Finite Group Databasebeta

← #17010586

Subgroups — #17010586

Order 4352.

1004 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 195 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#14539241C211yesmaximal, characteristic
2#14538906C212yesmaximal
3#14527924C214yesmaximal, complemented
4#14527862C214yesmaximal, complemented
5#14527510C211yesmaximal, characteristic, complemented
6#14527231C212yesmaximal, complemented
7C2×Q82×C17C211yesmaximal, characteristic, complemented
8C2×#10587628—22no—
9#11763359C2211yescharacteristic
10#11763201—22no—
11#11763188C2212yes—
12#11763183C2212yes—
13#11763122—22no—
14#11763104C2214yes—
15#11763009C2214yes—
16#11762948—22no—
17#11762907—24no—
18#11762866C2212yes—
19C4×#63223C2211yescharacteristic
20#11762558C2214yes—
21Q8⋊#3622C2218yes—
22Q82×C17—24no—
23Q82×C17—22no—
24Q82×C17C2214yes—
25C2×C17×C4⋊Q8—22no—
26C2×C17×C4⋊Q8—22no—
27C2×C17×C4⋊Q8C2211yescharacteristic
28C2×C4×Q8×C17—22no—
29C2×C4×Q8×C17C2212yes—
30C22×Q8×C17—22no—
31C22×Q8×C17D412yes—
32C22×Q8×C17D412yes—
33C2×#63252—21no—
34C17×C4⋊Q8—28no—
35C17×C4⋊Q8—28no—
36C17×C4⋊Q8—28no—
37C17×C4⋊Q8—28no—
38C17×C4⋊Q8—22no—
39C17×C4⋊Q8D412yes—
40C17×C4⋊Q8C2312yes—
41C4×Q8×C17—28no—
42C4×Q8×C17—28no—
43C4×Q8×C17—22no—
44C4×Q8×C17—22no—
45C4×Q8×C17D414yes—
46C4×Q8×C17C2314yes—
47C2×C4⋊C4×C17—24no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17—22no—
50C2×C4⋊C4×C17D412yes—
51C2×C42×C17—22no—
52C2×C42×C17—22no—
53C2×C42×C17C2311yescharacteristic
54C2×#63231—24no—
55#10587628—48no—
56C2×#63230—24no—
57C2×#63230—24no—
58#10587616—22no—
59C2×#63223—21no—
60#10587561—21no—
61#10587560C2311yescharacteristic
62C4×D68—22no—
63#10587555C2311yescharacteristic
64#10587506—24no—
65C22⋊#3622—22no—
66#10587480—24no—
67#10587478—24no—
68C4⋊#3622C2314yes—
69#10587474—21no—
70#10587474—22no—
71C4×#3622C2312yes—
72C2×Q8×C17—28no—
73C2×Q8×C17—28no—
74C2×Q8×C17—22no—
75C2×Q8×C17—22no—
76C2×Q8×C17—44no—
77C2×Q8×C17—84no—
78C2×Q8×C17C2×D414yes—
79C2×Q8×C17C2×D414yes—
80C22×C4×C17—22no—
81C22×C4×C17—22no—
82C22×C4×C17—22no—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D412yes—
85C2×C22⋊D17—41no—
86#63252—44no—
87C2×D68—21no—
88C2×D68—22no—
89C2×C4×D17—41no—
90C2×Dic34—21no—
91C4⋊C4×C17—216no—
92C4⋊C4×C17—24no—
93C4⋊C4×C17—22no—
94C4⋊C4×C17—24no—
95C4⋊C4×C17—24no—
96C4⋊C4×C17—24no—
97C4⋊C4×C17C2×D414yes—
98C42×C17—28no—
99C42×C17—24no—
100C42×C17—24no—
101C42×C17C2×D412yes—
102C42×C17C2×D411yescharacteristic
103C42×C17C2411yescharacteristic
104#63231—48no—
105#63230—48no—
106#63230—48no—
107#63227—42no—
108C4⋊Dic17—22no—
109#63223—24no—
110C2×#3622—22no—
111C2×#3622—24no—
112#18546—171nomaximal, Sylow
113C23×C17#12011yescharacteristic
114C22×D17—41no—
115Q8×C17—28no—
116Q8×C17—24no—
117Q8×C17—44no—
118Q8×C17—82no—
119Q8×C17—88no—
120Q8×C17#13718yescomplemented
121C2×C4×C17—28no—
122C2×C4×C17—24no—
123C2×C4×C17—22no—
124C2×C4×C17—24no—
125C2×C4×C17—24no—
126C2×C4×C17—24no—
127C2×C4×C17—44no—
128C2×C4×C17—44no—
129C2×C4×C17#12012yes—
130C2×C4×C17#12011yescharacteristic
131C2×C4×C17C22×D411yescharacteristic
132C2×C4×C17C22×D412yes—
133C22⋊D17—82no—
134C2×Dic17—41no—
135D68—42no—
136D68—44no—
137C4×D17—82no—
138Dic34—42no—
139#3622—24no—
140#3622—44no—
141C2×Q82D1711yescharacteristic, complemented
142#3152—171no—
143#3112—172no—
144#1633—174no—
145#1623—174no—
146#1558—171no—
147#1491—172no—
148C22×C17—22no—
149C22×C17#45312yes—
150C22×C17C2×#12011yescharacteristic
151C2×D17—82no—
152C2×D17—81no—
153C4×C17—28no—
154C4×C17—22no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—44no—
158C4×C17—44no—
159C4×C17C2×#13714yes—
160Dic17—81no—
161C2×#137—342no—
162Q82—22no—
163Q82—24no—
164Q82C2×D1714yes—
165#532—171no—
166C2×C4⋊Q8—22no—
167C2×C4⋊Q8—22no—
168C2×C4⋊Q8C2×D1711yescharacteristic
169C2×C4×Q8—22no—
170C2×C4×Q8C2×D1712yes—
171#497—342no—
172#494—172no—
173#492—172no—
174#480—342no—
175#477—174no—
176#460—174no—
177#448—342no—
178#440—344no—
179#432—172no—
180C4×M16—171no—
181#333—174no—
182Q8⋊C8—178no—
183C2×C17—22no—
184C2×C17#300412yes—
185C2×C17C2×#45311yescharacteristic
186D17—161no—
187C2×#41—341no—
188C22×Q8—22no—
189C22×Q8C22⋊D1712yes—
190C22×Q8C22⋊D1712yes—
191#137—688no—
192C2×Q16—344no—
193C2×SD16—344no—
194C2×SD16—344no—
195C2×M16—341no—
196C4⋊Q8—28no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8—22no—
201C4⋊Q8—171no—
202C4⋊Q8C22⋊D1712yes—
203C4⋊Q8C22×D1712yes—
204#127—171no—
205#124—341no—
206#121—342no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8C22⋊D1714yes—
212C4×Q8C22×D1714yes—
213C4×D4—342no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C22⋊D1712yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C22×D1711yescharacteristic
221#107—344no—
222C4⋊C8—174no—
223#103—344no—
224#102—344no—
225C22⋊C8—342no—
226#97—341no—
227#97—342no—
228C4×C8—172no—
229C17#1854611yescharacteristic, complemented, Sylow
230#41—684no—
231C2×Q8—28no—
232C2×Q8—28no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—84no—
237C2×Q8—341no—
238C2×Q8C2×C22⋊D1714yes—
239C2×Q8C2×C22⋊D1714yes—
240C2×D4—342no—
241C2×D4—341no—
242C2×D4—681no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—681no—
247C22×C4C2×C22⋊D1712yes—
248C22×C4C2×C22⋊D1711yescharacteristic
249Q16—688no—
250SD16—688no—
251SD16—688no—
252M16—344no—
253C2×C8—342no—
254C2×C8—344no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—342no—
262C4⋊C4C2×C22⋊D1714yes—
263C22⋊C4—682no—
264C42—28no—
265C42—24no—
266C42—24no—
267C42C2×C22⋊D1712yes—
268C42C2×C22⋊D1711yescharacteristic
269C42C23×D1711yescharacteristic
270C23—681no—
271C23#1058763911yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275Q8—82no—
276Q8—88no—
277Q8—682no—
278Q8#1058762818yescomplemented
279D4—682no—
280D4—684no—
281D4—1362no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—681no—
291C2×C4—1362no—
292C2×C4#1058763912yes—
293C2×C4#1058763911yescharacteristic
294C2×C4C22×C22⋊D1711yescharacteristic
295C2×C4C22×C22⋊D1712yes—
296C8—344no—
297C8—684no—
298C22—22no—
299C22—1362no—
300C22—1361no—
301C22#1176297712yes—
302C22C2×#1058763911yescharacteristic
303C4—28no—
304C4—22no—
305C4—21no—
306C4—21no—
307C4—44no—
308C4—44no—
309C4—1361no—
310C4C2×#1058762814yes—
311C2—22no—
312C2—2721no—
313C2#1453801812yes—
314C2C2×#1176297711yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×Q82×C17C2
socleC22×C17C2×#120
prime radical#11763359C22
hypercenterC2×Q82D17
hypocenterC17#18546
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #14527924C2semidirect
4. #14527862C2semidirect
5. #14527510C2semidirect
6. #14527231C2semidirect
7. C2×Q82×C17C2semidirect
120. Q8×C17#137semidirect
141. C2×Q82D17semidirect
229. C17#18546semidirect
278. Q8#10587628semidirect

Maximal subgroups

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

subgroupindex
#145392412
#145389062
#145279242
#145278622
#145275102
#145272312
C2×Q82×C172
#1854617