Finite Group Databasebeta

← #27537552

Subgroups — #27537552

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#20283632C211yesmaximal, characteristic, complemented
2#20283599C211yesmaximal, characteristic, complemented
3#20276614C214yesmaximal, complemented
4#20276607C214yesmaximal, complemented
5C2×#16810446C211yesmaximal, characteristic, complemented
6#20276464C211yesmaximal, characteristic, complemented
7C2×#16810433C211yesmaximal, characteristic, complemented
8#20276319C211yesmaximal, characteristic, complemented
9C53×#3414C211yesmaximal, characteristic
10#16811390—21no—
11#16811106—21no—
12C2×#13133157C2211yescharacteristic
13#16810745C2214yes—
14#16810740—22no—
15#16810735C2212yescomplemented
16#16810730C2212yescomplemented
17SD16×Dic53—22no—
18C2×#13133153C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20#16810640—21no—
21#16810624—21no—
22#16810578C2211yescharacteristic
23#16810446C2214yescomplemented
24#16810439C2212yescomplemented
25#16810436C2212yescomplemented
26#16810433C2214yes—
27#16810400C2214yescomplemented
28C2×#135×C53C2211yescharacteristic
29C2×#135×C53C2211yescharacteristic
30C53×#510C412yescomplemented
31C53×#510C2212yes—
32C53×#509C412yescomplemented
33C53×#509C2212yes—
34C53×#508C414yescomplemented
35C2×M32×C53C2211yescharacteristic
36#13133224—24no—
37#13133224—42no—
38C2×#11719917—21no—
39C2×#11719917—42no—
40C2×#11719917D411yescharacteristic
41Q8×Dic53—22no—
42D4×Dic53—22no—
43D4×Dic53—42no—
44#13133176—41no—
45#13133176D411yescharacteristic
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—41no—
48#13133157—21no—
49#13133157C2312yes—
50C2×#11719908—21no—
51#13133153—21no—
52#13133153C2312yes—
53#13133149—21no—
54#13133149C2312yes—
55#13133144—24no—
56#13133141—22no—
57#13133138—22no—
58#13133127C2312yes—
59C8×Dic53D412yes—
60C2×#41×C53—21no—
61C2×#41×C53—21no—
62#135×C53—21no—
63#135×C53—21no—
64#135×C53—21no—
65#135×C53—21no—
66#135×C53C2×C412yes—
67#135×C53C2×C412yes—
68#135×C53C2×C412yescomplemented
69#135×C53C2312yes—
70C2×Q16×C53C2×C411yescharacteristic, complemented
71C2×Q16×C53C2311yescharacteristic
72C2×SD16×C53—21no—
73C2×SD16×C53—21no—
74C2×D8×C53C2×C411yescharacteristic, complemented
75C2×D8×C53C2311yescharacteristic
76C22×C8×C53C2311yescharacteristic
77Q32×C53C2×C414yescomplemented
78SD32×C53C2×C414yescomplemented
79SD32×C53C2×C414yescomplemented
80D16×C53C2×C414yescomplemented
81M32×C53C2×C412yescomplemented
82M32×C53C2312yes—
83C2×C16×C53C2×C412yescomplemented
84#41×C53—22no—
85#41×C53—22no—
86#41×C53—22no—
87#41×C53—22no—
88#41×C53—41no—
89#41×C53—41no—
90#41×C53—41no—
91#41×C53—41no—
92C2×Q8×C53—21no—
93C2×Q8×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53C2×D411yescharacteristic
101C22×Dic53—42no—
102C22×Dic53—41no—
103Q16×C53—21no—
104Q16×C53—21no—
105Q16×C53C22×C412yes—
106Q16×C53C22×C412yes—
107SD16×C53—22no—
108SD16×C53—22no—
109SD16×C53—22no—
110SD16×C53—22no—
111D8×C53—21no—
112D8×C53—21no—
113D8×C53C22×C412yes—
114D8×C53C22×C412yes—
115C2×C8×C53C22×C411yescharacteristic
116C2×C8×C53C22×C411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53#4111yescharacteristic
119C2×C8×C53#4111yescharacteristic
120C2×C8×C53C2411yescharacteristic
121C22⋊Dic53—21no—
122C22⋊Dic53—42no—
123C4⋊Dic53—21no—
124C4⋊Dic53—22no—
125C4⋊Dic53—42no—
126C4⋊Dic53—41no—
127C4×Dic53—42no—
128C4×Dic53C2×D412yes—
129#11719917—21no—
130#11719917—21no—
131#11719917—22no—
132#11719917—44no—
133#11719917—41no—
134C2×#88322—44no—
135C2×#88322—41no—
136C2×#88322C2×D412yes—
137C16×C53C22×C414yes—
138#11719908—22no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53AGL(1,ℤ/8)11yescharacteristic
142Q8×C53—22no—
143Q8×C53—22no—
144Q8×C53—41no—
145Q8×C53—41no—
146D4×C53—22no—
147D4×C53—22no—
148D4×C53—42no—
149D4×C53—42no—
150D4×C53—42no—
151D4×C53—42no—
152D4×C53—41no—
153D4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53C22×D411yescharacteristic
166C2×Dic53—22no—
167C2×Dic53—42no—
168C2×Dic53—162no—
169C2×Dic53—161no—
170C8×C53C4×D411yescharacteristic
171C8×C53C4×D411yescharacteristic
172C8×C53C23×C411yescharacteristic
173C8×C53C2×#4111yescharacteristic
174#88322—22no—
175#88322—42no—
176#33469—531nomaximal, Sylow
177C22×C53—41no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—161no—
183C22×C53#44211yescharacteristic
184C22×C53#44211yescharacteristic
185C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
186C4×C53—42no—
187C4×C53—42no—
188C4×C53#44211yescharacteristic
189C4×C53#44211yescharacteristic
190C4×C53C2×C4×D411yescharacteristic
191C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
192Dic53—42no—
193Dic53—82no—
194#3414D5311yescharacteristic
195#2957—531no—
196#2952—531no—
197AGL(1,ℤ/16)—534no—
198#2180—534no—
199C2×#365—531no—
200#2148—531no—
201C2×#360—531no—
202#2110—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—82no—
207C2×C53—82no—
208C2×C53C2×#44211yescharacteristic
209C2×#135C2×D5311yescharacteristic
210C2×#135C2×D5311yescharacteristic
211#568—1061no—
212#520—1061no—
213#510Dic5312yescomplemented
214#510C2×D5312yes—
215#509Dic5312yescomplemented
216#509C2×D5312yes—
217#508Dic5314yescomplemented
218C2×M32—531no—
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#405—531no—
230#365—534no—
231#362—532no—
232#361—532no—
233#360—534no—
234#347—534no—
235C53#3346911yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—21no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135—21no—
242#135C2×Dic5312yes—
243#135C2×Dic5312yes—
244#135C2×Dic5312yescomplemented
245#135C22×D5312yes—
246C2×Q16C2×Dic5311yescharacteristic, complemented
247C2×Q16C22×D5311yescharacteristic
248C2×SD16—21no—
249C2×SD16—21no—
250C2×D8C2×Dic5311yescharacteristic, complemented
251C2×D8C22×D5311yescharacteristic
252#131—1064no—
253#131—2122no—
254C2×M16—531no—
255C2×M16—1061no—
256C2×M16—2122no—
257C22×C8C22×D5311yescharacteristic
258C4×Q8—1062no—
259C4×D4—1062no—
260C4×D4—2122no—
261#117—531no—
262#117—2121no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—2121no—
265Q32C2×Dic5314yescomplemented
266SD32C2×Dic5314yescomplemented
267SD32C2×Dic5314yescomplemented
268D16C2×Dic5314yescomplemented
269M32—532no—
270M32—1061no—
271M32C2×Dic5312yescomplemented
272M32C22×D5312yes—
273C2×C16—1061no—
274C2×C16C2×Dic5312yescomplemented
275#108—532no—
276#108—1061no—
277#106—532no—
278#106—1061no—
279#104—1064no—
280#103—1062no—
281#102—1062no—
282#97—532no—
283C4×C8—532no—
284#41—22no—
285#41—22no—
286#41—22no—
287#41—22no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—2122no—
301C22×C4—2121no—
302C22×C4D4×D5311yescharacteristic
303Q16—21no—
304Q16—21no—
305Q16C22×Dic5312yes—
306Q16C22×Dic5312yes—
307SD16—22no—
308SD16—22no—
309SD16—22no—
310SD16—22no—
311D8—21no—
312D8—21no—
313D8C22×Dic5312yes—
314D8C22×Dic5312yes—
315M16—1061no—
316M16—1061no—
317M16—1062no—
318M16—2124no—
319M16—2121no—
320C2×C8—532no—
321C2×C8—2124no—
322C2×C8—2121no—
323C2×C8D4×D5311yescharacteristic
324C2×C8#1171994711yescharacteristic
325C2×C8#1171994711yescharacteristic
326C2×C8C22×Dic5311yescharacteristic
327C2×C8C22×Dic5311yescharacteristic
328C2×C8C23×D5311yescharacteristic
329C4⋊C4—1061no—
330C4⋊C4—1062no—
331C4⋊C4—2122no—
332C4⋊C4—2121no—
333C22⋊C4—1061no—
334C22⋊C4—2122no—
335C42—532no—
336C42—2122no—
337C16—1062no—
338C16C22×Dic5314yes—
339C23—41no—
340C23—41no—
341C23#1313324111yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—41no—
345Q8—41no—
346D4—22no—
347D4—22no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—41no—
353D4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—8481no—
361C2×C4—8482no—
362C2×C4—1062no—
363C2×C4—2122no—
364C2×C4D4×Dic5311yescharacteristic
365C2×C4D4×Dic5311yescharacteristic
366C2×C4#1313324111yescharacteristic
367C2×C4#1313324111yescharacteristic
368C2×C4#1313324111yescharacteristic
369C2×C4C2×D4×D5311yescharacteristic
370C8—1062no—
371C8—2122no—
372C8D4×Dic5311yescharacteristic
373C8D4×Dic5311yescharacteristic
374C8C23×Dic5311yescharacteristic
375C8C2×#1171994711yescharacteristic
376C22—41no—
377C22—42no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—161no—
382C22#1681073511yescharacteristic
383C22#1681073511yescharacteristic
384C22C2×#1313324111yescharacteristic
385C4—42no—
386C4—42no—
387C4—4242no—
388C4—2122no—
389C4#1681073511yescharacteristic
390C4#1681073511yescharacteristic
391C4C2×D4×Dic5311yescharacteristic
392C4C2×#1313324111yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—82no—
398C2C2×#1681073511yescharacteristic
subgroupG/H
centerC2C2×#16810735
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3414C2
socleC2×C53C2×#442
cosocleC2×C8×C53C24
hypercenter#3414D53
hypocenterC53#33469
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #20283632C2semidirect
2. #20283599C2semidirect
3. #20276614C2semidirect
4. #20276607C2semidirect
5. C2×#16810446C2semidirect
6. #20276464C2semidirect
7. C2×#16810433C2semidirect
8. #20276319C2semidirect
15. #16810735C22semidirect
16. #16810730C22semidirect
23. #16810446C22semidirect
24. #16810439C22semidirect
25. #16810436C22semidirect
27. #16810400C22semidirect
30. C53×#510C4semidirect
32. C53×#509C4semidirect
34. C53×#508C4semidirect
68. #135×C53C2×C4semidirect
70. C2×Q16×C53C2×C4semidirect
74. C2×D8×C53C2×C4semidirect
77. Q32×C53C2×C4semidirect
78. SD32×C53C2×C4semidirect
79. SD32×C53C2×C4semidirect
80. D16×C53C2×C4semidirect
81. M32×C53C2×C4semidirect
83. C2×C16×C53C2×C4semidirect
213. #510Dic53semidirect
215. #509Dic53semidirect
217. #508Dic53semidirect
235. C53#33469semidirect
244. #135C2×Dic53semidirect
246. C2×Q16C2×Dic53semidirect
250. C2×D8C2×Dic53semidirect
265. Q32C2×Dic53semidirect
266. SD32C2×Dic53semidirect
267. SD32C2×Dic53semidirect
268. D16C2×Dic53semidirect
271. M32C2×Dic53semidirect
274. C2×C16C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202836322
#202835992
#202766142
#202766072
C2×#168104462
#202764642
C2×#168104332
#202763192
C53×#34142
#3346953