Finite Group Databasebeta

← #17010584

Subgroups — #17010584

Order 4352.

1004 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×#11763539C211yesmaximal, characteristic, complemented
2#14538902C212yesmaximal
3#14527922C214yesmaximal, complemented
4#14527860C214yesmaximal, complemented
5#14527510C211yesmaximal, characteristic, complemented
6#14527232C212yesmaximal, complemented
7C17×#3152C211yesmaximal, characteristic
8#11763539—24no—
9#11763539—22no—
10#11763539C2214yes—
11C2×#10587574—22no—
12C2×#10587630—22no—
13C2×C4×Dic34C2212yes—
14C2×#10587573—22no—
15#11763120—22no—
16#11763101C2214yes—
17#11763006C2214yes—
18#11762947—22no—
19#11762909—24no—
20#11762866C2212yes—
21#11762558C2214yes—
22#11762541C2218yes—
23C2×C17×#137—22no—
24C17×#532C2211yescharacteristic
25C2×C17×C4⋊Q8C2211yescharacteristic
26C17×#497—22no—
27C17×#494C2212yes—
28C17×#492C2212yes—
29C4×M16×C17C2211yescharacteristic
30C2×#41×C17—21no—
31C22×Q8×C17D412yes—
32C22×Dic34—22no—
33C22×Dic34D412yes—
34C17×#137—48no—
35C2×Q16×C17—24no—
36C2×SD16×C17—24no—
37C2×M16×C17—21no—
38C17×C4⋊Q8—22no—
39C17×C4⋊Q8D412yes—
40C17×C4⋊Q8C2311yescharacteristic
41C17×C4⋊Q8C2312yes—
42C17×#127C2311yescharacteristic
43C17×#124—21no—
44C17×#121—22no—
45C4×D4×C17—22no—
46C2×C4⋊C4×C17—22no—
47C2×C42×C17C2311yescharacteristic
48#10587630—28no—
49#10587630—28no—
50C2×#63229—24no—
51C2×C4⋊Dic17D412yes—
52C2×#63225—24no—
53C2×#63225—22no—
54C2×C4×Dic17—22no—
55C2×C4×Dic17—22no—
56#10587574—28no—
57#10587574—28no—
58#10587573—28no—
59#10587573—28no—
60C4×Dic34—22no—
61C4×Dic34—22no—
62C4×Dic34D414yes—
63C4×Dic34C2314yes—
64C17×#97—21no—
65C4×C8×C17C2312yes—
66#10587503—24no—
67C22⋊#3622—22no—
68#10587481—24no—
69#10587478—24no—
70C4⋊#3622C2314yes—
71#10587474—22no—
72#41×C17—44no—
73C2×Q8×C17—22no—
74C2×Q8×C17—21no—
75C2×Q8×C17—44no—
76C2×Q8×C17C2×D414yes—
77C2×D4×C17—21no—
78C2×D4×C17—22no—
79C2×D4×C17—41no—
80C22×C4×C17—22no—
81C22×C4×C17—41no—
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D411yescharacteristic
84C22×Dic17—22no—
85C22×Dic17—22no—
86C2×Dic34—22no—
87C2×Dic34—28no—
88C2×Dic34—28no—
89C2×Dic34—84no—
90C2×Dic34C2×D414yes—
91Q16×C17—48no—
92SD16×C17—48no—
93M16×C17—24no—
94C2×C8×C17—22no—
95C4⋊C4×C17—24no—
96C4⋊C4×C17—24no—
97C4⋊C4×C17—22no—
98C22⋊C4×C17—42no—
99C42×C17C2×D412yes—
100C42×C17C2×D411yescharacteristic
101C42×C17C2411yescharacteristic
102#63229—48no—
103C4⋊Dic17—22no—
104C4⋊Dic17C2×D414yes—
105#63225—216no—
106#63225—24no—
107#63225—24no—
108C4×Dic17—28no—
109C4×Dic17—24no—
110C4×Dic17—24no—
111C2×#3622—24no—
112#18546—171nomaximal, Sylow
113C23×C17—41no—
114C23×C17#12011yescharacteristic
115Q8×C17—44no—
116Q8×C17—42no—
117Q8×C17—82no—
118D4×C17—42no—
119D4×C17—44no—
120D4×C17—82no—
121C2×C4×C17—22no—
122C2×C4×C17—24no—
123C2×C4×C17—24no—
124C2×C4×C17—44no—
125C2×C4×C17—41no—
126C2×C4×C17—82no—
127C2×C4×C17#12012yes—
128C2×C4×C17#12011yescharacteristic
129C2×C4×C17C22×D412yes—
130C2×C4×C17C22×D411yescharacteristic
131C2×Dic17—28no—
132C2×Dic17—24no—
133C2×Dic17—24no—
134C2×Dic17—44no—
135Dic34—28no—
136Dic34—24no—
137Dic34—88no—
138Dic34#13718yescomplemented
139C8×C17—24no—
140#3622—44no—
141C2×Q82—171no—
142#3152D1711yescharacteristic
143#3112—172no—
144#1633—174no—
145#1623—174no—
146#1558—171no—
147#1491—172no—
148C22×C17—22no—
149C22×C17—82no—
150C22×C17—81no—
151C22×C17#45312yes—
152C22×C17C2×#12011yescharacteristic
153C4×C17—22no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—44no—
157C4×C17—81no—
158C4×C17C2×#13714yes—
159Dic17—28no—
160Dic17—44no—
161C2×#137—22no—
162Q82—174no—
163Q82—344no—
164Q82—342no—
165#532C2×D1711yescharacteristic
166C2×C4⋊Q8—342no—
167C2×C4⋊Q8—342no—
168C2×C4⋊Q8C2×D1711yescharacteristic
169C2×C4×Q8—172no—
170C2×C4×Q8—342no—
171#497—22no—
172#494C2×D1712yes—
173#492C2×D1712yes—
174#480—342no—
175#477—174no—
176#460—174no—
177#448—342no—
178#440—344no—
179#432—172no—
180C4×M16C2×D1711yescharacteristic
181#333—174no—
182Q8⋊C8—178no—
183C2×C17—22no—
184C2×C17—161no—
185C2×C17#300412yes—
186C2×C17C2×#45311yescharacteristic
187C2×#41—21no—
188C22×Q8—172no—
189C22×Q8—342no—
190C22×Q8C22⋊D1712yes—
191#137—48no—
192C2×Q16—24no—
193C2×SD16—24no—
194C2×SD16—344no—
195C2×M16—21no—
196C4⋊Q8—22no—
197C4⋊Q8—348no—
198C4⋊Q8—348no—
199C4⋊Q8—348no—
200C4⋊Q8—348no—
201C4⋊Q8C22⋊D1712yes—
202C4⋊Q8C22×D1711yescharacteristic
203C4⋊Q8C22×D1712yes—
204#127C22×D1711yescharacteristic
205#124—21no—
206#121—22no—
207C4×Q8—174no—
208C4×Q8—174no—
209C4×Q8—348no—
210C4×Q8—348no—
211C4×Q8—342no—
212C4×Q8—342no—
213C4×D4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—172no—
216C2×C4⋊C4—344no—
217C2×C4⋊C4—342no—
218C2×C42—342no—
219C2×C42—342no—
220C2×C42C22×D1711yescharacteristic
221#107—344no—
222C4⋊C8—174no—
223#103—344no—
224#102—344no—
225C22⋊C8—342no—
226#97—21no—
227#97—342no—
228C4×C8C22×D1712yes—
229C17#1854611yescharacteristic, complemented, Sylow
230#41—44no—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—44no—
234C2×Q8—174no—
235C2×Q8—348no—
236C2×Q8—348no—
237C2×Q8—342no—
238C2×Q8—1364no—
239C2×Q8C2×C22⋊D1714yes—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4D4×D1712yes—
248C22×C4C2×C22⋊D1711yescharacteristic
249Q16—48no—
250SD16—48no—
251SD16—688no—
252M16—24no—
253C2×C8—22no—
254C2×C8—344no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—174no—
259C4⋊C4—3416no—
260C4⋊C4—344no—
261C4⋊C4—342no—
262C4⋊C4—344no—
263C22⋊C4—42no—
264C42—348no—
265C42—344no—
266C42—344no—
267C42D4×D1712yes—
268C42C2×C22⋊D1711yescharacteristic
269C42C23×D1711yescharacteristic
270C23—41no—
271C23#1058762311yescharacteristic
272Q8—44no—
273Q8—42no—
274Q8—82no—
275Q8—178no—
276Q8—348no—
277Q8—344no—
278Q8—1368no—
279D4—42no—
280D4—44no—
281D4—82no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—348no—
289C2×C4—344no—
290C2×C4—344no—
291C2×C4—684no—
292C2×C4#1058762312yes—
293C2×C4#1058762311yescharacteristic
294C2×C4C2×D4×D1712yes—
295C2×C4C22×C22⋊D1711yescharacteristic
296C8—24no—
297C8—684no—
298C22—22no—
299C22—82no—
300C22—81no—
301C22#1176297612yes—
302C22C2×#1058762311yescharacteristic
303C4—22no—
304C4—21no—
305C4—21no—
306C4—44no—
307C4—81no—
308C4—348no—
309C4—684no—
310C4#137×D1714yes—
311C2—22no—
312C2—161no—
313C2#1453801612yes—
314C2C2×#1176297611yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3152C2
socleC22×C17C2×#120
prime radicalC17×#532C22
hypercenter#3152D17
hypocenterC17#18546
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#11763539C2semidirect
3. #14527922C2semidirect
4. #14527860C2semidirect
5. #14527510C2semidirect
6. #14527232C2semidirect
138. Dic34#137semidirect
229. C17#18546semidirect

Maximal subgroups

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

subgroupindex
C2×#117635392
#145389022
#145279222
#145278602
#145275102
#145272322
C17×#31522
#1854617