Finite Group Databasebeta

← #28301535

Subgroups — #28301535

Order 13568.

914 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290441C212yesmaximal, complemented
2C2×#16811466C212yesmaximal
3#20289553C212yesmaximal, complemented
4C2×C22⋊C4×Dic53C211yesmaximal, characteristic
5C2×C22⋊#88324C2214yes—
6C2×#13133272—22no—
7#16811489—22no—
8#16811466C418yescomplemented
9C24×Dic53C411yescharacteristic
10C22×C22⋊Dic53C2211yescharacteristic
11C22×C4×Dic53C411yescharacteristic
12C22×C4×Dic53C2211yescharacteristic
13C22⋊C4×Dic53—22no—
14C22⋊C4×Dic53—22no—
15C2×#13133124C411yescharacteristic
16C2×#13133124C2211yescharacteristic
17C22×C22⋊C4×C53C411yescharacteristic
18C22×#88324—24no—
19C22⋊#88324—24no—
20C22⋊#88324C2×C418yescomplemented
21#13133272—44no—
22C23×Dic53—22no—
23C23×Dic53—21no—
24C23×Dic53—21no—
25C23×Dic53—22no—
26C23×Dic53—42no—
27C23×Dic53—42no—
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53C2×C411yescharacteristic
30C23×Dic53C2×C411yescharacteristic
31C23×Dic53C2311yescharacteristic
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53C2×C414yes—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—42no—
38C2×C4×Dic53—42no—
39C2×C4×Dic53D412yes—
40C2×C4×Dic53D412yes—
41#13133124—22no—
42#13133124—22no—
43C25×C53C2×C411yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—22no—
49C24×C53—21no—
50C24×C53—21no—
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—41no—
54C24×C53—41no—
55C24×C53—41no—
56C24×C53—41no—
57C24×C53C4212yes—
58C24×C53M1612yes—
59C24×C53M1612yes—
60C24×C53C22×C411yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C22⋊C412yes—
70C22×C4×C53M1612yes—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—24no—
76C22×Dic53—24no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—48no—
84C22×Dic53—48no—
85C22×Dic53—42no—
86C22×Dic53—42no—
87C22×Dic53—42no—
88C22×Dic53—42no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53C4212yes—
94C22×Dic53C22⋊C411yescharacteristic
95C22×Dic53C22⋊C411yescharacteristic
96C22×Dic53M1612yes—
97C22×Dic53C22×C412yes—
98C22×Dic53C2×D412yes—
99C22×Dic53#4112yes—
100C2×#88324—28no—
101C2×#88324—44no—
102C2×#88324—44no—
103C22⋊C4×C53—28no—
104C22⋊Dic53—28no—
105C4×Dic53—24no—
106C4×Dic53—42no—
107C4×Dic53—42no—
108C4×Dic53—42no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—22no—
118C23×C53—22no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53#9714yescomplemented
136C23×C53C2×C4211yescharacteristic
137C23×C53C2×C22⋊C411yescharacteristic
138C23×C53#11711yescharacteristic
139C23×C53C2×M1611yescharacteristic
140C23×C53C2×M1611yescharacteristic
141C23×C53C2×M1611yescharacteristic
142C23×C53C2×M1611yescharacteristic
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×Dic53—24no—
154C2×Dic53—24no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—22no—
158C2×Dic53—21no—
159C2×Dic53—22no—
160C2×Dic53—21no—
161C2×Dic53—22no—
162C2×Dic53—22no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—48no—
166C2×Dic53—48no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53#9912yes—
176C2×Dic53#10012yes—
177C2×Dic53C4×D414yes—
178#88324—48no—
179#8152—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—24no—
193C22×C53—44no—
194C22×C53—44no—
195C22×C53—44no—
196C22×C53—44no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—42no—
200C22×C53—42no—
201C22×C53—42no—
202C22×C53—42no—
203C22×C53C4×C22⋊C411yescharacteristic
204C22×C53C2×#9711yescharacteristic
205C22×C53C4×M1611yescharacteristic
206C22×C53#40711yescharacteristic
207C22×C53#40711yescharacteristic
208C22×C53C2×#9911yescharacteristic
209C22×C53#41011yescharacteristic
210C22×C53C2×#10011yescharacteristic
211C22×C53#41311yescharacteristic
212C22×C53#43211yescharacteristic
213C22×C53#43211yescharacteristic
214C4×C53—42no—
215C4×C53—42no—
216Dic53—22no—
217Dic53—22no—
218Dic53—24no—
219Dic53—42no—
220Dic53—42no—
221C2×C4×C22⋊C4—531no—
222#1749—532no—
223C2×#323—532no—
224#1280—532no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—21no—
229C2×C53—44no—
230C2×C53#146211yescharacteristic
231C2×C53#146411yescharacteristic
232C2×C53#146511yescharacteristic
233C2×C53#146611yescharacteristic
234C2×C53#174911yescharacteristic
235C2×C53C4×#9911yescharacteristic
236C2×C53C4×#10011yescharacteristic
237C24×C4—531no—
238C22×C22⋊C4—531no—
239C22×C22⋊C4#615711yescharacteristic
240C22×C42—531no—
241C22×C42—531no—
242C2×C22⋊C8—534no—
243C2×#97—1062no—
244C4×C22⋊C4—1062no—
245C4×C22⋊C4—1062no—
246C2×#95—531no—
247C2×#95—531no—
248#336—1062no—
249#323—538no—
250C53#815211yescharacteristic, complemented, Sylow
251C25C2×#615711yescharacteristic
252C23×C4—531no—
253C23×C4—531no—
254C23×C4—531no—
255C23×C4—531no—
256C23×C4—1061no—
257C23×C4—1061no—
258C23×C4—1062no—
259C23×C4—1062no—
260C23×C4—2122no—
261C23×C4—2122no—
262C23×C4C2×#615711yescharacteristic
263C23×C4C2×#615711yescharacteristic
264C22×C8—1064no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—534no—
269C2×C22⋊C4—1062no—
270C2×C22⋊C4—1062no—
271C2×C42—532no—
272C2×C42—532no—
273C2×C42—1061no—
274C2×C42—1061no—
275C2×C42—2122no—
276C2×C42—2122no—
277C22⋊C8—538no—
278C22⋊C8—1064no—
279#97—2124no—
280#95—1062no—
281#95—1062no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24#1171993612yes—
291C24C4×#615712yes—
292C24#1171994012yes—
293C24C22×#615711yescharacteristic
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—531no—
303C22×C4—531no—
304C22×C4—532no—
305C22×C4—532no—
306C22×C4—532no—
307C22×C4—532no—
308C22×C4—532no—
309C22×C4—1062no—
310C22×C4—1062no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—1064no—
314C22×C4—1064no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—2122no—
320C22×C4—2122no—
321C22×C4—2121no—
322C22×C4—2121no—
323C22×C4—2128no—
324C22×C4—2128no—
325C22×C4—2122no—
326C22×C4—2122no—
327C22×C4—2121no—
328C22×C4—2121no—
329C22×C4—2121no—
330C22×C4—2121no—
331C22×C4#1171994012yes—
332C22×C4C22⋊#615712yes—
333C2×C8—1068no—
334C2×C8—2124no—
335C2×C8—2124no—
336C22⋊C4—28no—
337C22⋊C4—1068no—
338C42—1064no—
339C42—2122no—
340C42—2122no—
341C42—2122no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23#1313327214yescomplemented
369C23C2×C4×#615711yescharacteristic
370C23C2×C22⋊#615711yescharacteristic
371C23#1313329611yescharacteristic
372C23C2×#1171994011yescharacteristic
373C23C2×#1171994011yescharacteristic
374C23D53⋊M1611yescharacteristic
375C23C2×#1171993611yescharacteristic
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—534no—
387C2×C4—532no—
388C2×C4—532no—
389C2×C4—1064no—
390C2×C4—1064no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1062no—
394C2×C4—1061no—
395C2×C4—1062no—
396C2×C4—1061no—
397C2×C4—1062no—
398C2×C4—1062no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2128no—
402C2×C4—2128no—
403C2×C4—2122no—
404C2×C4—2122no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2122no—
410C2×C4—2122no—
411C8—2128no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—24no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22C4×C22⋊#615711yescharacteristic
436C22C2×#1313327211yescharacteristic
437C22C4×#1171994011yescharacteristic
438C22#1681159311yescharacteristic
439C22#1681159411yescharacteristic
440C22D53⋊#9911yescharacteristic
441C22#1681160311yescharacteristic
442C22D53⋊#10011yescharacteristic
443C22#1681161011yescharacteristic
444C22#1681164311yescharacteristic
445C22#1681164411yescharacteristic
446C4—42no—
447C4—42no—
448C4—1064no—
449C4—1062no—
450C4—1062no—
451C4—2122no—
452C4—2122no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—44no—
458C2#2028996111yescharacteristic
459C2#2028996711yescharacteristic
460C2#2028997111yescharacteristic
461C2#2028997511yescharacteristic
462C2#2029044111yescharacteristic
463C2#2029046211yescharacteristic
464C2#2029046511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#8152
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20290441C2semidirect
3. #20289553C2semidirect
8. #16811466C4semidirect
20. C22⋊#88324C2×C4semidirect
135. C23×C53#97semidirect
250. C53#8152semidirect
368. C23#13133272semidirect

Maximal subgroups

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

subgroupindex
#202904412
C2×#168114662
#202895532
C2×C22⋊C4×Dic532
#815253