Finite Group Databasebeta

← #27282160

Subgroups — #27282160

Order 13568.

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

normal subgroups only

This group has 482 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#20284659C211yesmaximal, characteristic
2#20274815C211yesmaximal, characteristic
3#20274790C211yesmaximal, characteristic
4#20273248C211yesmaximal, characteristic
5#20272623C211yesmaximal, characteristic
6#20272568C211yesmaximal, characteristic
7C53×#2332C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic
9#16810961—22no—
10#16810956C412yescomplemented
11#16810951C412yescomplemented
12#16810663—21no—
13#16810632—21no—
14#16810587C2211yescharacteristic
15#16810587C2211yescharacteristic
16#16810486C2211yescharacteristic
17#16810389—21no—
18#16810389—21no—
19#16810383—21no—
20#16810383—21no—
21#16810360—22no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C53×#391—21no—
24C53×#388—21no—
25C53×#388—21no—
26C53×#387—21no—
27C53×#387—21no—
28C53×#381C2211yescharacteristic
29C4×C4⋊C4×C53—21no—
30C4×C22⋊C4×C53C2211yescharacteristic
31C2×#11719917—21no—
32C2×#11719917—21no—
33#13133176—21no—
34#13133176—21no—
35#13133176—21no—
36#13133176—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—21no—
42#13133150—22no—
43#13133148—22no—
44#13133141—22no—
45#13133141—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322C2×C412yes—
50#13133124—21no—
51#13133124—21no—
52C22×Q8×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53C2×C412yes—
57C22⋊Q8×C53D412yes—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53D411yescharacteristic
69C2×C42×C53D411yescharacteristic
70#95×C53—21no—
71#95×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77C24×C53C2×D411yescharacteristic
78C2×Q8×C53—21no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53C22⋊C412yes—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—41no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—42no—
103C4⋊C4×C53—81no—
104C4⋊C4×C53C22⋊C412yes—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C42×C53—42no—
111C42×C53—41no—
112C42×C53—41no—
113C42×C53—81no—
114C22⋊Dic53—41no—
115C22⋊Dic53—41no—
116C22⋊Dic53—41no—
117C22⋊Dic53—41no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4×Dic53—41no—
123C4×Dic53—41no—
124#11719917—41no—
125#11719917—41no—
126#11719917—41no—
127#11719917—41no—
128C2×#88322—22no—
129C2×#88322—41no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53C4×D411yescharacteristic
135C23×C53#12011yescharacteristic
136C23×C53#12111yescharacteristic
137Q8×C53—42no—
138Q8×C53—81no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53C2×C22⋊C411yescharacteristic
166C2×C4×C53C4×D411yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12311yescharacteristic
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176#88322—42no—
177#88322—81no—
178#9526—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53#38611yescharacteristic
190C22×C53#45811yescharacteristic
191C22×C53#45811yescharacteristic
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—42no—
197C4×C53—42no—
198C4×C53—42no—
199C4×C53—81no—
200C4×C53—81no—
201C4×C53—81no—
202Dic53—81no—
203Dic53—81no—
204Dic53—161no—
205#3095—531no—
206#2332D5311yescharacteristic
207#1826—531no—
208#1820—531no—
209#1511—531no—
210#1306—531no—
211#1284—531no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—41no—
215C2×C53—41no—
216C2×C53#189011yescharacteristic
217C2×C53#190511yescharacteristic
218C2×C53#190611yescharacteristic
219C2×C22⋊Q8C2×D5311yescharacteristic
220C2×#117—531no—
221#485—1062no—
222#484—532no—
223#483—532no—
224#428—1061no—
225#419—1061no—
226#407—531no—
227#407—531no—
228#391—21no—
229#388—21no—
230#388—21no—
231#387—21no—
232#387—21no—
233#381—531no—
234#381C2×D5311yescharacteristic
235C4×C4⋊C4—21no—
236C4×C22⋊C4C2×D5311yescharacteristic
237#342—1061no—
238#342—1061no—
239#339—1061no—
240#339—1061no—
241#328—1062no—
242C53#952611yescharacteristic, complemented, Sylow
243C22×Q8C22⋊D5311yescharacteristic
244C23×C4C22×D5311yescharacteristic
245C2×M16—1061no—
246C2×M16—1061no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8C4×D5312yes—
250C22⋊Q8D21212yes—
251#117—1061no—
252#117—1061no—
253#117—1061no—
254#117—1061no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—531no—
259C2×C4⋊C4—531no—
260C2×C4⋊C4C22⋊D5311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4C22⋊D5311yescharacteristic
266C2×C22⋊C4C22⋊D5311yescharacteristic
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—1061no—
270C2×C42C22⋊D5311yescharacteristic
271C2×C42C22⋊D5311yescharacteristic
272#107—1062no—
273#106—1062no—
274#103—1062no—
275#103—1062no—
276C22⋊C8—532no—
277C22⋊C8—1061no—
278C22⋊C8—1061no—
279C22⋊C8—1061no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—1061no—
288#95—1061no—
289C24C2×C22⋊D5311yescharacteristic
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8#1171992112yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—1061no—
303C22×C4—1061no—
304C22×C4—2121no—
305C22×C4C2×C4×D5311yescharacteristic
306C22×C4C2×D21211yescharacteristic
307C22×C4#1171994511yescharacteristic
308C22×C4#1171994511yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310C22×C4C2×C22⋊D5311yescharacteristic
311M16—2121no—
312M16—2121no—
313M16—2121no—
314M16—2121no—
315C2×C8—1062no—
316C2×C8—2121no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—42no—
321C4⋊C4—81no—
322C4⋊C4—1061no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4#1171992112yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C22⋊C4—2121no—
335C22⋊C4—2121no—
336C42—41no—
337C42—41no—
338C42—42no—
339C42—81no—
340C42—2121no—
341C42—2121no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23C4×C22⋊D5311yescharacteristic
347C23#1313318611yescharacteristic
348C23#1313319111yescharacteristic
349Q8—42no—
350Q8—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—4241no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4C2×#1171992111yescharacteristic
385C2×C4C4×C22⋊D5311yescharacteristic
386C2×C4#1313319111yescharacteristic
387C2×C4#1313320011yescharacteristic
388C8—4241no—
389C8—2122no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22#1681050211yescharacteristic
401C22#1681081311yescharacteristic
402C22#1681081311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—8481no—
414C4—4241no—
415C4—4241no—
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#2027512811yescharacteristic
421C2#2027520911yescharacteristic
422C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2332C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2332D53
hypocenterC53#9526
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2332

Complements

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

NH ≅ G/Ntype
10. #16810956C4semidirect
11. #16810951C4semidirect
242. C53#9526semidirect

Maximal subgroups

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

subgroupindex
#202846592
#202748152
#202747902
#202732482
#202726232
#202725682
C53×#23322
#952653