Finite Group Databasebeta

← #27299644

Subgroups — #27299644

Order 13568.

952 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 135 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#20276410C211yesmaximal, characteristic, complemented
2#20276390C211yesmaximal, characteristic, complemented
3#20276356C212yesmaximal, complemented
4#20274690C212yesmaximal, complemented
5C53×#3486C211yesmaximal, characteristic
6#16810640—22no—
7#16810632—22no—
8#16810607C2212yes—
9#16810597C2212yes—
10#16810422—22no—
11#16810418—22no—
12#16810413—22no—
13#16810410—22no—
14C2×2−1+4×C53C2211yescharacteristic
15C23×Q8×C53C411yescharacteristic, complemented
16D4×Q8×C53—22no—
17D4×Q8×C53—22no—
18C53×#548—22no—
19C53×#548—22no—
20C53×#536—22no—
21C53×#536C411yescharacteristic, complemented
22C53×#536C2211yescharacteristic
23C53×#525—22no—
24C53×#525C411yescharacteristic, complemented
25C53×#525C2211yescharacteristic
26C53×#519C411yescharacteristic, complemented
27C2×#11719917—22no—
28#13133176—22no—
29#13133144—44no—
30#13133141—44no—
31#13133136—22no—
32#13133134—22no—
33#13133132—22no—
34#13133132—22no—
352−1+4×C53—44no—
36C2×#41×C53—22no—
37C2×#41×C53—21no—
38C2×#41×C53C2×C411yescharacteristic
39C2×#41×C53D412yes—
40C2×#41×C53C2311yescharacteristic
41C22×Q8×C53—22no—
42C22×Q8×C53—22no—
43C22×Q8×C53—21no—
44C22×Q8×C53—21no—
45C22×Q8×C53—44no—
46C22×Q8×C53C2×C411yescharacteristic
47C22×Q8×C53C2×C411yescharacteristic
48C22×Q8×C53D412yes—
49C23×C4×C53—21no—
50C23×C4×C53C2×C411yescharacteristic
51C4⋊Q8×C53—24no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53D412yes—
54#124×C53—24no—
55#124×C53—21no—
56#124×C53D412yes—
57#123×C53—22no—
58#123×C53—22no—
59#123×C53—21no—
60#123×C53D412yes—
61C22⋊Q8×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—24no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—22no—
68#121×C53—24no—
69#121×C53—21no—
70#121×C53—21no—
71#120×C53—21no—
72#120×C53D412yes—
73C4×Q8×C53—22no—
74C4×Q8×C53—22no—
75C4×D4×C53—24no—
76C4×D4×C53—24no—
77C4×D4×C53—22no—
78C4×D4×C53—22no—
79#117×C53—22no—
80#117×C53C2×C411yescharacteristic
81#117×C53C2×C411yescharacteristic
82C24×C53C22⋊C411yescharacteristic
83#41×C53—24no—
84#41×C53—42no—
85#41×C53—42no—
86#41×C53—42no—
87#41×C53—48no—
88#41×C53—44no—
89C2×Q8×C53—24no—
90C2×Q8×C53—24no—
91C2×Q8×C53—21no—
92C2×Q8×C53—21no—
93C2×Q8×C53—22no—
94C2×Q8×C53—24no—
95C2×Q8×C53—42no—
96C2×Q8×C53—44no—
97C2×Q8×C53—42no—
98C2×Q8×C53—48no—
99C2×Q8×C53—44no—
100C2×Q8×C53—44no—
101C2×Q8×C53—44no—
102C2×Q8×C53C22⋊C411yescharacteristic
103C2×Q8×C53C22⋊C412yes—
104C2×Q8×C53C22⋊C412yes—
105C2×Q8×C53C2×D411yescharacteristic
106C2×D4×C53—22no—
107C2×D4×C53—21no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53C22⋊C411yescharacteristic
111C2×D4×C53C2×D411yescharacteristic
112C2×D4×C53C2×D412yes—
113C22×C4×C53—21no—
114C22×C4×C53—24no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—44no—
122C22×C4×C53—42no—
123C22×C4×C53C22⋊C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×C4×C53C2×D412yes—
126C4⋊C4×C53—24no—
127C4⋊C4×C53—24no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—24no—
132C4⋊C4×C53—22no—
133C22⋊C4×C53—22no—
134C22⋊C4×C53—24no—
135C22⋊C4×C53—22no—
136C22⋊C4×C53—21no—
137C22⋊C4×C53—24no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—22no—
140C22⋊C4×C53C22⋊C412yes—
141C42×C53—24no—
142C42×C53—21no—
143C42×C53C22⋊C412yes—
144C22⋊Dic53—42no—
145C4⋊Dic53—42no—
146C4×Dic53—42no—
147#11719917—42no—
148#11719917—42no—
149C2×#88322—42no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—41no—
153C23×C53—42no—
154C23×C53C2×C22⋊C411yescharacteristic
155C23×C53#12012yes—
156Q8×C53—24no—
157Q8×C53—22no—
158Q8×C53—22no—
159Q8×C53—41no—
160Q8×C53—44no—
161Q8×C53—48no—
162Q8×C53—42no—
163Q8×C53—42no—
164Q8×C53#9914yes—
165D4×C53—22no—
166D4×C53—42no—
167D4×C53—41no—
168D4×C53—44no—
169D4×C53—44no—
170D4×C53—44no—
171D4×C53—42no—
172C2×C4×C53—24no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—22no—
177C2×C4×C53—21no—
178C2×C4×C53—22no—
179C2×C4×C53—21no—
180C2×C4×C53—22no—
181C2×C4×C53—22no—
182C2×C4×C53—44no—
183C2×C4×C53—44no—
184C2×C4×C53—48no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—41no—
190C2×C4×C53C2×C22⋊C411yescharacteristic
191C2×C4×C53C2×C22⋊C411yescharacteristic
192C2×C4×C53#12012yes—
193C2×Dic53—82no—
194#88322—82no—
195#12712—531nomaximal, Sylow
196C22×C53—22no—
197C22×C53—22no—
198C22×C53—21no—
199C22×C53—42no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—42no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53#37911yescharacteristic
208C4×C53—22no—
209C4×C53—24no—
210C4×C53—21no—
211C4×C53—42no—
212C4×C53—42no—
213C4×C53—41no—
214C4×C53—42no—
215C4×C53—41no—
216C4×C53C2×#9912yes—
217Dic53—162no—
218#3486D5311yescharacteristic
219#2135—531no—
220#2129—531no—
221#2120—532no—
222#1796—532no—
223C2×C53—21no—
224C2×C53—42no—
225C2×C53—42no—
226C2×C53—41no—
227C2×C53—81no—
228C2×C53#178111yescharacteristic
229C2×2−1+4C2×D5311yescharacteristic
230C23×Q8Dic5311yescharacteristic, complemented
231D4×Q8—22no—
232D4×Q8—22no—
233#548—22no—
234#548—22no—
235#536—22no—
236#536Dic5311yescharacteristic, complemented
237#536C2×D5311yescharacteristic
238#525—22no—
239#525Dic5311yescharacteristic, complemented
240#525C2×D5311yescharacteristic
241#519Dic5311yescharacteristic, complemented
242#421—1062no—
243#419—1062no—
244#413—532no—
245#410—532no—
246#356—1062no—
247#354—1062no—
248#352—1062no—
249AΣL(1,𝔽16)—1062no—
250C53#1271211yescharacteristic, complemented, Sylow
2512−1+4—44no—
252C2×#41—22no—
253C2×#41—21no—
254C2×#41C2×Dic5311yescharacteristic
255C2×#41C22⋊D5312yes—
256C2×#41C22×D5311yescharacteristic
257C22×Q8—21no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—21no—
261C22×Q8—44no—
262C22×Q8C2×Dic5311yescharacteristic
263C22×Q8C2×Dic5311yescharacteristic
264C22×Q8C22⋊D5312yes—
265C23×C4—21no—
266C23×C4C2×Dic5311yescharacteristic
267C2×M16—1062no—
268C4⋊Q8—24no—
269C4⋊Q8—21no—
270C4⋊Q8C22⋊D5312yes—
271#124—24no—
272#124—21no—
273#124C22⋊D5312yes—
274#123—22no—
275#123—21no—
276#123—22no—
277#123C22⋊D5312yes—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285#121—24no—
286#121—21no—
287#121—21no—
288#120—21no—
289#120C22⋊D5312yes—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296#117—22no—
297#117—1062no—
298#117C2×Dic5311yescharacteristic
299#117C2×Dic5311yescharacteristic
300#104—2124no—
301#103—2124no—
302#101—1062no—
303#100—1062no—
304#99—1062no—
305#99—1062no—
306C24C22⋊Dic5311yescharacteristic
307#41—24no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—48no—
312#41—44no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—48no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8C22⋊Dic5312yes—
327C2×Q8C22⋊Dic5311yescharacteristic
328C2×Q8C22⋊Dic5312yes—
329C2×Q8C2×C22⋊D5311yescharacteristic
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4C22⋊Dic5311yescharacteristic
335C2×D4C2×C22⋊D5311yescharacteristic
336C2×D4C2×C22⋊D5312yes—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4C22⋊Dic5311yescharacteristic
348C22×C4C22×Dic5311yescharacteristic
349C22×C4C2×C22⋊D5312yes—
350M16—2122no—
351M16—2122no—
352C2×C8—2122no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—2122no—
361C22⋊C4—22no—
362C22⋊C4—24no—
363C22⋊C4—22no—
364C22⋊C4—21no—
365C22⋊C4—24no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—2122no—
369C22⋊C4C22⋊Dic5312yes—
370C42—24no—
371C42—21no—
372C42—2122no—
373C42C22⋊Dic5312yes—
374C23—21no—
375C23—21no—
376C23—41no—
377C23—42no—
378C23C2×C22⋊Dic5311yescharacteristic
379C23#1313318612yes—
380Q8—24no—
381Q8—22no—
382Q8—22no—
383Q8—41no—
384Q8—44no—
385Q8—42no—
386Q8—48no—
387Q8—42no—
388Q8#1313313214yes—
389D4—22no—
390D4—42no—
391D4—41no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—42no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—4242no—
415C2×C4C2×C22⋊Dic5311yescharacteristic
416C2×C4C2×C22⋊Dic5311yescharacteristic
417C2×C4#1313318612yes—
418C8—4242no—
419C22—22no—
420C22—21no—
421C22—22no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22#1681047911yescharacteristic
431C4—22no—
432C4—24no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—41no—
437C4—42no—
438C4—41no—
439C4—8482no—
440C4C2×#1313313212yes—
441C2—21no—
442C2—42no—
443C2—42no—
444C2—41no—
445C2—81no—
446C2#2027463011yescharacteristic
subgroupG/H
centerC2#20274630
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
Fitting, prime radicalC53×#3486C2
socleC2×C53#1781
cosocleC2×#41×C53C23
hypercenter#3486D53
hypocenterC53#12712
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3486

Complements

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

NH ≅ G/Ntype
1. #20276410C2semidirect
2. #20276390C2semidirect
3. #20276356C2semidirect
4. #20274690C2semidirect
15. C23×Q8×C53C4semidirect
21. C53×#536C4semidirect
24. C53×#525C4semidirect
26. C53×#519C4semidirect
230. C23×Q8Dic53semidirect
236. #536Dic53semidirect
239. #525Dic53semidirect
241. #519Dic53semidirect
250. C53#12712semidirect

Maximal subgroups

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

subgroupindex
#202764102
#202763902
#202763562
#202746902
C53×#34862
#1271253