Finite Group Databasebeta

← #27302317

Subgroups — #27302317

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811006C211yesmaximal, characteristic
2C2×#16810428C211yesmaximal, characteristic, complemented
3C2×#16810426C211yesmaximal, characteristic, complemented
4C2×#16810424C211yesmaximal, characteristic
5C2×#16810398C211yesmaximal, characteristic, complemented
6#20274873C211yesmaximal, characteristic, complemented
7C2×C53×#357C211yesmaximal, characteristic
8C22×Dic212C2211yescharacteristic
9C22×#11719923—21no—
10C2×#13133204—21no—
11C22×#11719908C2211yescharacteristic
12#16811006—22no—
13#16811006C414yescomplemented
14C2×#13133151C2211yescharacteristic
15C2×#13133150—21no—
16C2×C8×Dic53C2211yescharacteristic
17#16810492—21no—
18#16810428—22no—
19#16810426—22no—
20#16810424—22no—
21#16810398—22no—
22C22×D8×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C53×#357—22no—
26C22×Dic106—21no—
27C2×Dic212—21no—
28C2×Dic212—21no—
29C2×Dic212—21no—
30C2×Dic212—21no—
31C2×Dic212—21no—
32C2×Dic212C2×C412yes—
33C2×#11719923—24no—
34C2×#11719923—42no—
35C2×#11719923—42no—
36C22×#88322D411yescharacteristic
37#13133204—24no—
38#13133204—42no—
39C2×C4⋊Dic53—21no—
40C2×#11719919—42no—
41C2×C22⋊Dic53—42no—
42C2×C4×Dic53D411yescharacteristic
43C2×#11719908—21no—
44C2×#11719908—21no—
45C2×#11719908D412yes—
46#13133151—21no—
47#13133151—21no—
48#13133150—42no—
49C8×Dic53C2×C412yes—
50C8×Dic53D412yes—
51C22×D4×C53—21no—
52C2×D8×C53—21no—
53C2×D8×C53—21no—
54C2×D8×C53—21no—
55C2×D8×C53—21no—
56C2×D8×C53—21no—
57C2×D8×C53C2×C412yes—
58C22×C8×C53C2311yescharacteristic
59C2×C4⋊C4×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53D412yes—
63#107×C53—21no—
64#107×C53—21no—
65C24×C53—41no—
66C2×D4×C53—22no—
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53D812yes—
77C2×Dic106—22no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83D8×C53—22no—
84D8×C53—22no—
85D8×C53—41no—
86D8×C53—41no—
87C2×C8×C53C22×C411yescharacteristic
88C2×C8×C53C2×D411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53#4111yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C22⋊Dic53—48no—
96#11719923—48no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99#11719919—84no—
100C4×Dic53C22⋊C412yes—
101C4×Dic53SD1612yes—
102C2×#88322C22⋊C412yes—
103C2×#88322D812yes—
104C2×#88322SD1612yes—
105Dic212—22no—
106Dic212—22no—
107Dic212—41no—
108Dic212—41no—
109C16×C53—22no—
110#11719908—22no—
111C23×C53—42no—
112C23×C53—81no—
113C23×C53—81no—
114C23×C53—82no—
115C23×C53—82no—
116C23×C53C2×D811yescharacteristic
117D4×C53—42no—
118D4×C53—42no—
119D4×C53—81no—
120D4×C53—81no—
121C2×C4×C53—81no—
122C2×C4×C53—82no—
123C2×C4×C53C2×C22⋊C411yescharacteristic
124C2×C4×C53#12411yescharacteristic
125C2×C4×C53#12711yescharacteristic
126C2×C4×C53C2×D811yescharacteristic
127C2×C4×C53C2×SD1611yescharacteristic
128C2×C4×C53C2×SD1611yescharacteristic
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—42no—
132C2×Dic53—81no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135C2×Dic53—82no—
136C2×Dic53#10214yes—
137Dic106—42no—
138Dic106—42no—
139Dic106—81no—
140Dic106—81no—
141C8×C53C4×D411yescharacteristic
142C8×C53C4×D411yescharacteristic
143C8×C53#12111yescharacteristic
144C8×C53#12111yescharacteristic
145#88322#10214yes—
146#13205—531nomaximal, Sylow
147C22×C53—84no—
148C22×C53—82no—
149C22×C53—162no—
150C22×C53—161no—
151C22×C53—161no—
152C22×C53C2×#10211yescharacteristic
153C22×C53#48611yescharacteristic
154C22×C53#49311yescharacteristic
155C22×C53#50912yes—
156C22×C53#51012yes—
157C4×C53—161no—
158C4×C53#39011yescharacteristic
159C4×C53C2×#10211yescharacteristic
160C4×C53#48611yescharacteristic
161C4×C53#49211yescharacteristic
162Dic53—24no—
163Dic53—82no—
164Dic53—161no—
165C2×#495—531no—
166C2×#358—531no—
167C2×#358—531no—
168C2×#357—531no—
169C2×#357D5311yescharacteristic
170C2×#346—531no—
171#1845—531no—
172C2×C53—162no—
173C2×C53#195711yescharacteristic
174C2×C53#213912yes—
175C2×C53#224312yes—
176C2×C53#225011yescharacteristic
177C2×C53#225111yescharacteristic
178C22×Q16—531no—
179C22×SD16—1061no—
180C22×D8C2×D5311yescharacteristic
181C2×#124—1061no—
182C22×C16—531no—
183C22×C16C2×D5311yescharacteristic
184#495—534no—
185#495—1062no—
186C2×#107—531no—
187C2×#107—1061no—
188C2×#107C2×D5311yescharacteristic
189C2×C4×C8—531no—
190#384—1061no—
191#358—1062no—
192#358—1062no—
193#357—22no—
194#357—1062no—
195#346—1062no—
196C53#1320511yescharacteristic, complemented, Sylow
197C22×Q8—1061no—
198C22×D4—21no—
199C2×Q16—532no—
200C2×Q16—1061no—
201C2×Q16—1061no—
202C2×Q16—1061no—
203C2×Q16—1061no—
204C2×Q16—1061no—
205C2×SD16—1064no—
206C2×SD16—2122no—
207C2×SD16—2122no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8C4×D5312yes—
214C22×C8—531no—
215C22×C8C22×D5311yescharacteristic
216#124—1064no—
217#124—2122no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—2122no—
221C2×C22⋊C4—2122no—
222C2×C42—531no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—532no—
226C2×C16—1061no—
227C2×C16—1061no—
228C2×C16C22⋊D5312yes—
229#107—21no—
230#107—21no—
231#107—1061no—
232#107—1061no—
233#107—2122no—
234C4×C8—532no—
235C4×C8—532no—
236C24—41no—
237C2×Q8—1062no—
238C2×Q8—2121no—
239C2×Q8—2121no—
240C2×Q8—2121no—
241C2×Q8—2121no—
242C2×Q8—2121no—
243C2×D4—22no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—532no—
251C22×C4—2121no—
252C22×C4—2121no—
253C22×C4D4×D5311yescharacteristic
254Q16—1062no—
255Q16—1062no—
256Q16—2121no—
257Q16—2121no—
258SD16—2128no—
259D8—22no—
260D8—22no—
261D8—41no—
262D8—41no—
263C2×C8—532no—
264C2×C8—532no—
265C2×C8—532no—
266C2×C8C2×C4×D5311yescharacteristic
267C2×C8#1171994511yescharacteristic
268C2×C8D4×D5311yescharacteristic
269C2×C8D4×D5311yescharacteristic
270C2×C8#1171994711yescharacteristic
271C2×C8C2×C22⋊D5311yescharacteristic
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—4244no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C22⋊C4—2128no—
278C42—532no—
279C42—532no—
280C16—22no—
281C16—1062no—
282C23—42no—
283C23—82no—
284C23—82no—
285C23—81no—
286C23—81no—
287C23D8×D5311yescharacteristic
288Q8—4241no—
289Q8—4241no—
290Q8—2122no—
291Q8—2122no—
292D4—42no—
293D4—42no—
294D4—81no—
295D4—81no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—534no—
299C2×C4—1062no—
300C2×C4—1062no—
301C2×C4—4242no—
302C2×C4—4241no—
303C2×C4—4241no—
304C2×C4—4241no—
305C2×C4—2122no—
306C2×C4C22⋊C4×D5311yescharacteristic
307C2×C4#1313320311yescharacteristic
308C2×C4#1313321311yescharacteristic
309C2×C4D8×D5311yescharacteristic
310C2×C4SD16×D5311yescharacteristic
311C2×C4SD16×D5311yescharacteristic
312C8—534no—
313C8#1313317711yescharacteristic
314C8C4×C22⋊D5311yescharacteristic
315C8#1313318911yescharacteristic
316C8#1313319011yescharacteristic
317C22—82no—
318C22—84no—
319C22—162no—
320C22—161no—
321C22—161no—
322C22#102×D5311yescharacteristic
323C22#1681096811yescharacteristic
324C22#1681099811yescharacteristic
325C22#1681105712yes—
326C22#1681106212yes—
327C4—161no—
328C4—8481no—
329C4—1064no—
330C4—4242no—
331C4#1681052311yescharacteristic
332C4#102×D5311yescharacteristic
333C4#1681096811yescharacteristic
334C4#1681099411yescharacteristic
335C2—162no—
336C2#2027551911yescharacteristic
337C2#2027642412yes—
338C2#2027695012yes—
339C2#2027697811yescharacteristic
340C2#2027698511yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#357C2
socleC23×C53C2×D8
prime radicalC22×D8×C53C22
cosocleC22×C8×C53C23
hypercenterC2×#357D53
hypocenterC53#13205
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#16810428C2semidirect
3. C2×#16810426C2semidirect
5. C2×#16810398C2semidirect
6. #20274873C2semidirect
13. #16811006C4semidirect
196. C53#13205semidirect

Maximal subgroups

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

subgroupindex
C2×#168110062
C2×#168104282
C2×#168104262
C2×#168104242
C2×#168103982
#202748732
C2×C53×#3572
#1320553