Finite Group Databasebeta

← #28310502

Subgroups — #28310502

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811586C211yesmaximal, characteristic
2#20290943C211yesmaximal, characteristic
3C2×#16811514C211yesmaximal, characteristic, complemented
4C2×#16811513C211yesmaximal, characteristic, complemented
5#20289755C211yesmaximal, characteristic, complemented
6#20289755C211yesmaximal, characteristic, complemented
7#20276324C211yesmaximal, characteristic
8C2×D53⋊M16C411yescharacteristic
9C2×#13133296C411yescharacteristic
10C2×#13133292C2211yescharacteristic
11C2×#13133292C2211yescharacteristic
12#16811586—22no—
13#16811586—22no—
14C2×#13133271C2211yescharacteristic
15C2×#13133271C2211yescharacteristic
16C2×C8×#6157C411yescharacteristic
17C2×C8×#6157C411yescharacteristic
18#16811518C412yescomplemented
19C22⋊#11719910C412yescomplemented
20#16811514—22no—
21#16811513—22no—
22C22×C8×D53C2211yescharacteristic
23C2×#13133156C2211yescharacteristic
24C2×#13133156C2211yescharacteristic
25C22⋊#11719908C411yescharacteristic, complemented
26#16810402C411yescharacteristic, complemented
27#16810402C411yescharacteristic, complemented
28C53×C22⋊C16C411yescharacteristic, complemented
29C2×#11719936—21no—
30D53⋊M16—22no—
31D53⋊M16—22no—
32C2×#11719940—21no—
33C2×D53⋊C8C2×C411yescharacteristic
34C2×D53⋊C8C2×C411yescharacteristic
35#13133296—22no—
36#13133296—22no—
37C2×C4⋊#6157—21no—
38C2×C22⋊#6157—21no—
39C2×C4×#6157C2×C411yescharacteristic
40C2×C4×#6157C2×C411yescharacteristic
41#13133292—21no—
42#13133292—21no—
43#13133292—21no—
44#13133292—21no—
45C2×#11719910C2×C412yes—
46C2×#11719910C2×C412yes—
47#13133271—21no—
48#13133271—21no—
49#13133271D412yes—
50#13133271D412yes—
51C8×#6157—22no—
52C8×#6157—22no—
53C22×C4×D53C2×C411yescharacteristic
54C22×#88322C2×C411yescharacteristic
55C2×C8×D53—21no—
56C2×C8×D53—21no—
57C2×C8×D53—21no—
58C2×C8×D53—21no—
59C2×C8×D53C2×C411yescharacteristic
60C2×C8×D53C2×C411yescharacteristic
61C2×C8×D53C2×C411yescharacteristic
62C2×C8×D53C2311yescharacteristic
63#13133156—21no—
64#13133156—21no—
65#13133156—21no—
66#13133156—21no—
67C2×#11719908C2×C411yescharacteristic
68C2×#11719908C2×C411yescharacteristic
69C22×C8×C53C2×C411yescharacteristic
70C2×C16×C53C2×C411yescharacteristic
71C2×C16×C53C2×C411yescharacteristic
72C23×D53M1611yescharacteristic
73C22×#6157—21no—
74C22×#6157—21no—
75C22×C4×C53C4211yescharacteristic
76C22×Dic53M1611yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53C4211yescharacteristic
82C2×C4×D53M1611yescharacteristic
83C2×C4×D53M1611yescharacteristic
84C2×C4×D53C22×C411yescharacteristic
85C22⋊#6157—24no—
86#11719940—24no—
87C2×#88324—21no—
88C2×#88324—21no—
89C4⋊#6157—24no—
90C4×#6157—21no—
91C4×#6157—21no—
92C4×#6157C22⋊C412yes—
93C4×#6157M1612yes—
94#11719936—24no—
95D53⋊C8—21no—
96D53⋊C8—21no—
97D53⋊C8C22⋊C412yes—
98D53⋊C8M1612yes—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53C4211yescharacteristic
102C2×C8×C53C22×C411yescharacteristic
103C2×#88322—21no—
104C2×#88322—21no—
105C2×#88322C4211yescharacteristic
106C2×#88322C22×C411yescharacteristic
107C8×D53—21no—
108C8×D53—21no—
109C8×D53—21no—
110C8×D53—21no—
111C8×D53—21no—
112C8×D53—21no—
113C8×D53C2×D411yescharacteristic
114C8×D53C2×D411yescharacteristic
115C8×D53#4111yescharacteristic
116C8×D53#4111yescharacteristic
117#11719910—22no—
118#11719910—22no—
119C16×C53—21no—
120C16×C53—21no—
121#11719908—21no—
122#11719908—21no—
123C23×C53#9711yescharacteristic
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53#9711yescharacteristic
129C22×D53C2×M1611yescharacteristic
130C2×#6157—22no—
131C2×#6157—22no—
132C2×#6157—41no—
133C2×#6157—41no—
134C2×#6157—41no—
135C2×#6157—41no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53#9711yescharacteristic
139C2×C4×C53C2×C4211yescharacteristic
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53#9711yescharacteristic
143C2×Dic53C2×M1611yescharacteristic
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53C2×C22⋊C411yescharacteristic
151C4×D53#11711yescharacteristic
152C4×D53C2×M1611yescharacteristic
153C4×D53C2×M1611yescharacteristic
154#88324—22no—
155#88324—22no—
156C8×C53—21no—
157C8×C53C4×D411yescharacteristic
158C8×C53C4×D411yescharacteristic
159#88322—21no—
160#88322C4×D411yescharacteristic
161#88322C4×D411yescharacteristic
162#12606—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53C2×#9711yescharacteristic
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—42no—
173C2×D53—42no—
174C2×D53#40711yescharacteristic
175C2×D53#43211yescharacteristic
176#6157—42no—
177#6157—42no—
178C4×C53—21no—
179C4×C53C4×C22⋊C411yescharacteristic
180C4×C53C4×M1611yescharacteristic
181Dic53—21no—
182Dic53#40711yescharacteristic
183Dic53#43211yescharacteristic
184C2×#405—531no—
185#2112—531no—
186#2112—531no—
187C2×#347—531no—
188C2×#347—531no—
189#1375—531no—
190#1375—531no—
191C2×C53—41no—
192C2×C53#174911yescharacteristic
193C2×C53#211011yescharacteristic
194C2×C53#211011yescharacteristic
195D53—21no—
196D53—21no—
197D53—41no—
198C22×M16—531no—
199C23×C8—531no—
200C2×#117—531no—
201C2×M32—531no—
202C2×M32—531no—
203C2×M32—531no—
204C2×M32—531no—
205#405—1062no—
206#405—1062no—
207C2×#97—531no—
208C2×#97—531no—
209C2×C4×C8—531no—
210C2×C4×C8—531no—
211C22⋊C16—532no—
212C22⋊C16—532no—
213C22⋊C16—531no—
214C22⋊C16—531no—
215C22⋊C16—531no—
216C22⋊C16#615711yescharacteristic, complemented
217#347—1062no—
218#347—1062no—
219C53#1260611yescharacteristic, complemented, Sylow
220C23×C4—531no—
221C2×M16—1062no—
222C2×M16—1062no—
223C2×M16—1061no—
224C2×M16—1061no—
225C22×C8—531no—
226C22×C8—531no—
227C22×C8—531no—
228C22×C8—531no—
229C22×C8—531no—
230C22×C8—531no—
231C22×C8—531no—
232C22×C8—1061no—
233C22×C8—1061no—
234C22×C8—1061no—
235C22×C8—1061no—
236C22×C8C2×#615711yescharacteristic
237#117—1062no—
238#117—1062no—
239C2×C4⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C42—531no—
242C2×C42—531no—
243M32—1061no—
244M32—1061no—
245M32—1061no—
246M32—1061no—
247M32—1061no—
248M32—1061no—
249M32—1061no—
250M32—1061no—
251C2×C16—532no—
252C2×C16—532no—
253C2×C16—531no—
254C2×C16—531no—
255C2×C16C2×#615711yescharacteristic
256C2×C16C2×#615711yescharacteristic
257#97—532no—
258#97—532no—
259#97—1061no—
260#97—1061no—
261C4×C8—1062no—
262C4×C8—1062no—
263C24—531no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4C4×#615711yescharacteristic
276M16—1064no—
277M16—1064no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—531no—
281C2×C8—531no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—532no—
286C2×C8—532no—
287C2×C8—531no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—1061no—
292C2×C8—1061no—
293C2×C8—1061no—
294C2×C8—1061no—
295C2×C8—1061no—
296C2×C8—1061no—
297C2×C8—1061no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8C4×#615711yescharacteristic
301C2×C8C22×#615711yescharacteristic
302C4⋊C4—1064no—
303C22⋊C4—1064no—
304C42—532no—
305C42—532no—
306C42—1061no—
307C42—1061no—
308C16—21no—
309C16—21no—
310C16—1062no—
311C16—1062no—
312C16—1061no—
313C16—1061no—
314C23—531no—
315C23—531no—
316C23—1061no—
317C23—1061no—
318C23—2121no—
319C23—2121no—
320C23#1313327111yescharacteristic
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—531no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1062no—
338C2×C4—1062no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4#1313327111yescharacteristic
344C2×C4C2×C4×#615711yescharacteristic
345C8—21no—
346C8—531no—
347C8—531no—
348C8—1062no—
349C8—1062no—
350C8—1061no—
351C8D4×#615711yescharacteristic
352C8D4×#615711yescharacteristic
353C22—21no—
354C22—21no—
355C22—21no—
356C22—531no—
357C22—531no—
358C22—1061no—
359C22—1061no—
360C22—1061no—
361C22—1061no—
362C22—1061no—
363C22—2122no—
364C22—2122no—
365C22C2×#1313327111yescharacteristic
366C4—21no—
367C4—531no—
368C4—531no—
369C4—1061no—
370C4—2122no—
371C4—2122no—
372C4C22⋊C4×#615711yescharacteristic
373C4M16×#615711yescharacteristic
374C2—41no—
375C2—1061no—
376C2—1061no—
377C2—2121no—
378C2#2029044311yescharacteristic
379C2#2029093411yescharacteristic
380C2#2029093511yescharacteristic
subgroupG/H
centerC2×C4C2×C4×#6157
commutatorC2×C4×C53C2×C42
FrattiniC2×C8C22×#6157
FittingC53×C22⋊C16C4
socleC22×C53C2×#97
prime radicalC23×D53M16
cosocleC2×C8×D53C23
hypercenterC22⋊C16#6157
hypocenterC53#12606
LCSG ▷ C2×C4×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
3. C2×#16811514C2semidirect
4. C2×#16811513C2semidirect
5. #20289755C2semidirect
6. #20289755C2semidirect
18. #16811518C4semidirect
19. C22⋊#11719910C4semidirect
25. C22⋊#11719908C4semidirect
26. #16810402C4semidirect
27. #16810402C4semidirect
28. C53×C22⋊C16C4semidirect
216. C22⋊C16#6157semidirect
219. C53#12606semidirect

Maximal subgroups

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

subgroupindex
C2×#168115862
#202909432
C2×#168115142
C2×#168115132
#202897552
#202763242
#1260653