Finite Group Databasebeta

← #27301166

Subgroups — #27301166

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133160C211yesmaximal, characteristic
2C2×C4×D424C211yesmaximal, characteristic
3C2×#16810427C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic
5#20272806C211yesmaximal, characteristic, complemented
6#20272793C211yesmaximal, characteristic, complemented
7C53×#1969C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C2×C4×D212—21no—
10C2×#13133160—22no—
11C2×#13133160—22no—
12C2×#13133160C414yescomplemented
13C22×#11719908C2211yescharacteristic
14C22×#11719908C2211yescharacteristic
15C4×D424—22no—
16C4×D424—22no—
17C2×#13133151C2211yescharacteristic
18C2×#13133139—21no—
19#16810427—22no—
20#16810423—22no—
21C22×Q16×C53C2211yescharacteristic
22C2×#107×C53C2211yescharacteristic
23C2×#103×C53—21no—
24C2×C4×C8×C53C2211yescharacteristic
25C53×#391—21no—
26C22×D212—21no—
27C22×C4×D53—41no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424C2×C412yes—
32C2×D424D412yes—
33C2×D424D412yes—
34C4×D212—24no—
35C4×D212—42no—
36C2×C4⋊Dic53—21no—
37C2×#11719921—41no—
38#13133160—28no—
39C2×#11719908—21no—
40C2×#11719908—21no—
41C2×#11719908—21no—
42C2×#11719908—21no—
43C2×#11719908—21no—
44C2×#11719908C2×C412yescomplemented
45#13133151—21no—
46#13133151D412yes—
47#13133139—24no—
48C22×Q8×C53—21no—
49C2×Q16×C53—21no—
50C2×Q16×C53—21no—
51C2×Q16×C53—21no—
52C2×Q16×C53—21no—
53C2×Q16×C53—21no—
54C2×Q16×C53C2×C412yescomplemented
55C22×C8×C53D411yescharacteristic
56C22×C8×C53C2311yescharacteristic
57C2×C4⋊C4×C53—21no—
58C2×C42×C53D411yescharacteristic
59#107×C53—21no—
60#107×C53—21no—
61#103×C53—42no—
62C4×C8×C53—21no—
63C4×C8×C53—21no—
64#95×C53—41no—
65#95×C53—41no—
66C23×D53—41no—
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
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—22no—
81C2×D212—22no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×C4×D53—44no—
86C2×C4×D53—82no—
87C2×C4×D53—82no—
88Q16×C53—22no—
89Q16×C53—22no—
90Q16×C53—41no—
91Q16×C53—41no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53C22×C411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53#4111yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C42×C53—21no—
104C42×C53—21no—
105#11719921—44no—
106C4⋊Dic53—22no—
107C4⋊Dic53—41no—
108D424—22no—
109D424—22no—
110D424—22no—
111D424C22⋊C414yes—
112#11719908—22no—
113#11719908—41no—
114C23×C53C2×D811yescharacteristic
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—81no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—81no—
122Q8×C53—42no—
123Q8×C53—42no—
124Q8×C53—81no—
125Q8×C53—81no—
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—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
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—42no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147D212—24no—
148D212—42no—
149D212—42no—
150D212—42no—
151C4×D53—88no—
152C8×C53—41no—
153C8×C53C2×C22⋊C411yescharacteristic
154C8×C53C4×D411yescharacteristic
155C8×C53#12111yescharacteristic
156C8×C53#12311yescharacteristic
157#13014—531nomaximal, Sylow
158C22×C53C4×D811yescharacteristic
159C22×C53#44711yescharacteristic
160C22×C53#45911yescharacteristic
161C22×C53C2×SD3211yescharacteristic
162C22×C53#50811yescharacteristic
163C22×C53#50911yescharacteristic
164C22×C53#51011yescharacteristic
165C2×D53—44no—
166C2×D53—82no—
167C2×D53—82no—
168C2×D53—82no—
169C2×D53—82no—
170C2×D53—82no—
171C2×D53—82no—
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—82no—
175C4×C53—161no—
176C4×C53#38611yescharacteristic
177C4×C53#44211yescharacteristic
178C4×C53#44911yescharacteristic
179C4×C53#46311yescharacteristic
180Dic53—82no—
181C22×SD32—531no—
182C2×C4×D8—531no—
183C2×#358—531no—
184C2×#357—531no—
185#1969D5311yescharacteristic, complemented
186#1380—531no—
187#1374—531no—
188C2×C53#187911yescharacteristic
189C2×C53C4×SD3211yescharacteristic
190C2×C53#217511yescharacteristic
191C2×C53#218711yescharacteristic
192C2×C53#218911yescharacteristic
193C2×C53#220811yescharacteristic
194C2×C53#221011yescharacteristic
195D53—84no—
196C22×Q16C2×D5311yescharacteristic
197C22×D8—531no—
198C2×C4×D4—1061no—
199C2×SD32—534no—
200C2×SD32—1062no—
201C2×SD32—1062no—
202C22×C16—531no—
203C22×C16—531no—
204C4×D8—1062no—
205C4×D8—1062no—
206C2×#107—531no—
207C2×#107C2×D5311yescharacteristic
208C2×#103—21no—
209C2×#102—1061no—
210C2×C4×C8C2×D5311yescharacteristic
211#391—21no—
212#358—1062no—
213#357—1062no—
214C53#1301411yescharacteristic, complemented, Sylow
215C22×Q8—21no—
216C22×D4—1061no—
217C23×C4—2121no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16C4×D5312yescomplemented
224C2×D8—532no—
225C2×D8—532no—
226C2×D8—532no—
227C2×D8—1061no—
228C2×D8—1061no—
229C2×D8—1061no—
230C22×C8C22⋊D5311yescharacteristic
231C22×C8C22×D5311yescharacteristic
232C4×D4—1064no—
233C4×D4—2122no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—1061no—
236C2×C22⋊C4—2121no—
237C2×C42C22⋊D5311yescharacteristic
238SD32—1068no—
239C2×C16—532no—
240C2×C16—1061no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16—1061no—
244C2×C16—1061no—
245#107—21no—
246#107—21no—
247#107—532no—
248#107—1061no—
249#103—42no—
250#102—1064no—
251C4×C8—21no—
252C4×C8—21no—
253#95—41no—
254#95—41no—
255C24—2121no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—1062no—
263C2×D4—1062no—
264C2×D4—1062no—
265C2×D4—2121no—
266C2×D4—2121no—
267C2×D4—2121no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—4242no—
271C22×C4—4242no—
272C22×C4—2124no—
273C22×C4—2121no—
274C22×C4#1171992211yescharacteristic
275C22×C4#1171992211yescharacteristic
276C22×C4C2×C22⋊D5311yescharacteristic
277Q16—22no—
278Q16—22no—
279Q16—41no—
280Q16—41no—
281D8—534no—
282D8—1062no—
283D8—1062no—
284D8—1062no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C2×C4×D5311yescharacteristic
289C2×C8#1171994511yescharacteristic
290C2×C8D4×D5311yescharacteristic
291C2×C8D4×D5311yescharacteristic
292C2×C8D4×D5311yescharacteristic
293C2×C8#1171994711yescharacteristic
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—1062no—
297C4⋊C4—2121no—
298C22⋊C4—2124no—
299C42—21no—
300C42—21no—
301C16—1062no—
302C16—2121no—
303C23—4241no—
304C23—4241no—
305C23—4241no—
306C23—4241no—
307C23—2122no—
308C23—2122no—
309C23—2122no—
310C23C2×#1171992211yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—81no—
314Q8—81no—
315D4—1064no—
316D4—2122no—
317D4—2122no—
318D4—2122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—4248no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4—2122no—
334C2×C4#1313316012yescomplemented
335C2×C4C4×C22⋊D5311yescharacteristic
336C2×C4#1313318511yescharacteristic
337C2×C4#1313318811yescharacteristic
338C2×C4#1313323711yescharacteristic
339C2×C4#1313324211yescharacteristic
340C2×C4#1313324311yescharacteristic
341C8—41no—
342C8C22⋊C4×D5311yescharacteristic
343C8#1313317711yescharacteristic
344C8#1313319011yescharacteristic
345C8#1313319911yescharacteristic
346C22—4242no—
347C22—4242no—
348C22—4242no—
349C22—4242no—
350C22—4242no—
351C22—4242no—
352C22—2124no—
353C22C4×#1171992211yescharacteristic
354C22#1681075611yescharacteristic
355C22#1681081511yescharacteristic
356C22C2×#1313316011yescharacteristic
357C22#1681105311yescharacteristic
358C22#1681105811yescharacteristic
359C22#1681106411yescharacteristic
360C4—22no—
361C4—41no—
362C4—82no—
363C4—161no—
364C4—4242no—
365C4#1681049911yescharacteristic
366C4#1681073311yescharacteristic
367C4#1681076611yescharacteristic
368C4#1681083511yescharacteristic
369C2—4244no—
370C2#2027506611yescharacteristic
371C2C4×#1313316011yescharacteristic
372C2#2027657711yescharacteristic
373C2#2027664311yescharacteristic
374C2#2027665311yescharacteristic
375C2#2027675711yescharacteristic
376C2#2027676711yescharacteristic
subgroupG/H
centerC23C2×#11719922
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1969C2
socleC23×C53C2×D8
prime radicalC22×D424C22
cosocleC22×C8×C53C23
hypercenter#1969D53
hypocenterC53#13014
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1969

Complements

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

NH ≅ G/Ntype
3. C2×#16810427C2semidirect
5. #20272806C2semidirect
6. #20272793C2semidirect
7. C53×#1969C2semidirect
12. C2×#13133160C4semidirect
44. C2×#11719908C2×C4semidirect
54. C2×Q16×C53C2×C4semidirect
137. C2×C4×C53SD32semidirect
185. #1969D53semidirect
214. C53#13014semidirect
223. C2×Q16C4×D53semidirect
334. 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×C4×D4242
C2×#168104272
C2×#168104232
#202728062
#202727932
C53×#19692
#1301453