Finite Group Databasebeta

← #28314256

Subgroups — #28314256

Order 13568.

1034 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291391C212yesmaximal, complemented
2#20289998C218yesmaximal, complemented
3#20289965C212yesmaximal, complemented
4#20289962C212yesmaximal
5C22×#13133212C211yesmaximal, characteristic
6#16811644C2214yescomplemented
7#16811643—22no—
8#16811640C2214yescomplemented
9#16811639—22no—
10#16811626—22no—
11#16811593C2214yes—
12C4×#11719940—22no—
13#16811469C2218yescomplemented
14#16811467C2218yes—
15C2×#13133212—22no—
16C2×#13133212—22no—
17C2×#13133212—22no—
18C2×#13133212—22no—
19C2×#13133212C414yes—
20C2×#13133212C2214yes—
21C22×C4⋊Dic53C411yescharacteristic
22C22×#11719919C2211yescharacteristic
23C22×#11719921C411yescharacteristic
24C22×#11719921C411yescharacteristic
25C22×#11719921C2211yescharacteristic
26C22×C4×Dic53C2211yescharacteristic
27C22×C4⋊C4×C53C411yescharacteristic
28C2×#11719940—24no—
29#13133284—24no—
30C4⋊#88324—24no—
31C22⋊#88324—24no—
32C22⋊#88324C2318yes—
33#13133272—24no—
34C4×#88324—24no—
35C24×D53C2×C411yescharacteristic
36C23×Dic53C2×C411yescharacteristic
37C23×Dic53C2×C411yescharacteristic
38C23×Dic53C2311yescharacteristic
39#13133212—24no—
40#13133212—24no—
41#13133212—24no—
42#13133212—24no—
43#13133212—24no—
44#13133212—24no—
45#13133212—24no—
46#13133212—24no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53C2×C414yes—
51C2×#11719919—22no—
52C2×#11719919—22no—
53C2×#11719919C2×C412yes—
54C2×#11719919C2312yes—
55C2×#11719921—22no—
56C2×#11719921—21no—
57C2×#11719921—21no—
58C2×#11719921—22no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×#11719921—22no—
62C2×#11719921—22no—
63C2×#11719921C2×C414yes—
64C2×#11719921C2×C414yes—
65C2×#11719921C2×C412yes—
66C2×#11719921C2312yes—
67C2×C4×Dic53C2×C412yes—
68C2×C4×Dic53D412yes—
69C2×C4×Dic53D412yes—
70C2×C4×Dic53D412yes—
71C2×C4×Dic53D412yes—
72C2×C4×Dic53C2312yes—
73C23×C4×C53C2×C411yescharacteristic
74C23×C4×C53C2×C411yescharacteristic
75C23×C4×C53C2×C411yescharacteristic
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53C2×C414yes—
80C24×C53C22×C411yescharacteristic
81C23×D53—21no—
82C23×D53—21no—
83C23×D53—42no—
84C23×D53—42no—
85C23×D53—42no—
86C23×D53C22×C412yes—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—42no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53C22⋊C412yes—
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C22×C412yes—
98C22×C4×C53C22×C412yes—
99C22×C4×C53C22×C412yes—
100C22×Dic53—21no—
101C22×Dic53—21no—
102C22×Dic53—42no—
103C22×Dic53—42no—
104C22×Dic53—42no—
105C22×Dic53C22⋊C412yes—
106C22×Dic53C22⋊C411yescharacteristic
107C22×Dic53C22⋊C411yescharacteristic
108C22×Dic53C22×C412yes—
109C22×Dic53C22×C412yes—
110C22×Dic53C22×C411yescharacteristic
111C22×Dic53C2×D412yes—
112C22×Dic53C2×D412yes—
113C22×Dic53C2411yescharacteristic
114#11719940—48no—
115C2×#88324—28no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53—24no—
118#11719921—24no—
119#11719921—24no—
120#11719921—24no—
121#11719921—24no—
122#11719921—24no—
123#11719921—24no—
124C4⋊Dic53—24no—
125C4⋊Dic53—24no—
126#11719919—24no—
127#11719919—24no—
128C4×Dic53C22⋊C412yes—
129C4×Dic53C22⋊C412yes—
130C4×Dic53C22⋊C412yes—
131C4×Dic53C22⋊C412yes—
132C4×Dic53C2×D412yes—
133C4×Dic53C2×D412yes—
134C4×Dic53C2×D412yes—
135C4×Dic53C2×D412yes—
136C23×C53—22no—
137C23×C53—22no—
138C23×C53—22no—
139C23×C53C2×C22⋊C411yescharacteristic
140C23×C53C2×C22⋊C411yescharacteristic
141C23×C53C23×C411yescharacteristic
142C22×D53—22no—
143C22×D53—24no—
144C22×D53—42no—
145C22×D53—42no—
146C22×D53—42no—
147C22×D53—42no—
148C22×D53—42no—
149C22×D53—42no—
150C22×D53—42no—
151C22×D53—42no—
152C22×D53—88no—
153C2×C4×C53—24no—
154C2×C4×C53—22no—
155C2×C4×C53—22no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53C2×C22⋊C412yes—
166C2×C4×C53C2×C22⋊C412yes—
167C2×Dic53—24no—
168C2×Dic53—24no—
169C2×Dic53—22no—
170C2×Dic53—24no—
171C2×Dic53—42no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—41no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53C2×C22⋊C412yes—
180C2×Dic53C2×C22⋊C412yes—
181C2×Dic53C2×C22⋊C412yes—
182C2×Dic53C22×D412yes—
183#88324—48no—
184#17398—531nomaximal, Sylow
185C22×C53—28no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53#41012yes—
193C22×C53#41312yes—
194C22×C53C22×C22⋊C411yescharacteristic
195C22×C53#56812yes—
196C2×D53—42no—
197C2×D53—42no—
198C2×D53—42no—
199C2×D53—84no—
200C2×D53—84no—
201C2×D53—84no—
202C4×C53—22no—
203C4×C53—22no—
204C4×C53—42no—
205C4×C53—42no—
206Dic53—22no—
207Dic53—22no—
208Dic53—22no—
209Dic53—22no—
210Dic53—42no—
211C22×#126—531no—
212#2920—532no—
213#1473—538no—
214#1463—532no—
215#1462—532no—
216C2×C53—22no—
217C2×C53—22no—
218C2×C53—22no—
219C2×C53#287911yescharacteristic
220C2×C53C2×#41011yescharacteristic
221C2×C53C2×#41311yescharacteristic
222D53—82no—
223C2×#126—534no—
224C2×#126—534no—
225C2×#126—1062no—
226C2×#126—1062no—
227C2×#126—1062no—
228C2×#126—1062no—
229C22×C4⋊C4—531no—
230C22×C4⋊C4—531no—
231C22×C4⋊C4#615711yescharacteristic
232C22×C22⋊C4—531no—
233C22×C22⋊C4—531no—
234C22×C22⋊C4—531no—
235C22×C42—531no—
236#432—534no—
237#432—1062no—
238#431—534no—
239#431—1062no—
240#423—1062no—
241#407—534no—
242C4×M16—1062no—
243#324—538no—
244#323—538no—
245C53#1739811yescharacteristic, complemented, Sylow
246C25—531no—
247C23×C4—531no—
248C23×C4—531no—
249C23×C4—531no—
250C23×C4C2×#615711yescharacteristic
251C23×C4C2×#615711yescharacteristic
252C23×C4C2×#615711yescharacteristic
253C2×M16—1064no—
254#126—1064no—
255#126—1064no—
256#126—1064no—
257#126—1064no—
258#126—1064no—
259#126—1064no—
260#126—1064no—
261#126—1064no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—532no—
266C2×C4⋊C4—532no—
267C2×C4⋊C4—534no—
268C2×C4⋊C4—1062no—
269C2×C4⋊C4—1062no—
270C2×C4⋊C4—1062no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4C2×#615714yes—
274C2×C22⋊C4—532no—
275C2×C22⋊C4—534no—
276C2×C22⋊C4—532no—
277C2×C22⋊C4—534no—
278C2×C22⋊C4—1062no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1061no—
281C2×C22⋊C4—1062no—
282C2×C22⋊C4—1062no—
283C2×C22⋊C4—1062no—
284C2×C22⋊C4—1061no—
285C2×C22⋊C4—1061no—
286C2×C42—532no—
287C2×C42—532no—
288C2×C42—532no—
289C2×C42—532no—
290C2×C42—532no—
291C2×C42—532no—
292C4⋊C8—1064no—
293C4⋊C8—1064no—
294C22⋊C8—538no—
295C22⋊C8—1064no—
296#97—1064no—
297C4×C8—1064no—
298C24—532no—
299C24—1061no—
300C24—1061no—
301C24—2122no—
302C24—2122no—
303C24—2122no—
304C24C22×#615711yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—532no—
313C22×C4—532no—
314C22×C4—532no—
315C22×C4—531no—
316C22×C4—531no—
317C22×C4—532no—
318C22×C4—531no—
319C22×C4—531no—
320C22×C4—532no—
321C22×C4—1061no—
322C22×C4—1061no—
323C22×C4—2122no—
324C22×C4—2122no—
325C22×C4—2122no—
326C22×C4C22⋊#615712yes—
327C22×C4C22⋊#615711yescharacteristic
328C22×C4C22⋊#615711yescharacteristic
329C22×C4C22×#615712yes—
330C22×C4C22×#615712yes—
331C22×C4C22×#615712yes—
332M16—2128no—
333C2×C8—1068no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—1064no—
337C4⋊C4—1064no—
338C4⋊C4—1064no—
339C4⋊C4—1064no—
340C22⋊C4—1064no—
341C22⋊C4—1064no—
342C22⋊C4—1064no—
343C22⋊C4—1064no—
344C22⋊C4—1064no—
345C22⋊C4—1064no—
346C42—532no—
347C42—532no—
348C42—532no—
349C42—532no—
350C42—532no—
351C42—532no—
352C42—532no—
353C42—532no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—1064no—
358C23—1062no—
359C23—4248no—
360C23—2122no—
361C23—2122no—
362C23—2122no—
363C23—2122no—
364C23—2122no—
365C23—2122no—
366C23—2122no—
367C23—2122no—
368C23C2×C22⋊#615711yescharacteristic
369C23C2×C22⋊#615711yescharacteristic
370C23C23×#615711yescharacteristic
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—532no—
384C2×C4—532no—
385C2×C4—532no—
386C2×C4—532no—
387C2×C4—1064no—
388C2×C4—1064no—
389C2×C4—1062no—
390C2×C4—1064no—
391C2×C4—2122no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2122no—
396C2×C4—2121no—
397C2×C4—2122no—
398C2×C4—2122no—
399C2×C4C2×C22⋊#615712yes—
400C2×C4C2×C22⋊#615712yes—
401C8—2128no—
402C22—28no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—4244no—
410C22—4244no—
411C22—4244no—
412C22—2122no—
413C22—2122no—
414C22—2122no—
415C22#1681160212yes—
416C22#1681161012yes—
417C22C22×C22⋊#615711yescharacteristic
418C22#1681170312yes—
419C4—22no—
420C4—22no—
421C4—42no—
422C4—42no—
423C4—1062no—
424C4—1062no—
425C4—1062no—
426C4—1062no—
427C4—2122no—
428C2—22no—
429C2—22no—
430C2—22no—
431C2—4242no—
432C2#2029130211yescharacteristic
433C2C2×#1681160211yescharacteristic
434C2C2×#1681161011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC22×Dic53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53#17398
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20291391C2semidirect
2. #20289998C2semidirect
3. #20289965C2semidirect
6. #16811644C22semidirect
8. #16811640C22semidirect
13. #16811469C22semidirect
245. C53#17398semidirect

Maximal subgroups

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

subgroupindex
#202913912
#202899982
#202899652
#202899622
C22×#131332122
#1739853