Finite Group Databasebeta

← #27537541

Subgroups — #27537541

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287851C211yesmaximal, characteristic
2C2×#16810744C211yesmaximal, characteristic, complemented
3#20283515C211yesmaximal, characteristic, complemented
4#20276612C214yesmaximal, complemented
5#20276605C214yesmaximal, complemented
6#20276462C211yesmaximal, characteristic, complemented
7#20276450C211yesmaximal, characteristic, complemented
8C2×#16810400C211yesmaximal, characteristic, complemented
9C53×#2155C211yesmaximal, characteristic, complemented
10C2×#13133239C2211yescharacteristic
11C2×#13133239C2211yescharacteristic
12C2×#13133225—21no—
13#16811091—21no—
14#16811066C412yescomplemented
15#16811066C2212yes—
16#16811061C412yescomplemented
17#16811061C2212yes—
18#16811055C414yescomplemented
19C2×#13133157C2211yescharacteristic
20#16810744—22no—
21#16810744C2214yes—
22#16810734C2212yescomplemented
23#16810729C2212yescomplemented
24#16810722—22no—
25C2×#13133143—21no—
26#16810617—21no—
27C2×#13133127C2211yescharacteristic
28#16810438C2212yescomplemented
29#16810435C2212yescomplemented
30#16810432C2214yes—
31#16810400C2214yescomplemented
32C2×M32×C53C2211yescharacteristic
33C2×#108×C53C2211yescharacteristic
34C53×#427C2211yescharacteristic
35C53×#365C2214yescomplemented
36C2×#11719945—21no—
37C2×#11719945—21no—
38#13133239—21no—
39#13133239—21no—
40#13133239—21no—
41#13133239—21no—
42#13133239C2×C412yes—
43#13133239C2×C412yes—
44#13133239C2×C412yescomplemented
45#13133239C2312yes—
46C2×Dic212C2×C411yescharacteristic, complemented
47C2×Dic212C2311yescharacteristic
48C2×#11719913—21no—
49C2×#11719913—21no—
50C2×D424C2×C411yescharacteristic, complemented
51C2×D424C2311yescharacteristic
52#13133225—24no—
53#13133225—42no—
54C2×#11719912—41no—
55C22×#88322D411yescharacteristic
56C2×C8×D53—41no—
57#13133182—22no—
58#13133179—22no—
59#13133177—42no—
60#13133174—41no—
61C2×C4×Dic53D411yescharacteristic
62C42×D53—41no—
63Dic424C2×C414yescomplemented
64#13133162C2×C414yescomplemented
65#13133162C2×C414yescomplemented
66D848C2×C414yescomplemented
67#13133157—21no—
68#13133157C2312yes—
69C2×#11719908—21no—
70#13133143—24no—
71#13133140—22no—
72#13133137—22no—
73#13133127D412yes—
74#13133127C2312yes—
75C2×M16×C53—21no—
76C22×C8×C53C2311yescharacteristic
77#117×C53—21no—
78M32×C53C2×C412yescomplemented
79M32×C53C2312yes—
80C2×C16×C53C2×C412yescomplemented
81#108×C53—21no—
82#108×C53C2312yes—
83#107×C53—21no—
84#106×C53C2312yes—
85C22×C4×C53C2×D411yescharacteristic
86C2×C22⋊D53—41no—
87C2×C22⋊D53—41no—
88C22×Dic53—21no—
89#11719945—22no—
90#11719945—22no—
91#11719945—22no—
92#11719945—22no—
93#11719945—41no—
94#11719945—41no—
95#11719945—41no—
96#11719945—41no—
97C2×D212—21no—
98C2×D212—21no—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×C4×D53—42no—
102C2×Dic106—21no—
103C2×Dic106—21no—
104M16×C53—22no—
105M16×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53C22×C411yescharacteristic
108C2×C8×C53C22×C411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C2×C8×C53C2411yescharacteristic
113C4⋊C4×C53—22no—
114C22⋊C4×C53—41no—
115C42×C53—41no—
116#11719921—42no—
117#11719919—42no—
118C4×Dic53—21no—
119C4×Dic53—42no—
120C4×Dic53C2×D412yes—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322C2×D412yes—
124Dic212—21no—
125Dic212—21no—
126Dic212C22×C412yes—
127Dic212C22×C412yes—
128D424—21no—
129D424—21no—
130D424C22×C412yes—
131D424C22×C412yes—
132#11719913—22no—
133#11719913—22no—
134#11719913—22no—
135#11719913—22no—
136#11719912—44no—
137C8×D53—44no—
138C16×C53C22×C414yes—
139#11719908—22no—
140C23×C53AGL(1,ℤ/8)11yescharacteristic
141C22×D53—41no—
142C22×D53—41no—
143C2×C4×C53—42no—
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53C22×D411yescharacteristic
150C22⋊D53—42no—
151C22⋊D53—42no—
152C22⋊D53—42no—
153C22⋊D53—42no—
154C2×Dic53—22no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—81no—
158D212—22no—
159D212—22no—
160D212—41no—
161D212—41no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—42no—
166C4×D53—84no—
167Dic106—22no—
168Dic106—22no—
169Dic106—41no—
170Dic106—41no—
171C8×C53—42no—
172C8×C53C4×D411yescharacteristic
173C8×C53C4×D411yescharacteristic
174C8×C53C23×C411yescharacteristic
175C8×C53C2×#4111yescharacteristic
176#88322—22no—
177#33468—531nomaximal, Sylow
178C22×C53—41no—
179C22×C53#44211yescharacteristic
180C22×C53#44211yescharacteristic
181C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
182C2×D53—42no—
183C2×D53—42no—
184C2×D53—82no—
185C2×D53—82no—
186C4×C53—82no—
187C4×C53#44211yescharacteristic
188C4×C53#44211yescharacteristic
189C4×C53C2×C4×D411yescharacteristic
190C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
191Dic53—42no—
192Dic53—42no—
193Dic53—42no—
194#3414—531no—
195C2×#444—531no—
196#2943—531no—
197AGL(1,ℤ/16)—534no—
198#2180—534no—
199#2155D5311yescharacteristic, complemented
200#2148—531no—
201#2145—531no—
202C2×#347—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53C2×#44211yescharacteristic
207D53—82no—
208D53—82no—
209C2×#135—531no—
210C2×#135—531no—
211C2×#131—1061no—
212C4×#41—1061no—
213#510—532no—
214#510—532no—
215#509—532no—
216#509—532no—
217#508—534no—
218C2×M32—531no—
219C2×M32C2×D5311yescharacteristic
220#444—534no—
221#444—1062no—
222#442—532no—
223#441—532no—
224#440—1062no—
225C2×#108C2×D5311yescharacteristic
226#427C2×D5311yescharacteristic
227C2×#104—1061no—
228#416—1061no—
229C2×#97—531no—
230#365C2×D5314yescomplemented
231#362—532no—
232#361—532no—
233#360—534no—
234#347—534no—
235C53#3346811yescharacteristic, complemented, Sylow
236C2×#41—1061no—
237C2×#41—1061no—
238#135—532no—
239#135—532no—
240#135—532no—
241#135—532no—
242#135—1061no—
243#135—1061no—
244#135—1061no—
245#135—1061no—
246C2×Q16—531no—
247C2×Q16—531no—
248C2×SD16—1061no—
249C2×SD16—1061no—
250C2×D8—531no—
251C2×D8—531no—
252#131—1064no—
253#131—2122no—
254C2×M16—21no—
255C2×M16—2121no—
256C22×C8—531no—
257C22×C8—2121no—
258C22×C8C22×D5311yescharacteristic
259C4×Q8—1062no—
260C4×D4—1062no—
261C4×D4—2122no—
262#117—21no—
263#117—2121no—
264C2×C42—531no—
265C2×C42—2121no—
266Q32—534no—
267SD32—534no—
268SD32—534no—
269D16—534no—
270M32—532no—
271M32—1061no—
272M32C4×D5312yescomplemented
273M32C22×D5312yes—
274C2×C16—1061no—
275C2×C16C4×D5312yescomplemented
276#108—21no—
277#108C22×D5312yes—
278#107—21no—
279#106C22×D5312yes—
280#104—1064no—
281#103—1062no—
282#102—1062no—
283#97—532no—
284#97—532no—
285#41—1062no—
286#41—1062no—
287#41—1062no—
288#41—1062no—
289#41—2121no—
290#41—2121no—
291#41—2121no—
292#41—2121no—
293C2×Q8—1061no—
294C2×Q8—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—2121no—
298C2×D4—2121no—
299C22×C4—1061no—
300C22×C4—2122no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4D4×D5311yescharacteristic
304Q16—532no—
305Q16—532no—
306Q16—1061no—
307Q16—1061no—
308SD16—1062no—
309SD16—1062no—
310SD16—1062no—
311SD16—1062no—
312D8—532no—
313D8—532no—
314D8—1061no—
315D8—1061no—
316M16—22no—
317M16—41no—
318M16—2124no—
319C2×C8—41no—
320C2×C8—532no—
321C2×C8—1061no—
322C2×C8—1061no—
323C2×C8—2124no—
324C2×C8C2×C4×D5311yescharacteristic
325C2×C8C2×C4×D5311yescharacteristic
326C2×C8D4×D5311yescharacteristic
327C2×C8#1171994911yescharacteristic
328C2×C8#1171994911yescharacteristic
329C2×C8C23×D5311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—2122no—
332C22⋊C4—41no—
333C22⋊C4—2122no—
334C42—41no—
335C42—532no—
336C42—1061no—
337C42—2122no—
338C16—1062no—
339C16C2×C4×D5314yes—
340C23—2121no—
341C23—2121no—
342C23#1313324411yescharacteristic
343Q8—1062no—
344Q8—1062no—
345Q8—2121no—
346Q8—2121no—
347D4—1062no—
348D4—1062no—
349D4—2122no—
350D4—2122no—
351D4—2122no—
352D4—2122no—
353D4—2121no—
354D4—2121no—
355C2×C4—42no—
356C2×C4—1062no—
357C2×C4—4244no—
358C2×C4—4241no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4#1313317911yescharacteristic
366C2×C4#1313317911yescharacteristic
367C2×C4#1313324411yescharacteristic
368C2×C4#1313324411yescharacteristic
369C2×C4#1313324411yescharacteristic
370C2×C4C2×D4×D5311yescharacteristic
371C8—42no—
372C8—1062no—
373C8#1313317911yescharacteristic
374C8#1313317911yescharacteristic
375C8C22×C4×D5311yescharacteristic
376C8C2×#1171994911yescharacteristic
377C22—41no—
378C22—4242no—
379C22—4242no—
380C22—2122no—
381C22—2122no—
382C22#1681073411yescharacteristic
383C22#1681073411yescharacteristic
384C22C2×#1313324411yescharacteristic
385C4—82no—
386C4—2122no—
387C4—2122no—
388C4—2122no—
389C4#1681073411yescharacteristic
390C4#1681073411yescharacteristic
391C4C2×#1313317911yescharacteristic
392C4C2×#1313324411yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—4242no—
397C2—4242no—
398C2C2×#1681073411yescharacteristic
subgroupG/H
centerC4#16810734
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2155C2
socleC2×C53C2×#442
prime radical#20287851C2
cosocleC2×C8×C53C24
hypercenter#2155D53
hypocenterC53#33468
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2155

Complements

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

NH ≅ G/Ntype
2. C2×#16810744C2semidirect
3. #20283515C2semidirect
4. #20276612C2semidirect
5. #20276605C2semidirect
6. #20276462C2semidirect
7. #20276450C2semidirect
8. C2×#16810400C2semidirect
9. C53×#2155C2semidirect
14. #16811066C4semidirect
16. #16811061C4semidirect
18. #16811055C4semidirect
22. #16810734C22semidirect
23. #16810729C22semidirect
28. #16810438C22semidirect
29. #16810435C22semidirect
31. #16810400C22semidirect
35. C53×#365C22semidirect
44. #13133239C2×C4semidirect
46. C2×Dic212C2×C4semidirect
50. C2×D424C2×C4semidirect
63. Dic424C2×C4semidirect
64. #13133162C2×C4semidirect
65. #13133162C2×C4semidirect
66. D848C2×C4semidirect
78. M32×C53C2×C4semidirect
80. C2×C16×C53C2×C4semidirect
199. #2155D53semidirect
230. #365C2×D53semidirect
235. C53#33468semidirect
272. M32C4×D53semidirect
275. C2×C16C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202878512
C2×#168107442
#202835152
#202766122
#202766052
#202764622
#202764502
C2×#168104002
C53×#21552
#3346853