Finite Group Databasebeta

← #27287247

Subgroups — #27287247

Order 13568.

466 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279721C211yesmaximal, characteristic
2#20273287C211yesmaximal, characteristic
3C2×#16810360C211yesmaximal, characteristic
4#20272593C211yesmaximal, characteristic
5#20272582C211yesmaximal, characteristic
6#20272563C211yesmaximal, characteristic
7C53×#1996C211yesmaximal, characteristic
8C2×#13133194—21no—
9C2×C22⋊#88322C2211yescharacteristic
10C2×C22⋊#88322C2211yescharacteristic
11#16810555—21no—
12#16810544—21no—
13#16810529—21no—
14#16810521—21no—
15#16810498—21no—
16#16810483C2211yescharacteristic
17C2×#13133124C2211yescharacteristic
18#16810360—21no—
19#16810360—21no—
20#16810360—21no—
21#16810360—21no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C53×#380C2211yescharacteristic
24C4×C22⋊C4×C53C2211yescharacteristic
25C53×#342—21no—
26C53×#342—21no—
27C22×Dic106—21no—
28C23×Dic53—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133194—42no—
32#13133194—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C22⋊#88322—21no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124D411yescharacteristic
48#13133124D411yescharacteristic
49#13133124D411yescharacteristic
50#13133124D411yescharacteristic
51C22×Q8×C53Q811yescharacteristic, complemented
52C23×C4×C53C2311yescharacteristic
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53Q811yescharacteristic, complemented
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53D411yescharacteristic
64C2×C42×C53—21no—
65#95×C53—21no—
66#95×C53—21no—
67C24×C53C2×D411yescharacteristic
68C2×Q8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—81no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—41no—
76C22×C4×C53—81no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×Q811yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—42no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—81no—
91C2×Dic106—42no—
92C2×Dic106—42no—
93C2×Dic106—81no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—81no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53SD1612yes—
106C22⋊C4×C53Q1612yes—
107C42×C53—81no—
108C22⋊Dic53—41no—
109C22⋊Dic53—41no—
110C4⋊Dic53—42no—
111C4⋊Dic53—42no—
112C4⋊Dic53—81no—
113C4⋊Dic53—81no—
114#11719919—42no—
115#11719919—42no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—81no—
119C2×#88322—81no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53#12011yescharacteristic
123C23×C53C22⋊Q811yescharacteristic
124C23×C53#12411yescharacteristic
125C23×C53C2×SD1611yescharacteristic
126C23×C53C2×Q1611yescharacteristic
127C23×C53AGL(1,ℤ/8)11yescharacteristic
128C23×C53#13711yescharacteristic
129Q8×C53—81no—
130Q8×C53—81no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53C22⋊Q811yescharacteristic
151C2×C4×C53C22⋊Q811yescharacteristic
152C2×C4×C53#12411yescharacteristic
153C2×C4×C53#12411yescharacteristic
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—
165C2×Dic53—41no—
166C2×Dic53—82no—
167C2×Dic53—82no—
168Dic106—84no—
169#88322—81no—
170#88322—81no—
171#10375—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53#39311yescharacteristic
183C22×C53#45011yescharacteristic
184C22×C53#45111yescharacteristic
185C22×C53#47511yescharacteristic
186C22×C53#47711yescharacteristic
187C22×C53#48711yescharacteristic
188C22×C53#48911yescharacteristic
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—42no—
194C4×C53—42no—
195C4×C53—81no—
196C4×C53—81no—
197C4×C53—161no—
198Dic53—42no—
199Dic53—81no—
200Dic53—81no—
201Dic53—81no—
202#2549—531no—
203#1996D5311yescharacteristic
204#1520—531no—
205C2×#328—531no—
206#1294—531no—
207#1289—531no—
208#1282—531no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—21no—
215C2×C53#202411yescharacteristic
216C2×C53#203111yescharacteristic
217C2×C53#203211yescharacteristic
218C2×C22⋊Q8—1061no—
219C2×C22⋊Q8C2×D5311yescharacteristic
220C2×C22⋊C8—531no—
221C2×C22⋊C8—531no—
222#398—1061no—
223#395—1061no—
224#391—1061no—
225#389—1061no—
226#385—1061no—
227#380—531no—
228#380C2×D5311yescharacteristic
229C4×C22⋊C4C2×D5311yescharacteristic
230C2×#95—531no—
231#342—21no—
232#342—21no—
233#328—1061no—
234#328—1061no—
235#328—1061no—
236#328—1061no—
237C53#1037511yescharacteristic, complemented, Sylow
238C22×Q8—1061no—
239C22×Q8Dic10611yescharacteristic, complemented
240C23×C4—1061no—
241C23×C4C22×D5311yescharacteristic
242C22×C8—1061no—
243C22×C8—1061no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—2122no—
249C22⋊Q8—2122no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4Dic10611yescharacteristic, complemented
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—1061no—
262C2×C22⋊C4C22⋊D5311yescharacteristic
263C2×C22⋊C4C22⋊D5311yescharacteristic
264C2×C42—21no—
265C22⋊C8—1061no—
266C22⋊C8—1061no—
267C22⋊C8—1061no—
268C22⋊C8—1061no—
269#95—21no—
270#95—21no—
271#95—531no—
272#95—531no—
273#95—531no—
274#95—531no—
275#95—1061no—
276#95—1061no—
277#95—1061no—
278C24C2×C22⋊D5311yescharacteristic
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—81no—
282C2×Q8—4241no—
283C2×Q8—2122no—
284C2×Q8—2122no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—81no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—4241no—
296C22×C4—2122no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4C2×Dic10611yescharacteristic
300C22×C4#1171994511yescharacteristic
301C22×C4D4×D5311yescharacteristic
302C22×C4D4×D5311yescharacteristic
303C22×C4#1171994711yescharacteristic
304C22×C4#1171994711yescharacteristic
305C2×C8—4241no—
306C2×C8—4241no—
307C2×C8—2121no—
308C2×C8—2121no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4—4241no—
315C4⋊C4—4241no—
316C4⋊C4—2122no—
317C4⋊C4—2122no—
318C4⋊C4—2122no—
319C4⋊C4—2122no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C22⋊C4#1171992412yes—
329C22⋊C4#1171992512yes—
330C42—81no—
331C23—41no—
332C23—41no—
333C23#1313318511yescharacteristic
334C23#1313319611yescharacteristic
335C23#1313320411yescharacteristic
336C23C2×#1171992411yescharacteristic
337C23C2×#1171992511yescharacteristic
338C23#1313324211yescharacteristic
339C23#1313324711yescharacteristic
340Q8—81no—
341Q8—81no—
342Q8—4244no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—1062no—
363C2×C4—4242no—
364C2×C4—4242no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4#1313319411yescharacteristic
377C2×C4#1313319411yescharacteristic
378C2×C4#1313320311yescharacteristic
379C2×C4#1313320311yescharacteristic
380C8—4241no—
381C8—4241no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#1681053511yescharacteristic
393C22#1681077311yescharacteristic
394C22#1681077811yescharacteristic
395C22#1681090811yescharacteristic
396C22#1681091811yescharacteristic
397C22#1681097311yescharacteristic
398C22#1681098211yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—161no—
408C4—4241no—
409C4—4241no—
410C4—4241no—
411C4—2122no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2#2027585611yescharacteristic
419C2#2027588911yescharacteristic
420C2#2027589211yescharacteristic
subgroupG/H
centerC22#16810535
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1996C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#1996D53
hypocenterC53#10375
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
51. C22×Q8×C53Q8semidirect
59. C2×C4⋊C4×C53Q8semidirect
237. C53#10375semidirect
239. C22×Q8Dic106semidirect
258. C2×C4⋊C4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202797212
#202732872
C2×#168103602
#202725932
#202725822
#202725632
C53×#19962
#1037553