Finite Group Databasebeta

← #17023651

Subgroups — #17023651

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#14539529C211yesmaximal, characteristic, complemented
2#14539525C212yesmaximal, complemented
3#14539158C212yesmaximal
4#14528446C212yesmaximal, complemented
5#14528445C212yesmaximal, complemented
6#14528443C212yesmaximal, complemented
7#14528395C212yesmaximal, complemented
8#14527087C211yesmaximal, characteristic, complemented
9C17×#3138C211yesmaximal, characteristic
10#11763366C2212yes—
11#11763207C2211yescharacteristic, complemented
12#11763205C2212yescomplemented
13#11763189C2211yescharacteristic, complemented
14#11763188C2212yescomplemented
15#11763173—24no—
16#11763161C2212yescomplemented
17#11763160C2212yescomplemented
18#11763159C2212yescomplemented
19#11763155C2212yes—
20#11763155C2212yes—
21#11763155C2212yes—
22#11763155C2212yes—
23#11763067—22no—
24#11763065—24no—
25#11762917—22no—
26#11762914—24no—
27#11762814—24no—
28C4×#63223C2212yes—
29#11762531C2212yescomplemented
30#11762530C2211yescharacteristic, complemented
31C8×#3622C2211yescharacteristic, complemented
32C17×#533C2211yescharacteristic
33C2×C17×#127C2211yescharacteristic
34C17×#490—22no—
35C17×#486C2212yes—
36C17×#486C2212yes—
37C17×#418—22no—
38C4×M16×C17C2211yescharacteristic
39C22×D4×C17—22no—
40C22×D4×C17—21no—
41C2×M16×C17D412yes—
42C17×C4⋊Q8C2312yes—
43C17×#127—22no—
44C17×#127—21no—
45C17×#127C2312yes—
46C17×#125—21no—
47C17×C22⋊Q8—22no—
48C4×Q8×C17—22no—
49C17×#117—21no—
50C2×C42×C17C2311yescharacteristic
51C2×#63231—24no—
52#10587612—22no—
53#10587610—22no—
54#10587610—22no—
55C2×#63223D414yes—
56C2×Dic68—22no—
57#10587556—22no—
58#10587555C2312yes—
59#10587555C2312yes—
60C4×Dic34—22no—
61C4×Dic34—22no—
62C17×#102—24no—
63C17×#102—24no—
64C17×#97D412yes—
65C4×C8×C17C2311yescharacteristic
66C4×C8×C17C2311yescharacteristic
67#10587506—24no—
68#10587503—24no—
69#10587503—24no—
70#10587503—24no—
71#10587503—24no—
72#10587488—22no—
73#10587487—24no—
74#10587481—24no—
75#10587479—24no—
76#10587474D414yes—
77C4×#3622C2312yes—
78C4×#3622C2312yes—
79C24×C17—41no—
80C2×Q8×C17—22no—
81C2×D4×C17—22no—
82C2×D4×C17—22no—
83C2×D4×C17—22no—
84C2×D4×C17—44no—
85C2×D4×C17—48no—
86C2×D4×C17—41no—
87C2×D4×C17—42no—
88C22×C4×C17C2×D412yes—
89C22×C4×C17C2×D411yescharacteristic
90C2×Dic34—22no—
91C2×Dic34—22no—
92M16×C17—42no—
93C2×C8×C17C2×D412yes—
94C2×C8×C17C2×D412yes—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—42no—
98C22⋊C4×C17—41no—
99C42×C17—41no—
100C42×C17C2×D412yes—
101C42×C17C2×D411yescharacteristic
102C42×C17C2411yescharacteristic
103C22⋊Dic17—42no—
104#63231—48no—
105C4⋊Dic17—22no—
106C4⋊Dic17—22no—
107C4⋊Dic17—22no—
108C4⋊Dic17—22no—
109#63225—42no—
110#63225—42no—
111C4×Dic17—42no—
112#63223—44no—
113C2×#3622C2×D414yes—
114C2×#3622C2×D414yes—
115Dic68—44no—
116#19753—171nomaximal, Sylow
117C23×C17—42no—
118C23×C17—84no—
119C23×C17—82no—
120C23×C17#12711yescharacteristic
121Q8×C17—44no—
122D4×C17—44no—
123D4×C17—44no—
124D4×C17—44no—
125D4×C17—84no—
126D4×C17—82no—
127C2×C4×C17—22no—
128C2×C4×C17—24no—
129C2×C4×C17—42no—
130C2×C4×C17#12711yescharacteristic
131C2×C4×C17#12712yes—
132C2×C4×C17C22×D412yes—
133C2×C4×C17C22×D411yescharacteristic
134C2×Dic17—42no—
135C2×Dic17—42no—
136Dic34—44no—
137Dic34—44no—
138C8×C17—22no—
139C8×C17—22no—
140#3622—24no—
141#3622—24no—
142#3185—171no—
143#3185—172no—
144#3138—172no—
145#3138D1711yescharacteristic
146#1720—172no—
147#1720—172no—
148#1720—172no—
149#1711—172no—
150#1455—171no—
151C22×C17—22no—
152C22×C17—21no—
153C22×C17—82no—
154C22×C17—82no—
155C22×C17—82no—
156C22×C17—164no—
157C22×C17C2×#12711yescharacteristic
158C4×C17—22no—
159C4×C17—21no—
160C4×C17—82no—
161C4×C17#57514yes—
162C4×C17#57512yes—
163Dic17—82no—
164Dic17—82no—
165#533—172no—
166#533C2×D1711yescharacteristic
167C2×#127C2×D1711yescharacteristic
168#498—172no—
169#498—171no—
170#494—172no—
171#494—171no—
172#490—22no—
173#490—344no—
174#487—172no—
175#487—172no—
176#487—172no—
177#486—172no—
178#486—172no—
179#486—172no—
180#486—172no—
181#486C2×D1712yes—
182#486C2×D1712yes—
183#470—344no—
184#470—342no—
185#441—344no—
186#441—342no—
187#418—22no—
188#418—344no—
189C4×M16—172no—
190C4×M16C2×D1711yescharacteristic
191#322—171no—
192#322—172no—
193C82—171no—
194C2×C17—41no—
195C2×C17—162no—
196C2×C17#315312yes—
197C2×C17#315311yescharacteristic
198C22×D4—22no—
199C22×D4—21no—
200C2×Q16—344no—
201C2×Q16—342no—
202C2×M16—174no—
203C2×M16C22⋊D1712yes—
204C4⋊Q8—172no—
205C4⋊Q8—172no—
206C4⋊Q8C22×D1712yes—
207#127—22no—
208#127—21no—
209#127C22×D1712yes—
210#125—21no—
211#125—342no—
212C22⋊Q8—22no—
213C22⋊Q8—342no—
214C22⋊Q8—342no—
215C4×Q8—22no—
216C4×Q8—342no—
217C4×Q8—342no—
218#117—21no—
219#117—342no—
220C2×C42C22×D1711yescharacteristic
221#106—344no—
222#106—342no—
223#103—344no—
224#103—344no—
225#103—344no—
226#102—24no—
227#102—24no—
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—41no—
240C2×Q8—22no—
241C2×Q8—342no—
242C2×Q8—342no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4D4×D1712yes—
251C22×C4C2×C22⋊D1711yescharacteristic
252Q16—684no—
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—42no—
263C4⋊C4—342no—
264C4⋊C4—342no—
265C4⋊C4—342no—
266C4⋊C4—342no—
267C4⋊C4—682no—
268C4⋊C4—682no—
269C22⋊C4—41no—
270C22⋊C4—682no—
271C42—41no—
272C42—682no—
273C42D4×D1712yes—
274C42C2×C22⋊D1711yescharacteristic
275C42C23×D1711yescharacteristic
276C23—42no—
277C23—84no—
278C23—82no—
279C23#1058762611yescharacteristic
280Q8—44no—
281Q8—684no—
282Q8—684no—
283D4—44no—
284D4—44no—
285D4—44no—
286D4—84no—
287D4—82no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4#1058762611yescharacteristic
294C2×C4#1058762612yes—
295C2×C4C2×D4×D1712yes—
296C2×C4C22×C22⋊D1711yescharacteristic
297C8—22no—
298C8—22no—
299C8—344no—
300C8—344no—
301C22—22no—
302C22—21no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—164no—
307C22C2×#1058762611yescharacteristic
308C4—22no—
309C4—21no—
310C4—82no—
311C4—1362no—
312C4—1362no—
313C4#1176360514yes—
314C4#1176360712yes—
315C2—41no—
316C2—162no—
317C2#1453924612yes—
318C2#1453924811yescharacteristic
subgroupG/H
centerC22C2×#10587626
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3138C2
socleC22×C17C2×#127
prime radicalC2×C17×#127C22
hypercenter#3138D17
hypocenterC17#19753
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #14539529C2semidirect
2. #14539525C2semidirect
4. #14528446C2semidirect
5. #14528445C2semidirect
6. #14528443C2semidirect
7. #14528395C2semidirect
8. #14527087C2semidirect
11. #11763207C22semidirect
12. #11763205C22semidirect
13. #11763189C22semidirect
14. #11763188C22semidirect
16. #11763161C22semidirect
17. #11763160C22semidirect
18. #11763159C22semidirect
29. #11762531C22semidirect
30. #11762530C22semidirect
31. C8×#3622C22semidirect
238. C17#19753semidirect

Maximal subgroups

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

subgroupindex
#145395292
#145395252
#145391582
#145284462
#145284452
#145284432
#145283952
#145270872
C17×#31382
#1975317