no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12711yescharacteristic
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C2×SD1612yes—
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53#13712yes—
162C2×Dic53—24no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168D212—44no—
169D212—42no—
170D212—42no—
171D212—44no—
172D212—82no—
173D212—81no—
174D212—81no—
175D212—81no—
176D212—81no—
177Dic106—81no—
178Dic106—81no—
179Dic106—81no—
180Dic106—81no—
181C8×C53—42no—
182C8×C53—81no—
183#88322—42no—
184#10051—531nomaximal, Sylow
185C22×C53#39211yescharacteristic
186C22×C53#44711yescharacteristic
187C22×C53#44812yes—
188C22×C53#44911yescharacteristic
189C22×C53#45412yes—
190C2×D53—82no—
191C2×D53—82no—
192C2×D53—82no—
193C2×D53—162no—
194C2×D53—162no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—42no—
198C4×C53—81no—
199C4×C53#44711yescharacteristic
200C4×C53#44911yescharacteristic
201C4×C53#46112yes—
202C4×C53#46611yescharacteristic
203C4×C53#46911yescharacteristic
204C4×C53#49211yescharacteristic
205C4×C53#49711yescharacteristic
206Dic53—44no—
207Dic53—82no—
208C2×#486—531no—
209C2×#460—531no—
210#2546—531no—
211#1982—531no—
212C2×#332—531no—
213C2×Q8⋊C8D5311yescharacteristic, complemented
214#1279—531no—
215C2×C53#162612yes—
216C2×C53#162712yes—
217C2×C53#199911yescharacteristic
218C2×C53#200311yescharacteristic
219C2×C53#201311yescharacteristic
220D53—162no—
221C22×SD16—1061no—
222C2×C4⋊Q8—531no—
223C2×C4⋊Q8—1061no—
224C2×#127—531no—
225C2×C4×Q8C2×D5311yescharacteristic
226#486—1062no—
227#486—1062no—
228#460—1062no—
229#460—1062no—
230C2×C4⋊C8—531no—
231C2×C4⋊C8C2×D5311yescharacteristic
232C2×#102—1062no—
233C2×#102—1061no—
234C2×#102—1062no—
235C2×C4×C8C2×D5311yescharacteristic
236#398—1062no—
237#391—1061no—
238#389—1062no—
239C4×C4⋊C4—531no—
240#332—1062no—
241Q8⋊C8—22no—
242C53#1005111yescharacteristic, complemented, Sylow
243C22×Q8—1061no—
244C22×Q8D21211yescharacteristic
245C22×D4—1061no—
246C22×D4—1061no—
247C22×D4—1061no—
248C2×SD16—1064no—
249C2×SD16—2122no—
250C2×SD16—2122no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—1061no—
254C4⋊Q8—1061no—
255C4⋊Q8—1061no—
256C4⋊Q8—1061no—
257C4⋊Q8—1061no—
258C4⋊Q8—2122no—
259C4⋊Q8—2122no—
260#127—532no—
261#127—1061no—
262#127—1061no—
263#127—1061no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—532no—
270C2×C4⋊C4—532no—
271C2×C4⋊C4—1064no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4D21211yescharacteristic
274C2×C42—21no—
275C2×C42—1061no—
276C2×C42C22×D5311yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—532no—
280C4⋊C8—1061no—
281#102—1064no—
282#102—1068no—
283#102—2124no—
284C4×C8—21no—
285C4×C8C22⋊D5312yes—
286#95—1064no—
287#95—1061no—
288C24—2121no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—2121no—
295C2×Q8—2121no—
296C2×Q8—2121no—
297C2×Q8—2121no—
298C2×Q8—2121no—
299C2×Q8—2121no—
300C2×Q8D42412yes—
301C2×D4—1062no—
302C2×D4—1062no—
303C2×D4—1062no—
304C2×D4—2124no—
305C2×D4—2121no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4—2121no—
309C2×D4—2122no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2122no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—1062no—
317C22×C4—1062no—
318C22×C4C2×D21211yescharacteristic
319C22×C4D4×D5311yescharacteristic
320C22×C4D4×D5311yescharacteristic
321SD16—2128no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—1062no—
327C2×C8—2121no—
328C2×C8—2121no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—534no—
334C4⋊C4—1062no—
335C4⋊C4—1062no—
336C4⋊C4—1062no—
337C4⋊C4—2128no—
338C4⋊C4—2121no—
339C4⋊C4—2121no—
340C42—41no—
341C42—41no—
342C42—2121no—
343C42—2121no—
344C42#1171994511yescharacteristic
345C42D4×D5311yescharacteristic
346C42D4×D5311yescharacteristic
347C42C2×C22⋊D5311yescharacteristic
348C23—4241no—
349C23—4242no—
350C23—4241no—
351C23—4242no—
352C23—2122no—
353C23#1313318711yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—4241no—
359Q8—4241no—
360Q8—4241no—
361Q8—4241no—
362D4—4242no—
363D4—4241no—
364D4—4241no—
365D4—4241no—
366D4—4241no—
367D4—2124no—
368D4—2122no—
369D4—2122no—
370D4—2124no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—1064no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4#1313318511yescharacteristic
384C2×C4#1313318711yescharacteristic
385C2×C4#1313319011yescharacteristic
386C2×C4#1313319011yescharacteristic
387C2×C4#1313319111yescharacteristic
388C2×C4#1313321311yescharacteristic
389C2×C4C2×D42411yescharacteristic
390C2×C4SD16×D5312yes—
391C2×C4#1313323911yescharacteristic
392C2×C4#1313324511yescharacteristic
393C2×C4#1313324511yescharacteristic
394C2×C4#1313324912yes—
395C8—42no—
396C8—81no—
397C8—2122no—
398C22—8482no—
399C22—8482no—
400C22—4242no—
401C22—4242no—
402C22—4242no—
403C22#1681053211yescharacteristic
404C22#1681075911yescharacteristic
405C22#1681076312yes—
406C22#1681077011yescharacteristic
407C22#1681079712yes—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—81no—
412C4—4242no—
413C4—2124no—
414C4#1681075911yescharacteristic
415C4#1681077011yescharacteristic
416C4#1681082712yes—
417C4#1681085911yescharacteristic
418C4#1681087611yescharacteristic
419C4#1681099411yescharacteristic
420C4#1681101611yescharacteristic
421C2—8482no—
422C2#2027381312yes—
423C2#2027382012yes—
424C2#2027571311yescharacteristic
425C2#2027573611yescharacteristic
426C2#2027579911yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenterC2×Q8⋊C8D53
hypocenterC53#10051
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #20279700C2semidirect
5. C2×#16810368C2semidirect
6. #20272555C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
127. C4⋊Dic53SD16semidirect
213. C2×Q8⋊C8D53semidirect
242. C53#10051semidirect

Maximal subgroups

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

subgroupindex
C2×#168109702
C2×#168108232
#202797002
#202756412
C2×#168103682
#202725552
C2×C53×Q8⋊C82
#1005153