Finite Group Databasebeta

← #27301156

Subgroups — #27301156

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133160C211yesmaximal, characteristic
2C2×#16810427C211yesmaximal, characteristic, complemented
3C2×#16810423C211yesmaximal, characteristic
4#20275581C211yesmaximal, characteristic
5#20272806C211yesmaximal, characteristic, complemented
6#20272793C211yesmaximal, characteristic, complemented
7C2×C4×Q16×C53C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C2×#13133160—22no—
10C2×#13133160—22no—
11C2×#13133160C414yescomplemented
12C22×#11719908C2211yescharacteristic
13C22×#11719908C2211yescharacteristic
14C2×#13133151C2211yescharacteristic
15C2×#13133139—21no—
16#16810525—21no—
17#16810427—22no—
18#16810423—22no—
19C22×Q16×C53C2211yescharacteristic
20C2×C4×Q8×C53—21no—
21C4×Q16×C53—22no—
22C4×Q16×C53—22no—
23C2×#107×C53C2211yescharacteristic
24C2×#103×C53—21no—
25C2×C4×C8×C53C2211yescharacteristic
26C22×D212—21no—
27C2×D424—21no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424C2×C412yes—
33C2×C4⋊Dic53—21no—
34C2×#11719921—41no—
35C2×#11719921—41no—
36#13133160—28no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40C2×#11719908—21no—
41C2×#11719908—21no—
42C2×#11719908C2×C412yes—
43#13133151—21no—
44#13133151—21no—
45#13133139—42no—
46C22×Q8×C53—21no—
47C2×Q16×C53—21no—
48C2×Q16×C53—21no—
49C2×Q16×C53—21no—
50C2×Q16×C53C2×C412yes—
51C2×Q16×C53D412yes—
52C2×Q16×C53D412yes—
53C22×C8×C53D411yescharacteristic
54C22×C8×C53C2311yescharacteristic
55C4×Q8×C53—24no—
56C4×Q8×C53—42no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—41no—
59C2×C42×C53—41no—
60C2×C42×C53D411yescharacteristic
61#107×C53—21no—
62#107×C53D412yes—
63#103×C53—24no—
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66C23×D53—41no—
67C2×Q8×C53—22no—
68C2×Q8×C53—22no—
69C2×Q8×C53—22no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53D811yescharacteristic
76C22×C4×C53D811yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—41no—
79C2×D212—22no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85Q16×C53—22no—
86Q16×C53—22no—
87Q16×C53—22no—
88Q16×C53C22⋊C414yescomplemented
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53C22×C411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53#4111yescharacteristic
97C2×C8×C53#4111yescharacteristic
98C4⋊C4×C53—22no—
99C4⋊C4×C53—44no—
100C4⋊C4×C53—41no—
101C42×C53—21no—
102C42×C53—21no—
103C42×C53—44no—
104#11719921—82no—
105#11719921—82no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108D424—22no—
109D424—22no—
110D424—41no—
111D424—41no—
112#11719908—22no—
113#11719908—41no—
114C23×C53C2×D811yescharacteristic
115C22×D53—42no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53—81no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—81no—
122Q8×C53—24no—
123Q8×C53—42no—
124Q8×C53—42no—
125Q8×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53SD3212yescomplemented
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147D212—42no—
148D212—42no—
149D212—81no—
150D212—81no—
151C8×C53—41no—
152C8×C53C2×C22⋊C411yescharacteristic
153C8×C53C4×D411yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12311yescharacteristic
156#13013—531nomaximal, Sylow
157C22×C53C4×D811yescharacteristic
158C22×C53#44711yescharacteristic
159C22×C53#45911yescharacteristic
160C22×C53C2×SD3211yescharacteristic
161C22×C53#50811yescharacteristic
162C22×C53#50911yescharacteristic
163C22×C53#51011yescharacteristic
164C2×D53—82no—
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—161no—
168C2×D53—161no—
169C2×D53—161no—
170C2×D53—161no—
171C4×C53—22no—
172C4×C53—44no—
173C4×C53—41no—
174C4×C53—82no—
175C4×C53#38611yescharacteristic
176C4×C53#44211yescharacteristic
177C4×C53#44911yescharacteristic
178C4×C53#46311yescharacteristic
179Dic53—161no—
180C22×SD32—531no—
181C2×C4×Q16D5311yescharacteristic, complemented
182C2×#358—531no—
183C2×#357—531no—
184#1970—531no—
185#1380—531no—
186#1374—531no—
187C2×C53#187911yescharacteristic
188C2×C53C4×SD3211yescharacteristic
189C2×C53#217511yescharacteristic
190C2×C53#218511yescharacteristic
191C2×C53#218611yescharacteristic
192C2×C53#220711yescharacteristic
193C2×C53#221111yescharacteristic
194D53—162no—
195C22×Q16C2×D5311yescharacteristic
196C22×D8—531no—
197C2×C4×Q8—21no—
198C2×SD32—534no—
199C2×SD32—1062no—
200C2×SD32—1062no—
201C22×C16—531no—
202C22×C16—531no—
203C4×Q16—22no—
204C4×Q16—22no—
205C2×#107—531no—
206C2×#107C2×D5311yescharacteristic
207C2×#103—21no—
208C2×#102—1061no—
209C2×C4×C8C2×D5311yescharacteristic
210#390—1061no—
211#358—1062no—
212#357—1062no—
213C53#1301311yescharacteristic, complemented, Sylow
214C22×Q8—21no—
215C22×D4—1061no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D5312yes—
220C2×Q16D21212yes—
221C2×Q16C22⋊D5312yes—
222C2×D8—532no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8—1061no—
228C22×C8C22⋊D5311yescharacteristic
229C22×C8C22×D5311yescharacteristic
230C4×Q8—24no—
231C4×Q8—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—1061no—
235C2×C22⋊C4—2121no—
236C2×C22⋊C4—2121no—
237C2×C42—41no—
238C2×C42C22⋊D5311yescharacteristic
239SD32—1068no—
240C2×C16—532no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16—1061no—
244C2×C16—1061no—
245C2×C16—1061no—
246#107—21no—
247#107—1061no—
248#107—1061no—
249#107C22⋊D5312yes—
250#103—24no—
251#102—2122no—
252C4×C8—21no—
253C4×C8—21no—
254C24—2121no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—1062no—
262C2×D4—2121no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—2121no—
270C22×C4#1171992211yescharacteristic
271C22×C4#1171992211yescharacteristic
272C22×C4C2×C22⋊D5311yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16#1171992114yescomplemented
277D8—1062no—
278D8—1062no—
279D8—2121no—
280D8—2121no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C2×C4×D5311yescharacteristic
285C2×C8C2×D21211yescharacteristic
286C2×C8#1171994511yescharacteristic
287C2×C8#1171994511yescharacteristic
288C2×C8C2×C22⋊D5311yescharacteristic
289C2×C8C2×C22⋊D5311yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—41no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C22⋊C4—4242no—
296C22⋊C4—4242no—
297C42—21no—
298C42—21no—
299C42—44no—
300C16—1062no—
301C16—2121no—
302C23—4241no—
303C23—4241no—
304C23—4241no—
305C23—4241no—
306C23—4241no—
307C23—4241no—
308C23—2122no—
309C23C2×#1171992211yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—4241no—
315D4—4241no—
316D4—2122no—
317D4—2122no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4#1313316012yescomplemented
333C2×C4C4×C22⋊D5311yescharacteristic
334C2×C4#1313318611yescharacteristic
335C2×C4#1313319111yescharacteristic
336C2×C4#1313323711yescharacteristic
337C2×C4#1313324211yescharacteristic
338C2×C4#1313324311yescharacteristic
339C8—41no—
340C8C2×#1171992111yescharacteristic
341C8C4×C22⋊D5311yescharacteristic
342C8#1313319111yescharacteristic
343C8#1313320011yescharacteristic
344C22—8481no—
345C22—8481no—
346C22—8481no—
347C22—8481no—
348C22—4242no—
349C22—4242no—
350C22—4242no—
351C22C4×#1171992211yescharacteristic
352C22#1681075711yescharacteristic
353C22#1681081811yescharacteristic
354C22C2×#1313316011yescharacteristic
355C22#1681105311yescharacteristic
356C22#1681105811yescharacteristic
357C22#1681106411yescharacteristic
358C4—22no—
359C4—44no—
360C4—41no—
361C4—82no—
362C4—8481no—
363C4#1681050211yescharacteristic
364C4#1681073311yescharacteristic
365C4#1681076811yescharacteristic
366C4#1681083811yescharacteristic
367C2—8482no—
368C2#2027506311yescharacteristic
369C2C4×#1313316011yescharacteristic
370C2#2027657711yescharacteristic
371C2#2027663211yescharacteristic
372C2#2027663811yescharacteristic
373C2#2027675511yescharacteristic
374C2#2027677711yescharacteristic
subgroupG/H
centerC23C2×#11719922
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C4×Q16×C53C2
socleC23×C53C2×D8
prime radicalC22×D424C22
cosocleC22×C8×C53C23
hypercenterC2×C4×Q16D53
hypocenterC53#13013
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#16810427C2semidirect
5. #20272806C2semidirect
6. #20272793C2semidirect
7. C2×C4×Q16×C53C2semidirect
11. C2×#13133160C4semidirect
88. Q16×C53C22⋊C4semidirect
137. C2×C4×C53SD32semidirect
181. C2×C4×Q16D53semidirect
213. C53#13013semidirect
276. Q16#11719921semidirect
332. C2×C4#13133160semidirect

Maximal subgroups

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

subgroupindex
C22×#131331602
C2×#168104272
C2×#168104232
#202755812
#202728062
#202727932
C2×C4×Q16×C532
#1301353