Finite Group Databasebeta

← #27301240

Subgroups — #27301240

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811058C211yesmaximal, characteristic
2C2×#16810433C211yesmaximal, characteristic, complemented
3#20276430C211yesmaximal, characteristic, complemented
4#20275591C211yesmaximal, characteristic, complemented
5#20272820C211yesmaximal, characteristic, complemented
6#20272799C211yesmaximal, characteristic, complemented
7C53×#2943C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9#16811058—22no—
10#16811058C414yescomplemented
11C2×#13133160—21no—
12C2×#13133158—21no—
13C2×#13133157C2211yescharacteristic
14C2×#13133157C2211yescharacteristic
15C2×#13133149C2211yescharacteristic
16C2×#13133139—21no—
17#16810525—21no—
18#16810433—22no—
19#16810423—22no—
20C2×#135×C53C2211yescharacteristic
21C4×#41×C53—21no—
22C53×#442—21no—
23C53×#441—21no—
24C53×#440—22no—
25C53×#427C2211yescharacteristic
26C53×#416—21no—
27C2×#97×C53C2211yescharacteristic
28C22×D212—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424C2×C412yescomplemented
35C2×C4⋊Dic53—21no—
36C2×#11719921—42no—
37#13133160—24no—
38#13133158—24no—
39#13133157—21no—
40#13133157—21no—
41#13133157—21no—
42#13133157C2×C412yescomplemented
43C2×#11719908—21no—
44C2×#11719908—21no—
45#13133149—21no—
46#13133149—21no—
47#13133139—42no—
48C2×#41×C53—21no—
49#135×C53—21no—
50#135×C53—21no—
51#135×C53C2×C412yes—
52#135×C53D412yes—
53C2×Q16×C53D411yescharacteristic
54C2×SD16×C53—21no—
55C2×D8×C53D411yescharacteristic
56C22×C8×C53D411yescharacteristic
57C22×C8×C53C2311yescharacteristic
58C4×Q8×C53—22no—
59C4×D4×C53—22no—
60C4×D4×C53—42no—
61#117×C53—21no—
62#117×C53—41no—
63C2×C42×C53—41no—
64C2×C42×C53D411yescharacteristic
65#107×C53D412yes—
66#106×C53—21no—
67#103×C53—22no—
68#102×C53—22no—
69#97×C53—21no—
70#97×C53—21no—
71C23×D53—41no—
72#41×C53—22no—
73#41×C53—22no—
74#41×C53—41no—
75#41×C53—41no—
76C2×Q8×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—41no—
79C22×C4×C53—42no—
80C22×C4×C53—41no—
81C22×C4×C53D812yes—
82C22×C4×C53C2×D411yescharacteristic
83C22×Dic53—41no—
84C2×D212—22no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90Q16×C53—21no—
91Q16×C53C22⋊C412yescomplemented
92SD16×C53—22no—
93SD16×C53—22no—
94D8×C53—21no—
95D8×C53C22⋊C412yescomplemented
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53C22×C411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53#4111yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C4⋊C4×C53—22no—
106C4⋊C4×C53—42no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—41no—
109C42×C53—21no—
110C42×C53—21no—
111C42×C53—42no—
112C42×C53—41no—
113#11719921—84no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116D424—22no—
117D424—22no—
118D424—41no—
119D424—41no—
120#11719908—22no—
121#11719908—41no—
122C23×C53—41no—
123C23×C53C2×D811yescharacteristic
124C22×D53—42no—
125C22×D53—82no—
126C22×D53—82no—
127C22×D53—81no—
128C22×D53—81no—
129Q8×C53—22no—
130Q8×C53—41no—
131D4×C53—22no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—41no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—84no—
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×Dic53—82no—
149C2×Dic53—81no—
150D212—42no—
151D212—42no—
152D212—81no—
153D212—81no—
154C8×C53—41no—
155C8×C53C2×C22⋊C411yescharacteristic
156C8×C53C4×D411yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12311yescharacteristic
159#13025—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—42no—
162C22×C53—82no—
163C22×C53C4×D811yescharacteristic
164C22×C53#44711yescharacteristic
165C22×C53#45911yescharacteristic
166C2×D53—84no—
167C2×D53—82no—
168C2×D53—161no—
169C2×D53—161no—
170C2×D53—162no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53—82no—
174C4×C53#38611yescharacteristic
175C4×C53#44211yescharacteristic
176C4×C53#44911yescharacteristic
177C4×C53#46311yescharacteristic
178Dic53—161no—
179C2×#509—531no—
180#2943D5311yescharacteristic, complemented
181C2×#360—531no—
182#2140—531no—
183#1972—531no—
184#1386—531no—
185#1377—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—82no—
189C2×C53#187911yescharacteristic
190C2×C53#219012yes—
191D53—162no—
192C2×#135C2×D5311yescharacteristic
193C22×D8—531no—
194C4×#41—21no—
195#509—534no—
196#509—1062no—
197C2×SD32—1061no—
198C2×D16—1061no—
199C2×M32—531no—
200C2×M32—531no—
201#442—21no—
202#441—21no—
203#440—22no—
204#427C2×D5311yescharacteristic
205C2×#106—531no—
206#416—21no—
207C2×#102—1061no—
208C2×#97C2×D5311yescharacteristic
209#390—1061no—
210#360—1062no—
211#357—1062no—
212C53#1302511yescharacteristic, complemented, Sylow
213C2×#41—21no—
214C22×D4—1061no—
215#135—21no—
216#135—21no—
217#135C4×D5312yes—
218#135D21212yes—
219C2×Q16C22⋊D5311yescharacteristic
220C2×SD16—21no—
221C2×D8—532no—
222C2×D8—1061no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8C22⋊D5311yescharacteristic
228C22×C8C22⋊D5311yescharacteristic
229C22×C8C22×D5311yescharacteristic
230C4×Q8—22no—
231C4×D4—22no—
232C4×D4—42no—
233#117—21no—
234#117—41no—
235C2×C4⋊C4—1061no—
236C2×C22⋊C4—2122no—
237C2×C42—41no—
238C2×C42C22⋊D5311yescharacteristic
239SD32—1064no—
240D16—1064no—
241M32—532no—
242M32—1061no—
243M32—1061no—
244M32—1061no—
245C2×C16—1061no—
246C2×C16—1061no—
247#107C22⋊D5312yes—
248#106—21no—
249#106—1061no—
250#106—1061no—
251#103—22no—
252#102—22no—
253#102—2122no—
254#97—21no—
255#97—21no—
256C24—2121no—
257#41—22no—
258#41—22no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—1062no—
265C2×D4—2121no—
266C2×D4—2121no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—2121no—
273C22×C4#1171992212yes—
274C22×C4C2×C22⋊D5311yescharacteristic
275Q16—21no—
276Q16#1171992112yescomplemented
277SD16—22no—
278SD16—22no—
279D8—21no—
280D8—1062no—
281D8—1062no—
282D8—2121no—
283D8—2121no—
284D8#1171992112yescomplemented
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C2×C4×D5311yescharacteristic
289C2×C8C2×D21211yescharacteristic
290C2×C8#1171994511yescharacteristic
291C2×C8#1171994511yescharacteristic
292C2×C8C2×C22⋊D5311yescharacteristic
293C2×C8C2×C22⋊D5311yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—2121no—
297C4⋊C4—2121no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—4244no—
301C42—21no—
302C42—21no—
303C42—42no—
304C42—41no—
305C16—1062no—
306C16—2121no—
307C23—41no—
308C23—4242no—
309C23—4242no—
310C23—4241no—
311C23—4241no—
312C23—2122no—
313C23C2×#1171992211yescharacteristic
314Q8—22no—
315Q8—41no—
316D4—22no—
317D4—42no—
318D4—42no—
319D4—41no—
320D4—4241no—
321D4—4241no—
322D4—2122no—
323D4—2122no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—84no—
331C2×C4—4242no—
332C2×C4—4241no—
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—22no—
345C22—42no—
346C22—82no—
347C22—8481no—
348C22—8481no—
349C22—8482no—
350C22—4244no—
351C22—4242no—
352C22C4×#1171992211yescharacteristic
353C22#1681075711yescharacteristic
354C22#1681081811yescharacteristic
355C4—42no—
356C4—42no—
357C4—82no—
358C4—8481no—
359C4#1681050211yescharacteristic
360C4#1681073311yescharacteristic
361C4#1681076811yescharacteristic
362C4#1681083811yescharacteristic
363C2—21no—
364C2—21no—
365C2—82no—
366C2—8482no—
367C2#2027506311yescharacteristic
368C2#2027666112yes—
subgroupG/H
centerC22#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2943C2
socleC22×C53#447
prime radicalC2×#16811058C2
cosocleC22×C8×C53C23
hypercenter#2943D53
hypocenterC53#13025
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #2943

Complements

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

NH ≅ G/Ntype
2. C2×#16810433C2semidirect
3. #20276430C2semidirect
4. #20275591C2semidirect
5. #20272820C2semidirect
6. #20272799C2semidirect
7. C53×#2943C2semidirect
10. #16811058C4semidirect
34. C2×D424C2×C4semidirect
42. #13133157C2×C4semidirect
91. Q16×C53C22⋊C4semidirect
95. D8×C53C22⋊C4semidirect
180. #2943D53semidirect
212. C53#13025semidirect
276. Q16#11719921semidirect
284. D8#11719921semidirect

Maximal subgroups

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

subgroupindex
C2×#168110582
C2×#168104332
#202764302
#202755912
#202728202
#202727992
C53×#29432
#1302553