Finite Group Databasebeta

← #27299556

Subgroups — #27299556

Order 13568.

1220 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#16810413C214yesmaximal, complemented
2#20274636C211yesmaximal, characteristic
3#20274630C211yesmaximal, characteristic
4C22×#123×C53C211yesmaximal, characteristic
5C2×#13133134—21no—
6C2×#13133134C2212yes—
7C2×#13133132—21no—
8C2×#13133132C2212yes—
9#16810587—21no—
10#16810479—21no—
11#16810413—28no—
12C23×D4×C53C2211yescharacteristic
13C24×C4×C53C411yescharacteristic, complemented
14C2×#123×C53—21no—
15C2×#123×C53—21no—
16C2×#123×C53—28no—
17C2×#123×C53C412yescomplemented
18C2×#123×C53C2212yes—
19C22×C4⋊C4×C53—21no—
20C22×C22⋊C4×C53—21no—
21C22×C22⋊C4×C53C411yescharacteristic, complemented
22C2×#11719917—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—42no—
25#13133134—24no—
26#13133134—42no—
27#13133132—24no—
28#13133132—42no—
29C22⋊#88322—42no—
30C25×C53C2×C411yescharacteristic
31C25×C53D411yescharacteristic
32C22×D4×C53—24no—
33C22×D4×C53—42no—
34C22×D4×C53—42no—
35C22×D4×C53C2×C411yescharacteristic
36C22×D4×C53D411yescharacteristic
37C22×D4×C53D411yescharacteristic
38C22×D4×C53C2311yescharacteristic
39C23×C4×C53—21no—
40C23×C4×C53—21no—
41C23×C4×C53—21no—
42C23×C4×C53—24no—
43C23×C4×C53—42no—
44C23×C4×C53—42no—
45C23×C4×C53C2×C412yes—
46C23×C4×C53D411yescharacteristic
47#123×C53—28no—
48#123×C53—28no—
49#123×C53—24no—
50#123×C53—24no—
51#123×C53—24no—
52#123×C53D418yes—
53C2×C4⋊C4×C53—28no—
54C2×C4⋊C4×C53—22no—
55C2×C4⋊C4×C53—22no—
56C2×C22⋊C4×C53—28no—
57C2×C22⋊C4×C53—24no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—22no—
61C2×C22⋊C4×C53C2×C412yes—
62C24×C53—22no—
63C24×C53—22no—
64C24×C53—22no—
65C24×C53—22no—
66C24×C53—22no—
67C24×C53—41no—
68C24×C53—44no—
69C24×C53—44no—
70C24×C53C22⋊C411yescharacteristic
71C24×C53C22⋊C411yescharacteristic
72C24×C53C22⋊C411yescharacteristic
73C24×C53C22×C411yescharacteristic
74C24×C53C2×D411yescharacteristic
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53—22no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—416no—
87C2×D4×C53C2×D414yes—
88C22×C4×C53—24no—
89C22×C4×C53—24no—
90C22×C4×C53—24no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—41no—
97C22×C4×C53—44no—
98C22×C4×C53—416no—
99C22×C4×C53—42no—
100C22×C4×C53—42no—
101C22×C4×C53—42no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53C22⋊C411yescharacteristic
106C22×C4×C53C22⋊C414yescomplemented
107C22×C4×C53C2×D411yescharacteristic
108C22×Dic53—42no—
109C4⋊C4×C53—28no—
110C4⋊C4×C53—28no—
111C22⋊C4×C53—28no—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—28no—
115C22⋊C4×C53C22⋊C414yescomplemented
116C22⋊Dic53—42no—
117C22⋊Dic53—42no—
118C22⋊Dic53—84no—
119#11719917—42no—
120#11719917—42no—
121C2×#88322—42no—
122C23×C53—24no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—28no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—44no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—48no—
139C23×C53—48no—
140C23×C53—44no—
141C23×C53—42no—
142C23×C53—42no—
143C23×C53—48no—
144C23×C53—44no—
145C23×C53—44no—
146C23×C53—82no—
147C23×C53—84no—
148C23×C53#9912yes—
149C23×C53#9912yes—
150C23×C53C2×C22⋊C411yescharacteristic
151C23×C53C2×C22⋊C412yes—
152C23×C53#12012yes—
153D4×C53—44no—
154D4×C53—44no—
155D4×C53—44no—
156D4×C53—44no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—24no—
160C2×C4×C53—24no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—44no—
165C2×C4×C53—48no—
166C2×C4×C53—44no—
167C2×C4×C53—48no—
168C2×C4×C53—48no—
169C2×C4×C53—44no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—44no—
173C2×C4×C53#12012yes—
174C2×Dic53—82no—
175C2×Dic53—84no—
176#88322—82no—
177#12688—531nomaximal, Sylow
178C22×C53—24no—
179C22×C53—24no—
180C22×C53—22no—
181C22×C53—22no—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—42no—
190C22×C53—416no—
191C22×C53—42no—
192C22×C53—42no—
193C22×C53—42no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—42no—
197C22×C53—41no—
198C22×C53—81no—
199C22×C53—88no—
200C22×C53—81no—
201C22×C53#35214yescomplemented
202C22×C53#37911yescharacteristic
203C22×C53C2×#9911yescharacteristic
204C22×C53C2×#9911yescharacteristic
205C4×C53—22no—
206C4×C53—44no—
207C4×C53—41no—
208C4×C53—44no—
209Dic53—162no—
210C22×#123D5311yescharacteristic
211C2×#352—534no—
212#1783—531no—
213#1781—531no—
214C2×C53—21no—
215C2×C53—24no—
216C2×C53—21no—
217C2×C53—42no—
218C2×C53—42no—
219C2×C53—42no—
220C2×C53—81no—
221C2×C53#178111yescharacteristic
222C2×C53C2×#35212yes—
223C23×D4C2×D5311yescharacteristic
224C24×C4Dic5311yescharacteristic, complemented
225C2×#123—21no—
226C2×#123—21no—
227C2×#123—28no—
228C2×#123Dic5312yescomplemented
229C2×#123C2×D5312yes—
230C22×C4⋊C4—21no—
231C22×C22⋊C4—21no—
232C22×C22⋊C4Dic5311yescharacteristic, complemented
233C2×#100—532no—
234C2×#100—1061no—
235C2×#99—532no—
236C2×#99—1061no—
237#407—1061no—
238#379—1061no—
239#352—1068no—
240C53#1268811yescharacteristic, complemented, Sylow
241C25C2×Dic5311yescharacteristic
242C25C22⋊D5311yescharacteristic
243C22×D4—24no—
244C22×D4—42no—
245C22×D4—42no—
246C22×D4C2×Dic5311yescharacteristic
247C22×D4C22⋊D5311yescharacteristic
248C22×D4C22⋊D5311yescharacteristic
249C22×D4C22×D5311yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—24no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4C2×Dic5312yes—
257C23×C4C22⋊D5311yescharacteristic
258C2×M16—1062no—
259#123—28no—
260#123—28no—
261#123—24no—
262#123—24no—
263#123—24no—
264#123C22⋊D5318yes—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—1062no—
274C2×C22⋊C4—2122no—
275C2×C22⋊C4C2×Dic5312yes—
276#100—1064no—
277#100—2122no—
278#99—1064no—
279#99—2122no—
280C22⋊C8—2122no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—41no—
287C24—44no—
288C24—44no—
289C24C22⋊Dic5311yescharacteristic
290C24C22⋊Dic5311yescharacteristic
291C24C22⋊Dic5311yescharacteristic
292C24C22×Dic5311yescharacteristic
293C24C2×C22⋊D5311yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—416no—
306C2×D4C2×C22⋊D5314yes—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—44no—
317C22×C4—416no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—2122no—
325C22×C4C22⋊Dic5311yescharacteristic
326C22×C4C22⋊Dic5314yescomplemented
327C22×C4C2×C22⋊D5311yescharacteristic
328M16—2122no—
329M16—2122no—
330C2×C8—2122no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C22⋊C4—24no—
334C22⋊C4—28no—
335C22⋊C4—22no—
336C22⋊C4—28no—
337C22⋊C4—4244no—
338C22⋊C4—2122no—
339C22⋊C4—2122no—
340C22⋊C4C22⋊Dic5314yescomplemented
341C23—21no—
342C23—24no—
343C23—21no—
344C23—28no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—48no—
358C23—48no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—48no—
363C23—44no—
364C23—44no—
365C23—82no—
366C23—84no—
367C23#1313313212yes—
368C23#1313313212yes—
369C23C2×C22⋊Dic5311yescharacteristic
370C23C2×C22⋊Dic5312yes—
371C23#1313318612yes—
372D4—44no—
373D4—44no—
374D4—44no—
375D4—44no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—4242no—
393C2×C4—4244no—
394C2×C4#1313318612yes—
395C8—4242no—
396C22—24no—
397C22—24no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—416no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—81no—
417C22—88no—
418C22—81no—
419C22#1681041314yescomplemented
420C22#1681047911yescharacteristic
421C22C2×#1313313211yescharacteristic
422C22C2×#1313313211yescharacteristic
423C4—22no—
424C4—44no—
425C4—41no—
426C4—44no—
427C4—8482no—
428C2—21no—
429C2—21no—
430C2—24no—
431C2—42no—
432C2—42no—
433C2—42no—
434C2—81no—
435C2#2027463011yescharacteristic
436C2C2×#1681041312yes—
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC22×#123×C53C2
socleC22×C53C2×#99
prime radicalC23×D4×C53C22
cosocleC22×D4×C53C23
hypercenterC22×#123D53
hypocenterC53#12688
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16810413C2semidirect
13. C24×C4×C53C4semidirect
17. C2×#123×C53C4semidirect
21. C22×C22⋊C4×C53C4semidirect
106. C22×C4×C53C22⋊C4semidirect
115. C22⋊C4×C53C22⋊C4semidirect
201. C22×C53#352semidirect
224. C24×C4Dic53semidirect
228. C2×#123Dic53semidirect
232. C22×C22⋊C4Dic53semidirect
240. C53#12688semidirect
326. C22×C4C22⋊Dic53semidirect
340. C22⋊C4C22⋊Dic53semidirect
419. C22#16810413semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
#202746362
#202746302
C22×#123×C532
#1268853