Finite Group Databasebeta

← #27284864

Subgroups — #27284864

Order 13568.

548 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 101 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#20284754C211yesmaximal, characteristic, complemented
2#20284512C211yesmaximal, characteristic
3#20273278C211yesmaximal, characteristic, complemented
4#20273144C211yesmaximal, characteristic, complemented
5#20273106C211yesmaximal, characteristic
6#20272573C211yesmaximal, characteristic, complemented
7C53×#2580C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810958—21no—
10#16810923—22no—
11#16810918—21no—
12#16810913—21no—
13#16810903—21no—
14#16810833—21no—
15#16810779—21no—
16C4×#11719925—22no—
17#16810676C2211yescharacteristic
18#16810676C2211yescharacteristic
19#16810663—21no—
20#16810647C2211yescharacteristic
21#16810628—21no—
22#16810359—22no—
23Q8⋊#88322—22no—
24D4⋊#88322—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C2×C4×D4×C53C2211yescharacteristic
27C53×#399—21no—
28C53×#393—21no—
29C53×#390—21no—
30C53×#389—21no—
31C53×#385—22no—
32C53×#382C2211yescharacteristic
33C2×#11719925—22no—
34C2×#11719917—21no—
35#13133217D411yescharacteristic
36#13133217D411yescharacteristic
37#13133209—21no—
38#13133196—21no—
39#13133196—21no—
40C4×Dic106—21no—
41C4×Dic106—21no—
42#13133176—21no—
43#13133150—22no—
44#13133150—22no—
45#13133148—22no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322D412yes—
50#13133141—22no—
51#13133140—22no—
52#13133138—22no—
53#13133138—22no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56C4×#88322—21no—
57C4×#88322D412yes—
58C22×Q8×C53D411yescharacteristic
59C22×D4×C53D411yescharacteristic
60C23×C4×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53D411yescharacteristic
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C22⋊C4×C53—22no—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—21no—
77C2×C42×C53—21no—
78C2×C42×C53C2311yescharacteristic
79#95×C53—22no—
80#95×C53—21no—
81#95×C53D412yes—
82C24×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—42no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53Q1612yescomplemented
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—82no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C2×Dic106—21no—
103C2×Dic106—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—82no—
113C22⋊C4×C53—44no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—82no—
117C42×C53—41no—
118C42×C53—41no—
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C22⋊Dic53—41no—
124#11719925—44no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129#11719919—41no—
130#11719919—41no—
131C4×Dic53—41no—
132#11719917—41no—
133#11719917—41no—
134C2×#88322—22no—
135C2×#88322—22no—
136C2×#88322—41no—
137C23×C53—22no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53#12011yescharacteristic
143Q8×C53—22no—
144Q8×C53—22no—
145Q8×C53—41no—
146Q8×C53—41no—
147D4×C53—22no—
148D4×C53—22no—
149D4×C53—41no—
150D4×C53—41no—
151C2×C4×C53—22no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—82no—
165C2×C4×C53—82no—
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12011yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12711yescharacteristic
172C2×C4×C53C2×Q1611yescharacteristic
173C2×C4×C53#13511yescharacteristic
174C2×C4×C53#13711yescharacteristic
175C2×C4×C53#13711yescharacteristic
176C2×Dic53—41no—
177C2×Dic53—41no—
178Dic106—42no—
179Dic106—81no—
180#88322—42no—
181#88322—42no—
182#88322—81no—
183#10002—531nomaximal, Sylow
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—44no—
187C22×C53—42no—
188C22×C53—82no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53#39211yescharacteristic
192C22×C53#45111yescharacteristic
193C22×C53#45211yescharacteristic
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—42no—
199C4×C53—82no—
200C4×C53—81no—
201C4×C53—81no—
202C4×C53#45111yescharacteristic
203C4×C53#45211yescharacteristic
204C4×C53#46711yescharacteristic
205C4×C53#47011yescharacteristic
206Dic53—81no—
207Dic53—81no—
208#3104—531no—
209#3077—531no—
210#2580D5311yescharacteristic
211#1518—531no—
212#1487—531no—
213#1478—531no—
214#1286—531no—
215C2×C53—21no—
216C2×C53—21no—
217C2×C53—82no—
218C2×C53#200111yescharacteristic
219C2×C53#200811yescharacteristic
220C2×C53#201011yescharacteristic
221#533—531no—
222C2×C4×Q8C2×D5311yescharacteristic
223C2×C4×D4C2×D5311yescharacteristic
224#484—1061no—
225#478—1062no—
226#477—1061no—
227#476—1061no—
228#474—1061no—
229#462—1061no—
230#451—1061no—
231C4×Q16—1062no—
232#431—531no—
233#431—531no—
234#428—1061no—
235#423—531no—
236#418—1061no—
237#399—21no—
238#393—21no—
239#390—21no—
240#389—21no—
241#385—22no—
242#382C2×D5311yescharacteristic
243#328—1062no—
244Q8⋊C8—1062no—
245D4⋊C8—1062no—
246C53#1000211yescharacteristic, complemented, Sylow
247C22×Q8C22⋊D5311yescharacteristic
248C22×D4C22⋊D5311yescharacteristic
249C23×C4—21no—
250C2×Q16—1062no—
251C2×M16—1061no—
252C4⋊Q8—531no—
253C4⋊Q8—531no—
254#125—1061no—
255C22⋊Q8—1061no—
256C22⋊Q8—1061no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—1061no—
262C4×Q8—1061no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267#117—1061no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C22⋊D5311yescharacteristic
272C2×C4⋊C4C22⋊D5311yescharacteristic
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C42—21no—
277C2×C42C22×D5311yescharacteristic
278#107—1062no—
279#107—1062no—
280#106—1062no—
281C4⋊C8—532no—
282C4⋊C8—1061no—
283C4⋊C8—1061no—
284C4⋊C8—1061no—
285#103—1062no—
286#103—1062no—
287#102—1062no—
288#102—1062no—
289C22⋊C8—1061no—
290C22⋊C8—1061no—
291C4×C8—532no—
292C4×C8—1061no—
293#95—22no—
294#95—21no—
295#95C22⋊D5312yes—
296C24—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×Q8—1061no—
301C2×Q8—1061no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4#1171992512yescomplemented
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4C2×C22⋊D5311yescharacteristic
316C22×C4C2×C22⋊D5311yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318Q16—2124no—
319M16—2121no—
320M16—2121no—
321C2×C8—1062no—
322C2×C8—1062no—
323C2×C8—2121no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—82no—
333C4⋊C4—1061no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C4⋊C4—2121no—
338C4⋊C4—2121no—
339C22⋊C4—44no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—82no—
343C22⋊C4—2121no—
344C42—41no—
345C42—41no—
346C42—2121no—
347C42D4×D5311yescharacteristic
348C42D4×D5311yescharacteristic
349C42D4×D5311yescharacteristic
350C42#1171994711yescharacteristic
351C23—22no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23#1313318611yescharacteristic
357Q8—22no—
358Q8—22no—
359Q8—41no—
360Q8—41no—
361Q8—4241no—
362Q8—2122no—
363D4—22no—
364D4—22no—
365D4—41no—
366D4—41no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4#1313318511yescharacteristic
385C2×C4#1313318511yescharacteristic
386C2×C4#1313318811yescharacteristic
387C2×C4#1313318911yescharacteristic
388C2×C4#1313318911yescharacteristic
389C2×C4#1313321311yescharacteristic
390C2×C4C2×#1171992511yescharacteristic
391C2×C4#1313323711yescharacteristic
392C2×C4#1313324711yescharacteristic
393C2×C4#1313324811yescharacteristic
394C8—4241no—
395C8—2122no—
396C8—2122no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—42no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22#1681053111yescharacteristic
405C22#1681077911yescharacteristic
406C22#1681078511yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—82no—
413C4—81no—
414C4—81no—
415C4—4241no—
416C4—4241no—
417C4#1681077811yescharacteristic
418C4#1681078311yescharacteristic
419C4#1681086211yescharacteristic
420C4#1681087911yescharacteristic
421C2—21no—
422C2—21no—
423C2—82no—
424C2#2027572311yescharacteristic
425C2#2027576811yescharacteristic
426C2#2027578011yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2580C2
socleC22×C53#392
prime radicalC22×D4×C53D4
hypercenter#2580D53
hypocenterC53#10002
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. #20284754C2semidirect
3. #20273278C2semidirect
4. #20273144C2semidirect
6. #20272573C2semidirect
91. C2×D4×C53Q16semidirect
246. C53#10002semidirect
307. C2×D4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202847542
#202845122
#202732782
#202731442
#202731062
#202725732
C53×#25802
#1000253