Finite Group Databasebeta

← #27303440

Subgroups — #27303440

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811008C211yesmaximal, characteristic
2C2×#16810427C211yesmaximal, characteristic, complemented
3C2×#16810425C211yesmaximal, characteristic
4C2×#16810423C211yesmaximal, characteristic
5#20275449C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C2×C53×#358C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C22×D424C2211yescharacteristic
10C2×#13133206—21no—
11C22×#11719908C2211yescharacteristic
12#16811008—22no—
13#16811008—22no—
14C2×#13133151C2211yescharacteristic
15C2×C8×Dic53C2211yescharacteristic
16#16810522—21no—
17#16810427—22no—
18#16810425—22no—
19#16810423—22no—
20#16810385—21no—
21C22×Q16×C53C2211yescharacteristic
22C22×C16×C53C2211yescharacteristic
23C2×#107×C53C2211yescharacteristic
24C53×#358—22no—
25C22×D212—21no—
26C2×#11719924—24no—
27C2×#11719924—42no—
28C2×#11719924—42no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424D412yes—
35C22×#88322D411yescharacteristic
36#13133206—24no—
37#13133206—42no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—41no—
40C2×#11719921—41no—
41C2×#11719921—41no—
42C2×C4×Dic53D411yescharacteristic
43C2×#11719908—21no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46#13133151—21no—
47#13133151—21no—
48C8×Dic53—21no—
49C8×Dic53—21no—
50#13133124—41no—
51C22×Q8×C53—21no—
52C2×Q16×C53—21no—
53C2×Q16×C53—21no—
54C2×Q16×C53—21no—
55C2×Q16×C53—21no—
56C2×Q16×C53—21no—
57C2×Q16×C53D412yes—
58C22×C8×C53C2311yescharacteristic
59C2×C4⋊C4×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53—21no—
63#107×C53—21no—
64#107×C53D412yes—
65C23×D53—41no—
66C2×Q8×C53—22no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—41no—
76C22×Dic53D811yescharacteristic
77C22×Dic53D811yescharacteristic
78C2×D212—22no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84Q16×C53—22no—
85Q16×C53—22no—
86Q16×C53—41no—
87Q16×C53—41no—
88C2×C8×C53C2×D411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53#4111yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C4⋊C4×C53—22no—
95C4⋊C4×C53—41no—
96#11719924—48no—
97#11719921—44no—
98#11719921—44no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53—82no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C2×#88322—21no—
105C2×#88322—21no—
106C2×#88322SD1612yes—
107D424—22no—
108D424—22no—
109D424—41no—
110D424—41no—
111C16×C53—41no—
112#11719908—41no—
113C23×C53C2×D811yescharacteristic
114C22×D53—42no—
115C22×D53—81no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53—81no—
119C22×D53—81no—
120C22×D53—81no—
121Q8×C53—42no—
122Q8×C53—42no—
123Q8×C53—81no—
124Q8×C53—81no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12411yescharacteristic
134C2×C4×C53C2×SD1611yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146D212—42no—
147D212—42no—
148D212—81no—
149D212—81no—
150C8×C53#12111yescharacteristic
151C8×C53#12111yescharacteristic
152C8×C53#12311yescharacteristic
153C8×C53#12311yescharacteristic
154#88322—22no—
155#13408—531nomaximal, Sylow
156C22×C53#44711yescharacteristic
157C22×C53#45911yescharacteristic
158C22×C53#48611yescharacteristic
159C22×C53#50811yescharacteristic
160C22×C53#50811yescharacteristic
161C22×C53#50911yescharacteristic
162C22×C53#51011yescharacteristic
163C2×D53—82no—
164C2×D53—82no—
165C2×D53—82no—
166C2×D53—161no—
167C2×D53—161no—
168C2×D53—161no—
169C2×D53—161no—
170C4×C53—82no—
171C4×C53—82no—
172C4×C53#39411yescharacteristic
173C4×C53#45011yescharacteristic
174C4×C53#46011yescharacteristic
175C4×C53#49011yescharacteristic
176Dic53—41no—
177Dic53—41no—
178Dic53—161no—
179C2×#495—531no—
180C2×#358—531no—
181C2×#358D5311yescharacteristic, complemented
182C2×#357—531no—
183C2×#357—531no—
184#1942—531no—
185#1374—531no—
186C2×C53#204111yescharacteristic
187C2×C53#218611yescharacteristic
188C2×C53#218911yescharacteristic
189C2×C53#221011yescharacteristic
190C2×C53#221111yescharacteristic
191C2×C53#224211yescharacteristic
192C2×C53#224311yescharacteristic
193D53—162no—
194C22×Q16C2×D5311yescharacteristic
195C22×SD16—1061no—
196C22×D8—531no—
197C2×#124—1061no—
198C22×C16—531no—
199C22×C16C2×D5311yescharacteristic
200#495—1062no—
201#495—1062no—
202C2×#107—531no—
203C2×#107C2×D5311yescharacteristic
204C2×C4×C8—531no—
205#389—1061no—
206#358—22no—
207#358—1062no—
208#357—1062no—
209#357—1062no—
210#340—1061no—
211C53#1340811yescharacteristic, complemented, Sylow
212C22×Q8—21no—
213C22×D4—1061no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16D21212yes—
220C2×SD16—1064no—
221C2×SD16—2122no—
222C2×SD16—2122no—
223C2×D8—532no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8—1061no—
228C2×D8—1061no—
229C22×C8—531no—
230C22×C8C22×D5311yescharacteristic
231#124—1064no—
232#124—2122no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—2121no—
236C2×C22⋊C4—2121no—
237C2×C22⋊C4—2121no—
238C2×C42—531no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—1061no—
243C2×C16—1061no—
244C2×C16—1061no—
245#107—21no—
246#107—1061no—
247#107—1061no—
248#107C22⋊D5312yes—
249C4×C8—1061no—
250C4×C8—1061no—
251#95—2121no—
252C24—2121no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—1062no—
260C2×D4—2121no—
261C2×D4—2121no—
262C2×D4—2121no—
263C2×D4—2121no—
264C2×D4—2121no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—2121no—
270C22×C4D4×D5311yescharacteristic
271Q16—22no—
272Q16—22no—
273Q16—41no—
274Q16—41no—
275SD16—2128no—
276D8—1062no—
277D8—1062no—
278D8—2121no—
279D8—2121no—
280C2×C8—532no—
281C2×C8—1061no—
282C2×C8—1061no—
283C2×C8C2×D21211yescharacteristic
284C2×C8#1171994511yescharacteristic
285C2×C8D4×D5311yescharacteristic
286C2×C8#1171994711yescharacteristic
287C2×C8#1171994711yescharacteristic
288C2×C8C2×C22⋊D5311yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—4242no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C22⋊C4—2124no—
295C22⋊C4—2124no—
296C42—1061no—
297C42—1061no—
298C16—41no—
299C16—2121no—
300C23—4241no—
301C23—4241no—
302C23—4241no—
303C23—4241no—
304C23—4241no—
305C23—4241no—
306C23—2122no—
307C23D8×D5311yescharacteristic
308Q8—42no—
309Q8—42no—
310Q8—81no—
311Q8—81no—
312D4—4241no—
313D4—4241no—
314D4—2122no—
315D4—2122no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—4241no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4#1313318711yescharacteristic
332C2×C4#1313318911yescharacteristic
333C2×C4#1313320311yescharacteristic
334C2×C4SD16×D5311yescharacteristic
335C2×C4#1313324111yescharacteristic
336C2×C4#1313324411yescharacteristic
337C8—1062no—
338C8#1313318811yescharacteristic
339C8#1313319111yescharacteristic
340C8#1313319911yescharacteristic
341C8#1313320111yescharacteristic
342C22—8481no—
343C22—8481no—
344C22—8481no—
345C22—8481no—
346C22—4242no—
347C22—4242no—
348C22—4242no—
349C22#1681075811yescharacteristic
350C22#1681081611yescharacteristic
351C22#1681096811yescharacteristic
352C22#1681105411yescharacteristic
353C22#1681105611yescharacteristic
354C22#1681106011yescharacteristic
355C22#1681106511yescharacteristic
356C4—82no—
357C4—82no—
358C4—8481no—
359C4—2121no—
360C4—2121no—
361C4#1681053911yescharacteristic
362C4#1681077511yescharacteristic
363C4#1681082111yescharacteristic
364C4#1681098611yescharacteristic
365C2—8482no—
366C2#2027594811yescharacteristic
367C2#2027664111yescharacteristic
368C2#2027665811yescharacteristic
369C2#2027677211yescharacteristic
370C2#2027677911yescharacteristic
371C2#2027694811yescharacteristic
372C2#2027695311yescharacteristic
subgroupG/H
centerC23D8×D53
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC2×C53×#358C2
socleC23×C53C2×D8
prime radicalC22×D424C22
hypercenterC2×#358D53
hypocenterC53#13408
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×#16810427C2semidirect
5. #20275449C2semidirect
6. #20272792C2semidirect
7. C2×C53×#358C2semidirect
181. C2×#358D53semidirect
211. C53#13408semidirect

Maximal subgroups

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

subgroupindex
C2×#168110082
C2×#168104272
C2×#168104252
C2×#168104232
#202754492
#202727922
C2×C53×#3582
#1340853