Finite Group Databasebeta

← #28303885

Subgroups — #28303885

Order 13568.

684 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×D53⋊(C4⋊C8)C211yesmaximal, characteristic
2C4×#13133274C211yesmaximal, characteristic
3C2×Q8⋊#88324C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5#20277253C211yesmaximal, characteristic
6C22×D53⋊C8—21no—
7C2×#13133284—21no—
8D53⋊(C4⋊C8)—24no—
9C2×C4⋊#88324C2212yes—
10C2×#13133274—21no—
11C2×C4×#88324C2212yes—
12#16811489—21no—
13Q8⋊#88324—24no—
14C2×#13133193C411yescharacteristic
15C2×Q8×Dic53C2211yescharacteristic
16C2×C42×D53C2211yescharacteristic
17#16810521C411yescharacteristic
18#16810510—21no—
19#16810497—21no—
20#16810480C411yescharacteristic
21C4⋊C4×Dic53C2211yescharacteristic
22#16810474—21no—
23#16810470—21no—
24C53×#391C411yescharacteristic
25C2×D53⋊C8—24no—
26C2×D53⋊C8—42no—
27C2×D53⋊C8—42no—
28C22×#88324—22no—
29C22×#88324—22no—
30#13133284—24no—
31C4⋊#88324—22no—
32C4⋊#88324D414yes—
33#13133274—24no—
34C4×#88324—22no—
35C4×#88324—22no—
36C22×C4×D53C2×C411yescharacteristic
37C22×C4×D53D411yescharacteristic
38C22×C4×D53D411yescharacteristic
39#13133193—22no—
40#13133193C814yescomplemented
41Q8×Dic53—22no—
42Q8×Dic53—22no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53C2×C411yescharacteristic
45C2×C4⋊Dic53C2×C411yescharacteristic
46C2×#11719919—21no—
47C2×#11719919—21no—
48C2×#11719921—21no—
49C2×#11719921—21no—
50C2×C4×Dic53—21no—
51C2×C4×Dic53—21no—
52C2×C4×Dic53D411yescharacteristic
53C2×C4×Dic53D411yescharacteristic
54C2×C4×Dic53C2311yescharacteristic
55C42×D53—22no—
56C42×D53—22no—
57#13133124—21no—
58#13133124—21no—
59#13133124—21no—
60C22×Q8×C53C2×C411yescharacteristic
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C42×C53C2×C411yescharacteristic
63#95×C53—21no—
64#95×C53—21no—
65C23×D53C22⋊C411yescharacteristic
66C2×Q8×C53—21no—
67C2×Q8×C53—22no—
68C2×Q8×C53—22no—
69C2×Q8×C53C2×C812yescomplemented
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C2×D411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—41no—
89C2×C4×D53C2×C812yes—
90C2×C4×D53M1612yes—
91C2×C4×D53SD1612yes—
92C2×C4×D53Q1612yes—
93C2×#88324—24no—
94C2×#88324—24no—
95C2×#88324—42no—
96C2×#88324—42no—
97C2×#88324—42no—
98C2×#88324—42no—
99D53⋊C8—48no—
100C4⋊C4×C53—22no—
101C42×C53—22no—
102#11719921—24no—
103#11719921—42no—
104C4⋊Dic53—21no—
105C4⋊Dic53—22no—
106C4⋊Dic53—42no—
107C4⋊Dic53C2×C812yescomplemented
108#11719919—24no—
109#11719919—42no—
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×C53C2×C22⋊C411yescharacteristic
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53C22⋊C812yes—
125C22×D53#10312yes—
126C22×D53#10412yes—
127Q8×C53—24no—
128Q8×C53—42no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53#10312yes—
141C2×C4×C53C4×D412yes—
142C2×C4×C53C22×C811yescharacteristic
143C2×C4×C53C2×M1611yescharacteristic
144C2×C4×C53#13112yes—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—22no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53C22⋊C812yes—
156C2×Dic53#12011yescharacteristic
157C2×Dic53#12112yes—
158C2×Dic53#12311yescharacteristic
159C2×Dic53C2×SD1611yescharacteristic
160C2×Dic53C2×Q1611yescharacteristic
161C2×Dic53AGL(1,ℤ/8)11yescharacteristic
162C2×Dic53#13711yescharacteristic
163C4×D53—24no—
164C4×D53—24no—
165C4×D53—22no—
166C4×D53—22no—
167C4×D53—22no—
168C4×D53—22no—
169C4×D53—44no—
170#88324—44no—
171#88324—44no—
172#8937—531nomaximal, Sylow
173C22×C53#38611yescharacteristic
174C22×C53C2×C22⋊C811yescharacteristic
175C22×C53#40811yescharacteristic
176C22×C53C2×#10311yescharacteristic
177C22×C53#41711yescharacteristic
178C22×C53C2×#10411yescharacteristic
179C22×C53#42111yescharacteristic
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—22no—
186C2×D53Q8⋊C814yescomplemented
187C4×C53—22no—
188C4×C53—42no—
189C4×C53—41no—
190C4×C53—81no—
191C4×C53C8×D412yes—
192C4×C53#43512yes—
193Dic53—22no—
194Dic53—22no—
195Dic53—42no—
196Dic53—81no—
197Dic53#45011yescharacteristic
198Dic53#45111yescharacteristic
199Dic53#48311yescharacteristic
200Dic53#48411yescharacteristic
201C22×C4⋊C8—531no—
202C4×C22⋊Q8—531no—
203C4×C22⋊C8—531no—
204C2×Q8⋊C8—532no—
205#1279—532no—
206C2×C53C2×Q8⋊C811yescharacteristic
207C2×C53#147711yescharacteristic
208C2×C53#148111yescharacteristic
209C2×C53#148311yescharacteristic
210C2×C53#186911yescharacteristic
211C2×C53#187111yescharacteristic
212C2×C53#187211yescharacteristic
213D53—24no—
214C23×C8—1061no—
215C2×C22⋊Q8—531no—
216C2×C4×Q8—531no—
217C22×C42—531no—
218C2×C4⋊C8—532no—
219C2×C4⋊C8—1064no—
220C2×C4⋊C8—1061no—
221C2×C22⋊C8—1061no—
222C2×C4×C8—532no—
223#391#615711yescharacteristic
224#389—531no—
225#387—1061no—
226#385—1061no—
227#380—531no—
228C4×C4⋊C4—531no—
229C4×C4⋊C4—1061no—
230C4×C22⋊C4—1061no—
231#336—1061no—
232Q8⋊C8—1064no—
233C53#893711yescharacteristic, complemented, Sylow
234C22×Q8C2×#615711yescharacteristic
235C23×C4—531no—
236C23×C4—531no—
237C23×C4—531no—
238C22×C8—1062no—
239C22×C8—1062no—
240C22×C8—1064no—
241C22×C8—2122no—
242C22×C8—2122no—
243C22⋊Q8—534no—
244C22⋊Q8—1062no—
245C4×Q8—1062no—
246C4×Q8—1062no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4C2×#615711yescharacteristic
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C42—531no—
256C2×C42—531no—
257C2×C42—531no—
258C2×C42—1062no—
259C2×C42—1062no—
260C2×C42—1061no—
261C2×C42—1061no—
262C2×C42C2×#615711yescharacteristic
263C4⋊C8—534no—
264C4⋊C8—1062no—
265C4⋊C8—1064no—
266C22⋊C8—1064no—
267C4×C8—1062no—
268C4×C8—1062no—
269#95—21no—
270#95—21no—
271#95—1061no—
272#95—1061no—
273#95—1061no—
274C24—531no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8D53⋊C812yescomplemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—532no—
285C22×C4—532no—
286C22×C4—532no—
287C22×C4—532no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1062no—
293C22×C4—1062no—
294C22×C4—1062no—
295C22×C4—1062no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4—2121no—
299C22×C4C22⋊#615711yescharacteristic
300C22×C4C22⋊#615711yescharacteristic
301C22×C4C22×#615711yescharacteristic
302C2×C8—1064no—
303C2×C8—1064no—
304C2×C8—2122no—
305C2×C8—2122no—
306C2×C8—2122no—
307C2×C8—2122no—
308C2×C8—2128no—
309C4⋊C4—22no—
310C4⋊C4—532no—
311C4⋊C4—1064no—
312C4⋊C4—1061no—
313C4⋊C4—1062no—
314C4⋊C4—2122no—
315C4⋊C4—2122no—
316C22⋊C4—1064no—
317C22⋊C4—2122no—
318C42—22no—
319C42—532no—
320C42—532no—
321C42—1061no—
322C42—1062no—
323C42—1061no—
324C42—2121no—
325C42—2121no—
326C42—2121no—
327C42—2121no—
328C23—532no—
329C23—532no—
330C23—532no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23—1061no—
335C23C2×C22⋊#615711yescharacteristic
336Q8—24no—
337Q8—42no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
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—1062no—
358C2×C4—1062no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1064no—
366C2×C4—1064no—
367C2×C4—1062no—
368C2×C4—2124no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4#1313328012yes—
375C2×C4D4×#615712yes—
376C2×C4C2×D53⋊C811yescharacteristic
377C2×C4D53⋊M1611yescharacteristic
378C2×C4#1313330412yes—
379C8—2124no—
380C8—2124no—
381C22—534no—
382C22—1062no—
383C22—1062no—
384C22—1062no—
385C22—1062no—
386C22—1062no—
387C22—1062no—
388C22#1681157111yescharacteristic
389C22D53⋊(C22⋊C8)11yescharacteristic
390C22#1681159711yescharacteristic
391C22C2×#1313328011yescharacteristic
392C22#1681161411yescharacteristic
393C22D53⋊#10411yescharacteristic
394C22#1681162211yescharacteristic
395C4—22no—
396C4—42no—
397C4—41no—
398C4—81no—
399C4—531no—
400C4—531no—
401C4—531no—
402C4—531no—
403C4—1062no—
404C4—1062no—
405C4—4241no—
406C4—2122no—
407C4#1681165012yes—
408C4#1681165512yes—
409C2—1064no—
410C2D53⋊(Q8⋊C8)11yescharacteristic
411C2#2029001311yescharacteristic
412C2#2029002411yescharacteristic
413C2#2029003111yescharacteristic
414C2#2029068911yescharacteristic
415C2#2029069711yescharacteristic
416C2#2029070011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#8937
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C2×Q8⋊#88324C2semidirect
4. #20289551C2semidirect
40. #13133193C8semidirect
69. C2×Q8×C53C2×C8semidirect
107. C4⋊Dic53C2×C8semidirect
186. C2×D53Q8⋊C8semidirect
233. C53#8937semidirect
278. C2×Q8D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×D53⋊(C4⋊C8)2
C4×#131332742
C2×Q8⋊#883242
#202895512
#202772532
#893753