Finite Group Databasebeta

← #28301716

Subgroups — #28301716

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133281C211yesmaximal, characteristic
2#20290428C211yesmaximal, characteristic, complemented
3#20290007C211yesmaximal, characteristic
4#20289925C211yesmaximal, characteristic, complemented
5#20289546C211yesmaximal, characteristic, complemented
6#20289542C211yesmaximal, characteristic, complemented
7C8×#11719947C211yesmaximal, characteristic
8#16811642C2211yescharacteristic
9C2×#13133281C412yes—
10C4×#11719936C2211yescharacteristic
11C4×D53⋊C8C2211yescharacteristic
12C42×#6157C2211yescharacteristic
13#16811493C412yes—
14(C2×#6157)⋊C8—21no—
15(C2×#6157)⋊C8—21no—
16#16811473C412yescomplemented
17D4⋊#88324C412yescomplemented
18#88324⋊C8—21no—
19#16811464—21no—
20C4×#11719947C2211yescharacteristic
21C8×Dic106C411yescharacteristic, complemented
22#16810747C411yescharacteristic, complemented
23D4×#88322C411yescharacteristic, complemented
24C8×C22⋊D53—21no—
25#16810686—21no—
26#16810675C2211yescharacteristic
27#16810673—21no—
28C2×C8×Dic53—21no—
29C4×C8×D53C2211yescharacteristic
30C8×D4×C53C411yescharacteristic, complemented
31C2×#11719936C2×C412yes—
32C2×D53⋊C8—21no—
33C2×C4×#6157—21no—
34C2×C4×#6157—21no—
35C2×C4×#6157C2×C412yes—
36C4⋊#88324C2×C412yes—
37#13133281—24no—
38#13133274—21no—
39#13133272—21no—
40C4×#88324—21no—
41C4×#88324—21no—
42C4×#88324—21no—
43C4×#88324C2×C412yes—
44C2×#11719947C2×C411yescharacteristic
45C22×#88322—21no—
46C2×C8×D53D411yescharacteristic
47C2×C8×D53D411yescharacteristic
48C4×Dic106C2×C411yescharacteristic
49#13133182C2×C411yescharacteristic
50D4×Dic53C2×C411yescharacteristic
51C4×C22⋊D53—21no—
52#13133177—21no—
53#13133175C2×C411yescharacteristic
54#13133173—21no—
55C2×C4×Dic53—21no—
56C42×D53C2311yescharacteristic
57C4⋊#88322C2×C411yescharacteristic
58#13133146—21no—
59#13133146—21no—
60C22⋊#88322—21no—
61#13133129—21no—
62C4×#88322C2×C411yescharacteristic
63C8×Dic53—21no—
64C8×Dic53—21no—
65C8×Dic53—21no—
66C8×Dic53D411yescharacteristic
67C8×Dic53D411yescharacteristic
68C22×C8×C53—21no—
69C4×D4×C53C2×C411yescharacteristic
70C4⋊C8×C53C2×C411yescharacteristic
71C22⋊C8×C53—21no—
72C4×C8×C53C2×C411yescharacteristic
73C22×#6157—22no—
74C2×D4×C53C4211yescharacteristic
75C22×C4×C53—21no—
76C2×C22⋊D53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79#11719947—22no—
80#11719947M1614yes—
81C2×C4×D53C22×C411yescharacteristic
82C2×C4×D53C2×D411yescharacteristic
83C2×C4×D53#4111yescharacteristic
84C2×Dic106C4211yescharacteristic
85C2×#88324—22no—
86C2×#88324—22no—
87C2×#88324—22no—
88C4×#6157—24no—
89C4×#6157—22no—
90C4×#6157—22no—
91C4×#6157—41no—
92C4×#6157—41no—
93#11719936—22no—
94#11719936—22no—
95D53⋊C8—41no—
96D53⋊C8—41no—
97C2×C8×C53—21no—
98C2×C8×C53—42no—
99C2×C8×C53C22⋊C411yescharacteristic
100C2×C8×C53C22⋊C411yescharacteristic
101C4⋊C4×C53C4211yescharacteristic
102C22⋊C4×C53—21no—
103C42×C53C22×C411yescharacteristic
104C22⋊Dic53—21no—
105#11719921—21no—
106C4⋊Dic53C4211yescharacteristic
107#11719919—21no—
108#11719919—21no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53C22×C411yescharacteristic
113C4×Dic53C2×D411yescharacteristic
114C4×Dic53#4111yescharacteristic
115C2×#88322—21no—
116C2×#88322—42no—
117C2×#88322C22⋊C411yescharacteristic
118C2×#88322C22⋊C411yescharacteristic
119C8×D53—41no—
120C8×D53—41no—
121C23×C53—21no—
122C22×D53C4×D411yescharacteristic
123C2×#6157—24no—
124C2×#6157—42no—
125C2×#6157—42no—
126D4×C53—21no—
127D4×C53#9712yes—
128C2×C4×C53—21no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53C2×C4211yescharacteristic
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53#11711yescharacteristic
134C22⋊D53—24no—
135C2×Dic53—21no—
136C2×Dic53—24no—
137C2×Dic53—21no—
138C2×Dic53—42no—
139C2×Dic53C4×D411yescharacteristic
140C2×Dic53C4×D411yescharacteristic
141C2×Dic53C4×D411yescharacteristic
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—22no—
145C4×D53C2×M1612yes—
146C4×D53#13112yes—
147Dic106—21no—
148Dic106#9712yes—
149#88324—42no—
150#88324—42no—
151#88324—42no—
152C8×C53—21no—
153C8×C53—21no—
154C8×C53—41no—
155#88322—21no—
156#88322—21no—
157#88322—41no—
158#8193—531nomaximal, Sylow
159C22×C53—22no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53C4×C22⋊C411yescharacteristic
163C2×D53—21no—
164C2×D53—21no—
165C2×D53#43512yes—
166#6157—44no—
167C4×C53—21no—
168C4×C53—41no—
169C4×C53C2×#9711yescharacteristic
170C4×C53#40511yescharacteristic
171C4×C53#42111yescharacteristic
172C4×C53#42111yescharacteristic
173Dic53—22no—
174Dic53—21no—
175Dic53—41no—
176Dic53#43512yes—
177Dic53#44312yes—
178C8×#41—531no—
179C4×#104—531no—
180#1744—531no—
181#1476—531no—
182#1451—531no—
183#1278—531no—
184#1276—531no—
185C2×C53—42no—
186C2×C53#175211yescharacteristic
187C2×C53#175911yescharacteristic
188C2×C53#176411yescharacteristic
189D53—22no—
190C4×#41—531no—
191C8×Q8—531no—
192C8×Q8—531no—
193C8×D4—531no—
194C8×D4—1061no—
195C8×D4—1061no—
196C8×D4#615711yescharacteristic, complemented
197#431—531no—
198#431—531no—
199#431—1061no—
200C2×#104—532no—
201C4×M16—531no—
202C2×C4×C8—531no—
203C2×C4×C8—531no—
204C2×C4×C8—1061no—
205C43—531no—
206#339—532no—
207#336—1061no—
208#336—1061no—
209Q8⋊C8—532no—
210D4⋊C8—532no—
211#322—1061no—
212#322—1061no—
213C53#819311yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C2×M16—532no—
216C22×C8—21no—
217C22×C8—531no—
218C22×C8—531no—
219C22×C8—1061no—
220C22×C8—1061no—
221C4×Q8—531no—
222C4×Q8—531no—
223C4×D4—531no—
224C4×D4—1061no—
225C4×D4—1061no—
226C4×D4C2×#615711yescharacteristic
227#117—531no—
228#117—1061no—
229C2×C42—531no—
230C2×C42—532no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—1061no—
234C4⋊C8—532no—
235C4⋊C8—531no—
236C4⋊C8—1061no—
237C4⋊C8—1061no—
238C4⋊C8C2×#615711yescharacteristic
239#104—1064no—
240C22⋊C8—21no—
241C22⋊C8—1061no—
242C22⋊C8—1061no—
243C22⋊C8—1061no—
244#97—1061no—
245C4×C8—531no—
246C4×C8—531no—
247C4×C8—532no—
248C4×C8—531no—
249C4×C8—1061no—
250C4×C8—1061no—
251C4×C8—1061no—
252C4×C8—1061no—
253C4×C8—1061no—
254C4×C8—1061no—
255C4×C8C2×#615711yescharacteristic
256#41—534no—
257#41—1062no—
258C2×Q8—531no—
259C2×D4—1061no—
260C2×D4C4×#615711yescharacteristic
261C22×C4—21no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1062no—
268M16—1062no—
269M16—1062no—
270C2×C8—21no—
271C2×C8—42no—
272C2×C8—531no—
273C2×C8—531no—
274C2×C8—1062no—
275C2×C8—1062no—
276C2×C8—1062no—
277C2×C8—1061no—
278C2×C8—2121no—
279C2×C8—2121no—
280C2×C8—2122no—
281C2×C8—2121no—
282C2×C8—2121no—
283C2×C8C22⋊#615711yescharacteristic
284C2×C8C22⋊#615711yescharacteristic
285C4⋊C4—531no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4C4×#615711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C42—531no—
293C42—531no—
294C42—531no—
295C42—1061no—
296C42—1061no—
297C42—1062no—
298C42—1062no—
299C42—1064no—
300C42—1061no—
301C42—2121no—
302C42—2121no—
303C42C22×#615711yescharacteristic
304C23—21no—
305C23—531no—
306Q8—532no—
307Q8—1061no—
308D4—21no—
309D4—1064no—
310D4#1313327112yes—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—532no—
317C2×C4—532no—
318C2×C4—531no—
319C2×C4—1062no—
320C2×C4—1064no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1064no—
325C2×C4—1061no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4C2×C4×#615711yescharacteristic
330C2×C4C2×C22⋊#615711yescharacteristic
331C2×C4#1313329611yescharacteristic
332C8—21no—
333C8—21no—
334C8—41no—
335C8—1061no—
336C8—1061no—
337C8—2122no—
338C8—2122no—
339C8—2122no—
340C8—2121no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—532no—
345C22—1061no—
346C22—1061no—
347C22C4×C22⋊#615711yescharacteristic
348C4—21no—
349C4—41no—
350C4—532no—
351C4—532no—
352C4—1062no—
353C4—1061no—
354C4—2124no—
355C4—2121no—
356C4C2×#1313327111yescharacteristic
357C4#1681158611yescharacteristic
358C4#1681162011yescharacteristic
359C4#1681162111yescharacteristic
360C2—42no—
361C2—1062no—
362C2#2029045111yescharacteristic
363C2#2029047811yescharacteristic
364C2#2029048911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC42C22×#6157
FittingC8×D4×C53C4
socleC22×C53C4×C22⋊C4
prime radicalC2×#11719947C2×C4
cosocleC42×D53C23
hypercenterC8×D4#6157
hypocenterC53#8193
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
2. #20290428C2semidirect
4. #20289925C2semidirect
5. #20289546C2semidirect
6. #20289542C2semidirect
16. #16811473C4semidirect
17. D4⋊#88324C4semidirect
21. C8×Dic106C4semidirect
22. #16810747C4semidirect
23. D4×#88322C4semidirect
30. C8×D4×C53C4semidirect
196. C8×D4#6157semidirect
213. C53#8193semidirect

Maximal subgroups

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

subgroupindex
C4×#131332812
#202904282
#202900072
#202899252
#202895462
#202895422
C8×#117199472
#819353