Finite Group Databasebeta

← #27299240

Subgroups — #27299240

Order 13568.

636 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 123 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#20277369C211yesmaximal, characteristic
2C2×#16810409C212yesmaximal
3C4×#13133134C211yesmaximal, characteristic, complemented
4#20272832C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic, complemented
6C2×#13133185C2211yescharacteristic
7C22×#11719921C411yescharacteristic, complemented
8C2×#13133134C2212yes—
9C4×#11719917—21no—
10#16810523—21no—
11#16810514—21no—
12#16810513—21no—
13#16810503C2211yescharacteristic
14#16810502—21no—
15#16810495C411yescharacteristic, complemented
16#16810478C411yescharacteristic, complemented
17C4×C22⋊Dic53C411yescharacteristic, complemented
18#16810409C418yescomplemented
19#16810392—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#99×C53C2212yes—
22C4×C22⋊C4×C53—21no—
23C53×#342—21no—
24C24×D53C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C23×Dic53C2×C411yescharacteristic
27C2×#11719917—22no—
28#13133185—22no—
29#13133185C2×C414yes—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53C2×C411yescharacteristic
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—41no—
40C2×#11719921—41no—
41C2×C4×Dic53—21no—
42#13133134—22no—
43#13133134C2×C414yes—
44#13133128—41no—
45C4×#88322—41no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×D4×C53C2311yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C4×D4×C53—41no—
53C4×D4×C53—41no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53—41no—
59C2×C42×C53D411yescharacteristic
60#99×C53—22no—
61#99×C53C2×C414yes—
62#95×C53—41no—
63C24×C53C22×C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C23×D53—21no—
66C23×D53—21no—
67C23×D53—41no—
68C23×D53—41no—
69C23×D53—81no—
70C23×D53—81no—
71C23×D53C4212yescomplemented
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53C22×C412yes—
75C2×D4×C53#4112yes—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—42no—
81C22×C4×C53—81no—
82C22×C4×C53—81no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C2×C22⋊D53—24no—
89C2×C22⋊D53—81no—
90C2×C22⋊D53—81no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—41no—
95C22×Dic53—81no—
96C4⋊C4×C53—41no—
97C22⋊C4×C53—24no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—42no—
101C42×C53—21no—
102C42×C53—21no—
103C42×C53—82no—
104C22⋊Dic53—21no—
105C22⋊Dic53—41no—
106C22⋊Dic53—81no—
107C22⋊Dic53C4212yescomplemented
108#11719921—24no—
109#11719921—42no—
110#11719921—41no—
111#11719921—41no—
112#11719921—81no—
113#11719921—81no—
114#11719919—81no—
115C4×Dic53—81no—
116#11719917—24no—
117#11719917—42no—
118C2×#88322—42no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53C2×C4211yescharacteristic
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53#11711yescharacteristic
130C23×C53C4×D412yes—
131C22×D53—22no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—42no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—84no—
143C22×D53—82no—
144C22×D53—82no—
145D4×C53—42no—
146D4×C53—42no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—44no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53—82no—
162C2×C4×C53—82no—
163C2×C4×C53#9911yescharacteristic
164C2×C4×C53#9911yescharacteristic
165C2×C4×C53C4×D412yes—
166C22⋊D53—84no—
167C2×Dic53—22no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—82no—
177#88322—44no—
178#12622—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—22no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—42no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—42no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—81no—
193C22×C53C4×C22⋊C411yescharacteristic
194C22×C53C2×#9911yescharacteristic
195C22×C53#41011yescharacteristic
196C2×D53—42no—
197C2×D53—42no—
198C2×D53—42no—
199C2×D53—81no—
200C2×D53—81no—
201C2×D53—84no—
202C2×D53—81no—
203C2×D53—81no—
204C2×D53—84no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—22no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53—84no—
212Dic53—42no—
213Dic53—81no—
214Dic53—81no—
215#2321—531no—
216C2×AΣL(1,𝔽16)—532no—
217C4×#100—531no—
218C4×#99D5311yescharacteristic, complemented
219#1390—532no—
220C2×C53—21no—
221C2×C53—21no—
222C2×C53—42no—
223C2×C53—41no—
224C2×C53—41no—
225C2×C53C4×#9911yescharacteristic
226C2×C53#212111yescharacteristic
227C2×C53#212211yescharacteristic
228D53—82no—
229C2×#120—531no—
230C2×C4×D4C2×D5311yescharacteristic
231C22×C22⋊C4—531no—
232C2×#100—532no—
233C2×#99C2×D5312yes—
234C4×M16—1061no—
235#390—1061no—
236#388—1061no—
237#388—1061no—
238#386—531no—
239#386—1061no—
240#385—531no—
241#379—531no—
242C4×C22⋊C4—21no—
243C4×C22⋊C4—531no—
244AΣL(1,𝔽16)—538no—
245#343—1061no—
246#342—21no—
247C53#1262211yescharacteristic, complemented, Sylow
248C25—531no—
249C22×D4—1061no—
250C22×D4C22×D5311yescharacteristic
251C23×C4—531no—
252C23×C4C4×D5311yescharacteristic
253C23×C4C22⋊D5311yescharacteristic
254C2×M16—1062no—
255#120—534no—
256#120—1062no—
257C4×D4—41no—
258C4×D4—41no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4D21211yescharacteristic
261C2×C22⋊C4—22no—
262C2×C22⋊C4—531no—
263C2×C22⋊C4—531no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4—1061no—
266C2×C22⋊C4—1061no—
267C2×C22⋊C4—1061no—
268C2×C22⋊C4—1061no—
269C2×C22⋊C4—1061no—
270C2×C22⋊C4—2121no—
271C2×C22⋊C4—2121no—
272C2×C22⋊C4C4×D5311yescharacteristic
273C2×C22⋊C4C22⋊D5311yescharacteristic
274C2×C42—41no—
275C2×C42—1061no—
276C2×C42D21211yescharacteristic
277#100—534no—
278#100—1062no—
279#99—22no—
280#99C4×D5314yes—
281#97—2121no—
282C4×C8—2121no—
283#95—41no—
284#95—1061no—
285#95—1061no—
286#95—1061no—
287C24—532no—
288C24—1061no—
289C24—1061no—
290C24—4241no—
291C24—4241no—
292C24—2121no—
293C24—2121no—
294C24C2×C4×D5311yescharacteristic
295C24C2×C22⋊D5311yescharacteristic
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—1064no—
299C2×D4—4241no—
300C2×D4—4241no—
301C2×D4C2×C4×D5312yes—
302C2×D4#1171994512yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—4241no—
314C22×C4—2121no—
315C22×C4#1171992111yescharacteristic
316C22×C4#1171992111yescharacteristic
317C22×C4#1171992111yescharacteristic
318C22×C4#1171992111yescharacteristic
319C22×C4C2×D21211yescharacteristic
320M16—1064no—
321M16—2122no—
322C2×C8—2122no—
323C4⋊C4—41no—
324C4⋊C4—4241no—
325C22⋊C4—24no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—42no—
329C22⋊C4—532no—
330C22⋊C4—1061no—
331C22⋊C4—1064no—
332C22⋊C4—4241no—
333C22⋊C4—4241no—
334C22⋊C4—4241no—
335C22⋊C4—2122no—
336C22⋊C4—2121no—
337C22⋊C4—2121no—
338C22⋊C4—2121no—
339C42—21no—
340C42—21no—
341C42—82no—
342C42—4241no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—1062no—
352C23—4242no—
353C23—4242no—
354C23—4244no—
355C23—4242no—
356C23—4242no—
357C23—2122no—
358C23—2121no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2122no—
363C23—2121no—
364C23—2121no—
365C23C42×D5311yescharacteristic
366C23C2×#1171992111yescharacteristic
367C23#1313317411yescharacteristic
368C23C4×C22⋊D5312yes—
369D4—42no—
370D4—42no—
371D4—4244no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—1062no—
389C2×C4—4242no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4#1313313111yescharacteristic
399C2×C4#1313313111yescharacteristic
400C2×C4C4×D21212yes—
401C8—2124no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—4241no—
417C22—4241no—
418C22—4244no—
419C22—4244no—
420C22—4241no—
421C22—4241no—
422C22—2122no—
423C22—2122no—
424C22—2122no—
425C22C4×#1171992111yescharacteristic
426C22C2×#1313313111yescharacteristic
427C22#1681059611yescharacteristic
428C4—22no—
429C4—21no—
430C4—21no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—84no—
435C4—4241no—
436C4—4241no—
437C4—2122no—
438C2—21no—
439C2—21no—
440C2—42no—
441C2—41no—
442C2—41no—
443C2—4242no—
444C2C4×#1313313111yescharacteristic
445C2#2027635911yescharacteristic
446C2#2027636411yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radicalC2×#13133185C22
cosocleC22×D4×C53C23
hypercenterC4×#99D53
hypocenterC53#12622
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. C4×#13133134C2semidirect
4. #20272832C2semidirect
5. C4×#99×C53C2semidirect
7. C22×#11719921C4semidirect
15. #16810495C4semidirect
16. #16810478C4semidirect
17. C4×C22⋊Dic53C4semidirect
18. #16810409C4semidirect
71. C23×D53C42semidirect
107. C22⋊Dic53C42semidirect
218. C4×#99D53semidirect
247. C53#12622semidirect

Maximal subgroups

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

subgroupindex
#202773692
C2×#168104092
C4×#131331342
#202728322
C4×#99×C532
#1262253