Finite Group Databasebeta

← #27299239

Subgroups — #27299239

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810410C212yesmaximal
2C4×#13133134C211yesmaximal, characteristic, complemented
3C4×#13133132C211yesmaximal, characteristic, complemented
4#20272833C212yesmaximal, complemented
5C53×#2321C211yesmaximal, characteristic
6C2×#13133134C2212yes—
7C2×#13133132C2212yes—
8C4×#11719917—21no—
9C4×C22⋊Dic53—21no—
10#16810410C418yescomplemented
11#16810392—21no—
12#16810390—21no—
13C2×#120×C53C2211yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C22×C22⋊C4×C53C411yescharacteristic, complemented
16C53×#390—21no—
17C53×#388—21no—
18C53×#388—21no—
19C53×#386—21no—
20C53×#386C2211yescharacteristic
21C53×#385C411yescharacteristic, complemented
22C53×#379C411yescharacteristic, complemented
23C4×C22⋊C4×C53C411yescharacteristic, complemented
24C2×#11719917—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—41no—
27#13133134—22no—
28#13133134C2×C414yes—
29#13133132—22no—
30#13133132C2×C414yes—
31#13133128—41no—
32C4×#88322—41no—
33#13133124—41no—
34C25×C53C2×C411yescharacteristic
35C22×D4×C53—21no—
36C22×D4×C53C2311yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40#120×C53—22no—
41#120×C53C2×C414yes—
42C4×D4×C53—41no—
43C4×D4×C53—41no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—41no—
53C2×C22⋊C4×C53—41no—
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53—21no—
59C2×C42×C53D411yescharacteristic
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63C24×C53—21no—
64C24×C53—21no—
65C24×C53—41no—
66C24×C53—41no—
67C24×C53—81no—
68C24×C53—81no—
69C24×C53C4212yescomplemented
70C24×C53C22×C411yescharacteristic
71C24×C53C2×D411yescharacteristic
72C2×D4×C53—24no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—81no—
76C2×D4×C53—81no—
77C2×D4×C53C22×C412yes—
78C2×D4×C53#4112yes—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—41no—
87C22×C4×C53—81no—
88C22×C4×C53—81no—
89C22×C4×C53—81no—
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×Dic53—42no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—81no—
98C22⋊C4×C53—21no—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—81no—
107C22⋊C4×C53—81no—
108C22⋊C4×C53—81no—
109C22⋊C4×C53C4212yescomplemented
110C42×C53—21no—
111C42×C53—21no—
112C42×C53—81no—
113C22⋊Dic53—24no—
114C22⋊Dic53—42no—
115C4×Dic53—82no—
116#11719917—24no—
117#11719917—42no—
118C2×#88322—42no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—22no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—42no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—42no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—82no—
137C23×C53—82no—
138C23×C53—82no—
139C23×C53—82no—
140C23×C53—84no—
141C23×C53C2×C4211yescharacteristic
142C23×C53C2×C22⋊C411yescharacteristic
143C23×C53#11711yescharacteristic
144C23×C53C4×D412yes—
145D4×C53—42no—
146D4×C53—42no—
147D4×C53—84no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—22no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×C4×C53—81no—
170C2×C4×C53—82no—
171C2×C4×C53#9911yescharacteristic
172C2×C4×C53#9911yescharacteristic
173C2×C4×C53C4×D412yes—
174C2×Dic53—44no—
175C2×Dic53—82no—
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—42no—
189C22×C53—42no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—42no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—84no—
199C22×C53—84no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53C4×C22⋊C411yescharacteristic
203C22×C53C2×#9911yescharacteristic
204C22×C53#41011yescharacteristic
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—22no—
208C4×C53—42no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53—81no—
213C4×C53—81no—
214Dic53—84no—
215#2321D5311yescharacteristic
216C2×AΣL(1,𝔽16)—532no—
217C4×#100—531no—
218C4×#99—531no—
219#1390—532no—
220C2×C53—21no—
221C2×C53—21no—
222C2×C53—42no—
223C2×C53—41no—
224C2×C53—41no—
225C2×C53—82no—
226C2×C53C4×#9911yescharacteristic
227C2×C53#212111yescharacteristic
228C2×C53#212211yescharacteristic
229C2×#120C2×D5311yescharacteristic
230C2×C4×D4C2×D5311yescharacteristic
231C22×C22⋊C4Dic5311yescharacteristic, complemented
232C2×#100—532no—
233C2×#99—532no—
234C4×M16—1061no—
235#390—21no—
236#388—21no—
237#388—21no—
238#386—21no—
239#386C2×D5311yescharacteristic
240#385Dic5311yescharacteristic, complemented
241#379Dic5311yescharacteristic, complemented
242C4×C22⋊C4—1061no—
243C4×C22⋊C4Dic5311yescharacteristic, complemented
244AΣL(1,𝔽16)—538no—
245#343—1061no—
246#342—1061no—
247C53#1262211yescharacteristic, complemented, Sylow
248C25C2×Dic5311yescharacteristic
249C22×D4—21no—
250C22×D4C22×D5311yescharacteristic
251C23×C4C2×Dic5311yescharacteristic
252C23×C4C2×Dic5311yescharacteristic
253C23×C4C22⋊D5311yescharacteristic
254C2×M16—1062no—
255#120—22no—
256#120C4×D5314yes—
257C4×D4—41no—
258C4×D4—41no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D5311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—1062no—
270C2×C22⋊C4C2×Dic5311yescharacteristic
271C2×C22⋊C4C2×Dic5311yescharacteristic
272C2×C22⋊C4C2×Dic5311yescharacteristic
273C2×C22⋊C4C22⋊D5311yescharacteristic
274C2×C42—21no—
275C2×C42—2121no—
276C2×C42C22⋊D5311yescharacteristic
277#100—534no—
278#100—1062no—
279#99—534no—
280#99—1062no—
281#97—2121no—
282C4×C8—2121no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—2121no—
287C24—21no—
288C24—21no—
289C24—41no—
290C24—41no—
291C24—81no—
292C24—81no—
293C24C4×Dic5312yescomplemented
294C24C22×Dic5311yescharacteristic
295C24C2×C22⋊D5311yescharacteristic
296C2×D4—24no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4C2×C4×D5312yes—
302C2×D4#1171994512yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—2122no—
315C22×C4C22⋊Dic5311yescharacteristic
316C22×C4C22⋊Dic5311yescharacteristic
317C22×C4C22⋊Dic5311yescharacteristic
318C22×C4C22⋊Dic5311yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320M16—1064no—
321M16—2122no—
322C2×C8—2122no—
323C4⋊C4—41no—
324C4⋊C4—81no—
325C22⋊C4—21no—
326C22⋊C4—24no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—81no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C22⋊C4—1064no—
337C22⋊C4—2122no—
338C22⋊C4C4×Dic5312yescomplemented
339C42—21no—
340C42—21no—
341C42—81no—
342C42—4242no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—84no—
365C23C2×C4×Dic5311yescharacteristic
366C23C2×C22⋊Dic5311yescharacteristic
367C23#1313317611yescharacteristic
368C23C4×C22⋊D5312yes—
369D4—42no—
370D4—42no—
371D4—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—4242no—
396C2×C4—4242no—
397C2×C4—2124no—
398C2×C4#1313313211yescharacteristic
399C2×C4#1313313211yescharacteristic
400C2×C4C4×C22⋊D5312yes—
401C8—2124no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—84no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22C4×C22⋊Dic5311yescharacteristic
426C22C2×#1313313211yescharacteristic
427C22#1681059711yescharacteristic
428C4—22no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—81no—
437C4—4244no—
438C2—21no—
439C2—21no—
440C2—42no—
441C2—41no—
442C2—41no—
443C2—82no—
444C2C4×#1313313211yescharacteristic
445C2#2027636111yescharacteristic
446C2#2027636511yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC53×#2321C2
socleC22×C53C2×#99
prime radicalC2×#120×C53C22
cosocleC22×D4×C53C23
hypercenter#2321D53
hypocenterC53#12622
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2321

Complements

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

NH ≅ G/Ntype
2. C4×#13133134C2semidirect
3. C4×#13133132C2semidirect
4. #20272833C2semidirect
10. #16810410C4semidirect
15. C22×C22⋊C4×C53C4semidirect
21. C53×#385C4semidirect
22. C53×#379C4semidirect
23. C4×C22⋊C4×C53C4semidirect
69. C24×C53C42semidirect
109. C22⋊C4×C53C42semidirect
231. C22×C22⋊C4Dic53semidirect
240. #385Dic53semidirect
241. #379Dic53semidirect
243. C4×C22⋊C4Dic53semidirect
247. C53#12622semidirect
293. C24C4×Dic53semidirect
338. C22⋊C4C4×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104102
C4×#131331342
C4×#131331322
#202728332
C53×#23212
#1262253