Finite Group Databasebeta

← #27299714

Subgroups — #27299714

Order 13568.

708 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 127 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#20276361C212yesmaximal
2C2×#16810413C212yesmaximal, complemented
3#20274826C211yesmaximal, characteristic, complemented
4#20274823C211yesmaximal, characteristic, complemented
5C53×#3491C211yesmaximal, characteristic
6C2×#13133134C2212yes—
7C2×#13133132C2212yes—
8#16810413—22no—
9#16810413—22no—
10#16810413—22no—
11#16810410—22no—
12#16810393—22no—
13#16810389—22no—
14C53×#552—22no—
15C53×#552—22no—
16C53×#551—22no—
17C53×#551—22no—
18C53×#543C411yescharacteristic, complemented
19C53×#539—22no—
20C53×#534C2211yescharacteristic
21C53×#529—22no—
22C2×#123×C53C411yescharacteristic, complemented
23C2×#123×C53C2211yescharacteristic
24C53×#518C411yescharacteristic, complemented
25C53×#518C2211yescharacteristic
26C2×#117×C53C411yescharacteristic, complemented
27C2×#11719917—22no—
28C2×C22⋊Dic53—22no—
29#13133134—22no—
30#13133134—22no—
31#13133132—22no—
32#13133132—22no—
33C22×D4×C53C2×C411yescharacteristic
34C22×D4×C53C2311yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36#126×C53—22no—
37#126×C53—22no—
38#126×C53—22no—
39#126×C53—22no—
40#125×C53—22no—
41#125×C53—22no—
42#125×C53—22no—
43#124×C53—22no—
44#123×C53—22no—
45#123×C53—22no—
46#123×C53—22no—
47#123×C53—22no—
48#123×C53—21no—
49#123×C53—21no—
50#123×C53—21no—
51#123×C53D412yes—
52#123×C53D412yes—
53#123×C53D412yes—
54C22⋊Q8×C53—22no—
55C22⋊Q8×C53—22no—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53—22no—
59#121×C53—22no—
60#120×C53—21no—
61#120×C53D412yes—
62C4×Q8×C53—22no—
63C4×D4×C53—22no—
64C4×D4×C53—22no—
65C4×D4×C53—22no—
66C4×D4×C53—22no—
67C4×D4×C53—22no—
68#117×C53—22no—
69#117×C53—22no—
70#117×C53—22no—
71#117×C53—22no—
72#117×C53—21no—
73#117×C53D412yes—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C22⋊C4×C53—21no—
78C2×C22⋊C4×C53—21no—
79C2×C22⋊C4×C53—21no—
80C2×C22⋊C4×C53C2×C411yescharacteristic
81C2×C22⋊C4×C53C2×C411yescharacteristic
82C2×C22⋊C4×C53C2×C411yescharacteristic
83C2×C22⋊C4×C53C2×C411yescharacteristic
84C2×C22⋊C4×C53D412yes—
85C2×C42×C53—21no—
86C24×C53C22⋊C411yescharacteristic
87C24×C53C22×C411yescharacteristic
88C24×C53C2×D411yescharacteristic
89C2×Q8×C53—22no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—22no—
93C2×D4×C53—42no—
94C2×D4×C53—42no—
95C2×D4×C53C2×D412yes—
96C2×D4×C53C2×D412yes—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—42no—
106C22×C4×C53C22⋊C411yescharacteristic
107C22×C4×C53C22⋊C411yescharacteristic
108C22×C4×C53C22⋊C411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×Dic53—42no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—22no—
123C22⋊C4×C53—22no—
124C22⋊C4×C53—22no—
125C22⋊C4×C53—22no—
126C22⋊C4×C53—21no—
127C22⋊C4×C53—21no—
128C22⋊C4×C53—21no—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—22no—
131C22⋊C4×C53—22no—
132C22⋊C4×C53—22no—
133C22⋊C4×C53—22no—
134C22⋊C4×C53—22no—
135C22⋊C4×C53C22⋊C412yes—
136C22⋊C4×C53C22⋊C412yes—
137C22⋊C4×C53C22⋊C412yes—
138C22⋊C4×C53C22⋊C412yes—
139C42×C53—22no—
140C42×C53—22no—
141C42×C53—22no—
142C42×C53—22no—
143C22⋊Dic53—42no—
144C22⋊Dic53—42no—
145#11719917—42no—
146#11719917—42no—
147C2×#88322—42no—
148C23×C53—22no—
149C23×C53—22no—
150C23×C53—21no—
151C23×C53—41no—
152C23×C53—81no—
153C23×C53—81no—
154C23×C53—81no—
155C23×C53C2×C22⋊C411yescharacteristic
156C23×C53C2×C22⋊C411yescharacteristic
157C23×C53C2×C22⋊C411yescharacteristic
158C23×C53#12012yes—
159Q8×C53—82no—
160D4×C53—42no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—42no—
164D4×C53—82no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—22no—
172C2×C4×C53—21no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—81no—
179C2×C4×C53—82no—
180C2×C4×C53—82no—
181C2×C4×C53—81no—
182C2×C4×C53—81no—
183C2×C4×C53—81no—
184C2×C4×C53#12012yes—
185C2×Dic53—82no—
186C2×Dic53—162no—
187#88322—82no—
188#12726—531nomaximal, Sylow
189C22×C53—22no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—42no—
195C22×C53—41no—
196C22×C53—42no—
197C22×C53—41no—
198C22×C53—82no—
199C22×C53—82no—
200C22×C53—161no—
201C22×C53#37911yescharacteristic
202C22×C53#41012yes—
203C4×C53—22no—
204C4×C53—42no—
205C4×C53—42no—
206C4×C53—42no—
207C4×C53—42no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53—41no—
211Dic53—162no—
212#3491D5311yescharacteristic
213#2121—532no—
214C2×#352—532no—
215#1829—531no—
216#1828—531no—
217C2×C53—21no—
218C2×C53—22no—
219C2×C53—42no—
220C2×C53—41no—
221C2×C53—41no—
222C2×C53—81no—
223C2×C53#178211yescharacteristic
224#552—22no—
225#552—22no—
226#551—22no—
227#551—22no—
228#543Dic5311yescharacteristic, complemented
229#539—22no—
230#534C2×D5311yescharacteristic
231#529—22no—
232C2×#123Dic5311yescharacteristic, complemented
233C2×#123C2×D5311yescharacteristic
234#518Dic5311yescharacteristic, complemented
235#518C2×D5311yescharacteristic
236C2×#117Dic5311yescharacteristic, complemented
237C2×#100—532no—
238C2×#99—532no—
239#352—1062no—
240#352—1062no—
241#352—1062no—
242AΣL(1,𝔽16)—1062no—
243#344—1062no—
244#342—1062no—
245C53#1272611yescharacteristic, complemented, Sylow
246C22×D4C2×Dic5311yescharacteristic
247C22×D4C22×D5311yescharacteristic
248C23×C4C2×Dic5311yescharacteristic
249C2×M16—1062no—
250#126—22no—
251#126—22no—
252#126—22no—
253#126—22no—
254#125—22no—
255#125—22no—
256#125—22no—
257#124—22no—
258#123—22no—
259#123—22no—
260#123—22no—
261#123—21no—
262#123—21no—
263#123—21no—
264#123—22no—
265#123C22⋊D5312yes—
266#123C22⋊D5312yes—
267#123C22⋊D5312yes—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273#121—22no—
274#120—21no—
275#120C22⋊D5312yes—
276C4×Q8—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282#117—22no—
283#117—22no—
284#117—21no—
285#117—22no—
286#117—22no—
287#117C22⋊D5312yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—1062no—
295C2×C22⋊C4C2×Dic5311yescharacteristic
296C2×C22⋊C4C2×Dic5311yescharacteristic
297C2×C22⋊C4C2×Dic5311yescharacteristic
298C2×C22⋊C4C2×Dic5311yescharacteristic
299C2×C22⋊C4C22⋊D5312yes—
300C2×C42—21no—
301#100—1062no—
302#100—1062no—
303#99—1062no—
304#99—1062no—
305C24C22⋊Dic5311yescharacteristic
306C24C22×Dic5311yescharacteristic
307C24C2×C22⋊D5311yescharacteristic
308C2×Q8—22no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—22no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4C2×C22⋊D5312yes—
315C2×D4C2×C22⋊D5312yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—2122no—
326C22×C4C22⋊Dic5311yescharacteristic
327C22×C4C22⋊Dic5311yescharacteristic
328C22×C4C22⋊Dic5311yescharacteristic
329C22×C4C2×C22⋊D5311yescharacteristic
330M16—2122no—
331M16—2122no—
332C2×C8—2122no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—2122no—
358C22⋊C4—2122no—
359C22⋊C4C22⋊Dic5312yes—
360C22⋊C4C22⋊Dic5312yes—
361C22⋊C4C22⋊Dic5312yes—
362C22⋊C4C22⋊Dic5312yes—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C23—22no—
368C23—22no—
369C23—21no—
370C23—41no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C2×C22⋊Dic5311yescharacteristic
375C23C2×C22⋊Dic5311yescharacteristic
376C23C2×C22⋊Dic5311yescharacteristic
377C23#1313318612yes—
378Q8—82no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—82no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—8482no—
404C2×C4—4242no—
405C2×C4#1313318612yes—
406C8—4242no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22#1681047911yescharacteristic
420C22#1681059712yes—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—8482no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—41no—
434C2—41no—
435C2—81no—
436C2#2027463311yescharacteristic
subgroupG/H
centerC2#20274633
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#3491C2
socleC2×C53#1782
prime radicalC53×#534C22
cosocleC22×D4×C53C23
hypercenter#3491D53
hypocenterC53#12726
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3491

Complements

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

NH ≅ G/Ntype
2. C2×#16810413C2semidirect
3. #20274826C2semidirect
4. #20274823C2semidirect
18. C53×#543C4semidirect
22. C2×#123×C53C4semidirect
24. C53×#518C4semidirect
26. C2×#117×C53C4semidirect
228. #543Dic53semidirect
232. C2×#123Dic53semidirect
234. #518Dic53semidirect
236. C2×#117Dic53semidirect
245. C53#12726semidirect

Maximal subgroups

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

subgroupindex
#202763612
C2×#168104132
#202748262
#202748232
C53×#34912
#1272653