Finite Group Databasebeta

← #27277258

Subgroups — #27277258

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#11719945C211yesmaximal, characteristic
2#20274732C212yesmaximal
3C4×#13133151C211yesmaximal, characteristic, complemented
4#20272616C212yesmaximal, complemented
5C53×#1567C211yesmaximal, characteristic, complemented
6C4×#11719945C2211yescharacteristic
7C8×Dic106C411yescharacteristic, complemented
8C8×Dic106C411yescharacteristic, complemented
9C8×D212C411yescharacteristic, complemented
10C8×D212C411yescharacteristic, complemented
11C8×C22⋊D53—21no—
12C8×C22⋊D53—21no—
13#16810676C2211yescharacteristic
14#16810674—21no—
15C2×#13133151—21no—
16C2×#13133143—22no—
17C4×C8×D53—21no—
18C4×C4⋊Dic53C2212yes—
19#16810386—21no—
20#16810384—22no—
21C53×#423C2212yes—
22C2×C4×C8×C53C2211yescharacteristic
23C53×#335—21no—
24C53×#335—21no—
25C2×#11719945D411yescharacteristic
26C2×C8×D53—21no—
27C2×C8×D53—21no—
28C4×Dic106C2×C411yescharacteristic
29C4×Dic106Q811yescharacteristic
30C4×D212C2×C411yescharacteristic
31C4×D212Q811yescharacteristic
32C4×C22⋊D53—21no—
33C4×C22⋊D53—21no—
34#13133176D411yescharacteristic
35#13133174—21no—
36C2×C4⋊Dic53—22no—
37C2×C4×Dic53—22no—
38C42×D53—21no—
39#13133151—24no—
40C4⋊#88322C2×C411yescharacteristic
41C4⋊#88322C2×C411yescharacteristic
42#13133146—21no—
43#13133146—21no—
44#13133143—44no—
45C22⋊#88322—21no—
46#13133129—21no—
47#13133129—21no—
48C4×#88322—21no—
49C8×Dic53—21no—
50C8×Dic53—21no—
51#13133124—22no—
52C2×M16×C53—22no—
53C22×C8×C53D411yescharacteristic
54C22×C8×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C4⋊C8×C53—22no—
58C4×C8×C53C2×C411yescharacteristic
59C4×C8×C53C2×C411yescharacteristic
60C4×C8×C53D411yescharacteristic
61C4×C8×C53D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53#4111yescharacteristic
65C2×C22⋊D53—21no—
66C22×Dic53—42no—
67#11719945—21no—
68#11719945—21no—
69#11719945D812yes—
70#11719945Q1612yes—
71C2×D212C4⋊C411yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×Dic106C4⋊C411yescharacteristic
76M16×C53—42no—
77M16×C53—42no—
78C2×C8×C53—22no—
79C2×C8×C53—41no—
80C2×C8×C53—42no—
81C2×C8×C53C22⋊C411yescharacteristic
82C2×C8×C53C22⋊C411yescharacteristic
83C2×C8×C53C22⋊C411yescharacteristic
84C2×C8×C53C22⋊C411yescharacteristic
85C42×C53C22×C411yescharacteristic
86C42×C53C2×D411yescharacteristic
87C42×C53C2×Q811yescharacteristic
88C42×C53#4111yescharacteristic
89C22⋊Dic53—21no—
90#11719921—21no—
91#11719921—21no—
92C4⋊Dic53—24no—
93C4⋊Dic53—42no—
94C4⋊Dic53C4⋊C411yescharacteristic
95C4⋊Dic53C4⋊C411yescharacteristic
96#11719919—21no—
97#11719919—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—21no—
101C4×Dic53—42no—
102C4×Dic53—42no—
103C2×#88322—21no—
104C2×#88322—21no—
105C8×D53—42no—
106C8×D53—42no—
107C23×C53#12111yescharacteristic
108C22×D53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53C2×C22⋊C411yescharacteristic
113C2×C4×C53C2×C4⋊C411yescharacteristic
114C2×C4×C53#11711yescharacteristic
115C2×C4×C53#12111yescharacteristic
116C2×C4×C53C22⋊Q811yescharacteristic
117C2×C4×C53C22⋊Q811yescharacteristic
118C2×C4×C53C2×D811yescharacteristic
119C2×C4×C53C2×Q1611yescharacteristic
120C2×C4×C53#13511yescharacteristic
121C2×C4×C53#13511yescharacteristic
122C22⋊D53—22no—
123C22⋊D53—22no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—44no—
128C2×Dic53—82no—
129C2×Dic53—82no—
130D212—21no—
131D212#10712yescomplemented
132C4×D53—22no—
133C4×D53—22no—
134C4×D53—42no—
135C4×D53—41no—
136C4×D53—41no—
137Dic106—21no—
138Dic106#10712yescomplemented
139C8×C53—21no—
140C8×C53—21no—
141C8×C53—21no—
142C8×C53—82no—
143C8×C53#10412yescomplemented
144#88322—41no—
145#88322—41no—
146#8667—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53#38011yescharacteristic
150C22×C53#46611yescharacteristic
151C22×C53#46711yescharacteristic
152C2×D53—22no—
153C2×D53—41no—
154C2×D53—41no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53C2×#10411yescharacteristic
159C4×C53#42111yescharacteristic
160C4×C53C2×#10711yescharacteristic
161C4×C53#42711yescharacteristic
162C4×C53#46611yescharacteristic
163C4×C53#46711yescharacteristic
164Dic53—22no—
165Dic53—41no—
166Dic53—41no—
167Dic53—84no—
168C8×#41—531no—
169#1806—532no—
170C4×#107—531no—
171#1567D5311yescharacteristic, complemented
172#1303—532no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53#180611yescharacteristic
176C2×C53#180811yescharacteristic
177C2×C53#181411yescharacteristic
178D53—42no—
179C4×#41—531no—
180C8×Q8—531no—
181C8×Q8—531no—
182C8×D4—531no—
183C8×D4—531no—
184C8×D4—1061no—
185C8×D4—1061no—
186#431—531no—
187#431—1061no—
188C2×#107—1061no—
189#423C2×D5312yes—
190C2×#104—1062no—
191C2×C4×C8—1061no—
192C2×C4×C8C2×D5311yescharacteristic
193C4×C4⋊C4—532no—
194#340—1061no—
195#339—1062no—
196#335—21no—
197#335—21no—
198C53#866711yescharacteristic, complemented, Sylow
199C2×#41—531no—
200C2×M16—22no—
201C22×C8—1061no—
202C22×C8—1061no—
203C22×C8D21211yescharacteristic
204C22×C8D21211yescharacteristic
205C4×Q8—531no—
206C4×Q8—531no—
207C4×D4—531no—
208C4×D4—531no—
209C4×D4—1061no—
210C4×D4—1061no—
211#117—531no—
212#117—1061no—
213C2×C4⋊C4—1062no—
214C2×C42—1061no—
215C2×C42—1062no—
216C2×C42C22×D5311yescharacteristic
217#107—1064no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—531no—
221C4⋊C8—531no—
222C4⋊C8—1061no—
223C4⋊C8—1061no—
224#104—2124no—
225C22⋊C8—1061no—
226C22⋊C8—1061no—
227C22⋊C8—1061no—
228C4×C8—1061no—
229C4×C8—1061no—
230C4×C8—1061no—
231C4×C8C4×D5311yescharacteristic
232C4×C8C4×D5311yescharacteristic
233C4×C8C22⋊D5311yescharacteristic
234C4×C8C22⋊D5311yescharacteristic
235#95—1062no—
236#41—532no—
237#41—532no—
238#41—1061no—
239#41—1061no—
240C2×Q8—531no—
241C2×D4—531no—
242C2×D4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—2122no—
247C22×C4C2×D21211yescharacteristic
248C22×C4D4×D5311yescharacteristic
249C22×C4#1171994911yescharacteristic
250M16—42no—
251M16—42no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—1061no—
256C2×C8—1061no—
257C2×C8—2122no—
258C2×C8—2122no—
259C2×C8#1171992111yescharacteristic
260C2×C8#1171992111yescharacteristic
261C2×C8#1171992111yescharacteristic
262C2×C8#1171992111yescharacteristic
263C4⋊C4—531no—
264C4⋊C4—531no—
265C4⋊C4—1064no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—2122no—
269C22⋊C4—1061no—
270C22⋊C4—1061no—
271C22⋊C4—1061no—
272C42—1061no—
273C42—1061no—
274C42—1061no—
275C42—2122no—
276C42—2122no—
277C42C2×C4×D5311yescharacteristic
278C42#1171994711yescharacteristic
279C42Q8×D5311yescharacteristic
280C42C2×C22⋊D5311yescharacteristic
281C23—1061no—
282C23#1313319211yescharacteristic
283Q8—532no—
284Q8—1061no—
285D4—532no—
286D4—1062no—
287D4—1062no—
288D4—1061no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—1061no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—4242no—
298C2×C4—4242no—
299C2×C4—2122no—
300C2×C4—2124no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4C2×#1171992111yescharacteristic
304C2×C4C4⋊C4×D5311yescharacteristic
305C2×C4#1313317511yescharacteristic
306C2×C4#1313318811yescharacteristic
307C2×C4#1313319311yescharacteristic
308C2×C4#1313319711yescharacteristic
309C2×C4D8×D5311yescharacteristic
310C2×C4Q16×D5311yescharacteristic
311C2×C4#1313323611yescharacteristic
312C2×C4#1313324011yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—82no—
317C8—2121no—
318C8—2121no—
319C8#1313314312yescomplemented
320C22—21no—
321C22—21no—
322C22—1062no—
323C22—2121no—
324C22—2121no—
325C22#1681048011yescharacteristic
326C22#1681086011yescharacteristic
327C22#1681086511yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—1062no—
332C4—4244no—
333C4—2121no—
334C4—2121no—
335C4C2×#1313314311yescharacteristic
336C4#1681063911yescharacteristic
337C4#107×D5311yescharacteristic
338C4#1681066011yescharacteristic
339C4#1681085611yescharacteristic
340C4#1681086111yescharacteristic
341C2—21no—
342C2—21no—
343C2—2122no—
344C2#2027473211yescharacteristic
345C2#2027474111yescharacteristic
346C2#2027477011yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1567C2
socleC22×C53#380
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1567D53
hypocenterC53#8667
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1567

Complements

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

NH ≅ G/Ntype
3. C4×#13133151C2semidirect
4. #20272616C2semidirect
5. C53×#1567C2semidirect
7. C8×Dic106C4semidirect
8. C8×Dic106C4semidirect
9. C8×D212C4semidirect
10. C8×D212C4semidirect
131. D212#107semidirect
138. Dic106#107semidirect
143. C8×C53#104semidirect
171. #1567D53semidirect
198. C53#8667semidirect
319. C8#13133143semidirect

Maximal subgroups

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

subgroupindex
C8×#117199452
#202747322
C4×#131331512
#202726162
C53×#15672
#866753