Finite Group Databasebeta

← #27275735

Subgroups — #27275735

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133134C211yesmaximal, characteristic
2#20273317C212yesmaximal, complemented
3#20273162C212yesmaximal, complemented
4#20273056C211yesmaximal, characteristic
5C2×C53×#552C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×#13133134—21no—
8C4×#11719917C2212yes—
9#16810392—21no—
10#16810365—24no—
11Q8⋊#88322—24no—
12#16810347—21no—
13#16810347—21no—
14C53×#552—24no—
15C53×#552—24no—
16C2×#126×C53—21no—
17C2×#125×C53—21no—
18C2×#125×C53C2211yescharacteristic
19C2×#123×C53—21no—
20C2×C22⋊Q8×C53C411yescharacteristic
21C2×C22⋊Q8×C53C411yescharacteristic
22C2×C4×Q8×C53C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C2×#117×C53—21no—
25C2×#117×C53C411yescharacteristic
26C22×C4⋊C4×C53C411yescharacteristic
27C2×#11719917—22no—
28C4⋊#88322—22no—
29#13133134—24no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128—22no—
33C4×#88322—22no—
34C22×Q8×C53C2×C411yescharacteristic
35C22×D4×C53D411yescharacteristic
36C23×C4×C53—21no—
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39#126×C53—28no—
40#125×C53—28no—
41#125×C53—22no—
42#125×C53D414yes—
43#123×C53—28no—
44C22⋊Q8×C53—22no—
45C22⋊Q8×C53—22no—
46C22⋊Q8×C53—22no—
47C22⋊Q8×C53—22no—
48C4×Q8×C53—22no—
49C4×Q8×C53D414yes—
50C4×D4×C53—22no—
51C4×D4×C53—22no—
52#117×C53—28no—
53#117×C53—22no—
54#117×C53—22no—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53C2×C411yescharacteristic
64C2×C4⋊C4×C53C2×C411yescharacteristic
65C2×C4⋊C4×C53C2×C411yescharacteristic
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53D411yescharacteristic
72C2×C42×C53—21no—
73C2×C42×C53—21no—
74C2×C42×C53C2311yescharacteristic
75C24×C53C22⋊C411yescharacteristic
76C24×C53C22⋊C411yescharacteristic
77C2×Q8×C53—21no—
78C2×Q8×C53—42no—
79C2×Q8×C53C22⋊C412yes—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—41no—
83C22×C4×C53—24no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C22⋊C411yescharacteristic
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C4⋊C4×C53—24no—
102C4⋊C4×C53—24no—
103C4⋊C4×C53—24no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53C22⋊C412yes—
112C4⋊C4×C53C22⋊C412yes—
113C4⋊C4×C53C22⋊C412yes—
114C22⋊C4×C53—24no—
115C22⋊C4×C53—24no—
116C22⋊C4×C53—24no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—22no—
119C42×C53—24no—
120C42×C53—24no—
121C42×C53C2×D412yes—
122C42×C53C2×D412yes—
123#11719917—24no—
124#11719917—42no—
125C2×#88322—42no—
126C2×#88322—42no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53#10412yes—
137C23×C53C2×C22⋊C411yescharacteristic
138Q8×C53—24no—
139Q8×C53—42no—
140D4×C53—44no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—48no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53C2×C22⋊C411yescharacteristic
165C2×C4×C53C2×C22⋊C411yescharacteristic
166C2×C4×C53#12012yes—
167C2×C4×C53#12012yes—
168#88322—44no—
169#88322—82no—
170#8371—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—22no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#37911yescharacteristic
184C22×C53C2×#10411yescharacteristic
185C22×C53#42111yescharacteristic
186C4×C53—22no—
187C4×C53—21no—
188C4×C53—22no—
189C4×C53—42no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—42no—
193C4×C53—42no—
194C4×C53—42no—
195C4×C53#41312yes—
196C2×#552D5311yescharacteristic
197C4×#100—531no—
198#1528—532no—
199#1491—532no—
200#1466—531no—
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—42no—
204C2×C53—81no—
205C2×C53#178511yescharacteristic
206C2×C53#178911yescharacteristic
207C2×C53#179611yescharacteristic
208#552—24no—
209#552—24no—
210C2×#126—21no—
211C2×#125—21no—
212C2×#125C2×D5311yescharacteristic
213C2×#123—21no—
214C2×C22⋊Q8Dic5311yescharacteristic
215C2×C22⋊Q8Dic5311yescharacteristic
216C2×C4×Q8C2×D5311yescharacteristic
217C2×C4×D4C2×D5311yescharacteristic
218C2×#117—21no—
219C2×#117Dic5311yescharacteristic
220C22×C4⋊C4Dic5311yescharacteristic
221#432—532no—
222C2×#100—1061no—
223C4×M16—532no—
224#343—1061no—
225#330—1064no—
226Q8⋊C8—1064no—
227#323—1061no—
228#323—1061no—
229C53#837111yescharacteristic, complemented, Sylow
230C22×Q8C2×Dic5311yescharacteristic
231C22×D4C22⋊D5311yescharacteristic
232C23×C4—21no—
233C23×C4C2×Dic5311yescharacteristic
234C23×C4C2×Dic5311yescharacteristic
235C2×M16—1062no—
236#126—28no—
237#125—28no—
238#125—22no—
239#125C22⋊D5314yes—
240#123—28no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C4×Q8—22no—
246C4×Q8C22⋊D5314yes—
247C4×D4—22no—
248C4×D4—22no—
249#117—28no—
250#117—22no—
251#117—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C2×Dic5311yescharacteristic
261C2×C4⋊C4C2×Dic5311yescharacteristic
262C2×C4⋊C4C2×Dic5311yescharacteristic
263C2×C4⋊C4C22⋊D5311yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4C22⋊D5311yescharacteristic
268C2×C22⋊C4C22⋊D5311yescharacteristic
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C22×D5311yescharacteristic
272C4⋊C8—1062no—
273#100—1064no—
274C22⋊C8—1062no—
275#97—1062no—
276#97—1062no—
277C4×C8—1062no—
278C24C22⋊Dic5311yescharacteristic
279C24C22⋊Dic5311yescharacteristic
280C2×Q8—21no—
281C2×Q8—42no—
282C2×Q8C22⋊Dic5312yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C22⋊Dic5311yescharacteristic
300C22×C4C22⋊Dic5311yescharacteristic
301C22×C4C22×Dic5311yescharacteristic
302C22×C4C2×C22⋊D5311yescharacteristic
303C22×C4C2×C22⋊D5311yescharacteristic
304M16—1064no—
305M16—2122no—
306C2×C8—2122no—
307C2×C8—2122no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4C22⋊Dic5312yes—
319C4⋊C4C22⋊Dic5312yes—
320C4⋊C4C22⋊Dic5312yes—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C42—24no—
327C42—24no—
328C42C2×C22⋊D5312yes—
329C42C2×C22⋊D5312yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#1313314412yes—
340C23C2×C22⋊Dic5311yescharacteristic
341Q8—24no—
342Q8—42no—
343D4—44no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4C2×C22⋊Dic5311yescharacteristic
368C2×C4C2×C22⋊Dic5311yescharacteristic
369C2×C4#1313318612yes—
370C2×C4#1313318612yes—
371C8—4242no—
372C8—2124no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22#1681047911yescharacteristic
386C22C2×#1313314411yescharacteristic
387C22#1681064011yescharacteristic
388C4—22no—
389C4—21no—
390C4—22no—
391C4—42no—
392C4—41no—
393C4—41no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4#1681060712yes—
398C2—21no—
399C2—21no—
400C2—42no—
401C2—81no—
402C2#2027464311yescharacteristic
403C2#2027465911yescharacteristic
404C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#552C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenterC2×#552D53
hypocenterC53#8371
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
2. #20273317C2semidirect
3. #20273162C2semidirect
229. C53#8371semidirect

Maximal subgroups

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

subgroupindex
C4×#131331342
#202733172
#202731622
#202730562
C2×C53×#5522
#837153