Finite Group Databasebeta

← #28303906

Subgroups — #28303906

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811626C211yesmaximal, characteristic
2C4×C22⋊#88324C211yesmaximal, characteristic
3C2×#16811473C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5#20277243C211yesmaximal, characteristic
6C22×#11719940—21no—
7#16811626—24no—
8C2×#13133284—21no—
9C2×C4⋊#88324C2212yes—
10C2×C22⋊#88324—21no—
11C2×C4×#88324C2212yes—
12#16811489—21no—
13#16811473—24no—
14C2×#13133196C411yescharacteristic
15C2×#13133182C2211yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17#16810528C411yescharacteristic
18#16810519C411yescharacteristic
19#16810505—21no—
20#16810495—21no—
21C4⋊C4×Dic53C2211yescharacteristic
22#16810474—21no—
23C22⋊C4×Dic53—21no—
24C53×#380C411yescharacteristic
25C2×#11719940—24no—
26C2×#11719940—42no—
27C2×#11719940—42no—
28C22×#88324—22no—
29C22×#88324—22no—
30#13133284—24no—
31C4⋊#88324—22no—
32C4⋊#88324D414yes—
33C22⋊#88324—24no—
34C4×#88324—22no—
35C4×#88324—22no—
36C22×Dic106C2×C411yescharacteristic
37C23×Dic53D411yescharacteristic
38C23×Dic53D411yescharacteristic
39#13133196—22no—
40#13133196—22no—
41#13133182—22no—
42#13133182—22no—
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×#11719919—21no—
47C2×C22⋊Dic53—21no—
48C2×C4×Dic53—22no—
49C2×C4×Dic53—22no—
50C2×C4×Dic53—21no—
51C2×C4×Dic53—21no—
52C2×C4×Dic53C2×C411yescharacteristic
53C2×C4×Dic53D411yescharacteristic
54C2×C4×Dic53D411yescharacteristic
55C2×C4×Dic53C2311yescharacteristic
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59#13133124—21no—
60C23×C4×C53C2×C411yescharacteristic
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—21no—
64#95×C53—21no—
65C24×C53C22⋊C411yescharacteristic
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—22no—
81C22×Dic53—22no—
82C22×Dic53—41no—
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C22⋊C411yescharacteristic
85C22×Dic53SD1612yes—
86C22×Dic53Q1612yes—
87C22×Dic53C2×D411yescharacteristic
88C22×Dic53C2×D411yescharacteristic
89C2×Dic106—22no—
90C2×Dic106—22no—
91C2×Dic106—21no—
92C2×Dic106M1612yescomplemented
93#11719940—48no—
94C2×#88324—24no—
95C2×#88324—24no—
96C2×#88324—42no—
97C2×#88324—42no—
98C2×#88324—42no—
99C2×#88324—42no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C22⋊C4×C53—42no—
103C22⋊Dic53—24no—
104C4⋊Dic53—21no—
105C4⋊Dic53M1612yescomplemented
106#11719919—24no—
107#11719919—42no—
108#11719919—42no—
109C4×Dic53—22no—
110C4×Dic53—21no—
111C4×Dic53—22no—
112C4×Dic53—21no—
113C4×Dic53—41no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116C4×Dic53—41no—
117C4×Dic53C2×D412yes—
118C4×Dic53#4112yes—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53#10312yes—
126C23×C53C2×C22⋊C411yescharacteristic
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53C4×D412yes—
139C2×C4×C53C2×M1611yescharacteristic
140C2×C4×C53C2×M1611yescharacteristic
141C2×C4×C53#13112yes—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—22no—
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—44no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53#10312yes—
160C2×Dic53#10412yes—
161C2×Dic53#12011yescharacteristic
162C2×Dic53#12112yes—
163C2×Dic53#12311yescharacteristic
164C2×Dic53C2×SD1611yescharacteristic
165C2×Dic53C2×Q1611yescharacteristic
166C2×Dic53AGL(1,ℤ/8)11yescharacteristic
167C2×Dic53#13711yescharacteristic
168Dic106—24no—
169Dic106—42no—
170#88324—44no—
171#88324—44no—
172#8942—531nomaximal, Sylow
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53#38611yescharacteristic
181C22×C53#40711yescharacteristic
182C22×C53#40811yescharacteristic
183C22×C53C2×#10311yescharacteristic
184C22×C53#41711yescharacteristic
185C22×C53C2×#10411yescharacteristic
186C22×C53#42111yescharacteristic
187C4×C53—41no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53#43512yes—
191C4×C53#43612yes—
192Dic53—22no—
193Dic53—22no—
194Dic53—22no—
195Dic53—42no—
196Dic53—42no—
197Dic53#45011yescharacteristic
198Dic53#45111yescharacteristic
199Dic53#48311yescharacteristic
200Dic53#48411yescharacteristic
201C2×#423—531no—
202C4×C22⋊Q8—531no—
203C4×C22⋊C8—531no—
204C2×Q8⋊C8—532no—
205#1279—532no—
206C2×C53—42no—
207C2×C53#148711yescharacteristic
208C2×C53#148911yescharacteristic
209C2×C53#149311yescharacteristic
210C2×C53#149511yescharacteristic
211C2×C53#187011yescharacteristic
212C2×C53#187111yescharacteristic
213C2×C53#187211yescharacteristic
214C22×M16—1061no—
215C2×C22⋊Q8—531no—
216C2×C4×Q8—531no—
217C22×C42—531no—
218#423—1064no—
219C2×C4⋊C8—532no—
220C2×C4⋊C8—1061no—
221C2×C22⋊C8—1061no—
222C2×C4×C8—532no—
223#391—531no—
224#389—531no—
225#387—1061no—
226#385—1061no—
227#380#615711yescharacteristic
228C4×C4⋊C4—531no—
229C4×C4⋊C4—1061no—
230C4×C22⋊C4—1061no—
231#336—1061no—
232Q8⋊C8—1064no—
233C53#894211yescharacteristic, complemented, Sylow
234C22×Q8—531no—
235C23×C4—531no—
236C23×C4—531no—
237C23×C4C2×#615711yescharacteristic
238C2×M16—1064no—
239C2×M16—2122no—
240C2×M16—2122no—
241C22×C8—1062no—
242C22×C8—1062no—
243C22⋊Q8—1062no—
244C22⋊Q8—1062no—
245C4×Q8—1062no—
246C4×Q8—1062no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4C2×#615711yescharacteristic
252C2×C4⋊C4C2×#615711yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—1061no—
255C2×C42—531no—
256C2×C42—531no—
257C2×C42—531no—
258C2×C42—531no—
259C2×C42—1062no—
260C2×C42—1062no—
261C2×C42—1061no—
262C2×C42—1061no—
263C4⋊C8—534no—
264C4⋊C8—1062no—
265C4⋊C8—1064no—
266C22⋊C8—1064no—
267C4×C8—1062no—
268C4×C8—1062no—
269#95—21no—
270#95—1061no—
271#95—1061no—
272#95—1061no—
273#95—1061no—
274C24C22⋊#615711yescharacteristic
275C2×Q8—532no—
276C2×Q8—1061no—
277C2×Q8—1062no—
278C2×Q8—1062no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—531no—
286C22×C4—532no—
287C22×C4—532no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1062no—
296C22×C4—1062no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—2121no—
300C22×C4C22⋊#615711yescharacteristic
301C22×C4C22×#615711yescharacteristic
302M16—2128no—
303C2×C8—1064no—
304C2×C8—1064no—
305C2×C8—2122no—
306C2×C8—2122no—
307C2×C8—2122no—
308C2×C8—2122no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—532no—
312C4⋊C4—1064no—
313C4⋊C4—1061no—
314C4⋊C4—2122no—
315C4⋊C4—2122no—
316C22⋊C4—42no—
317C22⋊C4—1064no—
318C42—532no—
319C42—532no—
320C42—1061no—
321C42—1061no—
322C42—1062no—
323C42—1062no—
324C42—2121no—
325C42—2121no—
326C42—2121no—
327C42—2121no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#1313328012yes—
335C23C2×C22⋊#615711yescharacteristic
336Q8—1064no—
337Q8—2122no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—531no—
350C2×C4—531no—
351C2×C4—531no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—531no—
355C2×C4—532no—
356C2×C4—532no—
357C2×C4—532no—
358C2×C4—1062no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4—1062no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1062no—
369C2×C4—1062no—
370C2×C4—2124no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4D4×#615712yes—
376C2×C4D53⋊M1611yescharacteristic
377C2×C4D53⋊M1611yescharacteristic
378C2×C4#1313330412yes—
379C8—2124no—
380C8—2124no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22#1681157111yescharacteristic
389C22#1681159511yescharacteristic
390C22#1681159611yescharacteristic
391C22C2×#1313328011yescharacteristic
392C22#1681161411yescharacteristic
393C22D53⋊#10411yescharacteristic
394C22#1681162111yescharacteristic
395C4—41no—
396C4—81no—
397C4—81no—
398C4—531no—
399C4—531no—
400C4—531no—
401C4—531no—
402C4—1062no—
403C4—1062no—
404C4—1062no—
405C4—2122no—
406C4—2122no—
407C4#1681165512yes—
408C4#1681165612yes—
409C2—42no—
410C2#2029004511yescharacteristic
411C2#2029005211yescharacteristic
412C2#2029006311yescharacteristic
413C2#2029007011yescharacteristic
414C2#2029069411yescharacteristic
415C2#2029069911yescharacteristic
416C2#2029070011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#8942
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#16811473C2semidirect
4. #20289551C2semidirect
92. C2×Dic106M16semidirect
105. C4⋊Dic53M16semidirect
233. C53#8942semidirect

Maximal subgroups

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

subgroupindex
C2×#168116262
C4×C22⋊#883242
C2×#168114732
#202895512
#202772432
#894253