Finite Group Databasebeta

← #27284857

Subgroups — #27284857

Order 13568.

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

normal subgroups only

This group has 568 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#20284723C211yesmaximal, characteristic
2#20284520C211yesmaximal, characteristic, complemented
3#20273278C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic
5#20273110C211yesmaximal, characteristic, complemented
6#20272573C211yesmaximal, characteristic, complemented
7C53×#2536C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810953—21no—
10#16810930—22no—
11#16810918—21no—
12#16810913—21no—
13#16810908—21no—
14#16810828—21no—
15#16810774—21no—
16C4×#11719923—22no—
17#16810676C2211yescharacteristic
18#16810676C2211yescharacteristic
19#16810663—21no—
20#16810647C2211yescharacteristic
21#16810632—21no—
22#16810359—22no—
23Q8⋊#88322—22no—
24D4⋊#88322—22no—
25C2×C22⋊Q8×C53—21no—
26C2×C4×Q8×C53C2211yescharacteristic
27C2×C4×D4×C53C2211yescharacteristic
28C53×#398—21no—
29C53×#394—21no—
30C53×#388—22no—
31C53×#382C2211yescharacteristic
32C53×#380—21no—
33C2×#11719923—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#13133148—22no—
45#13133148—22no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322D412yes—
50#13133141—22no—
51#13133141—22no—
52#13133140—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—
61C22⋊Q8×C53—42no—
62C22⋊Q8×C53—42no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×Q8×C53—21no—
66C4×Q8×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C4×D4×C53—21no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53D411yescharacteristic
75C2×C4⋊C4×C53D411yescharacteristic
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—22no—
78C2×C22⋊C4×C53—21no—
79C2×C42×C53—21no—
80C2×C42×C53C2311yescharacteristic
81#95×C53—22no—
82#95×C53—21no—
83#95×C53D412yes—
84C24×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53—21no—
89C2×Q8×C53—21no—
90C2×Q8×C53SD1612yescomplemented
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—42no—
94C22×C4×C53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C2×Dic106—21no—
108C2×Dic106—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—44no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C22⋊C4×C53—44no—
119C22⋊C4×C53—44no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C22⋊Dic53—41no—
129#11719923—44no—
130C4⋊Dic53—21no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134#11719919—41no—
135#11719919—41no—
136C4×Dic53—41no—
137#11719917—41no—
138#11719917—41no—
139C2×#88322—22no—
140C2×#88322—22no—
141C2×#88322—41no—
142C23×C53—22no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53#12011yescharacteristic
150Q8×C53—22no—
151Q8×C53—22no—
152Q8×C53—41no—
153Q8×C53—41no—
154D4×C53—22no—
155D4×C53—22no—
156D4×C53—41no—
157D4×C53—41no—
158C2×C4×C53—22no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53—81no—
172C2×C4×C53—81no—
173C2×C4×C53—81no—
174C2×C4×C53—81no—
175C2×C4×C53#12011yescharacteristic
176C2×C4×C53#12011yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12711yescharacteristic
181C2×C4×C53C2×SD1611yescharacteristic
182C2×C4×C53#13511yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C53#13711yescharacteristic
185C2×Dic53—41no—
186C2×Dic53—41no—
187Dic106—42no—
188Dic106—81no—
189#88322—42no—
190#88322—42no—
191#88322—81no—
192#10001—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—42no—
196C22×C53—42no—
197C22×C53—42no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53#39211yescharacteristic
203C22×C53#45011yescharacteristic
204C22×C53#45211yescharacteristic
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—42no—
210C4×C53—82no—
211C4×C53—81no—
212C4×C53—81no—
213C4×C53#44811yescharacteristic
214C4×C53#44911yescharacteristic
215C4×C53#46511yescharacteristic
216C4×C53#46811yescharacteristic
217Dic53—81no—
218Dic53—81no—
219#3101—531no—
220#3078—531no—
221#2536D5311yescharacteristic
222#1518—531no—
223#1486—531no—
224#1479—531no—
225#1286—531no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—82no—
229C2×C53#200011yescharacteristic
230C2×C53#200911yescharacteristic
231C2×C53#201011yescharacteristic
232#533—531no—
233C2×C22⋊Q8—21no—
234C2×C4×Q8C2×D5311yescharacteristic
235C2×C4×D4C2×D5311yescharacteristic
236#483—1061no—
237#479—1062no—
238#477—1061no—
239#476—1061no—
240#475—1061no—
241#461—1061no—
242#450—1061no—
243C4×SD16—1062no—
244#431—531no—
245#431—531no—
246#428—1061no—
247#423—531no—
248#419—1061no—
249#398—21no—
250#394—21no—
251#388—22no—
252#382C2×D5311yescharacteristic
253#380—21no—
254#328—1062no—
255Q8⋊C8—1062no—
256D4⋊C8—1062no—
257C53#1000111yescharacteristic, complemented, Sylow
258C22×Q8C22⋊D5311yescharacteristic
259C22×D4C22⋊D5311yescharacteristic
260C23×C4—21no—
261C2×SD16—1062no—
262C2×M16—1061no—
263C4⋊Q8—531no—
264C4⋊Q8—531no—
265#125—1061no—
266C22⋊Q8—42no—
267C22⋊Q8—42no—
268C22⋊Q8—1061no—
269C22⋊Q8—1061no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—1061no—
275C4×Q8—1061no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280#117—1061no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C22⋊D5311yescharacteristic
285C2×C4⋊C4C22⋊D5311yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C42—21no—
290C2×C42C22×D5311yescharacteristic
291#107—1062no—
292#106—1062no—
293#106—1062no—
294C4⋊C8—532no—
295C4⋊C8—1061no—
296C4⋊C8—1061no—
297C4⋊C8—1061no—
298#103—1062no—
299#103—1062no—
300#103—1062no—
301#102—1062no—
302C22⋊C8—1061no—
303C22⋊C8—1061no—
304C4×C8—532no—
305C4×C8—1061no—
306#95—22no—
307#95—21no—
308#95C22⋊D5312yes—
309C24—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—1061no—
316C2×Q8—1061no—
317C2×Q8#1171992312yescomplemented
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—42no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×C22⋊D5311yescharacteristic
332C22×C4C2×C22⋊D5311yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334SD16—2124no—
335M16—2121no—
336M16—2121no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—2121no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—44no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—1061no—
350C4⋊C4—1061no—
351C4⋊C4—1061no—
352C4⋊C4—1061no—
353C4⋊C4—2121no—
354C4⋊C4—2121no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—2121no—
360C42—41no—
361C42—41no—
362C42—2121no—
363C42D4×D5311yescharacteristic
364C42D4×D5311yescharacteristic
365C42D4×D5311yescharacteristic
366C42#1171994711yescharacteristic
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23#1313318611yescharacteristic
375Q8—22no—
376Q8—22no—
377Q8—41no—
378Q8—41no—
379Q8—4241no—
380Q8—2122no—
381D4—22no—
382D4—22no—
383D4—41no—
384D4—41no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4#1313318511yescharacteristic
405C2×C4#1313318511yescharacteristic
406C2×C4#1313318811yescharacteristic
407C2×C4#1313318911yescharacteristic
408C2×C4#1313318911yescharacteristic
409C2×C4#1313321311yescharacteristic
410C2×C4C2×#1171992311yescharacteristic
411C2×C4#1313323711yescharacteristic
412C2×C4#1313324311yescharacteristic
413C2×C4#1313324711yescharacteristic
414C8—4241no—
415C8—2122no—
416C8—2122no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#1681053111yescharacteristic
427C22#1681077411yescharacteristic
428C22#1681078511yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—81no—
437C4—4241no—
438C4—4241no—
439C4#1681076111yescharacteristic
440C4#1681076611yescharacteristic
441C4#1681085011yescharacteristic
442C4#1681086711yescharacteristic
443C2—21no—
444C2—21no—
445C2—82no—
446C2#2027571811yescharacteristic
447C2#2027577511yescharacteristic
448C2#2027578011yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2536C2
socleC22×C53#392
prime radicalC22×D4×C53D4
hypercenter#2536D53
hypocenterC53#10001
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #20284520C2semidirect
3. #20273278C2semidirect
5. #20273110C2semidirect
6. #20272573C2semidirect
90. C2×Q8×C53SD16semidirect
257. C53#10001semidirect
317. C2×Q8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202847232
#202845202
#202732782
#202731402
#202731102
#202725732
C53×#25362
#1000153