Finite Group Databasebeta

← #17007117

Subgroups — #17007117

Order 4352.

970 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 183 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#14538910C212yesmaximal, complemented
2#14532068C211yesmaximal, characteristic
3#14527775C214yesmaximal, complemented
4#14527741C214yesmaximal, complemented
5#14527296C212yesmaximal, complemented
6#14527125C211yesmaximal, characteristic
7C17×#3501C211yesmaximal, characteristic
8C2×C4×Dic34C2211yescharacteristic
9#11763137C2214yes—
10#11763091—22no—
11#11763016—22no—
12#11762964C2214yes—
13#11762914—24no—
14#11762866C2212yes—
15#11762747—21no—
16#11762734C2212yes—
17#11762723—21no—
18#11762721C2212yes—
19C4×C22⋊Dic17—22no—
20#11762655C2211yescharacteristic
21#11762546C2218yes—
22#11762533C2214yes—
23C17×#554—24no—
24C17×#554—22no—
25D4×Q8×C17—22no—
26C17×#544—21no—
27C17×#543C2214yes—
28C17×#533C2212yes—
29C2×C17×C22⋊Q8—22no—
30C17×#518—21no—
31C2×C4×D4×C17C2211yescharacteristic
32C22×Q8×C17—22no—
33C22×D4×C17D411yescharacteristic
34C23×C4×C17D412yes—
35C22×Dic34C2311yescharacteristic
36C17×C4⋊Q8—24no—
37C17×C4⋊Q8D412yes—
38C17×C4⋊Q8D412yes—
39C17×#125—24no—
40C17×#125—22no—
41C17×C22⋊Q8—24no—
42C17×C22⋊Q8—24no—
43C17×C22⋊Q8—28no—
44C17×C22⋊Q8—22no—
45C17×C22⋊Q8—24no—
46C17×C22⋊Q8—28no—
47C17×C22⋊Q8C2314yes—
48C4×Q8×C17—22no—
49C4×Q8×C17D414yes—
50C4×D4×C17—24no—
51C4×D4×C17—24no—
52C4×D4×C17—22no—
53C4×D4×C17—22no—
54C17×#117—22no—
55C2×C4⋊C4×C17—22no—
56C2×C4⋊C4×C17—24no—
57C2×C4⋊C4×C17D411yescharacteristic
58C2×C22⋊C4×C17—24no—
59C2×C22⋊C4×C17C2312yes—
60C2×C42×C17C2311yescharacteristic
61C2×C22⋊Dic17—22no—
62C2×C22⋊Dic17—22no—
63C2×#63231—24no—
64C2×C4⋊Dic17C2311yescharacteristic
65C2×#63225—21no—
66C2×C4×Dic17—21no—
67C2×C4×Dic17—21no—
68C4×Dic34—22no—
69C4×Dic34—22no—
70#10587506—24no—
71#10587502—22no—
72#10587502—21no—
73#10587502—22no—
74#10587502C2312yes—
75C22⋊#3622C2314yes—
76#10587481—24no—
77#10587479—24no—
78C4⋊#3622—22no—
79#10587474—22no—
80C24×C17C2×D412yes—
81C2×Q8×C17—24no—
82C2×Q8×C17—22no—
83C2×Q8×C17—84no—
84C2×Q8×C17C2×D414yes—
85C2×D4×C17—24no—
86C2×D4×C17—22no—
87C22×C4×C17—24no—
88C22×C4×C17—24no—
89C22×C4×C17—22no—
90C22×C4×C17—22no—
91C22×C4×C17—44no—
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17C2×D412yes—
95C22×C4×C17C2411yescharacteristic
96C22×Dic17—21no—
97C22×Dic17—22no—
98C22×Dic17—21no—
99C2×Dic34—22no—
100C2×Dic34—42no—
101C4⋊C4×C17—24no—
102C4⋊C4×C17—28no—
103C4⋊C4×C17—24no—
104C4⋊C4×C17—22no—
105C4⋊C4×C17—22no—
106C4⋊C4×C17—22no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—24no—
109C4⋊C4×C17C2×D414yes—
110C22⋊C4×C17—28no—
111C22⋊C4×C17—24no—
112C22⋊C4×C17—24no—
113C42×C17—24no—
114C42×C17C2×D412yes—
115C42×C17C2×D412yes—
116C22⋊Dic17—44no—
117C22⋊Dic17—44no—
118#63231—48no—
119C4⋊Dic17—22no—
120#63225—42no—
121C4×Dic17—42no—
122C4×Dic17—41no—
123C4×Dic17—41no—
124C2×#3622—24no—
125#18181—171nomaximal, Sylow
126C23×C17—22no—
127C23×C17—22no—
128C23×C17—42no—
129C23×C17—42no—
130C23×C17#13714yes—
131C23×C17C22×D411yescharacteristic
132Q8×C17—44no—
133Q8×C17—44no—
134Q8×C17—84no—
135D4×C17—28no—
136C2×C4×C17—22no—
137C2×C4×C17—22no—
138C2×C4×C17—24no—
139C2×C4×C17—24no—
140C2×C4×C17—22no—
141C2×C4×C17—41no—
142C2×C4×C17—44no—
143C2×C4×C17—42no—
144C2×C4×C17—82no—
145C2×C4×C17—84no—
146C2×C4×C17—84no—
147C2×C4×C17#12012yes—
148C2×C4×C17#12011yescharacteristic
149C2×C4×C17#12011yescharacteristic
150C2×C4×C17C22×D412yes—
151C2×Dic17—22no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—42no—
155C2×Dic17—42no—
156C2×Dic17—41no—
157C2×Dic17—42no—
158C2×Dic17—42no—
159Dic34—24no—
160Dic34—42no—
161#3622—44no—
162#3501D1711yescharacteristic
163#3113—172no—
164#2398—171no—
165#1608—174no—
166#1602—174no—
167#1506—172no—
168#1466—171no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—24no—
172C22×C17—44no—
173C22×C17—42no—
174C22×C17—42no—
175C22×C17—82no—
176C22×C17C2×#12011yescharacteristic
177C22×C17C2×#13712yes—
178C4×C17—22no—
179C4×C17—21no—
180C4×C17—22no—
181C4×C17—42no—
182C4×C17—44no—
183C4×C17—44no—
184C4×C17#45512yes—
185Dic17—42no—
186Dic17—82no—
187Dic17—81no—
188#554—22no—
189#554—24no—
190D4×Q8—22no—
191#544—21no—
192#543C2×D1714yes—
193#533C2×D1712yes—
194C2×C22⋊Q8—22no—
195#518—21no—
196C2×C4×Q8—171no—
197C2×C4×D4C2×D1711yescharacteristic
198#483—174no—
199#475—342no—
200#462—342no—
201#451—174no—
202#441—344no—
203#432—172no—
204#400—341no—
205#397—172no—
206#394—341no—
207#393—172no—
208C4×C22⋊C4—342no—
209#376—171no—
210#328—178no—
211#323—174no—
212C2×C17—22no—
213C2×C17—44no—
214C2×C17#300411yescharacteristic
215C2×C17C2×#45511yescharacteristic
216C2×C17#302511yescharacteristic
217C22×Q8—22no—
218C22×Q8—171no—
219C22×D4C22⋊D1711yescharacteristic
220C23×C4C22⋊D1712yes—
221C2×Q16—344no—
222C4⋊Q8—24no—
223C4⋊Q8C22⋊D1712yes—
224C4⋊Q8C22⋊D1712yes—
225#125—24no—
226#125—22no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—28no—
230C22⋊Q8—22no—
231C22⋊Q8—24no—
232C22⋊Q8—28no—
233C22⋊Q8C22×D1714yes—
234C4×Q8—22no—
235C4×Q8—342no—
236C4×Q8—342no—
237C4×Q8C22⋊D1714yes—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—22no—
241C4×D4—22no—
242#117—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4—341no—
247C2×C4⋊C4C22⋊D1711yescharacteristic
248C2×C22⋊C4—24no—
249C2×C22⋊C4—342no—
250C2×C22⋊C4—342no—
251C2×C22⋊C4C22×D1712yes—
252C2×C42—341no—
253C2×C42—341no—
254C2×C42C22×D1711yescharacteristic
255#106—344no—
256C4⋊C8—342no—
257#103—344no—
258#103—344no—
259C22⋊C8—174no—
260#97—342no—
261#95—172no—
262#95—341no—
263#95—342no—
264#95—342no—
265C17#1818111yescharacteristic, complemented, Sylow
266C24C2×C22⋊D1712yes—
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—84no—
270C2×Q8—342no—
271C2×Q8—682no—
272C2×Q8C2×C22⋊D1714yes—
273C2×D4—24no—
274C2×D4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—44no—
280C22×C4—341no—
281C22×C4—342no—
282C22×C4—341no—
283C22×C4C2×C22⋊D1711yescharacteristic
284C22×C4C2×C22⋊D1711yescharacteristic
285C22×C4C2×C22⋊D1712yes—
286C22×C4C23×D1711yescharacteristic
287Q16—688no—
288C2×C8—344no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—342no—
298C4⋊C4—682no—
299C4⋊C4C2×C22⋊D1714yes—
300C22⋊C4—28no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—684no—
304C22⋊C4—684no—
305C42—24no—
306C42—682no—
307C42—681no—
308C42—681no—
309C42C2×C22⋊D1712yes—
310C42C2×C22⋊D1712yes—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23#1058762814yes—
316C23C22×C22⋊D1711yescharacteristic
317Q8—44no—
318Q8—44no—
319Q8—84no—
320Q8—344no—
321Q8—682no—
322D4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—342no—
335C2×C4—681no—
336C2×C4—681no—
337C2×C4—682no—
338C2×C4—682no—
339C2×C4—681no—
340C2×C4—682no—
341C2×C4—682no—
342C2×C4#1058763912yes—
343C2×C4#1058763911yescharacteristic
344C2×C4#1058763911yescharacteristic
345C2×C4C22×C22⋊D1712yes—
346C8—684no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22C2×#1058763911yescharacteristic
355C22C2×#1058762812yes—
356C4—22no—
357C4—21no—
358C4—22no—
359C4—42no—
360C4—44no—
361C4—44no—
362C4—682no—
363C4—1362no—
364C4—1361no—
365C4#1176298712yes—
366C2—22no—
367C2—44no—
368C2#1453801811yescharacteristic
369C2C2×#1176298711yescharacteristic
370C2#1453817611yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#3501C2
socleC22×C17C2×#120
prime radicalC22×D4×C17D4
hypercenter#3501D17
hypocenterC17#18181
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #14538910C2semidirect
3. #14527775C2semidirect
4. #14527741C2semidirect
5. #14527296C2semidirect
265. C17#18181semidirect

Maximal subgroups

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

subgroupindex
#145389102
#145320682
#145277752
#145277412
#145272962
#145271252
C17×#35012
#1818117