Finite Group Databasebeta

← #27279453

Subgroups — #27279453

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 410 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×C4×M16×C53C211yesmaximal, 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×M16×C53—21no—
21C22×C42×C53C2211yescharacteristic
22C4×M16×C53—24no—
23C2×#97×C53—21no—
24C2×C4×C8×C53C2212yes—
25C22×Dic106C2×C411yescharacteristic
26C22×#88322—22no—
27#13133196—22no—
28#13133196—22no—
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
52C2×M16×C53—24no—
53C2×M16×C53—24no—
54C2×M16×C53—24no—
55C22×C8×C53—22no—
56C2×C42×C53—22no—
57C2×C42×C53—22no—
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C2×C42×C53D411yescharacteristic
61C2×C42×C53C2311yescharacteristic
62#97×C53—24no—
63C4×C8×C53—22no—
64C4×C8×C53D414yes—
65C24×C53C22⋊C411yescharacteristic
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—22no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—22no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C2×Dic106—22no—
89C2×Dic106—22no—
90C2×Dic106—21no—
91C2×Dic106M1612yescomplemented
92M16×C53—216no—
93C2×C8×C53—24no—
94C2×C8×C53—24no—
95C2×C8×C53—24no—
96C42×C53—22no—
97C42×C53—21no—
98C42×C53—22no—
99C42×C53—21no—
100C42×C53C2×D412yes—
101C42×C53#4112yes—
102C22⋊Dic53—24no—
103C22⋊Dic53—42no—
104C4⋊Dic53—21no—
105C4⋊Dic53—22no—
106C4⋊Dic53—22no—
107C4⋊Dic53M1612yescomplemented
108#11719919—24no—
109#11719919—42no—
110#11719919—41no—
111#11719919—41no—
112C4×Dic53—42no—
113C4×Dic53—41no—
114C4×Dic53—41no—
115C2×#88322—42no—
116C2×#88322—42no—
117C2×#88322—42no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53#10412yes—
125C23×C53C2×C22⋊C411yescharacteristic
126C2×C4×C53—24no—
127C2×C4×C53—24no—
128C2×C4×C53—21no—
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—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53#10412yes—
143C2×C4×C53C4×D412yes—
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12112yes—
146C2×C4×C53#12311yescharacteristic
147C2×C4×C53C2×M1611yescharacteristic
148C2×C4×C53C2×M1611yescharacteristic
149C2×C4×C53#13112yes—
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13712yes—
152C2×C4×C53#13711yescharacteristic
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165Dic106—24no—
166Dic106—42no—
167C8×C53—28no—
168#88322—82no—
169#9077—531nomaximal, Sylow
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53#38611yescharacteristic
178C22×C53#40711yescharacteristic
179C22×C53#40811yescharacteristic
180C22×C53#41711yescharacteristic
181C22×C53#41911yescharacteristic
182C22×C53C2×#10411yescharacteristic
183C22×C53C2×#10411yescharacteristic
184C4×C53—21no—
185C4×C53—22no—
186C4×C53—21no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53#43512yes—
190C4×C53#43612yes—
191C4×C53#46812yes—
192C4×C53#47012yes—
193Dic53—42no—
194Dic53—42no—
195Dic53—81no—
196Dic53—81no—
197C2×C4×M16D5311yescharacteristic
198C4×C22⋊Q8—531no—
199#1797—531no—
200C2×Q8⋊C8—532no—
201#1279—532no—
202C2×C53—42no—
203C2×C53#148311yescharacteristic
204C2×C53#148511yescharacteristic
205C2×C53#148911yescharacteristic
206C2×C53#149111yescharacteristic
207C2×C53#187011yescharacteristic
208C2×C53#189411yescharacteristic
209C2×C53#189611yescharacteristic
210C22×M16—21no—
211C2×C22⋊Q8—531no—
212C2×C4×Q8—531no—
213C22×C42C2×D5311yescharacteristic
214C2×C4⋊C8—532no—
215C2×C22⋊C8—1061no—
216C4×M16—24no—
217C2×#97—21no—
218C2×C4×C8C2×D5312yes—
219#391—531no—
220#389—531no—
221#387—1061no—
222#385—1061no—
223#380—531no—
224C4×C4⋊C4—531no—
225C4×C4⋊C4—1061no—
226C4×C22⋊C4—1061no—
227#336—1061no—
228Q8⋊C8—1064no—
229C53#907711yescharacteristic, complemented, Sylow
230C22×Q8—531no—
231C23×C4C4×D5311yescharacteristic
232C23×C4D21211yescharacteristic
233C23×C4C22⋊D5311yescharacteristic
234C2×M16—24no—
235C2×M16—24no—
236C2×M16—24no—
237C22×C8—22no—
238C22×C8—1062no—
239C22⋊Q8—1062no—
240C22⋊Q8—1062no—
241C4×Q8—1062no—
242C4×Q8—1062no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—1061no—
254C2×C42—1061no—
255C2×C42C4×D5311yescharacteristic
256C2×C42D21211yescharacteristic
257C2×C42C22⋊D5311yescharacteristic
258C2×C42C22×D5311yescharacteristic
259C4⋊C8—1062no—
260C4⋊C8—1062no—
261C22⋊C8—2122no—
262#97—24no—
263C4×C8—22no—
264C4×C8C22⋊D5314yes—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268#95—1061no—
269#95—1061no—
270C24#1171992111yescharacteristic
271C2×Q8—532no—
272C2×Q8—1061no—
273C2×Q8—1062no—
274C2×Q8—1062no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4#1171992111yescharacteristic
292C22×C4#1171992111yescharacteristic
293C22×C4#1171992111yescharacteristic
294C22×C4C2×C4×D5311yescharacteristic
295C22×C4C2×D21211yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297M16—216no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—2122no—
302C2×C8—2122no—
303C2×C8—2122no—
304C4⋊C4—532no—
305C4⋊C4—1064no—
306C4⋊C4—1061no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C4⋊C4—2122no—
310C4⋊C4—2121no—
311C4⋊C4—2121no—
312C22⋊C4—1064no—
313C22⋊C4—2122no—
314C42—22no—
315C42—21no—
316C42—22no—
317C42—21no—
318C42—2122no—
319C42—2121no—
320C42—2121no—
321C42#1171994512yes—
322C42C2×C22⋊D5312yes—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#1313314512yes—
330C23C2×#1171992111yescharacteristic
331Q8—1064no—
332Q8—2122no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4#1313314512yes—
362C2×C4C4×C22⋊D5312yes—
363C2×C4#1313318611yescharacteristic
364C2×C4#1313319112yes—
365C2×C4#1313320011yescharacteristic
366C2×C4M16×D5311yescharacteristic
367C2×C4M16×D5311yescharacteristic
368C2×C4#1313322512yes—
369C2×C4#1313324511yescharacteristic
370C2×C4#1313325012yes—
371C2×C4#1313325011yescharacteristic
372C8—28no—
373C8—4242no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22#1681050211yescharacteristic
382C22#1681058611yescharacteristic
383C22#1681059111yescharacteristic
384C22#1681062511yescharacteristic
385C22#1681063311yescharacteristic
386C22C2×#1313314511yescharacteristic
387C22C2×#1313314511yescharacteristic
388C4—21no—
389C4—22no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—4241no—
394C4—4241no—
395C4—2122no—
396C4—2122no—
397C4#1681069212yes—
398C4#1681069912yes—
399C4#1681086912yes—
400C4#1681088112yes—
401C2—42no—
402C2#2027312911yescharacteristic
403C2#2027313711yescharacteristic
404C2#2027315511yescharacteristic
405C2#2027316311yescharacteristic
406C2#2027501211yescharacteristic
407C2#2027514811yescharacteristic
408C2#2027515811yescharacteristic
subgroupG/H
centerC22×C4C2×D212
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C4×M16×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×C4×M16D53
hypocenterC53#9077
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16810355C2semidirect
4. #20272555C2semidirect
91. C2×Dic106M16semidirect
107. C4⋊Dic53M16semidirect
229. C53#9077semidirect

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×C4×M16×C532
#907753