Finite Group Databasebeta

← #17898660

Subgroups — #17898660

Order 4352.

2612 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14543558C211yesmaximal, characteristic
2#14543497C212yesmaximal, complemented
3#14537700C216yesmaximal, complemented
4#14537693C216yesmaximal, complemented
5#14537686C216yesmaximal, complemented
6C2×#11762925C216yesmaximal, complemented
7#14537329C212yesmaximal, complemented
8#131×Dic17C211yesmaximal, characteristic, complemented
9C17×#2916C211yesmaximal, characteristic, complemented
10#11763645C2212yes—
11#11763626C414yescomplemented
12#11763621C418yescomplemented
13#11763614C414yescomplemented
14#11763609C2216yes—
15C2×#10587590C2216yes—
16#11763575C22112yescomplemented
17#11763574—28no—
18C2×#10587593C22112yescomplemented
19C2×#10587634C2211yescharacteristic
20#131×D17—28no—
21#41×Dic17C2211yescharacteristic
22#11762930C22124yescomplemented
23#11762925C2214yescomplemented
24#11762925C22112yescomplemented
25#11762925C22112yescomplemented
26#11762925C22112yescomplemented
27#11762825C22112yescomplemented
28C2×#10587496C22112yescomplemented
29#11762767C2213yescomplemented
30#11762764C2213yescomplemented
31M16×Dic17C2213yescomplemented
32C2×C8×Dic17C2213yescomplemented
33C17×#568C2212yes—
34C2×C17×#131C2211yescharacteristic
35C17×#430C2216yescomplemented
36C2×C17×#108C2216yescomplemented
37C2×#41×C17C2311yescharacteristic
38#10587697C2×C418yes—
39#10587696C2×C418yes—
40#41×D17—28no—
41#10587694—22no—
42#10587691—26no—
43C2×#63252C23112yes—
44C17×#131—28no—
45C17×#131C2×C414yes—
46C17×#131C2×C414yes—
47C2×M16×C17C23112yes—
48C2×M16×C17C2313yes—
49C22×C8×C17C2313yes—
50#10587634D414yes—
51#10587634D414yes—
52Q8×Dic17—21no—
53D4×Dic17—23no—
54#10587612C2313yes—
55C2×C4×Dic17C2313yes—
56C2×#63223C2313yes—
57C22×#3622C2313yes—
58#10587595C2×C4124yescomplemented
59#10587594C2×C4124yescomplemented
60#10587593—212no—
61#10587593—212no—
62#10587593C2318yes—
63#10587593C23124yes—
64M16×D17—224no—
65C2×Dic68C2×C416yescomplemented
66#10587590C2×C4112yescomplemented
67#10587590C2×C4112yescomplemented
68#10587590C2314yes—
69#10587590C23112yes—
70C2×D136C2×C416yescomplemented
71C2×#63219C2×C4112yescomplemented
72#10587587—224no—
73C2×#63218—212no—
74C2×C8×D17—212no—
75C17×#108C2314yes—
76C17×#108C23112yes—
77#10587496C2318yes—
78#10587496C23124yes—
79#10587486C2316yes—
80C8×Dic17C2311yescharacteristic
81C8×Dic17C2313yes—
82C8×Dic17C2313yes—
83C8×Dic17C2313yes—
84#41×C17C22×C414yes—
85#41×C17C2×D414yes—
86C2×Q8×C17#4111yescharacteristic
87C2×D4×C17#4113yes—
88C22×C4×C17C2413yes—
89C2×C22⋊D17—212no—
90C22×Dic17—23no—
91#63256—28no—
92Q8×D17—28no—
93#63254—224no—
94D4×D17—224no—
95#63252—212no—
96#63252—212no—
97#63252C22×C4124yes—
98#63252C2418yes—
99C2×D68C22×C4112yes—
100C2×C4×D17—212no—
101C2×Dic34C22×C4112yes—
102M16×C17—212no—
103M16×C17C22×C4112yes—
104M16×C17C2418yes—
105C2×C8×C17—212no—
106C2×C8×C17C22×C416yes—
107C2×C8×C17C22×C416yes—
108C2×C8×C17C2411yescharacteristic
109C2×C8×C17C2413yes—
110C22⋊Dic17—23no—
111C4⋊Dic17—23no—
112C4×Dic17—23no—
113C4×Dic17C2411yescharacteristic
114C4×Dic17C2413yes—
115#63223C2×D4112yes—
116C2×#3622C2×D416yes—
117C2×#3622C2×D416yes—
118C2×#3622C2411yescharacteristic
119C2×#3622C2413yes—
120Dic68C22×C414yes—
121Dic68C22×C4112yes—
122D136C22×C414yes—
123D136C22×C4112yes—
124#63219C22×C418yes—
125#63219C22×C4124yes—
126#63218—28no—
127#63218—224no—
128C8×D17—28no—
129C8×D17—224no—
130#60183—171nomaximal, Sylow
131C23×C17C2×#4113yes—
132C22×D17—212no—
133Q8×C17C4×D414yes—
134D4×C17C4×D4112yes—
135C2×C4×C17C23×C416yes—
136C2×C4×C17C22×D416yes—
137C2×C4×C17C2×#4113yes—
138C2×C4×C17C2511yescharacteristic
139C22⋊D17—224no—
140C22⋊D17—28no—
141C2×Dic17—212no—
142C2×Dic17—21no—
143C2×Dic17—23no—
144C2×Dic17—46no—
145D68—212no—
146D68C23×C418yes—
147C4×D17—224no—
148C4×D17—28no—
149Dic34—212no—
150Dic34C23×C418yes—
151C8×C17—28no—
152C8×C17C23×C411yescharacteristic
153C8×C17C23×C411yescharacteristic
154C8×C17C23×C416yes—
155#3622C22×D411yescharacteristic
156#3622C22×D416yes—
157#3622C22×D411yescharacteristic
158#3584—171no—
159#3573—172no—
160#2957—176no—
161#2956—176no—
162#2955—176no—
163C2×#443—176no—
164#2916D1711yescharacteristic, complemented
165#2901—172no—
166C4×#131—171no—
167C22×C17—43no—
168C22×C17C2×C4×D416yes—
169C22×C17C22×#4111yescharacteristic
170C2×D17—28no—
171C2×D17—812no—
172C4×C17C2×C4×D416yes—
173C4×C17C24×C411yescharacteristic
174C4×C17C23×D411yescharacteristic
175Dic17—22no—
176Dic17—28no—
177Dic17—43no—
178#585—172no—
179#578—174no—
180#577—178no—
181#576—174no—
182#575—176no—
183C2×#135—176no—
184#568—1712no—
185#568—348no—
186#568C2×D1712yes—
187C2×#131—171no—
188C2×#131—1712no—
189C2×#131—348no—
190C2×#131C2×D1711yescharacteristic
191C4×#41—171no—
192#444—1724no—
193#443—174no—
194#443—1712no—
195#443—1712no—
196#443—1712no—
197#430C2×D1716yescomplemented
198C2×#108C2×D1716yescomplemented
199#421—1712no—
200C2×#104—1712no—
201#405—173no—
202#405—173no—
203C4×M16—173no—
204C2×C4×C8—173no—
205C2×C17—21no—
206C2×C17—26no—
207C2×C17C22×C4×D411yescharacteristic
208D17—48no—
2092−1+4—178no—
2102−1+4—342no—
211H(5,𝔽2)—178no—
212H(5,𝔽2)—346no—
213C2×#41—1712no—
214C2×#41—348no—
215C2×#41C22×D1711yescharacteristic
216#137—1724no—
217AGL(1,ℤ/8)—1724no—
218#135—1712no—
219#135—1712no—
220#135—174no—
221#135—1712no—
222C2×Q16—176no—
223C2×SD16—1712no—
224C2×D8—176no—
225#131—28no—
226#131—174no—
227#131—174no—
228#131—178no—
229#131—1724no—
230#131—3424no—
231#131—3412no—
232#131—3412no—
233#131C4×D1714yes—
234#131C4×D1714yes—
235C2×M16—173no—
236C2×M16—3412no—
237C2×M16—3424no—
238C2×M16C22×D17112yes—
239C2×M16C22×D1713yes—
240C22×C8—173no—
241C22×C8—3412no—
242C22×C8C22×D1713yes—
243C4×Q8—341no—
244C4×D4—343no—
245#117—173no—
246C2×C42—173no—
247#108C22×D1714yes—
248#108C22×D17112yes—
249#104—178no—
250#104—1724no—
251#97—176no—
252C4×C8—171no—
253C4×C8—173no—
254C4×C8—173no—
255C4×C8—173no—
256C17#6018311yescharacteristic, complemented, Sylow
257#41—1724no—
258#41—178no—
259#41—348no—
260#41—3424no—
261#41—3412no—
262#41—3412no—
263#41C2×C4×D1714yes—
264#41D4×D1714yes—
265C2×Q8—1712no—
266C2×Q8—348no—
267C2×Q8#6325611yescharacteristic
268C2×D4—1712no—
269C2×D4—3412no—
270C2×D4—3424no—
271C2×D4#6325613yes—
272C22×C4—3412no—
273C22×C4—343no—
274C22×C4C23×D1713yes—
275Q16—174no—
276Q16—1712no—
277SD16—178no—
278SD16—1724no—
279D8—174no—
280D8—1712no—
281M16—212no—
282M16—1712no—
283M16—348no—
284M16—3424no—
285M16C2×C4×D17112yes—
286M16C23×D1718yes—
287C2×C8—212no—
288C2×C8—176no—
289C2×C8—176no—
290C2×C8—171no—
291C2×C8—173no—
292C2×C8—348no—
293C2×C8—3424no—
294C2×C8C2×C4×D1716yes—
295C2×C8C2×C4×D1716yes—
296C2×C8C23×D1711yescharacteristic
297C2×C8C23×D1713yes—
298C4⋊C4—343no—
299C22⋊C4—343no—
300C42—171no—
301C42—173no—
302C42—343no—
303C23—3412no—
304C23C2×#6325613yes—
305Q8—178no—
306Q8—3412no—
307Q8#1058757914yes—
308D4—178no—
309D4—3424no—
310D4—348no—
311D4—3412no—
312D4#10587579112yes—
313C2×C4—3412no—
314C2×C4—341no—
315C2×C4—3424no—
316C2×C4—348no—
317C2×C4—343no—
318C2×C4—686no—
319C2×C4C22×C4×D1716yes—
320C2×C4C2×D4×D1716yes—
321C2×C4C2×#6325613yes—
322C2×C4C24×D1711yescharacteristic
323C8—28no—
324C8—171no—
325C8—171no—
326C8—176no—
327C8C22×C4×D1711yescharacteristic
328C8C22×C4×D1711yescharacteristic
329C8C22×C4×D1716yes—
330C22—43no—
331C22—348no—
332C22—13612no—
333C22C2×#1058757916yes—
334C22C22×#6325611yescharacteristic
335C4—342no—
336C4—348no—
337C4—683no—
338C4C2×#1058757916yes—
339C4C23×C4×D1711yescharacteristic
340C4C22×D4×D1711yescharacteristic
341C2—21no—
342C2—26no—
343C2—688no—
344C2C22×#1058757911yescharacteristic
subgroupG/H
centerC4C22×D4×D17
commutatorC4×C17C24×C4
FrattiniC2×C4C24×D17
FittingC17×#2916C2
socleC2×C17C22×C4×D4
prime radical#14543558C2
cosocleC2×C4×C17C25
hypercenter#2916D17
hypocenterC17#60183
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2916

Complements

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

NH ≅ G/Ntype
2. #14543497C2semidirect
3. #14537700C2semidirect
4. #14537693C2semidirect
5. #14537686C2semidirect
6. C2×#11762925C2semidirect
7. #14537329C2semidirect
8. #131×Dic17C2semidirect
9. C17×#2916C2semidirect
11. #11763626C4semidirect
12. #11763621C4semidirect
13. #11763614C4semidirect
16. #11763575C22semidirect
18. C2×#10587593C22semidirect
22. #11762930C22semidirect
23. #11762925C22semidirect
24. #11762925C22semidirect
25. #11762925C22semidirect
26. #11762925C22semidirect
27. #11762825C22semidirect
28. C2×#10587496C22semidirect
29. #11762767C22semidirect
30. #11762764C22semidirect
31. M16×Dic17C22semidirect
32. C2×C8×Dic17C22semidirect
35. C17×#430C22semidirect
36. C2×C17×#108C22semidirect
58. #10587595C2×C4semidirect
59. #10587594C2×C4semidirect
65. C2×Dic68C2×C4semidirect
66. #10587590C2×C4semidirect
67. #10587590C2×C4semidirect
70. C2×D136C2×C4semidirect
71. C2×#63219C2×C4semidirect
164. #2916D17semidirect
197. #430C2×D17semidirect
198. C2×#108C2×D17semidirect
256. C17#60183semidirect

Maximal subgroups

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

subgroupindex
#145435582
#145434972
#145377002
#145376932
#145376862
C2×#117629252
#145373292
#131×Dic172
C17×#29162
#6018317