Finite Group Databasebeta

← #17010406

Subgroups — #17010406

Order 4352.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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#14539234C211yesmaximal, characteristic
2#14538906C212yesmaximal, complemented
3#14527905C214yesmaximal, complemented
4#14527843C214yesmaximal, complemented
5#14527498C211yesmaximal, characteristic, complemented
6#14527231C212yesmaximal, complemented
7C2×C17×#559C211yesmaximal, characteristic, complemented
8C2×#10587618—22no—
9#11763359C2211yescharacteristic
10#11763196—22no—
11#11763185C2212yes—
12#11763181C2212yes—
13#11763122—22no—
14#11763104C2214yescomplemented
15#11763009C2214yescomplemented
16#11762948—22no—
17#11762907—24no—
18#11762866C2212yes—
19C4×#63223C2211yescharacteristic
20#11762553C2214yes—
21Q8⋊#3622C2218yescomplemented
22C17×#559—24no—
23C17×#559—22no—
24C17×#559C2214yes—
25C2×C17×#127C2211yescharacteristic
26C2×C17×#124—22no—
27C2×C17×#123—22no—
28C2×C4×Q8×C17C2212yes—
29C2×C4×D4×C17—22no—
30C22×Q8×C17D412yes—
31C22×D4×C17—22no—
32C22×D4×C17D412yes—
33C23×C4×C17—22no—
34C2×#63252—21no—
35C17×#127—22no—
36C17×#127D412yes—
37C17×#127C2312yes—
38C17×#124—28no—
39C17×#124—28no—
40C17×#123—28no—
41C17×#123—28no—
42C4×Q8×C17—22no—
43C4×Q8×C17—22no—
44C4×Q8×C17D414yes—
45C4×Q8×C17C2314yes—
46C4×D4×C17—28no—
47C4×D4×C17—28no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17D412yes—
50C2×C22⋊C4×C17—24no—
51C2×C22⋊C4×C17—22no—
52C2×C42×C17—22no—
53C2×C42×C17C2311yescharacteristic
54C2×#63230—24no—
55C2×#63229—24no—
56#10587618—48no—
57C2×#63228—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—
72C24×C17—22no—
73C2×Q8×C17—22no—
74C2×Q8×C17—44no—
75C2×Q8×C17C2×D414yes—
76C2×D4×C17—24no—
77C2×D4×C17—22no—
78C2×D4×C17—24no—
79C2×D4×C17—48no—
80C2×D4×C17—44no—
81C2×D4×C17C2×D414yes—
82C22×C4×C17—22no—
83C22×C4×C17—22no—
84C22×C4×C17—28no—
85C22×C4×C17—84no—
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D412yes—
88C2×C22⋊D17—41no—
89#63252—44no—
90C2×D68—21no—
91C2×D68—22no—
92C2×C4×D17—41no—
93C2×Dic34—21no—
94C4⋊C4×C17—24no—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—24no—
97C4⋊C4×C17C2×D414yes—
98C22⋊C4×C17—28no—
99C22⋊C4×C17—216no—
100C42×C17—24no—
101C42×C17—24no—
102C42×C17C2×D412yes—
103C42×C17C2×D411yescharacteristic
104C42×C17C2411yescharacteristic
105#63230—48no—
106#63229—48no—
107#63228—48no—
108#63227—42no—
109C4⋊Dic17—22no—
110#63223—24no—
111C2×#3622—22no—
112C2×#3622—24no—
113#18526—171nomaximal, Sylow
114C23×C17—24no—
115C23×C17—48no—
116C23×C17—44no—
117C23×C17#12011yescharacteristic
118C22×D17—41no—
119Q8×C17—24no—
120Q8×C17—24no—
121Q8×C17—44no—
122D4×C17—44no—
123D4×C17—48no—
124D4×C17—48no—
125D4×C17—82no—
126C2×C4×C17—22no—
127C2×C4×C17—24no—
128C2×C4×C17—24no—
129C2×C4×C17—24no—
130C2×C4×C17—44no—
131C2×C4×C17—44no—
132C2×C4×C17—88no—
133C2×C4×C17#12012yes—
134C2×C4×C17#12011yescharacteristic
135C2×C4×C17C22×D411yescharacteristic
136C2×C4×C17C22×D412yes—
137C22⋊D17—82no—
138C2×Dic17—41no—
139D68—42no—
140D68—44no—
141C4×D17—82no—
142Dic34—42no—
143#3622—24no—
144#3622—44no—
145C2×#559D1711yescharacteristic, complemented
146#3151—171no—
147#3112—172no—
148#1630—174no—
149#1620—174no—
150#1555—171no—
151#1491—172no—
152C22×C17—22no—
153C22×C17—48no—
154C22×C17—44no—
155C22×C17—88no—
156C22×C17#45312yes—
157C22×C17C2×#12011yescharacteristic
158C2×D17—82no—
159C2×D17—81no—
160C4×C17—22no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—44no—
164C4×C17—44no—
165C4×C17#57514yes—
166Dic17—81no—
167C2×AGL(1,ℤ/8)—342no—
168#559—22no—
169#559—24no—
170#559C2×D1714yes—
171#532—171no—
172C2×#127C2×D1711yescharacteristic
173C2×#124—22no—
174C2×#123—22no—
175C2×C4×Q8C2×D1712yes—
176C2×C4×D4—22no—
177#496—342no—
178#493—172no—
179#492—172no—
180#480—342no—
181#477—174no—
182#460—174no—
183#448—342no—
184#440—344no—
185#432—172no—
186C4×M16—171no—
187#331—174no—
188Q8⋊C8—178no—
189C2×C17—22no—
190C2×C17—84no—
191C2×C17#300512yes—
192C2×C17C2×#45311yescharacteristic
193D17—161no—
194C2×#41—341no—
195C22×Q8C22⋊D1712yes—
196C22×D4—22no—
197C22×D4C22⋊D1712yes—
198C23×C4—22no—
199AGL(1,ℤ/8)—688no—
200C2×SD16—344no—
201C2×SD16—344no—
202C2×D8—344no—
203C2×M16—341no—
204C4⋊Q8—171no—
205#127—22no—
206#127—171no—
207#127C22⋊D1712yes—
208#127C22×D1712yes—
209#124—28no—
210#124—28no—
211#124—341no—
212#123—28no—
213#123—28no—
214#121—342no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8C22⋊D1714yes—
218C4×Q8C22×D1714yes—
219C4×D4—28no—
220C4×D4—28no—
221C4×D4—342no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C22⋊D1712yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C42—22no—
227C2×C42C22×D1711yescharacteristic
228#107—344no—
229C4⋊C8—174no—
230#103—344no—
231#102—344no—
232C22⋊C8—342no—
233#97—341no—
234#97—342no—
235C4×C8—172no—
236C17#1852611yescharacteristic, complemented, Sylow
237C24—22no—
238#41—684no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—341no—
242C2×Q8C2×C22⋊D1714yes—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—342no—
249C2×D4—341no—
250C2×D4—681no—
251C2×D4C2×C22⋊D1714yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—84no—
256C22×C4—681no—
257C22×C4C2×C22⋊D1712yes—
258C22×C4C2×C22⋊D1711yescharacteristic
259SD16—688no—
260SD16—688no—
261D8—688no—
262M16—344no—
263C2×C8—342no—
264C2×C8—344no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—342no—
269C4⋊C4C2×C22⋊D1714yes—
270C22⋊C4—216no—
271C22⋊C4—28no—
272C22⋊C4—682no—
273C42—24no—
274C42—24no—
275C42C2×C22⋊D1712yes—
276C42C2×C22⋊D1711yescharacteristic
277C42C23×D1711yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—681no—
282C23#1058763911yescharacteristic
283Q8—24no—
284Q8—24no—
285Q8—44no—
286Q8—682no—
287D4—44no—
288D4—48no—
289D4—48no—
290D4—82no—
291D4—682no—
292D4—684no—
293D4—1362no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—88no—
301C2×C4—681no—
302C2×C4—1362no—
303C2×C4#1058763912yes—
304C2×C4#1058763911yescharacteristic
305C2×C4C22×C22⋊D1711yescharacteristic
306C2×C4C22×C22⋊D1712yes—
307C8—344no—
308C8—684no—
309C22—22no—
310C22—48no—
311C22—44no—
312C22—88no—
313C22—1362no—
314C22—1361no—
315C22#1176297712yes—
316C22C2×#1058763911yescharacteristic
317C4—22no—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—44no—
322C4—1361no—
323C4#1176360714yes—
324C2—22no—
325C2—84no—
326C2—2721no—
327C2#1453802612yes—
328C2C2×#1176297711yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#559C2
socleC22×C17C2×#120
prime radical#14539234C2
hypercenterC2×#559D17
hypocenterC17#18526
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #14538906C2semidirect
3. #14527905C2semidirect
4. #14527843C2semidirect
5. #14527498C2semidirect
6. #14527231C2semidirect
7. C2×C17×#559C2semidirect
14. #11763104C22semidirect
15. #11763009C22semidirect
21. Q8⋊#3622C22semidirect
145. C2×#559D17semidirect
236. C17#18526semidirect

Maximal subgroups

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

subgroupindex
#145392342
#145389062
#145279052
#145278432
#145274982
#145272312
C2×C17×#5592
#1852617