Finite Group Databasebeta

← #28310559

Subgroups — #28310559

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290943C211yesmaximal, characteristic
2C2×#16811514C211yesmaximal, characteristic, complemented
3C2×#16811513C211yesmaximal, characteristic, complemented
4#20290451C211yesmaximal, characteristic
5#20289755C211yesmaximal, characteristic, complemented
6#20289755C211yesmaximal, characteristic, complemented
7C2×M32×D53C211yesmaximal, characteristic
8C2×#13133292C2211yescharacteristic
9C2×#13133292C2211yescharacteristic
10D53⋊(C22⋊C8)C411yescharacteristic
11C2×#13133271C2211yescharacteristic
12C2×#13133271C2211yescharacteristic
13C4×C22⋊#6157C411yescharacteristic
14#16811518C412yescomplemented
15C22⋊#11719910C412yescomplemented
16#16811514—22no—
17#16811513—22no—
18(C2×#6157)⋊C8C411yescharacteristic
19(C2×#6157)⋊C8C411yescharacteristic
20C22×C8×D53C2211yescharacteristic
21C2×#13133157C411yescharacteristic, complemented
22C2×#13133156C2211yescharacteristic
23C2×#13133156C2211yescharacteristic
24M32×D53—22no—
25M32×D53—22no—
26C2×C16×D53C411yescharacteristic, complemented
27C2×C16×D53C411yescharacteristic, complemented
28C2×M32×C53C411yescharacteristic, complemented
29C2×D53⋊C8C2×C411yescharacteristic
30C2×D53⋊C8C2×C411yescharacteristic
31C2×C22⋊#6157—21no—
32C2×C4×#6157C2×C411yescharacteristic
33C2×C4×#6157C2×C411yescharacteristic
34#13133292—21no—
35#13133292—21no—
36#13133292—21no—
37#13133292—21no—
38C2×#11719910C2×C412yes—
39C2×#11719910C2×C412yes—
40#13133274—21no—
41C22⋊#88324—21no—
42#13133271—21no—
43#13133271—21no—
44#13133271—21no—
45#13133271—21no—
46(C2×#6157)⋊C4—21no—
47C22×C4×D53C2×C411yescharacteristic
48C22×#88322C2×C411yescharacteristic
49C2×C8×D53—21no—
50C2×C8×D53—21no—
51C2×C8×D53—21no—
52C2×C8×D53—21no—
53C2×C8×D53C2×C411yescharacteristic
54C2×C8×D53C2×C411yescharacteristic
55C2×C8×D53C2×C411yescharacteristic
56C2×C8×D53C2311yescharacteristic
57#13133157—22no—
58#13133156—21no—
59#13133156—21no—
60#13133156D412yes—
61#13133156D412yes—
62C2×#11719908C2×C411yescharacteristic
63C2×#11719908C2×C411yescharacteristic
64C16×D53—22no—
65C16×D53—22no—
66C22×C8×C53C2×C411yescharacteristic
67M32×C53—22no—
68C2×C16×C53C2×C411yescharacteristic
69C2×C16×C53C2×C411yescharacteristic
70C23×D53M1611yescharacteristic
71C22×#6157—21no—
72C22×#6157—21no—
73C22×C4×C53C4211yescharacteristic
74C22×Dic53M1611yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53C4211yescharacteristic
80C2×C4×D53M1611yescharacteristic
81C2×C4×D53M1611yescharacteristic
82C2×C4×D53C22×C411yescharacteristic
83C22⋊#6157—42no—
84C2×#88324—21no—
85C2×#88324—21no—
86C4×#6157—21no—
87C4×#6157—21no—
88C4×#6157—21no—
89C4×#6157—21no—
90D53⋊C8—21no—
91D53⋊C8—21no—
92D53⋊C8—21no—
93D53⋊C8—21no—
94C2×C8×C53—21no—
95C2×C8×C53—21no—
96C2×C8×C53C4211yescharacteristic
97C2×C8×C53C22×C411yescharacteristic
98C2×#88322—21no—
99C2×#88322—21no—
100C2×#88322C4211yescharacteristic
101C2×#88322C22×C411yescharacteristic
102C8×D53—21no—
103C8×D53—21no—
104C8×D53—21no—
105C8×D53—21no—
106C8×D53—21no—
107C8×D53—21no—
108C8×D53C2×D411yescharacteristic
109C8×D53C2×D411yescharacteristic
110C8×D53#4111yescharacteristic
111C8×D53#4111yescharacteristic
112#11719910—22no—
113#11719910—22no—
114C16×C53—21no—
115C16×C53C22⋊C412yescomplemented
116#11719908—21no—
117#11719908C22⋊C412yescomplemented
118C23×C53#9711yescharacteristic
119C22×D53—22no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—41no—
123C22×D53#9711yescharacteristic
124C22×D53C2×M1611yescharacteristic
125C2×#6157—41no—
126C2×#6157—41no—
127C2×#6157—41no—
128C2×#6157—41no—
129C2×#6157—41no—
130C2×#6157—41no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53#9711yescharacteristic
134C2×C4×C53C2×C4211yescharacteristic
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53#9711yescharacteristic
138C2×Dic53C2×M1611yescharacteristic
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53C4×D411yescharacteristic
146C4×D53C4×D411yescharacteristic
147C4×D53#13111yescharacteristic
148C4×D53#13111yescharacteristic
149#88324—41no—
150#88324—41no—
151C8×C53—21no—
152C8×C53C2×C22⋊C411yescharacteristic
153C8×C53#11711yescharacteristic
154#88322—21no—
155#88322C4×D411yescharacteristic
156#88322C4×D411yescharacteristic
157#12616—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53C2×#9711yescharacteristic
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—42no—
167C2×D53—42no—
168C2×D53#34712yescomplemented
169C2×D53#43511yescharacteristic
170C2×D53#43511yescharacteristic
171#6157—81no—
172#6157—81no—
173C4×C53—21no—
174C4×C53C4×C22⋊C411yescharacteristic
175C4×C53#40511yescharacteristic
176Dic53—21no—
177Dic53#43511yescharacteristic
178Dic53#43511yescharacteristic
179C22×M32—531no—
180#2112—531no—
181C2×#347—531no—
182C2×#347—531no—
183#1752—531no—
184#1375—531no—
185#1375—531no—
186C2×C53—41no—
187C2×C53#175211yescharacteristic
188C2×C53C2×#34711yescharacteristic
189C2×C53#211011yescharacteristic
190D53—21no—
191D53—21no—
192D53—22no—
193C23×C8—531no—
194C2×M32—531no—
195C2×M32—531no—
196C2×M32—531no—
197C2×M32—531no—
198C2×M32—531no—
199C2×M32—1062no—
200C2×M32—1062no—
201C2×M32#615711yescharacteristic, complemented
202C22×C16—531no—
203C22×C16—531no—
204C2×C22⋊C8—531no—
205C2×#97—531no—
206C2×#97—531no—
207C4×C22⋊C4—531no—
208C22⋊C16—532no—
209C22⋊C16—532no—
210#347—1062no—
211#347—1062no—
212#336—531no—
213#336—531no—
214C53#1261611yescharacteristic, complemented, Sylow
215C23×C4—531no—
216C22×C8—531no—
217C22×C8—531no—
218C22×C8—531no—
219C22×C8—531no—
220C22×C8—531no—
221C22×C8—531no—
222C22×C8—531no—
223C22×C8—1061no—
224C22×C8—1061no—
225C22×C8—1061no—
226C22×C8—1061no—
227C22×C8C2×#615711yescharacteristic
228C2×C22⋊C4—1061no—
229C2×C42—531no—
230C2×C42—531no—
231M32—22no—
232M32—532no—
233M32—532no—
234M32—1061no—
235M32—1061no—
236M32—1062no—
237M32—1061no—
238M32—1061no—
239M32—1061no—
240M32—1061no—
241C2×C16—531no—
242C2×C16—532no—
243C2×C16—532no—
244C2×C16—531no—
245C2×C16—1062no—
246C2×C16—1062no—
247C2×C16C2×#615711yescharacteristic
248C2×C16C2×#615711yescharacteristic
249C22⋊C8—1061no—
250C22⋊C8—1061no—
251#97—1061no—
252#97—1061no—
253#97—1061no—
254#97—1061no—
255#95—1061no—
256C24—531no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4C4×#615711yescharacteristic
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—531no—
272C2×C8—531no—
273C2×C8—531no—
274C2×C8—531no—
275C2×C8—531no—
276C2×C8—531no—
277C2×C8—1061no—
278C2×C8—1061no—
279C2×C8—1061no—
280C2×C8—1061no—
281C2×C8—1061no—
282C2×C8—1061no—
283C2×C8—1061no—
284C2×C8—1061no—
285C2×C8—1061no—
286C2×C8—1061no—
287C2×C8—1061no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8C4×#615711yescharacteristic
292C2×C8C22×#615711yescharacteristic
293C22⋊C4—2122no—
294C42—1061no—
295C42—1061no—
296C42—1061no—
297C42—1061no—
298C16—21no—
299C16—532no—
300C16—1062no—
301C16—1062no—
302C16—1061no—
303C16C22⋊#615712yescomplemented
304C23—531no—
305C23—531no—
306C23—1062no—
307C23—1061no—
308C23—1061no—
309C23—2121no—
310C23#1313327111yescharacteristic
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4#1313327111yescharacteristic
334C2×C4C2×C4×#615711yescharacteristic
335C8—21no—
336C8—531no—
337C8—531no—
338C8—1061no—
339C8—2121no—
340C8—2121no—
341C8C2×C22⋊#615711yescharacteristic
342C8#1313329611yescharacteristic
343C22—21no—
344C22—21no—
345C22—21no—
346C22—532no—
347C22—531no—
348C22—531no—
349C22—1061no—
350C22—1061no—
351C22—1061no—
352C22—1061no—
353C22—2122no—
354C22—2122no—
355C22C2×#1313327111yescharacteristic
356C4—21no—
357C4—531no—
358C4—531no—
359C4—1061no—
360C4—4241no—
361C4—4241no—
362C4C4×C22⋊#615711yescharacteristic
363C4#1681158611yescharacteristic
364C2—41no—
365C2—1061no—
366C2—1061no—
367C2—1062no—
368C2#2029045111yescharacteristic
369C2#2029093011yescharacteristic
370C2#2029093511yescharacteristic
subgroupG/H
centerC2×C4C2×C4×#6157
commutatorC2×C4×C53C2×C42
FrattiniC2×C8C22×#6157
FittingC2×M32×C53C4
socleC22×C53C2×#97
prime radicalC23×D53M16
cosocleC2×C8×D53C23
hypercenterC2×M32#6157
hypocenterC53#12616
LCSG ▷ C2×C4×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×M32

Complements

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

NH ≅ G/Ntype
2. C2×#16811514C2semidirect
3. C2×#16811513C2semidirect
5. #20289755C2semidirect
6. #20289755C2semidirect
14. #16811518C4semidirect
15. C22⋊#11719910C4semidirect
21. C2×#13133157C4semidirect
26. C2×C16×D53C4semidirect
27. C2×C16×D53C4semidirect
28. C2×M32×C53C4semidirect
115. C16×C53C22⋊C4semidirect
117. #11719908C22⋊C4semidirect
168. C2×D53#347semidirect
201. C2×M32#6157semidirect
214. C53#12616semidirect
303. C16C22⋊#6157semidirect

Maximal subgroups

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

subgroupindex
#202909432
C2×#168115142
C2×#168115132
#202904512
#202897552
C2×M32×D532
#1261653