Finite Group Databasebeta

← #27537553

Subgroups — #27537553

Order 13568.

646 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
2#20283633C211yesmaximal, characteristic, complemented
3#20283600C211yesmaximal, characteristic, complemented
4#20276615C214yesmaximal, complemented
5#20276608C214yesmaximal, complemented
6C2×#16810446C211yesmaximal, characteristic, complemented
7#20276465C211yesmaximal, characteristic, complemented
8C2×#16810434C211yesmaximal, characteristic, complemented
9C53×#2110C211yesmaximal, characteristic, complemented
10C2×#13133239C2211yescharacteristic
11C2×#13133239C2211yescharacteristic
12#16811389—21no—
13#16811103—21no—
14#16811066C412yescomplemented
15#16811066C2212yes—
16#16811061C412yescomplemented
17#16811061C2212yes—
18#16811055C414yescomplemented
19#16810746C2214yes—
20#16810741—22no—
21#16810736C2212yescomplemented
22#16810731C2212yescomplemented
23C4×#11719913—22no—
24C2×#13133153C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26#16810641—21no—
27#16810625—21no—
28#16810446C2214yescomplemented
29#16810440C2212yescomplemented
30#16810437C2212yescomplemented
31#16810434C2214yes—
32C2×M32×C53C2211yescharacteristic
33C2×M32×C53C2211yescharacteristic
34C53×#405C2211yescharacteristic
35C53×#347C2214yescomplemented
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—42no—
53#13133223—24no—
54C2×#11719917—21no—
55M16×D53—42no—
56C4×Dic106—22no—
57C4×D212—22no—
58#13133177—42no—
59#13133175—41no—
60C2×C4⋊Dic53—21no—
61C4⋊C4×D53—41no—
62Dic424C2×C414yescomplemented
63#13133162C2×C414yescomplemented
64#13133162C2×C414yescomplemented
65D848C2×C414yescomplemented
66#13133153—21no—
67#13133153C2312yes—
68#13133149—21no—
69#13133149C2312yes—
70#13133145—24no—
71#13133142—22no—
72#13133139—22no—
73C2×M16×C53D411yescharacteristic
74C22×C8×C53C2311yescharacteristic
75#117×C53D411yescharacteristic
76M32×C53—21no—
77M32×C53C2×C412yescomplemented
78M32×C53C2312yes—
79M32×C53C2312yes—
80C2×C16×C53—21no—
81C2×C16×C53C2×C412yescomplemented
82#97×C53C2312yes—
83C4×C8×C53D412yes—
84C22×C4×C53C2×D411yescharacteristic
85C2×C22⋊D53—41no—
86C2×C22⋊D53—41no—
87C22×Dic53—41no—
88#11719945—22no—
89#11719945—22no—
90#11719945—22no—
91#11719945—22no—
92#11719945—41no—
93#11719945—41no—
94#11719945—41no—
95#11719945—41no—
96C2×D212—21no—
97C2×D212—21no—
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C4×D53—42no—
101C2×Dic106—21no—
102C2×Dic106—21no—
103M16×C53—21no—
104M16×C53—21no—
105C2×C8×C53C22×C411yescharacteristic
106C2×C8×C53C22×C411yescharacteristic
107C2×C8×C53C2×D412yes—
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53#4111yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C2×C8×C53C2411yescharacteristic
112C4⋊C4×C53—21no—
113C22⋊C4×C53—21no—
114C42×C53C2×D412yes—
115#11719921—42no—
116C4⋊Dic53—22no—
117C4⋊Dic53—41no—
118#11719919—42no—
119C4×Dic53—42no—
120#11719917—22no—
121#11719917—41no—
122C2×#88322—41no—
123Dic212—21no—
124Dic212—21no—
125Dic212C22×C412yes—
126Dic212C22×C412yes—
127D424—21no—
128D424—21no—
129D424C22×C412yes—
130D424C22×C412yes—
131#11719913—22no—
132#11719913—22no—
133#11719913—22no—
134#11719913—22no—
135#11719912—44no—
136C8×D53—44no—
137C16×C53—22no—
138C16×C53C22×C414yes—
139C23×C53AGL(1,ℤ/8)11yescharacteristic
140C22×D53—41no—
141C22×D53—41no—
142C2×C4×C53—22no—
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53C22×D411yescharacteristic
149C22⋊D53—42no—
150C22⋊D53—42no—
151C22⋊D53—42no—
152C22⋊D53—42no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—42no—
156C2×Dic53—161no—
157D212—22no—
158D212—22no—
159D212—41no—
160D212—41no—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—162no—
166Dic106—22no—
167Dic106—22no—
168Dic106—41no—
169Dic106—41no—
170C8×C53—22no—
171C8×C53C4×D411yescharacteristic
172C8×C53C4×D411yescharacteristic
173C8×C53C23×C411yescharacteristic
174C8×C53C2×#4111yescharacteristic
175#88322—42no—
176#33469—531nomaximal, Sylow
177C22×C53—41no—
178C22×C53#44211yescharacteristic
179C22×C53#44211yescharacteristic
180C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
181C2×D53—42no—
182C2×D53—42no—
183C2×D53—81no—
184C2×D53—81no—
185C2×D53—161no—
186C4×C53—42no—
187C4×C53#44211yescharacteristic
188C4×C53#44211yescharacteristic
189C4×C53C2×C4×D411yescharacteristic
190C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
191Dic53—42no—
192Dic53—42no—
193Dic53—82no—
194#3414—531no—
195#2957—531no—
196#2952—531no—
197AGL(1,ℤ/16)—534no—
198#2180—534no—
199C2×#365—531no—
200#2148—531no—
201C2×#360—531no—
202#2110D5311yescharacteristic, complemented
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53C2×#44211yescharacteristic
207D53—82no—
208D53—82no—
209C2×#135—531no—
210C2×#135—531no—
211#568—1061no—
212#520—1061no—
213#510—532no—
214#510—532no—
215#509—532no—
216#509—532no—
217#508—534no—
218C2×M32C2×D5311yescharacteristic
219C2×M32C2×D5311yescharacteristic
220#444—534no—
221#443—1062no—
222#442—532no—
223#441—532no—
224C4×SD16—1062no—
225C2×#108—531no—
226C2×#106—531no—
227#421—1061no—
228#417—1061no—
229#405C2×D5311yescharacteristic
230#365—534no—
231#362—532no—
232#361—532no—
233#360—534no—
234#347C2×D5314yescomplemented
235C53#3346911yescharacteristic, 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—1061no—
255C2×M16—2122no—
256C2×M16D21211yescharacteristic
257C22×C8C22×D5311yescharacteristic
258C4×Q8—1062no—
259C4×D4—1062no—
260C4×D4—2122no—
261#117—2121no—
262#117D21211yescharacteristic
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—2121no—
265Q32—534no—
266SD32—534no—
267SD32—534no—
268D16—534no—
269M32—21no—
270M32C4×D5312yescomplemented
271M32C22×D5312yes—
272M32C22×D5312yes—
273C2×C16—21no—
274C2×C16C4×D5312yescomplemented
275#108—532no—
276#108—1061no—
277#106—532no—
278#106—1061no—
279#104—1064no—
280#103—1062no—
281#102—1062no—
282#97C22×D5312yes—
283C4×C8D21212yes—
284#41—1062no—
285#41—1062no—
286#41—1062no—
287#41—1062no—
288#41—2121no—
289#41—2121no—
290#41—2121no—
291#41—2121no—
292C2×Q8—1061no—
293C2×Q8—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—2121no—
297C2×D4—2121no—
298C22×C4—2122no—
299C22×C4—2121no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4C2×D21211yescharacteristic
303Q16—532no—
304Q16—532no—
305Q16—1061no—
306Q16—1061no—
307SD16—1062no—
308SD16—1062no—
309SD16—1062no—
310SD16—1062no—
311D8—532no—
312D8—532no—
313D8—1061no—
314D8—1061no—
315M16—21no—
316M16—21no—
317M16—1062no—
318M16—2124no—
319M16—2121no—
320C2×C8—2124no—
321C2×C8—2121no—
322C2×C8C2×C4×D5311yescharacteristic
323C2×C8C2×C4×D5311yescharacteristic
324C2×C8C2×D21212yes—
325C2×C8C2×D21211yescharacteristic
326C2×C8#1171994511yescharacteristic
327C2×C8#1171994511yescharacteristic
328C2×C8C23×D5311yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—1062no—
331C4⋊C4—2122no—
332C4⋊C4—2121no—
333C22⋊C4—21no—
334C22⋊C4—2122no—
335C42—2122no—
336C42C2×D21212yes—
337C16—22no—
338C16C2×C4×D5314yes—
339C23—2121no—
340C23—2121no—
341C23#1313324511yescharacteristic
342Q8—1062no—
343Q8—1062no—
344Q8—2121no—
345Q8—2121no—
346D4—1062no—
347D4—1062no—
348D4—2122no—
349D4—2122no—
350D4—2122no—
351D4—2122no—
352D4—2121no—
353D4—2121no—
354C2×C4—22no—
355C2×C4—8482no—
356C2×C4—8481no—
357C2×C4—2122no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4C4×D21211yescharacteristic
365C2×C4C4×D21211yescharacteristic
366C2×C4#1313324511yescharacteristic
367C2×C4#1313324511yescharacteristic
368C2×C4#1313324511yescharacteristic
369C2×C4C22×D21211yescharacteristic
370C8—22no—
371C8—2122no—
372C8C4×D21211yescharacteristic
373C8C4×D21211yescharacteristic
374C8C22×C4×D5311yescharacteristic
375C8C2×#1171994511yescharacteristic
376C22—41no—
377C22—8481no—
378C22—4241no—
379C22—4241no—
380C22—2122no—
381C22—2122no—
382C22#1681073611yescharacteristic
383C22#1681073611yescharacteristic
384C22C2×#1313324511yescharacteristic
385C4—42no—
386C4—4242no—
387C4—2122no—
388C4—2122no—
389C4#1681073611yescharacteristic
390C4#1681073611yescharacteristic
391C4C2×C4×D21211yescharacteristic
392C4C2×#1313324511yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—4242no—
397C2—4242no—
398C2C2×#1681073611yescharacteristic
subgroupG/H
centerC2C2×#16810736
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2110C2
socleC2×C53C2×#442
prime radical#20287851C2
cosocleC2×C8×C53C24
hypercenter#2110D53
hypocenterC53#33469
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #20283633C2semidirect
3. #20283600C2semidirect
4. #20276615C2semidirect
5. #20276608C2semidirect
6. C2×#16810446C2semidirect
7. #20276465C2semidirect
8. C2×#16810434C2semidirect
9. C53×#2110C2semidirect
14. #16811066C4semidirect
16. #16811061C4semidirect
18. #16811055C4semidirect
21. #16810736C22semidirect
22. #16810731C22semidirect
28. #16810446C22semidirect
29. #16810440C22semidirect
30. #16810437C22semidirect
35. C53×#347C22semidirect
44. #13133239C2×C4semidirect
46. C2×Dic212C2×C4semidirect
50. C2×D424C2×C4semidirect
62. Dic424C2×C4semidirect
63. #13133162C2×C4semidirect
64. #13133162C2×C4semidirect
65. D848C2×C4semidirect
77. M32×C53C2×C4semidirect
81. C2×C16×C53C2×C4semidirect
202. #2110D53semidirect
234. #347C2×D53semidirect
235. C53#33469semidirect
270. M32C4×D53semidirect
274. C2×C16C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202878512
#202836332
#202836002
#202766152
#202766082
C2×#168104462
#202764652
C2×#168104342
C53×#21102
#3346953