Finite Group Databasebeta

← #27284871

Subgroups — #27284871

Order 13568.

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

normal subgroups only

This group has 500 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
2#20284520C211yesmaximal, characteristic
3#20273278C211yesmaximal, characteristic
4#20273144C211yesmaximal, characteristic
5#20273110C211yesmaximal, characteristic
6#20272573C211yesmaximal, characteristic
7C53×#2581C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810958—21no—
10#16810930—22no—
11#16810918—21no—
12#16810918—21no—
13#16810908—21no—
14#16810833—21no—
15#16810779—21no—
16C4×#11719925—22no—
17#16810676C2211yescharacteristic
18#16810676C2211yescharacteristic
19#16810663—21no—
20#16810647C2211yescharacteristic
21#16810632—21no—
22#16810359—22no—
23Q8⋊#88322—22no—
24Q8⋊#88322—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C2×C4×Q8×C53C2211yescharacteristic
27C53×#400—21no—
28C53×#395—21no—
29C53×#391—21no—
30C53×#389—21no—
31C53×#387—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#13133141—22no—
52#13133141—22no—
53#13133140—22no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56C4×#88322—21no—
57C4×#88322D412yes—
58C22×Q8×C53D411yescharacteristic
59C22×Q8×C53D411yescharacteristic
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×Q8×C53—21no—
66C4×Q8×C53—21no—
67C4×Q8×C53—21no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53D411yescharacteristic
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C42×C53—21no—
75C2×C42×C53—21no—
76C2×C42×C53C2311yescharacteristic
77#95×C53—22no—
78#95×C53—22no—
79#95×C53—22no—
80#95×C53—21no—
81#95×C53D412yes—
82C2×Q8×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53—21no—
89C2×Q8×C53—42no—
90C2×Q8×C53Q1612yescomplemented
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C2×Dic106—21no—
100C2×Dic106—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—44no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C42×C53—41no—
113C42×C53—41no—
114C42×C53—41no—
115C42×C53—41no—
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120C22⋊Dic53—41no—
121#11719925—44no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126#11719919—41no—
127#11719919—41no—
128C4×Dic53—41no—
129#11719917—41no—
130#11719917—41no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—41no—
134C23×C53#12011yescharacteristic
135Q8×C53—22no—
136Q8×C53—22no—
137Q8×C53—22no—
138Q8×C53—22no—
139Q8×C53—41no—
140Q8×C53—41no—
141Q8×C53—41no—
142Q8×C53—41no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53#12011yescharacteristic
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12711yescharacteristic
164C2×C4×C53C2×Q1611yescharacteristic
165C2×C4×C53#13511yescharacteristic
166C2×C4×C53#13711yescharacteristic
167C2×C4×C53#13711yescharacteristic
168C2×Dic53—41no—
169C2×Dic53—41no—
170Dic106—42no—
171Dic106—81no—
172#88322—42no—
173#88322—42no—
174#88322—81no—
175#10003—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53#39211yescharacteristic
179C22×C53#45111yescharacteristic
180C22×C53#45211yescharacteristic
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—42no—
186C4×C53—42no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53#45111yescharacteristic
191C4×C53#45211yescharacteristic
192C4×C53#46711yescharacteristic
193C4×C53#47011yescharacteristic
194Dic53—81no—
195Dic53—81no—
196#3104—531no—
197#3078—531no—
198#2581D5311yescharacteristic
199#1518—531no—
200#1487—531no—
201#1479—531no—
202#1286—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53#200111yescharacteristic
206C2×C53#200911yescharacteristic
207C2×C53#201011yescharacteristic
208#533—531no—
209C2×C4×Q8C2×D5311yescharacteristic
210C2×C4×Q8C2×D5311yescharacteristic
211#484—1061no—
212#479—1062no—
213#477—1061no—
214#477—1061no—
215#475—1061no—
216#462—1061no—
217#451—1061no—
218C4×Q16—1062no—
219#431—531no—
220#431—531no—
221#428—1061no—
222#423—531no—
223#419—1061no—
224#400—21no—
225#395—21no—
226#391—21no—
227#389—21no—
228#387—22no—
229#382C2×D5311yescharacteristic
230#328—1062no—
231Q8⋊C8—1062no—
232Q8⋊C8—1062no—
233C53#1000311yescharacteristic, complemented, Sylow
234C22×Q8C22⋊D5311yescharacteristic
235C22×Q8C22⋊D5311yescharacteristic
236C2×Q16—1062no—
237C2×M16—1061no—
238C4⋊Q8—531no—
239C4⋊Q8—531no—
240#125—1061no—
241C22⋊Q8—1061no—
242C22⋊Q8—1061no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—1061no—
252C4×Q8—1061no—
253#117—1061no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22⋊D5311yescharacteristic
259C2×C4⋊C4C22⋊D5311yescharacteristic
260C2×C42—21no—
261C2×C42—21no—
262C2×C42C22×D5311yescharacteristic
263#107—1062no—
264#107—1062no—
265#106—1062no—
266C4⋊C8—532no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270#103—1062no—
271#103—1062no—
272#103—1062no—
273#103—1062no—
274C22⋊C8—1061no—
275C22⋊C8—1061no—
276C4×C8—532no—
277C4×C8—1061no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—21no—
282#95C22⋊D5312yes—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—42no—
291C2×Q8—1061no—
292C2×Q8—1061no—
293C2×Q8#1171992512yescomplemented
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4C2×C22⋊D5311yescharacteristic
300C22×C4C2×C22⋊D5311yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302Q16—2124no—
303M16—2121no—
304M16—2121no—
305C2×C8—1062no—
306C2×C8—1062no—
307C2×C8—2121no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—1061no—
320C4⋊C4—1061no—
321C4⋊C4—1061no—
322C4⋊C4—1061no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C22⋊C4—2121no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—2121no—
331C42D4×D5311yescharacteristic
332C42D4×D5311yescharacteristic
333C42D4×D5311yescharacteristic
334C42#1171994711yescharacteristic
335C23#1313318611yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—22no—
339Q8—22no—
340Q8—41no—
341Q8—41no—
342Q8—41no—
343Q8—41no—
344Q8—4241no—
345Q8—2122no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4#1313318511yescharacteristic
364C2×C4#1313318511yescharacteristic
365C2×C4#1313318811yescharacteristic
366C2×C4#1313318911yescharacteristic
367C2×C4#1313318911yescharacteristic
368C2×C4#1313321311yescharacteristic
369C2×C4C2×#1171992511yescharacteristic
370C2×C4#1313323711yescharacteristic
371C2×C4#1313324711yescharacteristic
372C2×C4#1313324811yescharacteristic
373C8—4241no—
374C8—2122no—
375C8—2122no—
376C22—21no—
377C22—21no—
378C22#1681053111yescharacteristic
379C22#1681077911yescharacteristic
380C22#1681078511yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—4241no—
391C4—4241no—
392C4#1681077811yescharacteristic
393C4#1681078311yescharacteristic
394C4#1681086211yescharacteristic
395C4#1681087911yescharacteristic
396C2—21no—
397C2—21no—
398C2#2027572311yescharacteristic
399C2#2027577511yescharacteristic
400C2#2027578011yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2581C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#2581D53
hypocenterC53#10003
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
90. C2×Q8×C53Q16semidirect
233. C53#10003semidirect
293. C2×Q8#11719925semidirect

Maximal subgroups

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

subgroupindex
#202847542
#202845202
#202732782
#202731442
#202731102
#202725732
C53×#25812
#1000353