Finite Group Databasebeta

← #17043170

Subgroups — #17043170

Order 4352.

702 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539296C211yesmaximal, characteristic
2#14537717C212yesmaximal
3#14529759C212yesmaximal
4#14529532C211yesmaximal, characteristic
5#14529529C212yesmaximal
6#14528954C211yesmaximal, characteristic
7#14528666C211yesmaximal, characteristic
8#14528665C211yesmaximal, characteristic
9#14528662C212yesmaximal
10C4×C8×Dic17C211yesmaximal, characteristic
11C17×#1945C211yesmaximal, characteristic
12#11763362C2212yes—
13#11763213C412yescomplemented
14#11763210C412yescomplemented
15#11763209C412yescomplemented
16#11763206C412yescomplemented
17#11762938—24no—
18#11762933—24no—
19C2×#10587490C2211yescharacteristic
20C2×#10587493C2212yes—
21#11762847C2211yescharacteristic
22#11762845C2212yes—
23#11762835C2212yes—
24#11762832C2212yes—
25#11762767C2212yes—
26#11762765C2212yes—
27C2×C4×#3622C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32C42×Dic17C2211yescharacteristic
33#11762570C2212yes—
34#11762569C2212yes—
35#11762568C2212yes—
36C2×C17×#108C2211yescharacteristic
37C17×#427C2211yescharacteristic
38C17×#424C2212yes—
39C2×C4×C8×C17C2211yescharacteristic
40C17×#339C2212yes—
41C2×M16×C17C2312yes—
42C22×C8×C17C2311yescharacteristic
43C22×C8×C17C2311yescharacteristic
44C17×#117C2312yes—
45C2×C42×C17C2311yescharacteristic
46#10587612C2312yes—
47C2×C4×Dic17—22no—
48C2×C4×Dic17C2311yescharacteristic
49C2×C4×Dic17C2311yescharacteristic
50C2×#63223C2312yes—
51C22×#3622C2311yescharacteristic
52C22×#3622C2311yescharacteristic
53#10587576C2×C414yes—
54#10587574C2×C414yes—
55#10587564—24no—
56C4×Dic34—24no—
57C17×#108—22no—
58C17×#107C2×C412yes—
59C17×#106C2×C412yes—
60C17×C4⋊C8—24no—
61C4×C8×C17—22no—
62#10587493—24no—
63#10587490—22no—
64#10587489C2×C412yes—
65#10587488C2×C412yes—
66#10587486—22no—
67#10587485—24no—
68#10587485—24no—
69C8×Dic17—22no—
70C8×Dic17—22no—
71C8×Dic17C2×C411yescharacteristic
72C8×Dic17C2×C411yescharacteristic
73C8×Dic17C2×C411yescharacteristic
74C8×Dic17C2×C411yescharacteristic
75C8×Dic17D412yes—
76C8×Dic17D412yes—
77C8×Dic17Q814yescomplemented
78#10587479C2×C414yes—
79#10587478C2×C414yes—
80#10587474—22no—
81C4×#3622—22no—
82C4×#3622Q814yescomplemented
83C22×C4×C17—21no—
84C22×C4×C17C2411yescharacteristic
85C22×Dic17—21no—
86C22×Dic17#4112yes—
87C2×Dic34—24no—
88M16×C17—22no—
89M16×C17—22no—
90C2×C8×C17—21no—
91C2×C8×C17—21no—
92C2×C8×C17—22no—
93C2×C8×C17C22×C411yescharacteristic
94C2×C8×C17C22×C411yescharacteristic
95C2×C8×C17C2×D412yes—
96C2×C8×C17C2×Q812yes—
97C2×C8×C17#4112yes—
98C4⋊C4×C17C22×C414yes—
99C22⋊C4×C17—22no—
100C42×C17—22no—
101C42×C17—21no—
102C42×C17C2×Q812yes—
103C22⋊Dic17—22no—
104C4⋊Dic17—24no—
105C4⋊Dic17C22×C414yes—
106#63225—24no—
107#63225—24no—
108C4×Dic17—28no—
109C4×Dic17—22no—
110C4×Dic17—21no—
111C4×Dic17C22×C411yescharacteristic
112C4×Dic17C22×C411yescharacteristic
113C4×Dic17C2×D412yes—
114C4×Dic17C2×Q812yes—
115#63223—22no—
116#63223—22no—
117C2×#3622—22no—
118C2×#3622C22×C411yescharacteristic
119C2×#3622C22×C411yescharacteristic
120C2×#3622C2×D411yescharacteristic
121C2×#3622C2×D411yescharacteristic
122C2×#3622C2×Q812yes—
123C2×#3622C2×Q812yes—
124C2×#3622C2×Q812yes—
125C2×#3622#4112yes—
126#21732—171nomaximal, Sylow
127C23×C17C2×#4111yescharacteristic
128C2×C4×C17—24no—
129C2×C4×C17—22no—
130C2×C4×C17—24no—
131C2×C4×C17C23×C411yescharacteristic
132C2×C4×C17C22×D411yescharacteristic
133C2×C4×C17C22×Q811yescharacteristic
134C2×C4×C17C22×Q811yescharacteristic
135C2×C4×C17C2×#4111yescharacteristic
136C2×C4×C17C2×#4111yescharacteristic
137C2×Dic17—24no—
138C2×Dic17—22no—
139C2×Dic17—24no—
140C2×Dic17—24no—
141C2×Dic17C4×D412yes—
142C2×Dic17C4×D412yes—
143Dic34—48no—
144C8×C17—22no—
145C8×C17—42no—
146C8×C17C4×Q812yes—
147C8×C17C4×Q812yes—
148#3622—42no—
149#3622C4×Q812yes—
150#3622C4×Q812yes—
151#3622C4⋊Q812yes—
152#3622C4⋊Q812yes—
153#3160—171no—
154#2960—172no—
155#1994—172no—
156#1945—172no—
157#1945—171no—
158#1945D1711yescharacteristic
159#1848—171no—
160#1772—171no—
161#1772—171no—
162#1772—172no—
163C42×C8—171no—
164C22×C17—22no—
165C22×C17C2×C4×D411yescharacteristic
166C22×C17#53311yescharacteristic
167C22×C17#53311yescharacteristic
168C4×C17—24no—
169C4×C17—44no—
170C4×C17C2×C4×Q811yescharacteristic
171C4×C17C2×C4×Q811yescharacteristic
172C4×C17C2×C4⋊Q811yescharacteristic
173C4×C17#53211yescharacteristic
174Dic17—24no—
175Dic17—22no—
176Dic17—44no—
177Dic17#44314yes—
178#533—172no—
179#500—172no—
180#499—172no—
181#499—172no—
182#498—172no—
183#446—344no—
184C8×Q8—344no—
185C2×#108—172no—
186C2×#108—171no—
187C2×#108C2×D1711yescharacteristic
188#427—171no—
189#427—172no—
190#427C2×D1711yescharacteristic
191#424—172no—
192#424C2×D1712yes—
193#423—172no—
194#405—172no—
195#405—172no—
196C2×C4×C8—171no—
197C2×C4×C8—171no—
198C2×C4×C8—171no—
199C2×C4×C8—171no—
200C2×C4×C8—171no—
201C2×C4×C8C2×D1711yescharacteristic
202C43—171no—
203#339—172no—
204#339—172no—
205#339—172no—
206#339C2×D1712yes—
207C2×C17—21no—
208C2×C17—21no—
209C2×C17C4×C4⋊Q811yescharacteristic
210C2×C17C2×#44312yes—
211C2×M16—172no—
212C2×M16C22×D1712yes—
213C22×C8—171no—
214C22×C8—171no—
215C22×C8C22×D1711yescharacteristic
216C22×C8C22×D1711yescharacteristic
217C4⋊Q8—174no—
218#125—174no—
219C22⋊Q8—344no—
220C4×Q8—344no—
221#117—172no—
222#117C22×D1712yes—
223C2×C42—171no—
224C2×C42—171no—
225C2×C42—342no—
226C2×C42C22×D1711yescharacteristic
227#108—22no—
228#108—344no—
229#108—342no—
230#107—172no—
231#107—174no—
232#107C4×D1712yes—
233#106—172no—
234#106—174no—
235#106C4×D1712yes—
236C4⋊C8—24no—
237C4⋊C8—344no—
238C4⋊C8—344no—
239#97—342no—
240#97—342no—
241C4×C8—22no—
242C4×C8—171no—
243C4×C8—171no—
244C4×C8—171no—
245C4×C8—171no—
246C4×C8—172no—
247C4×C8—172no—
248C4×C8—174no—
249C4×C8—174no—
250C4×C8—342no—
251C4×C8—342no—
252C4×C8—342no—
253C17#2173211yescharacteristic, complemented, Sylow
254C2×Q8—344no—
255C22×C4—21no—
256C22×C4—172no—
257C22×C4—341no—
258C22×C4C23×D1711yescharacteristic
259M16—22no—
260M16—22no—
261M16—342no—
262M16—342no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—22no—
266C2×C8—171no—
267C2×C8—171no—
268C2×C8—171no—
269C2×C8—171no—
270C2×C8—172no—
271C2×C8—172no—
272C2×C8—172no—
273C2×C8—172no—
274C2×C8—342no—
275C2×C8C2×C4×D1711yescharacteristic
276C2×C8C2×C4×D1711yescharacteristic
277C2×C8D4×D1712yes—
278C2×C8#6325412yes—
279C2×C8Q8×D1712yes—
280C4⋊C4—174no—
281C4⋊C4—344no—
282C4⋊C4—344no—
283C4⋊C4—344no—
284C4⋊C4C2×C4×D1714yes—
285C22⋊C4—22no—
286C22⋊C4—342no—
287C42—22no—
288C42—21no—
289C42—171no—
290C42—171no—
291C42—172no—
292C42—172no—
293C42—348no—
294C42—342no—
295C42—341no—
296C42Q8×D1712yes—
297C23#41×D1711yescharacteristic
298Q8—688no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—172no—
303C2×C4—172no—
304C2×C4—344no—
305C2×C4—344no—
306C2×C4—342no—
307C2×C4—344no—
308C2×C4C22×C4×D1711yescharacteristic
309C2×C4C2×D4×D1711yescharacteristic
310C2×C4C2×#6325411yescharacteristic
311C2×C4C2×Q8×D1711yescharacteristic
312C2×C4C2×Q8×D1711yescharacteristic
313C2×C4#41×D1711yescharacteristic
314C8—22no—
315C8—42no—
316C8—172no—
317C8—172no—
318C8—172no—
319C8—172no—
320C8—682no—
321C8#1058757312yes—
322C8#1058757312yes—
323C22—22no—
324C22C4×D4×D1711yescharacteristic
325C22#1176336111yescharacteristic
326C22#1176336511yescharacteristic
327C4—24no—
328C4—44no—
329C4—174no—
330C4—344no—
331C4—342no—
332C4—684no—
333C4C4×Q8×D1711yescharacteristic
334C4C2×#1058757311yescharacteristic
335C4C4⋊Q8×D1711yescharacteristic
336C4#1176335411yescharacteristic
337C2—21no—
338C2—21no—
339C2#1453135511yescharacteristic
340C2D17×#44312yes—
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1945C2
socleC22×C17C2×C4×D4
prime radicalC23×C17C2×#41
cosocleC22×C4×C17C24
hypercenter#1945D17
hypocenterC17#21732
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1945

Complements

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

NH ≅ G/Ntype
13. #11763213C4semidirect
14. #11763210C4semidirect
15. #11763209C4semidirect
16. #11763206C4semidirect
77. C8×Dic17Q8semidirect
82. C4×#3622Q8semidirect
253. C17#21732semidirect

Maximal subgroups

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

subgroupindex
#145392962
#145377172
#145297592
#145295322
#145295292
#145289542
#145286662
#145286652
#145286622
C4×C8×Dic172
C17×#19452
#2173217