Finite Group Databasebeta

← #27299622

Subgroups — #27299622

Order 13568.

1000 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1000 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#20276361C214yesmaximal, complemented
2#20274643C211yesmaximal, characteristic, complemented
3#20274633C211yesmaximal, characteristic, complemented
4C53×#3485C211yesmaximal, characteristic
5C2×#13133136—21no—
6C2×#13133134C2212yescomplemented
7C2×#13133132—21no—
8C2×#13133132C2212yescomplemented
9#16810587—21no—
10#16810486—21no—
11#16810413—24no—
12#16810410—24no—
13C2×H(5,𝔽2)×C53C411yescharacteristic, complemented
14C22×#41×C53C2211yescharacteristic
15C53×#547—24no—
16C53×#546—24no—
17C53×#536—21no—
18C53×#534—21no—
19C53×#534C2212yes—
20C53×#525—21no—
21C2×#123×C53—21no—
22C2×#123×C53C412yescomplemented
23C53×#518C411yescharacteristic, complemented
24C2×#11719917—22no—
25C2×C22⋊Dic53—22no—
26#13133136—42no—
27#13133134—24no—
28#13133132—24no—
29#13133132—42no—
30C22⋊#88322—42no—
31#13133124—42no—
32H(5,𝔽2)×C53—42no—
33H(5,𝔽2)×C53—42no—
34C2×#41×C53—24no—
35C2×#41×C53—21no—
36C2×#41×C53—22no—
37C2×#41×C53—42no—
38C2×#41×C53—42no—
39C22×Q8×C53D411yescharacteristic
40C22×D4×C53—21no—
41C22×D4×C53—22no—
42C22×D4×C53C2×C411yescharacteristic
43C22×D4×C53C2×C412yes—
44C22×D4×C53D411yescharacteristic
45C22×D4×C53C2311yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C4⋊Q8×C53—22no—
50#124×C53—22no—
51#123×C53—24no—
52#123×C53—22no—
53#123×C53—24no—
54#123×C53—22no—
55#123×C53—22no—
56#123×C53—22no—
57#123×C53D414yes—
58C22⋊Q8×C53—22no—
59C22⋊Q8×C53—24no—
60#121×C53—22no—
61#121×C53—24no—
62#121×C53—24no—
63#120×C53—22no—
64#120×C53—22no—
65#120×C53D414yes—
66C4×D4×C53—24no—
67C4×D4×C53—24no—
68#117×C53—21no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C22⋊C4×C53—22no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53C2×C412yes—
75C24×C53—21no—
76C24×C53C22⋊C411yescharacteristic
77C24×C53C22⋊C412yes—
78C24×C53C22×C411yescharacteristic
79C24×C53C2×D411yescharacteristic
80#41×C53—42no—
81#41×C53—416no—
82#41×C53—48no—
83#41×C53—42no—
84C2×Q8×C53—21no—
85C2×Q8×C53—22no—
86C2×Q8×C53—22no—
87C2×Q8×C53—41no—
88C2×D4×C53—22no—
89C2×D4×C53—22no—
90C2×D4×C53—22no—
91C2×D4×C53—21no—
92C2×D4×C53—22no—
93C2×D4×C53—22no—
94C2×D4×C53—22no—
95C2×D4×C53—22no—
96C2×D4×C53—24no—
97C2×D4×C53—22no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—48no—
102C2×D4×C53—42no—
103C2×D4×C53—42no—
104C2×D4×C53—42no—
105C2×D4×C53—42no—
106C2×D4×C53C2×D414yes—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53C22⋊C411yescharacteristic
121C22×C4×C53C22⋊C411yescharacteristic
122C22×C4×C53C22⋊C411yescharacteristic
123C22×C4×C53C22⋊C412yes—
124C22×C4×C53C2×D411yescharacteristic
125C22×Dic53—42no—
126C4⋊C4×C53—24no—
127C4⋊C4×C53—24no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—24no—
130C22⋊C4×C53—24no—
131C22⋊C4×C53—22no—
132C22⋊C4×C53—24no—
133C22⋊C4×C53—22no—
134C22⋊C4×C53—24no—
135C22⋊C4×C53—24no—
136C22⋊C4×C53C22⋊C414yes—
137C42×C53—22no—
138C22⋊Dic53—42no—
139C22⋊Dic53—42no—
140#11719917—42no—
141#11719917—42no—
142C2×#88322—42no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—22no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—81no—
153C23×C53—81no—
154C23×C53—81no—
155C23×C53—82no—
156C23×C53—82no—
157C23×C53#9912yes—
158C23×C53C2×C22⋊C412yes—
159C23×C53C2×C22⋊C411yescharacteristic
160C23×C53#12012yes—
161Q8×C53—44no—
162Q8×C53—42no—
163D4×C53—28no—
164D4×C53—44no—
165D4×C53—42no—
166D4×C53—44no—
167D4×C53—44no—
168D4×C53—44no—
169D4×C53—42no—
170D4×C53—42no—
171D4×C53—84no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—22no—
178C2×C4×C53—21no—
179C2×C4×C53—24no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—44no—
185C2×C4×C53—44no—
186C2×C4×C53—44no—
187C2×C4×C53—44no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—82no—
191C2×C4×C53—82no—
192C2×C4×C53#9912yes—
193C2×C4×C53#12012yes—
194C2×Dic53—82no—
195C2×Dic53—84no—
196#88322—82no—
197#12706—531nomaximal, Sylow
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—41no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—42no—
205C22×C53—42no—
206C22×C53—42no—
207C22×C53—44no—
208C22×C53—42no—
209C22×C53—81no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53—81no—
213C22×C53—162no—
214C22×C53—162no—
215C22×C53#37911yescharacteristic
216C22×C53C2×#9911yescharacteristic
217C22×C53C2×#9911yescharacteristic
218C4×C53—22no—
219C4×C53—22no—
220C4×C53—44no—
221C4×C53—41no—
222C4×C53—41no—
223C4×C53—42no—
224Dic53—162no—
225#3485D5311yescharacteristic
226#2121—534no—
227#1785—531no—
228#1782—531no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—42no—
233C2×C53—42no—
234C2×C53—42no—
235C2×C53—82no—
236C2×C53#178111yescharacteristic
237C2×H(5,𝔽2)Dic5311yescharacteristic, complemented
238C22×#41C2×D5311yescharacteristic
239#547—24no—
240#546—24no—
241#536—21no—
242#534—21no—
243#534C2×D5312yes—
244#525—21no—
245C2×#123—21no—
246C2×#123Dic5312yescomplemented
247#518Dic5311yescharacteristic, complemented
248C2×#101—1061no—
249C2×#100—532no—
250C2×#99—532no—
251C2×#99—1061no—
252#407—1061no—
253#381—1061no—
254#352—1064no—
255AΣL(1,𝔽16)—1064no—
256C53#1270611yescharacteristic, complemented, Sylow
257H(5,𝔽2)—42no—
258H(5,𝔽2)—42no—
259C2×#41—24no—
260C2×#41—21no—
261C2×#41—22no—
262C2×#41—42no—
263C2×#41—42no—
264C22×Q8C22⋊D5311yescharacteristic
265C22×D4—21no—
266C22×D4—22no—
267C22×D4C2×Dic5311yescharacteristic
268C22×D4C2×Dic5312yes—
269C22×D4C22⋊D5311yescharacteristic
270C22×D4C22×D5311yescharacteristic
271C23×C4C2×Dic5311yescharacteristic
272C23×C4C22⋊D5311yescharacteristic
273C23×C4C22⋊D5311yescharacteristic
274C2×M16—1062no—
275C4⋊Q8—22no—
276#124—22no—
277#123—24no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—22no—
282#123—24no—
283#123C22⋊D5314yes—
284C22⋊Q8—22no—
285C22⋊Q8—24no—
286#121—22no—
287#121—24no—
288#121—24no—
289#120—22no—
290#120—22no—
291#120C22⋊D5314yes—
292C4×D4—24no—
293C4×D4—24no—
294#117—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—1062no—
301C2×C22⋊C4C2×Dic5312yes—
302#101—2122no—
303#100—1064no—
304#99—1064no—
305#99—2122no—
306C22⋊C8—2122no—
307#95—2122no—
308C24—21no—
309C24C22⋊Dic5311yescharacteristic
310C24C22⋊Dic5312yes—
311C24C22×Dic5311yescharacteristic
312C24C2×C22⋊D5311yescharacteristic
313#41—42no—
314#41—416no—
315#41—48no—
316#41—42no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—48no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×C22⋊D5314yes—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—2122no—
354C22×C4C22⋊Dic5311yescharacteristic
355C22×C4C22⋊Dic5311yescharacteristic
356C22×C4C22⋊Dic5311yescharacteristic
357C22×C4C22⋊Dic5312yes—
358C22×C4C2×C22⋊D5311yescharacteristic
359M16—2122no—
360M16—2122no—
361C2×C8—2122no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—2122no—
373C22⋊C4—2122no—
374C22⋊C4C22⋊Dic5314yes—
375C42—22no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—22no—
383C23—42no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—82no—
389C23—82no—
390C23#1313313212yes—
391C23C2×C22⋊Dic5312yes—
392C23C2×C22⋊Dic5311yescharacteristic
393C23#1313318612yes—
394Q8—44no—
395Q8—42no—
396D4—28no—
397D4—44no—
398D4—42no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—42no—
403D4—42no—
404D4—84no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—4242no—
426C2×C4—4244no—
427C2×C4#1313313212yes—
428C2×C4#1313318612yes—
429C8—4242no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—41no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—44no—
440C22—42no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—162no—
446C22—162no—
447C22#1681047911yescharacteristic
448C22C2×#1313313211yescharacteristic
449C22C2×#1313313211yescharacteristic
450C4—22no—
451C4—22no—
452C4—44no—
453C4—41no—
454C4—41no—
455C4—42no—
456C4—8482no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—42no—
461C2—42no—
462C2—42no—
463C2—82no—
464C2#2027463011yescharacteristic
subgroupG/H
centerC2#20274630
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC53×#3485C2
socleC2×C53#1781
cosocleC22×D4×C53C23
hypercenter#3485D53
hypocenterC53#12706
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #20276361C2semidirect
2. #20274643C2semidirect
3. #20274633C2semidirect
6. C2×#13133134C22semidirect
8. C2×#13133132C22semidirect
13. C2×H(5,𝔽2)×C53C4semidirect
22. C2×#123×C53C4semidirect
23. C53×#518C4semidirect
237. C2×H(5,𝔽2)Dic53semidirect
246. C2×#123Dic53semidirect
247. #518Dic53semidirect
256. C53#12706semidirect

Maximal subgroups

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

subgroupindex
#202763612
#202746432
#202746332
C53×#34852
#1270653