Finite Group Databasebeta

← #28310561

Subgroups — #28310561

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D53⋊M32C211yesmaximal, characteristic
2C2×#16811514C211yesmaximal, characteristic, complemented
3C2×#16811513C211yesmaximal, characteristic, complemented
4#20290451C211yesmaximal, characteristic
5#20289755C211yesmaximal, characteristic, complemented
6#20289755C211yesmaximal, characteristic, complemented
7#20276324C211yesmaximal, characteristic
8C2×#13133292C2211yescharacteristic
9C2×#13133292C2211yescharacteristic
10D53⋊M32—22no—
11D53⋊M32—22no—
12C2×#13133291C412yescomplemented
13C2×D53⋊C16C412yescomplemented
14D53⋊(C22⋊C8)C411yescharacteristic
15C2×#13133271C2211yescharacteristic
16C2×#13133271C2211yescharacteristic
17C4×C22⋊#6157C411yescharacteristic
18#16811514—22no—
19#16811513—22no—
20(C2×#6157)⋊C8C411yescharacteristic
21(C2×#6157)⋊C8C411yescharacteristic
22C22×C8×D53C2211yescharacteristic
23C2×#13133156C2211yescharacteristic
24C2×#13133156C2211yescharacteristic
25C22⋊#11719908C411yescharacteristic, complemented
26#16810402C411yescharacteristic, complemented
27#16810402C411yescharacteristic, complemented
28C53×C22⋊C16C411yescharacteristic, 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#13133292D412yes—
37#13133292D412yes—
38#13133291—24no—
39D53⋊C16—24no—
40C2×#11719910C2×C412yes—
41C2×#11719910C2×C412yes—
42#13133274—21no—
43C22⋊#88324—21no—
44#13133271—21no—
45#13133271—21no—
46#13133271—21no—
47#13133271—21no—
48(C2×#6157)⋊C4—21no—
49C22×C4×D53C2×C411yescharacteristic
50C22×#88322C2×C411yescharacteristic
51C2×C8×D53—21no—
52C2×C8×D53—21no—
53C2×C8×D53—21no—
54C2×C8×D53—21no—
55C2×C8×D53C2×C411yescharacteristic
56C2×C8×D53C2×C411yescharacteristic
57C2×C8×D53C2×C411yescharacteristic
58C2×C8×D53C2311yescharacteristic
59#13133156—21no—
60#13133156—21no—
61#13133156—21no—
62#13133156—21no—
63C2×#11719908C2×C411yescharacteristic
64C2×#11719908C2×C411yescharacteristic
65C22×C8×C53C2×C411yescharacteristic
66C2×C16×C53C2×C411yescharacteristic
67C2×C16×C53C2×C411yescharacteristic
68C23×D53M1611yescharacteristic
69C22×#6157—21no—
70C22×#6157—21no—
71C22×C4×C53C4211yescharacteristic
72C22×Dic53M1611yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53C4211yescharacteristic
78C2×C4×D53M1611yescharacteristic
79C2×C4×D53M1611yescharacteristic
80C2×C4×D53C22×C411yescharacteristic
81C22⋊#6157—42no—
82C2×#88324—21no—
83C2×#88324—21no—
84C4×#6157—21no—
85C4×#6157—21no—
86C4×#6157—21no—
87C4×#6157—21no—
88D53⋊C8—21no—
89D53⋊C8—21no—
90D53⋊C8—21no—
91D53⋊C8—21no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53C4211yescharacteristic
95C2×C8×C53C22×C411yescharacteristic
96C2×#88322—21no—
97C2×#88322—21no—
98C2×#88322C4211yescharacteristic
99C2×#88322C22×C411yescharacteristic
100C8×D53—21no—
101C8×D53—21no—
102C8×D53—21no—
103C8×D53—21no—
104C8×D53—21no—
105C8×D53—21no—
106C8×D53C2×D411yescharacteristic
107C8×D53C2×D411yescharacteristic
108C8×D53#4111yescharacteristic
109C8×D53#4111yescharacteristic
110#11719910—22no—
111#11719910C22⋊C414yescomplemented
112C16×C53—21no—
113C16×C53—21no—
114#11719908—21no—
115#11719908—21no—
116C23×C53#9711yescharacteristic
117C22×D53—22no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—41no—
121C22×D53#9711yescharacteristic
122C22×D53C2×M1611yescharacteristic
123C2×#6157—41no—
124C2×#6157—41no—
125C2×#6157—41no—
126C2×#6157—41no—
127C2×#6157—41no—
128C2×#6157—41no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53#9711yescharacteristic
132C2×C4×C53C2×C4211yescharacteristic
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53#9711yescharacteristic
136C2×Dic53C2×M1611yescharacteristic
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53C4×D411yescharacteristic
144C4×D53C4×D411yescharacteristic
145C4×D53#13111yescharacteristic
146C4×D53#13111yescharacteristic
147#88324—41no—
148#88324—41no—
149C8×C53—21no—
150C8×C53C4×D411yescharacteristic
151C8×C53C4×D411yescharacteristic
152#88322—21no—
153#88322C2×C22⋊C411yescharacteristic
154#88322#11711yescharacteristic
155#12616—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53C2×#9711yescharacteristic
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—42no—
165C2×D53—42no—
166C2×D53#34712yescomplemented
167C2×D53#43511yescharacteristic
168C2×D53#43511yescharacteristic
169#6157—81no—
170#6157—81no—
171C4×C53—21no—
172C4×C53C4×C22⋊C411yescharacteristic
173C4×C53#40511yescharacteristic
174Dic53—21no—
175Dic53#43511yescharacteristic
176Dic53#43511yescharacteristic
177C22×M32—531no—
178#2112—531no—
179C2×#347—531no—
180C2×#347—531no—
181#1752—531no—
182#1375—531no—
183#1375—531no—
184C2×C53—41no—
185C2×C53#175211yescharacteristic
186C2×C53C2×#34711yescharacteristic
187C2×C53#211011yescharacteristic
188D53—21no—
189D53—21no—
190D53—22no—
191C23×C8—531no—
192C2×M32—531no—
193C2×M32—531no—
194C2×M32—532no—
195C2×M32—531no—
196C2×M32—531no—
197C2×M32—1062no—
198C2×M32—1062no—
199C22×C16—532no—
200C2×C22⋊C8—531no—
201C2×#97—531no—
202C2×#97—531no—
203C4×C22⋊C4—531no—
204C22⋊C16—531no—
205C22⋊C16—531no—
206C22⋊C16—531no—
207C22⋊C16#615711yescharacteristic, complemented
208#347—1062no—
209#347—1062no—
210#336—531no—
211#336—531no—
212C53#1261611yescharacteristic, complemented, Sylow
213C23×C4—531no—
214C22×C8—531no—
215C22×C8—531no—
216C22×C8—531no—
217C22×C8—531no—
218C22×C8—531no—
219C22×C8—531no—
220C22×C8—531no—
221C22×C8—1061no—
222C22×C8—1061no—
223C22×C8—1061no—
224C22×C8—1061no—
225C22×C8C2×#615711yescharacteristic
226C2×C22⋊C4—1061no—
227C2×C42—531no—
228C2×C42—531no—
229M32—532no—
230M32—532no—
231M32—1061no—
232M32—1061no—
233M32—1061no—
234M32—1061no—
235M32—1064no—
236M32—1061no—
237M32—1061no—
238C2×C16—532no—
239C2×C16—532no—
240C2×C16—531no—
241C2×C16—531no—
242C2×C16—1064no—
243C2×C16C2×#615711yescharacteristic
244C2×C16C2×#615711yescharacteristic
245C22⋊C8—1061no—
246C22⋊C8—1061no—
247#97—1061no—
248#97—1061no—
249#97—1061no—
250#97—1061no—
251#95—1061no—
252C24—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—531no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4C4×#615711yescharacteristic
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8—531no—
271C2×C8—531no—
272C2×C8—531no—
273C2×C8—1061no—
274C2×C8—1061no—
275C2×C8—1061no—
276C2×C8—1061no—
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×C8C4×#615711yescharacteristic
288C2×C8C22×#615711yescharacteristic
289C22⋊C4—2122no—
290C42—1061no—
291C42—1061no—
292C42—1061no—
293C42—1061no—
294C16—21no—
295C16—21no—
296C16—534no—
297C16—1062no—
298C16—1061no—
299C16—1061no—
300C23—531no—
301C23—531no—
302C23—1062no—
303C23—1061no—
304C23—1061no—
305C23—2121no—
306C23#1313327111yescharacteristic
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4#1313327111yescharacteristic
330C2×C4C2×C4×#615711yescharacteristic
331C8—21no—
332C8—531no—
333C8—531no—
334C8—1061no—
335C8—2121no—
336C8—2121no—
337C8D4×#615711yescharacteristic
338C8D4×#615711yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—532no—
343C22—531no—
344C22—531no—
345C22—1061no—
346C22—1061no—
347C22—1061no—
348C22—1061no—
349C22—2122no—
350C22—2122no—
351C22C2×#1313327111yescharacteristic
352C4—21no—
353C4—531no—
354C4—531no—
355C4—1061no—
356C4—4241no—
357C4—4241no—
358C4C22⋊C4×#615711yescharacteristic
359C4#1681158911yescharacteristic
360C2—41no—
361C2—1061no—
362C2—1061no—
363C2—1062no—
364C2#2029045411yescharacteristic
365C2#2029093011yescharacteristic
366C2#2029093411yescharacteristic
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#12616
LCSG ▷ C2×C4×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#16811514C2semidirect
3. C2×#16811513C2semidirect
5. #20289755C2semidirect
6. #20289755C2semidirect
12. C2×#13133291C4semidirect
13. C2×D53⋊C16C4semidirect
25. C22⋊#11719908C4semidirect
26. #16810402C4semidirect
27. #16810402C4semidirect
28. C53×C22⋊C16C4semidirect
111. #11719910C22⋊C4semidirect
166. C2×D53#347semidirect
207. C22⋊C16#6157semidirect
212. C53#12616semidirect

Maximal subgroups

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

subgroupindex
C2×D53⋊M322
C2×#168115142
C2×#168115132
#202904512
#202897552
#202763242
#1261653