Finite Group Databasebeta

← #17023654

Subgroups — #17023654

Order 4352.

736 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539525C212yesmaximal, complemented
2#14539160C212yesmaximal
3#14539159C211yesmaximal, characteristic
4#14528447C212yesmaximal, complemented
5#14528442C212yesmaximal, complemented
6#14528441C212yesmaximal, complemented
7#14528397C212yesmaximal, complemented
8#14527087C211yesmaximal, characteristic, complemented
9C17×#3185C211yesmaximal, characteristic, complemented
10#11763366C2211yescharacteristic
11C2×#10587560C2211yescharacteristic
12#11763205C2212yescomplemented
13#11763188C2212yescomplemented
14#11763175—24no—
15#11763174—22no—
16#11763161C2212yescomplemented
17#11763159C2212yescomplemented
18#11763157C2212yes—
19#11763157C2212yes—
20#11763157C2212yes—
21#11763157C2212yes—
22#11763156C2212yes—
23#11763156C2212yes—
24#11763065—24no—
25#11762914—24no—
26#11762815—22no—
27#11762813—24no—
28C4×#63223C2212yes—
29#11762531C2212yescomplemented
30#11762530C2211yescharacteristic, complemented
31C8×#3622C2211yescharacteristic, complemented
32C17×#533C2212yes—
33C17×#498C2211yescharacteristic, complemented
34C17×#494C2211yescharacteristic, complemented
35C17×#487C2212yescomplemented
36C17×#470—22no—
37C17×#441—22no—
38C4×M16×C17C2211yescharacteristic
39C22×D68—22no—
40C22×D68—21no—
41C2×Q16×C17—22no—
42C2×M16×C17D412yes—
43C17×C4⋊Q8C2312yes—
44C17×C4⋊Q8C2312yes—
45C17×#125—22no—
46C17×C22⋊Q8—22no—
47C17×C22⋊Q8—22no—
48C4×Q8×C17—22no—
49C4×Q8×C17—22no—
50C17×#117—22no—
51C2×C42×C17C2311yescharacteristic
52C2×#63231—24no—
53#10587612—21no—
54#10587610—22no—
55C2×#63223D414yes—
56#10587560—21no—
57#10587560—22no—
58#10587560C2312yes—
59#10587556—21no—
60#10587555C2312yes—
61C4×Dic34—22no—
62C17×#106—22no—
63C17×#103—24no—
64C17×#97D412yes—
65C4×C8×C17C2311yescharacteristic
66C4×C8×C17C2311yescharacteristic
67#10587506—24no—
68#10587492—24no—
69#10587492—24no—
70#10587481—24no—
71#10587480—24no—
72#10587480—24no—
73#10587480—24no—
74#10587480—24no—
75#10587479—24no—
76#10587474D414yes—
77C4×#3622C2312yes—
78C4×#3622C2312yes—
79C23×D17—41no—
80C2×Q8×C17—22no—
81C2×Q8×C17—22no—
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D411yescharacteristic
84C2×D68—22no—
85C2×D68—22no—
86C2×D68—22no—
87C2×D68—44no—
88C2×D68—48no—
89C2×D68—42no—
90C2×D68—41no—
91C2×Dic34—22no—
92Q16×C17—44no—
93M16×C17—42no—
94C2×C8×C17C2×D412yes—
95C2×C8×C17C2×D412yes—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—22no—
99C4⋊C4×C17—22no—
100C4⋊C4×C17—42no—
101C4⋊C4×C17—42no—
102C22⋊C4×C17—42no—
103C42×C17—42no—
104C42×C17C2×D412yes—
105C42×C17C2×D411yescharacteristic
106C42×C17C2411yescharacteristic
107C22⋊Dic17—41no—
108#63231—48no—
109C4⋊Dic17—22no—
110C4⋊Dic17—22no—
111#63225—42no—
112C4×Dic17—41no—
113#63223—44no—
114C2×#3622C2×D414yes—
115C2×#3622C2×D414yes—
116#19753—171nomaximal, Sylow
117C23×C17#12711yescharacteristic
118C22×D17—42no—
119C22×D17—84no—
120C22×D17—82no—
121Q8×C17—44no—
122Q8×C17—44no—
123C2×C4×C17—22no—
124C2×C4×C17—24no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17#12712yes—
128C2×C4×C17#12711yescharacteristic
129C2×C4×C17C22×D412yes—
130C2×C4×C17C22×D411yescharacteristic
131C2×Dic17—42no—
132D68—44no—
133D68—44no—
134D68—44no—
135D68—84no—
136D68—82no—
137Dic34—44no—
138C8×C17—22no—
139C8×C17—22no—
140#3622—24no—
141#3622—24no—
142#3185—172no—
143#3185D1711yescharacteristic, complemented
144#3138—171no—
145#3138—172no—
146#1720—172no—
147#1720—172no—
148#1720—172no—
149#1711—172no—
150#1455—171no—
151C22×C17—22no—
152C22×C17—21no—
153C22×C17C2×#12711yescharacteristic
154C2×D17—82no—
155C2×D17—82no—
156C2×D17—82no—
157C2×D17—164no—
158C4×C17—22no—
159C4×C17—21no—
160C4×C17—82no—
161C4×C17—82no—
162C4×C17#57512yes—
163C4×C17#57514yes—
164Dic17—82no—
165#533—171no—
166#533C2×D1712yes—
167C2×#127—171no—
168#498—172no—
169#498C2×D1711yescharacteristic, complemented
170#494—172no—
171#494C2×D1711yescharacteristic, complemented
172#490—344no—
173#490—342no—
174#487—172no—
175#487—172no—
176#487C2×D1712yescomplemented
177#486—172no—
178#486—172no—
179#486—172no—
180#486—172no—
181#486—172no—
182#486—172no—
183#470—22no—
184#470—344no—
185#441—22no—
186#441—344no—
187#418—344no—
188#418—342no—
189C4×M16—172no—
190C4×M16C2×D1711yescharacteristic
191#322—171no—
192#322—172no—
193C82—171no—
194C2×C17—41no—
195C2×C17#315312yes—
196C2×C17#315311yescharacteristic
197D17—162no—
198C22×D4—342no—
199C22×D4—341no—
200C2×Q16—22no—
201C2×Q16—344no—
202C2×M16—174no—
203C2×M16C22⋊D1712yes—
204C4⋊Q8—172no—
205C4⋊Q8C22×D1712yes—
206C4⋊Q8C22×D1712yes—
207#127—172no—
208#127—342no—
209#127—341no—
210#125—22no—
211#125—341no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—342no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—342no—
218#117—22no—
219#117—341no—
220C2×C42C22×D1711yescharacteristic
221#106—22no—
222#106—344no—
223#103—24no—
224#103—344no—
225#103—344no—
226#102—344no—
227#102—344no—
228#102—344no—
229#102—344no—
230#102—344no—
231#102—344no—
232#97—174no—
233#97C22⋊D1712yes—
234C4×C8—172no—
235C4×C8—172no—
236C4×C8C22×D1711yescharacteristic
237C4×C8C22×D1711yescharacteristic
238C17#1975311yescharacteristic, complemented, Sylow
239C24—681no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—342no—
243C2×D4—342no—
244C2×D4—342no—
245C2×D4—342no—
246C2×D4—684no—
247C2×D4—688no—
248C2×D4—682no—
249C2×D4—681no—
250C22×C4D4×D1712yes—
251C22×C4C2×C22⋊D1711yescharacteristic
252Q16—44no—
253Q16—688no—
254M16—42no—
255M16—684no—
256C2×C8—174no—
257C2×C8—174no—
258C2×C8C2×C22⋊D1712yes—
259C2×C8C2×C22⋊D1712yes—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—342no—
267C4⋊C4—342no—
268C4⋊C4—682no—
269C22⋊C4—42no—
270C22⋊C4—681no—
271C42—42no—
272C42—681no—
273C42D4×D1712yes—
274C42C2×C22⋊D1711yescharacteristic
275C42C23×D1711yescharacteristic
276C23—682no—
277C23—1364no—
278C23—1362no—
279C23#1058762611yescharacteristic
280Q8—44no—
281Q8—44no—
282Q8—684no—
283D4—684no—
284D4—684no—
285D4—684no—
286D4—1364no—
287D4—1362no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—682no—
293C2×C4#1058762612yes—
294C2×C4#1058762611yescharacteristic
295C2×C4C2×D4×D1712yes—
296C2×C4C22×C22⋊D1711yescharacteristic
297C8—22no—
298C8—22no—
299C8—344no—
300C8—344no—
301C22—22no—
302C22—21no—
303C22—2724no—
304C22—1362no—
305C22—1362no—
306C22—1362no—
307C22C2×#1058762611yescharacteristic
308C4—22no—
309C4—21no—
310C4—82no—
311C4—82no—
312C4—1362no—
313C4#1176360712yes—
314C4#1176360814yes—
315C2—41no—
316C2—2722no—
317C2#1453925112yes—
318C2#1453925211yescharacteristic
subgroupG/H
centerC22C2×#10587626
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3185C2
socleC22×C17C2×#127
prime radicalC2×#10587560C22
hypercenter#3185D17
hypocenterC17#19753
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3185

Complements

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

NH ≅ G/Ntype
1. #14539525C2semidirect
4. #14528447C2semidirect
5. #14528442C2semidirect
6. #14528441C2semidirect
7. #14528397C2semidirect
8. #14527087C2semidirect
9. C17×#3185C2semidirect
12. #11763205C22semidirect
13. #11763188C22semidirect
16. #11763161C22semidirect
17. #11763159C22semidirect
29. #11762531C22semidirect
30. #11762530C22semidirect
31. C8×#3622C22semidirect
33. C17×#498C22semidirect
34. C17×#494C22semidirect
35. C17×#487C22semidirect
143. #3185D17semidirect
169. #498C2×D17semidirect
171. #494C2×D17semidirect
176. #487C2×D17semidirect
238. C17#19753semidirect

Maximal subgroups

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

subgroupindex
#145395252
#145391602
#145391592
#145284472
#145284422
#145284412
#145283972
#145270872
C17×#31852
#1975317