Finite Group Databasebeta

← #27302304

Subgroups — #27302304

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810998C211yesmaximal, characteristic
2C2×#16810425C211yesmaximal, characteristic, complemented
3C2×#16810424C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic, complemented
5C2×C16×Dic53C211yesmaximal, characteristic, complemented
6#20274873C211yesmaximal, characteristic, complemented
7C2×C53×#357C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×#11719922—21no—
10C2×#13133213—21no—
11C22×#11719908C2211yescharacteristic
12#16810998—22no—
13#16810998C414yescomplemented
14C2×#13133151C2211yescharacteristic
15C2×#13133150—21no—
16C2×C8×Dic53C2211yescharacteristic
17#16810492—21no—
18#16810425—22no—
19#16810424—22no—
20#16810423—22no—
21C16×Dic53—22no—
22C22×D8×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C53×#357—22no—
26C22×D212—21no—
27C22×C22⋊D53—42no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424C2×C412yes—
34C2×#11719922—24no—
35C2×#11719922—42no—
36C2×#11719922—42no—
37C22×#88322D411yescharacteristic
38#13133213—24no—
39#13133213—42no—
40C2×C4⋊Dic53—21no—
41C2×#11719919—42no—
42C2×C4×Dic53D411yescharacteristic
43C2×#11719908—21no—
44C2×#11719908—21no—
45C2×#11719908D412yescomplemented
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×C53D412yescomplemented
63#107×C53—21no—
64#107×C53—21no—
65C24×C53—41no—
66C23×D53—41no—
67C2×D4×C53—22no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C2×C22⋊D53—48no—
76C2×C22⋊D53—84no—
77C2×C22⋊D53—84no—
78C22×Dic53—41no—
79C22×Dic53D812yes—
80C2×D212—22no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86D8×C53—22no—
87D8×C53—22no—
88D8×C53—41no—
89D8×C53—41no—
90C2×C8×C53C22×C411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53#4111yescharacteristic
95C2×C8×C53#4111yescharacteristic
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98#11719922—48no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101#11719919—84no—
102C4×Dic53C22⋊C412yes—
103C4×Dic53SD1612yes—
104C2×#88322C22⋊C412yes—
105C2×#88322D812yes—
106C2×#88322SD1612yes—
107D424—22no—
108D424—22no—
109D424—41no—
110D424—41no—
111C16×C53—22no—
112#11719908—22no—
113C23×C53—42no—
114C23×C53—81no—
115C23×C53—81no—
116C23×C53—82no—
117C23×C53—82no—
118C23×C53C2×D811yescharacteristic
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C22×D53—82no—
123C22×D53—82no—
124D4×C53—42no—
125D4×C53—42no—
126D4×C53—81no—
127D4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53—82no—
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53#12411yescharacteristic
132C2×C4×C53#12711yescharacteristic
133C2×C4×C53C2×D811yescharacteristic
134C2×C4×C53C2×SD1611yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136C22⋊D53—816no—
137C2×Dic53—81no—
138C2×Dic53—82no—
139C2×Dic53#10214yes—
140C2×Dic53D1614yes—
141C2×Dic53SD3214yes—
142D212—42no—
143D212—42no—
144D212—81no—
145D212—81no—
146C8×C53C4×D411yescharacteristic
147C8×C53C4×D411yescharacteristic
148C8×C53#12111yescharacteristic
149C8×C53#12111yescharacteristic
150#88322#10214yes—
151#13202—531nomaximal, Sylow
152C22×C53—84no—
153C22×C53—82no—
154C22×C53—162no—
155C22×C53—161no—
156C22×C53—161no—
157C22×C53C2×#10211yescharacteristic
158C22×C53#48611yescharacteristic
159C22×C53#49311yescharacteristic
160C22×C53C2×D1612yes—
161C22×C53C2×SD3212yes—
162C2×D53—84no—
163C2×D53—82no—
164C2×D53—162no—
165C2×D53—161no—
166C2×D53—161no—
167C4×C53—161no—
168C4×C53#39011yescharacteristic
169C4×C53C2×#10211yescharacteristic
170C4×C53#48611yescharacteristic
171C4×C53#49211yescharacteristic
172Dic53—161no—
173Dic53#35718yescomplemented
174C2×#493—531no—
175C2×#357—531no—
176C2×#357—531no—
177C2×#357—531no—
178C2×#357D5311yescharacteristic, complemented
179C2×C4×C16—531no—
180#1845—531no—
181C2×C53—162no—
182C2×C53#195711yescharacteristic
183C2×C53C2×#35712yes—
184C2×C53#224012yes—
185C2×C53#224611yescharacteristic
186C2×C53#224811yescharacteristic
187D53—162no—
188C22×D8—531no—
189C22×D8—1061no—
190C22×D8C2×D5311yescharacteristic
191C2×#127—1061no—
192C22×C16—531no—
193C22×C16C2×D5311yescharacteristic
194#493—534no—
195#493—1062no—
196C2×#107—531no—
197C2×#107—1061no—
198C2×#107C2×D5311yescharacteristic
199C2×C4×C8—531no—
200#384—1061no—
201#357—22no—
202#357—1062no—
203#357—1062no—
204#357—1062no—
205C4×C16—1062no—
206C53#1320211yescharacteristic, complemented, Sylow
207C22×D4—21no—
208C22×D4—1061no—
209C22×D4—2122no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—532no—
216C2×D8—1064no—
217C2×D8—1061no—
218C2×D8—1061no—
219C2×D8—1061no—
220C2×D8—1061no—
221C2×D8—1061no—
222C2×D8—2122no—
223C2×D8—2122no—
224C2×D8C4×D5312yes—
225C22×C8—531no—
226C22×C8C22×D5311yescharacteristic
227#127—1064no—
228#127—2122no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—2122no—
232C2×C42—531no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—532no—
236C2×C16—1061no—
237C2×C16—1061no—
238C2×C16C22⋊D5312yescomplemented
239#107—21no—
240#107—21no—
241#107—1061no—
242#107—1061no—
243#107—2122no—
244C4×C8—532no—
245C4×C8—532no—
246C24—41no—
247C24—2121no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—1062no—
255C2×D4—4244no—
256C2×D4—4244no—
257C2×D4—2128no—
258C2×D4—2121no—
259C2×D4—2121no—
260C2×D4—2121no—
261C2×D4—2121no—
262C2×D4—2121no—
263C22×C4—41no—
264C22×C4—532no—
265C22×C4—2121no—
266C22×C4D4×D5311yescharacteristic
267D8—22no—
268D8—22no—
269D8—41no—
270D8—41no—
271D8—1062no—
272D8—1062no—
273D8—2128no—
274D8—2121no—
275D8—2121no—
276C2×C8—532no—
277C2×C8—532no—
278C2×C8—532no—
279C2×C8C2×C4×D5311yescharacteristic
280C2×C8#1171994511yescharacteristic
281C2×C8D4×D5311yescharacteristic
282C2×C8D4×D5311yescharacteristic
283C2×C8#1171994711yescharacteristic
284C2×C8C2×C22⋊D5311yescharacteristic
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—4244no—
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C42—532no—
291C42—532no—
292C16—22no—
293C16—1062no—
294C23—42no—
295C23—82no—
296C23—82no—
297C23—81no—
298C23—81no—
299C23—4242no—
300C23—4242no—
301C23—4241no—
302C23—4241no—
303C23—2122no—
304C23D8×D5311yescharacteristic
305D4—42no—
306D4—42no—
307D4—81no—
308D4—81no—
309D4—42416no—
310D4—4241no—
311D4—4241no—
312D4—2122no—
313D4—2122no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—534no—
317C2×C4—534no—
318C2×C4—534no—
319C2×C4—4242no—
320C2×C4—4241no—
321C2×C4C22⋊C4×D5311yescharacteristic
322C2×C4#1313320311yescharacteristic
323C2×C4#1313321311yescharacteristic
324C2×C4D8×D5311yescharacteristic
325C2×C4SD16×D5311yescharacteristic
326C2×C4SD16×D5311yescharacteristic
327C8—534no—
328C8#1313317711yescharacteristic
329C8C4×C22⋊D5311yescharacteristic
330C8#1313318911yescharacteristic
331C8#1313319011yescharacteristic
332C22—82no—
333C22—84no—
334C22—162no—
335C22—161no—
336C22—161no—
337C22—8482no—
338C22—8481no—
339C22—8481no—
340C22—4242no—
341C22—4244no—
342C22#102×D5311yescharacteristic
343C22#1681096811yescharacteristic
344C22#1681099811yescharacteristic
345C22D16×D5312yes—
346C22SD32×D5312yes—
347C4—161no—
348C4—538no—
349C4—8481no—
350C4#1681052311yescharacteristic
351C4#102×D5311yescharacteristic
352C4#1681096811yescharacteristic
353C4#1681099411yescharacteristic
354C2—162no—
355C2—8482no—
356C2#2027551911yescharacteristic
357C2#357×D5312yes—
358C2#2027693712yes—
359C2#2027696311yescharacteristic
360C2#2027696911yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#357C2
socleC23×C53C2×D8
prime radicalC2×#16810998C2
cosocleC22×C8×C53C23
hypercenterC2×#357D53
hypocenterC53#13202
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×#16810425C2semidirect
3. C2×#16810424C2semidirect
4. C2×#16810423C2semidirect
5. C2×C16×Dic53C2semidirect
6. #20274873C2semidirect
7. C2×C53×#357C2semidirect
13. #16810998C4semidirect
45. C2×#11719908D4semidirect
62. C2×C16×C53D4semidirect
173. Dic53#357semidirect
178. C2×#357D53semidirect
206. C53#13202semidirect
238. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109982
C2×#168104252
C2×#168104242
C2×#168104232
C2×C16×Dic532
#202748732
C2×C53×#3572
#1320253