Finite Group Databasebeta

← #27279433

Subgroups — #27279433

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133196C211yesmaximal, characteristic
2#20274695C211yesmaximal, characteristic
3C2×#16810355C212yesmaximal, complemented
4#20272555C212yesmaximal, complemented
5C2×C53×#431C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7C2×C4×Dic106C2211yescharacteristic
8C2×C4⋊#88322C2212yes—
9C2×C22⋊#88322—21no—
10#16810529C411yescharacteristic
11#16810521C411yescharacteristic
12#16810507—21no—
13#16810496—21no—
14#16810483C411yescharacteristic
15C4×#11719919—21no—
16C4×C4⋊Dic53C2211yescharacteristic
17C4×C22⋊Dic53—21no—
18#16810378—21no—
19#16810355—24no—
20C22×C42×C53C2211yescharacteristic
21C53×#431—24no—
22C2×C4⋊C8×C53—21no—
23C2×C22⋊C8×C53—21no—
24C2×C4×C8×C53C2212yes—
25C22×Dic106C2×C411yescharacteristic
26C22×#88322—22no—
27#13133196—22no—
28#13133196C814yescomplemented
29C4×Dic106—22no—
30C4×Dic106—22no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C4⋊#88322—22no—
42C4⋊#88322—22no—
43C22⋊#88322—42no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C22×C8×C53—22no—
53C2×C42×C53—22no—
54C2×C42×C53—22no—
55C2×C42×C53C2×C411yescharacteristic
56C2×C42×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—24no—
60C22⋊C8×C53—24no—
61C4×C8×C53—22no—
62C4×C8×C53D414yes—
63C24×C53C22⋊C411yescharacteristic
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—22no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C2×C812yes—
78C22×C4×C53M1612yes—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C2×Dic106—22no—
87C2×Dic106—22no—
88C2×Dic106—21no—
89C2×Dic106C2×C812yes—
90C2×C8×C53—24no—
91C2×C8×C53—42no—
92C2×C8×C53—42no—
93C42×C53—22no—
94C42×C53—21no—
95C42×C53—22no—
96C42×C53—21no—
97C42×C53C2×D412yes—
98C42×C53#4112yes—
99C22⋊Dic53—24no—
100C22⋊Dic53—42no—
101C4⋊Dic53—21no—
102C4⋊Dic53—22no—
103C4⋊Dic53—22no—
104C4⋊Dic53C2×C812yes—
105#11719919—24no—
106#11719919—42no—
107#11719919—41no—
108#11719919—41no—
109C4×Dic53—42no—
110C4×Dic53—41no—
111C4×Dic53—41no—
112C2×#88322—42no—
113C2×#88322—42no—
114C2×#88322—42no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53C22⋊C812yes—
122C23×C53C2×C22⋊C411yescharacteristic
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—24no—
128C2×C4×C53—24no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53C22⋊C812yes—
140C2×C4×C53C4×D412yes—
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12112yes—
143C2×C4×C53#12311yescharacteristic
144C2×C4×C53C22×C811yescharacteristic
145C2×C4×C53C2×M1611yescharacteristic
146C2×C4×C53#13112yes—
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13712yes—
149C2×C4×C53#13711yescharacteristic
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162Dic106—24no—
163Dic106—42no—
164C8×C53—44no—
165#88322—82no—
166#9073—531nomaximal, Sylow
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53#38611yescharacteristic
175C22×C53C2×C22⋊C811yescharacteristic
176C22×C53#40811yescharacteristic
177C22×C53#41711yescharacteristic
178C22×C53#41911yescharacteristic
179C22×C53#42111yescharacteristic
180C22×C53#42111yescharacteristic
181C4×C53—22no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53C8×D412yes—
187C4×C53#43512yes—
188C4×C53#46812yes—
189C4×C53#47012yes—
190Dic53—42no—
191Dic53—42no—
192Dic53—81no—
193Dic53—81no—
194C2×#431D5311yescharacteristic
195C4×C22⋊Q8—531no—
196#1797—531no—
197C2×Q8⋊C8—532no—
198#1279—532no—
199C2×C53—42no—
200C2×C53#147711yescharacteristic
201C2×C53#147911yescharacteristic
202C2×C53#149511yescharacteristic
203C2×C53#149711yescharacteristic
204C2×C53#186911yescharacteristic
205C2×C53#189411yescharacteristic
206C2×C53#189711yescharacteristic
207C2×C22⋊Q8—531no—
208C2×C4×Q8—531no—
209C22×C42C2×D5311yescharacteristic
210#431—24no—
211C2×C4⋊C8—21no—
212C2×C4⋊C8—532no—
213C2×C22⋊C8—21no—
214C2×C22⋊C8—1061no—
215C2×C4×C8C2×D5312yes—
216#391—531no—
217#389—531no—
218#387—1061no—
219#385—1061no—
220#380—531no—
221C4×C4⋊C4—531no—
222C4×C4⋊C4—1061no—
223C4×C22⋊C4—1061no—
224#336—1061no—
225Q8⋊C8—1064no—
226C53#907311yescharacteristic, complemented, Sylow
227C22×Q8—531no—
228C23×C4C4×D5311yescharacteristic
229C23×C4D21211yescharacteristic
230C23×C4C22⋊D5311yescharacteristic
231C22×C8—22no—
232C22×C8—1062no—
233C22⋊Q8—534no—
234C22⋊Q8—1062no—
235C4×Q8—1062no—
236C4×Q8—1062no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—1061no—
248C2×C42—1061no—
249C2×C42C4×D5311yescharacteristic
250C2×C42D21211yescharacteristic
251C2×C42C22⋊D5311yescharacteristic
252C2×C42C22×D5311yescharacteristic
253C4⋊C8—24no—
254C4⋊C8—1062no—
255C4⋊C8—1062no—
256C22⋊C8—24no—
257C22⋊C8—2122no—
258C4×C8—22no—
259C4×C8C22⋊D5314yes—
260#95—1061no—
261#95—1061no—
262#95—1061no—
263#95—1061no—
264#95—1061no—
265C24#1171992111yescharacteristic
266C2×Q8—532no—
267C2×Q8—1061no—
268C2×Q8—1062no—
269C2×Q8—1062no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4C8×D5312yes—
285C22×C4#1171991212yes—
286C22×C4#1171992111yescharacteristic
287C22×C4#1171992111yescharacteristic
288C22×C4#1171992111yescharacteristic
289C22×C4C2×C4×D5311yescharacteristic
290C22×C4C2×D21211yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—2122no—
296C2×C8—2122no—
297C2×C8—2122no—
298C4⋊C4—532no—
299C4⋊C4—1064no—
300C4⋊C4—1061no—
301C4⋊C4—1062no—
302C4⋊C4—1062no—
303C4⋊C4—2122no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—1064no—
307C22⋊C4—2122no—
308C42—22no—
309C42—21no—
310C42—22no—
311C42—21no—
312C42—2122no—
313C42—2121no—
314C42—2121no—
315C42#1171994512yes—
316C42C2×C22⋊D5312yes—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23#1313312912yes—
324C23C2×#1171992111yescharacteristic
325Q8—1064no—
326Q8—2122no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—1062no—
344C2×C4—1062no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4#1313312912yes—
356C2×C4C4×C22⋊D5312yes—
357C2×C4#1313318611yescharacteristic
358C2×C4#1313319112yes—
359C2×C4#1313320011yescharacteristic
360C2×C4C2×C8×D5311yescharacteristic
361C2×C4C2×#1171991211yescharacteristic
362C2×C4#1313322312yes—
363C2×C4#1313324511yescharacteristic
364C2×C4#1313325012yes—
365C2×C4#1313325011yescharacteristic
366C8—44no—
367C8—4242no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22#1681050211yescharacteristic
376C22C2×#1313312911yescharacteristic
377C22#1681058911yescharacteristic
378C22#1681062511yescharacteristic
379C22#1681063311yescharacteristic
380C22#1681064111yescharacteristic
381C22#1681064111yescharacteristic
382C4—22no—
383C4—21no—
384C4—21no—
385C4—41no—
386C4—41no—
387C4—4241no—
388C4—4241no—
389C4—2122no—
390C4—2122no—
391C4C8×C22⋊D5312yes—
392C4#1681069412yes—
393C4#1681086912yes—
394C4#1681088112yes—
395C2—42no—
396C2#2027310311yescharacteristic
397C2#2027311111yescharacteristic
398C2#2027318111yescharacteristic
399C2#2027318911yescharacteristic
400C2#2027500711yescharacteristic
401C2#2027514811yescharacteristic
402C2#2027516311yescharacteristic
subgroupG/H
centerC22×C4C2×D212
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC2×C53×#431C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#431D53
hypocenterC53#9073
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
3. C2×#16810355C2semidirect
4. #20272555C2semidirect
28. #13133196C8semidirect
226. C53#9073semidirect

Maximal subgroups

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

subgroupindex
C4×#131331962
#202746952
C2×#168103552
#202725552
C2×C53×#4312
#907353