Finite Group Databasebeta

← #17034896

Subgroups — #17034896

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531393C212yesmaximal
2#14528948C211yesmaximal, characteristic, complemented
3#14528944C211yesmaximal, characteristic, complemented
4#14528918C212yesmaximal, complemented
5#14528817C212yesmaximal
6#14528813C212yesmaximal
7#14528681C212yesmaximal, complemented
8C4×#10587489C212yesmaximal, complemented
9C2×C17×#405C211yesmaximal, characteristic
10C2×#10587489C2212yes—
11C2×#10587489C2212yes—
12C2×#10587489C2212yes—
13C2×#10587488C2212yes—
14#11762702—24no—
15#11762695—24no—
16#11762679C2212yes—
17#11762675C2212yes—
18#11762669C2214yes—
19C4×C4⋊Dic17C2214yes—
20#11762572C2214yes—
21#11762572C2214yes—
22C22×M16×C17C2211yescharacteristic
23C23×C8×C17C2211yescharacteristic
24C2×C17×#117C2211yescharacteristic
25C17×#405C418yescomplemented
26C2×C17×#97C2211yescharacteristic
27C2×C17×#97C2211yescharacteristic
28C2×C4×C8×C17C2211yescharacteristic
29C2×C4×C8×C17C2211yescharacteristic
30C23×C4×C17C2311yescharacteristic
31C2×M16×C17—22no—
32C2×M16×C17—22no—
33C2×M16×C17C2×C414yes—
34C22×C8×C17—22no—
35C22×C8×C17—22no—
36C22×C8×C17C2×C414yes—
37C22×C8×C17C2311yescharacteristic
38C22×C8×C17C2311yescharacteristic
39C22×C8×C17C2311yescharacteristic
40C22×C8×C17C2311yescharacteristic
41C17×#117C2×C414yes—
42C17×#117Q814yes—
43C2×C4⋊C4×C17D412yes—
44C2×C22⋊C4×C17D412yes—
45C2×C42×C17C2311yescharacteristic
46C2×C42×C17C2311yescharacteristic
47C2×C22⋊Dic17—24no—
48C2×C4⋊Dic17—24no—
49C2×C4⋊Dic17C2314yes—
50C2×C4⋊Dic17C2314yes—
51C2×C4×Dic17—24no—
52C17×#97C2×C414yes—
53C17×#97C2×C414yes—
54C4×C8×C17C2×C414yes—
55C4×C8×C17C2×C414yes—
56#10587502—24no—
57#10587502—24no—
58#10587502—24no—
59#10587489—24no—
60#10587489—24no—
61#10587489—24no—
62#10587488—24no—
63C24×C17C2×D411yescharacteristic
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17C22×C412yes—
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D412yes—
71C22×C4×C17C2×D412yes—
72C22×C4×C17C2×Q812yes—
73C22×C4×C17C2411yescharacteristic
74C22×Dic17—24no—
75C22×Dic17—24no—
76M16×C17—28no—
77C2×C8×C17—21no—
78C2×C8×C17—21no—
79C2×C8×C17—21no—
80C2×C8×C17—21no—
81C2×C8×C17—28no—
82C2×C8×C17C22×C412yes—
83C2×C8×C17C22×C412yes—
84C2×C8×C17C22×C412yes—
85C2×C8×C17C22×C412yes—
86C2×C8×C17#4112yes—
87C2×C8×C17#4112yes—
88C2×C8×C17#4112yes—
89C2×C8×C17#4112yes—
90C4⋊C4×C17C4⋊C418yes—
91C22⋊C4×C17C4⋊C418yes—
92C42×C17C22×C412yes—
93C42×C17C22×C412yes—
94C42×C17C2×Q812yes—
95C42×C17C2×Q812yes—
96C22⋊Dic17—84no—
97C4⋊Dic17—24no—
98C4⋊Dic17—24no—
99C4⋊Dic17—24no—
100C4⋊Dic17—24no—
101C4⋊Dic17—84no—
102C4×Dic17—84no—
103#20821—171nomaximal, Sylow
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—22no—
107C23×C17—22no—
108C23×C17C2×C4⋊C412yes—
109C23×C17C22×D411yescharacteristic
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17C2×C4⋊C414yes—
121C2×C4×C17C2×C4⋊C414yes—
122C2×C4×C17C2×C4⋊C412yes—
123C2×C4×C17C23×C411yescharacteristic
124C2×C4×C17C22×Q811yescharacteristic
125C2×C4×C17C2×#4111yescharacteristic
126C2×C4×C17C2×#4111yescharacteristic
127C2×C4×C17C2×#4111yescharacteristic
128C2×C4×C17C2×#4111yescharacteristic
129C2×Dic17—44no—
130C2×Dic17—44no—
131C2×Dic17—44no—
132C2×Dic17—44no—
133C2×Dic17—44no—
134C2×Dic17—44no—
135C8×C17—22no—
136C8×C17—22no—
137C8×C17#11714yes—
138C8×C17#11714yes—
139C2×#405D1711yescharacteristic
140#2314—172no—
141#1846—171no—
142#1845—171no—
143#1836—172no—
144#1809—172no—
145#1808—172no—
146#1776—172no—
147C4×#107—172no—
148C22×C17—24no—
149C22×C17—22no—
150C22×C17—42no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17C22×C4⋊C411yescharacteristic
154C22×C17#53211yescharacteristic
155C22×C17#53211yescharacteristic
156C22×C17#57612yes—
157C22×C17#57812yes—
158C4×C17—24no—
159C4×C17—24no—
160C4×C17—22no—
161C4×C17C2×#11711yescharacteristic
162C4×C17C2×#11711yescharacteristic
163C4×C17#53311yescharacteristic
164C4×C17#53311yescharacteristic
165Dic17—84no—
166Dic17—84no—
167C22×M16C2×D1711yescharacteristic
168C23×C8C2×D1711yescharacteristic
169C2×#117C2×D1711yescharacteristic
170C2×#107—172no—
171C2×#107—172no—
172C2×#107—172no—
173C2×#106—172no—
174#405Dic1718yescomplemented
175C2×#97C2×D1711yescharacteristic
176C2×#97C2×D1711yescharacteristic
177C2×C4×C8C2×D1711yescharacteristic
178C2×C4×C8C2×D1711yescharacteristic
179#388—344no—
180#387—344no—
181#384—172no—
182#382—172no—
183#380—174no—
184C4×C4⋊C4—174no—
185#340—174no—
186#340—174no—
187C2×C17—42no—
188C2×C17#228511yescharacteristic
189C2×C17#291312yes—
190C2×C17#317512yes—
191C2×C17#317611yescharacteristic
192C2×C17#317711yescharacteristic
193C23×C4C22×D1711yescharacteristic
194C2×M16—22no—
195C2×M16—22no—
196C2×M16C2×Dic1714yes—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×Dic1714yes—
200C22×C8C22×D1711yescharacteristic
201C22×C8C22×D1711yescharacteristic
202C22×C8C22×D1711yescharacteristic
203C22×C8C22×D1711yescharacteristic
204#117Dic3414yes—
205#117C2×Dic1714yes—
206C2×C4⋊C4—174no—
207C2×C4⋊C4—174no—
208C2×C4⋊C4—344no—
209C2×C4⋊C4D6812yes—
210C2×C22⋊C4—344no—
211C2×C22⋊C4D6812yes—
212C2×C42—344no—
213C2×C42C22×D1711yescharacteristic
214C2×C42C22×D1711yescharacteristic
215#107—344no—
216#107—344no—
217#107—344no—
218#106—344no—
219#97C2×Dic1714yes—
220#97C2×Dic1714yes—
221C4×C8C2×Dic1714yes—
222C4×C8C2×Dic1714yes—
223#95—344no—
224#95—344no—
225#95—344no—
226C17#2082111yescharacteristic, complemented, Sylow
227C24C2×D6811yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—344no—
233C22×C4—344no—
234C22×C4C2×Dic3412yes—
235C22×C4C2×D6811yescharacteristic
236C22×C4C2×D6812yes—
237C22×C4C2×D6812yes—
238C22×C4C22×Dic1712yes—
239C22×C4C23×D1711yescharacteristic
240M16—28no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—28no—
246C2×C8#6325212yes—
247C2×C8#6325212yes—
248C2×C8#6325212yes—
249C2×C8#6325212yes—
250C2×C8C22×Dic1712yes—
251C2×C8C22×Dic1712yes—
252C2×C8C22×Dic1712yes—
253C2×C8C22×Dic1712yes—
254C4⋊C4—344no—
255C4⋊C4—344no—
256C4⋊C4—344no—
257C4⋊C4—344no—
258C4⋊C4—1364no—
259C4⋊C4C4⋊Dic1718yes—
260C22⋊C4—1364no—
261C22⋊C4C4⋊Dic1718yes—
262C42—1364no—
263C42C2×Dic3412yes—
264C42C2×Dic3412yes—
265C42C22×Dic1712yes—
266C42C22×Dic1712yes—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23C2×C4⋊Dic1712yes—
272C23C22×D6811yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—684no—
284C2×C4—684no—
285C2×C4—684no—
286C2×C4—684no—
287C2×C4—684no—
288C2×C4—684no—
289C2×C4C2×C4⋊Dic1714yes—
290C2×C4C2×C4⋊Dic1714yes—
291C2×C4C2×C4⋊Dic1712yes—
292C2×C4C22×Dic3411yescharacteristic
293C2×C4C2×#6325211yescharacteristic
294C2×C4C2×#6325211yescharacteristic
295C2×C4C2×#6325211yescharacteristic
296C2×C4C2×#6325211yescharacteristic
297C2×C4C23×Dic1711yescharacteristic
298C8—22no—
299C8—22no—
300C8#1058761214yes—
301C8#1058761214yes—
302C22—22no—
303C22—24no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22C22×C4⋊Dic1711yescharacteristic
308C22#1176335911yescharacteristic
309C22#1176335911yescharacteristic
310C22#1176361412yes—
311C22#1176362612yes—
312C4—22no—
313C4—24no—
314C4—24no—
315C4—1364no—
316C4—1364no—
317C4C2×#1058761211yescharacteristic
318C4C2×#1058761211yescharacteristic
319C4#1176336611yescharacteristic
320C4#1176336611yescharacteristic
321C2—42no—
322C2#1453119111yescharacteristic
323C2#1453738412yes—
324C2#1453942312yes—
325C2#1453943411yescharacteristic
326C2#1453944311yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#405C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenterC2×#405D17
hypocenterC17#20821
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. #14528948C2semidirect
3. #14528944C2semidirect
4. #14528918C2semidirect
7. #14528681C2semidirect
8. C4×#10587489C2semidirect
25. C17×#405C4semidirect
174. #405Dic17semidirect
226. C17#20821semidirect

Maximal subgroups

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

subgroupindex
#145313932
#145289482
#145289442
#145289182
#145288172
#145288132
#145286812
C4×#105874892
C2×C17×#4052
#2082117