Finite Group Databasebeta

← #27294950

Subgroups — #27294950

Order 13568.

698 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 91 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#20274789C211yesmaximal, characteristic, complemented
2C2×#16810347C211yesmaximal, characteristic, complemented
3#20272736C212yesmaximal, complemented
4#20272712C212yesmaximal, complemented
5C53×#2404C211yesmaximal, characteristic
6C2×C22⋊#88322—21no—
7#16810479—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C22×#41×C53C411yescharacteristic, complemented
13C2×#120×C53—21no—
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#393C411yescharacteristic, complemented
16C53×#393C2211yescharacteristic
17C53×#392C411yescharacteristic, complemented
18C53×#392C2211yescharacteristic
19C53×#386—21no—
20C53×#386—21no—
21C53×#386—21no—
22C53×#381C411yescharacteristic, complemented
23C22×#88322—41no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—41no—
26C2×C22⋊Dic53—41no—
27C22⋊#88322—22no—
28C22⋊#88322—41no—
29C25×C53D411yescharacteristic
30C2×#41×C53—41no—
31C2×#41×C53—41no—
32C2×#41×C53—44no—
33C22×Q8×C53C2×C411yescharacteristic
34C22×D4×C53—21no—
35C22×D4×C53—21no—
36C22×D4×C53—21no—
37C22×D4×C53C2×C411yescharacteristic
38C23×C4×C53—21no—
39C23×C4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41#120×C53—44no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—41no—
48C2×C22⋊C4×C53—41no—
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—21no—
61C24×C53—22no—
62C24×C53—41no—
63C24×C53—42no—
64C24×C53—81no—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C22⋊C411yescharacteristic
67C24×C53C2×D411yescharacteristic
68#41×C53—44no—
69#41×C53—44no—
70#41×C53—44no—
71#41×C53—44no—
72C2×Q8×C53—42no—
73C2×Q8×C53—41no—
74C2×D4×C53—24no—
75C2×D4×C53—41no—
76C2×D4×C53—42no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—42no—
80C2×D4×C53—42no—
81C2×D4×C53—42no—
82C2×D4×C53—42no—
83C2×D4×C53—42no—
84C2×D4×C53—81no—
85C2×D4×C53—81no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53—81no—
98C22×C4×C53C22⋊C411yescharacteristic
99C22×C4×C53C22⋊C411yescharacteristic
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×Dic53—42no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—42no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—81no—
111C22⋊C4×C53—81no—
112C22⋊Dic53—42no—
113C22⋊Dic53—42no—
114C22⋊Dic53—82no—
115C22⋊Dic53—82no—
116C2×#88322—42no—
117C2×#88322—82no—
118C23×C53—22no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—22no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—41no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—81no—
143C23×C53—81no—
144C23×C53—84no—
145C23×C53—84no—
146C23×C53#9911yescharacteristic
147C23×C53#9911yescharacteristic
148C23×C53C2×C22⋊C411yescharacteristic
149C23×C53#13512yes—
150Q8×C53—42no—
151Q8×C53—42no—
152D4×C53—44no—
153D4×C53—42no—
154D4×C53—42no—
155D4×C53—44no—
156D4×C53—84no—
157D4×C53—84no—
158C2×C4×C53—22no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—48no—
167C2×C4×C53—42no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—81no—
180C2×Dic53—82no—
181C2×Dic53—82no—
182C2×Dic53—82no—
183#88322—82no—
184#11839—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—22no—
192C22×C53—21no—
193C22×C53—42no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—42no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—42no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—81no—
206C22×C53—84no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53—81no—
210C22×C53—84no—
211C22×C53—81no—
212C22×C53—81no—
213C22×C53AΣL(1,𝔽16)12yes—
214C22×C53C2×#9911yescharacteristic
215C22×C53#41611yescharacteristic
216C22×C53#42111yescharacteristic
217C4×C53—41no—
218C4×C53—42no—
219C4×C53—41no—
220C4×C53—41no—
221C4×C53—41no—
222C4×C53—81no—
223C4×C53—81no—
224Dic53—162no—
225#2404D5311yescharacteristic
226#1819—531no—
227C2×#323—531no—
228#1351—532no—
229#1343—532no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—22no—
233C2×C53—42no—
234C2×C53—41no—
235C2×C53—81no—
236C2×C53—82no—
237C2×C53#151611yescharacteristic
238C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
239C2×C53#212711yescharacteristic
240C22×#41Dic5311yescharacteristic, complemented
241C2×#120—21no—
242C22×C22⋊C4C2×D5311yescharacteristic
243C2×C22⋊C8—1061no—
244#393Dic5311yescharacteristic, complemented
245#393C2×D5311yescharacteristic
246#392Dic5311yescharacteristic, complemented
247#392C2×D5311yescharacteristic
248#386—21no—
249#386—21no—
250#386—21no—
251#381Dic5311yescharacteristic, complemented
252#379—1061no—
253#342—532no—
254#342—1061no—
255#323—532no—
256#323—1061no—
257C53#1183911yescharacteristic, complemented, Sylow
258C25C22⋊D5311yescharacteristic
259C2×#41—41no—
260C2×#41—41no—
261C2×#41—44no—
262C22×Q8C2×Dic5311yescharacteristic
263C22×D4—21no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4C2×Dic5311yescharacteristic
267C23×C4—21no—
268C23×C4C2×Dic5311yescharacteristic
269C23×C4C22⋊D5311yescharacteristic
270C22×C8—2121no—
271#120—44no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—1062no—
280C2×C22⋊C4—2121no—
281C2×C22⋊C4—2121no—
282C2×C22⋊C4C2×Dic5311yescharacteristic
283C2×C22⋊C4C22⋊D5311yescharacteristic
284C2×C22⋊C4C22⋊D5311yescharacteristic
285C2×C22⋊C4C22×D5311yescharacteristic
286C22⋊C8—1062no—
287C22⋊C8—2121no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95C2×Dic5311yescharacteristic
292#95C2×Dic5311yescharacteristic
293C24—21no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—41no—
298C24—42no—
299C24—81no—
300C24C22⋊Dic5311yescharacteristic
301C24C22⋊Dic5311yescharacteristic
302C24C2×C22⋊D5311yescharacteristic
303#41—44no—
304#41—44no—
305#41—44no—
306#41—44no—
307C2×Q8—42no—
308C2×Q8—41no—
309C2×D4—24no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—81no—
320C2×D4—81no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—81no—
333C22×C4—2122no—
334C22×C4C22⋊Dic5311yescharacteristic
335C22×C4C22⋊Dic5311yescharacteristic
336C22×C4C22×Dic5311yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338C2×C8—4242no—
339C2×C8—2122no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—4242no—
350C22⋊C4—4242no—
351C22⋊C4—2122no—
352C22⋊C4—2122no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—21no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—81no—
378C23—81no—
379C23—84no—
380C23—84no—
381C23#1313313211yescharacteristic
382C23#1313313211yescharacteristic
383C23C2×C22⋊Dic5311yescharacteristic
384C23#1313323712yes—
385Q8—42no—
386Q8—42no—
387D4—44no—
388D4—42no—
389D4—42no—
390D4—44no—
391D4—84no—
392D4—84no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—81no—
415C2×C4—4242no—
416C2×C4—4242no—
417C2×C4—4242no—
418C8—4242no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—81no—
440C22—84no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—84no—
445C22—81no—
446C22—81no—
447C22#1681041012yes—
448C22C2×#1313313211yescharacteristic
449C22#1681061911yescharacteristic
450C22#1681064011yescharacteristic
451C4—41no—
452C4—42no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—81no—
457C4—81no—
458C4—8482no—
459C2—21no—
460C2—21no—
461C2—22no—
462C2—42no—
463C2—41no—
464C2—81no—
465C2—82no—
466C2#2027327111yescharacteristic
467C2C2×#1681041011yescharacteristic
468C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
Fitting, prime radicalC53×#2404C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2404D53
hypocenterC53#11839
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. #20274789C2semidirect
2. C2×#16810347C2semidirect
3. #20272736C2semidirect
4. #20272712C2semidirect
12. C22×#41×C53C4semidirect
15. C53×#393C4semidirect
17. C53×#392C4semidirect
22. C53×#381C4semidirect
240. C22×#41Dic53semidirect
244. #393Dic53semidirect
246. #392Dic53semidirect
251. #381Dic53semidirect
257. C53#11839semidirect

Maximal subgroups

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

subgroupindex
#202747892
C2×#168103472
#202727362
#202727122
C53×#24042
#1183953