Finite Group Databasebeta

← #17066291

Subgroups — #17066291

Order 4352.

1158 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1158 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
1C2×#11763421C211yesmaximal, characteristic
2C2×#11762965C212yesmaximal
3#14529935C212yesmaximal, complemented
4#14529883C212yesmaximal, complemented
5#14529128C214yesmaximal, complemented
6#14529096C212yesmaximal
7#14528855C211yesmaximal, characteristic
8C17×#2310C211yesmaximal, characteristic
9C22×#63231C2214yes—
10#11763421—28no—
11C2×#10587555C2211yescharacteristic
12C2×#10587565—22no—
13C2×#10587556C2211yescharacteristic
14C2×#10587575—22no—
15C2×#10587610C2212yes—
16C2×C4×Dic34C2212yes—
17#11762965—28no—
18C2×#10587506C2214yes—
19C2×#10587481C2214yes—
20C2×C22⋊#3622C2212yes—
21#11762571C2214yes—
22#11762564C2212yes—
23#11762564C2212yes—
24C23×Q8×C17C2211yescharacteristic
25C2×C17×#117C2211yescharacteristic
26C2×C17×#117C2211yescharacteristic
27C17×#391—22no—
28C17×#391C2212yes—
29C17×#391C2212yes—
30C17×#381—22no—
31C22×Q8×C17—24no—
32C22×Q8×C17—21no—
33C22×Q8×C17—24no—
34C22×Q8×C17—42no—
35C22×Q8×C17C2312yes—
36C23×C4×C17—21no—
37C23×C4×C17C2311yescharacteristic
38C22×Dic34C2312yes—
39C17×#117—24no—
40C17×#117—42no—
41C2×C4⋊C4×C17D412yes—
42C2×C4⋊C4×C17C2312yes—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17D412yes—
45C2×C42×C17—21no—
46C2×C42×C17C2311yescharacteristic
47C2×C42×C17C2311yescharacteristic
48C2×C22⋊Dic17—22no—
49C2×#63231—28no—
50C2×#63231—28no—
51C2×#63231—44no—
52C2×C4⋊Dic17—22no—
53C2×C4⋊Dic17C2312yes—
54#10587610—24no—
55#10587610D418yes—
56C2×#63225—22no—
57C2×#63225—22no—
58C2×#63225—22no—
59C2×C4×Dic17—22no—
60C22×#3622C2314yes—
61#10587575—216no—
62#10587565—216no—
63#10587556—24no—
64#10587555—24no—
65C4×Dic34—24no—
66C4×Dic34—24no—
67C17×#95—24no—
68C17×#95—24no—
69#10587506—28no—
70C22⋊#3622—24no—
71#10587481—44no—
72C24×C17C2×D411yescharacteristic
73C2×Q8×C17—24no—
74C2×Q8×C17—22no—
75C2×Q8×C17—24no—
76C2×Q8×C17—24no—
77C2×Q8×C17—416no—
78C2×Q8×C17—42no—
79C2×Q8×C17—42no—
80C2×Q8×C17#4114yes—
81C22×C4×C17—22no—
82C22×C4×C17—22no—
83C22×C4×C17—24no—
84C22×C4×C17—22no—
85C22×C4×C17—22no—
86C22×C4×C17—42no—
87C22×C4×C17—42no—
88C22×C4×C17C2×D414yes—
89C22×C4×C17C2×D412yes—
90C22×C4×C17C2×D412yes—
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2411yescharacteristic
93C22×Dic17—22no—
94C22×Dic17—22no—
95C2×Dic34—24no—
96C2×Dic34—24no—
97C2×Dic34—22no—
98C2×Dic34C2×D414yes—
99C4⋊C4×C17—24no—
100C4⋊C4×C17—24no—
101C22⋊C4×C17—24no—
102C22⋊C4×C17—42no—
103C42×C17—22no—
104C42×C17—22no—
105C42×C17—42no—
106C22⋊Dic17—28no—
107#63231—416no—
108C4⋊Dic17—28no—
109C4⋊Dic17—22no—
110C4⋊Dic17C2×D414yes—
111#63225—28no—
112#63225—28no—
113#63225—28no—
114C4×Dic17—28no—
115C2×#3622—24no—
116C2×#3622—24no—
117C2×#3622—24no—
118#23943—171nomaximal, Sylow
119C23×C17—21no—
120C23×C17—21no—
121C23×C17—24no—
122C23×C17#12012yes—
123C23×C17C22×D411yescharacteristic
124Q8×C17—28no—
125Q8×C17—48no—
126Q8×C17—44no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—28no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—24no—
133C2×C4×C17—24no—
134C2×C4×C17—24no—
135C2×C4×C17—48no—
136C2×C4×C17—42no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—44no—
140C2×C4×C17#12012yes—
141C2×C4×C17C22×D412yes—
142C2×C4×C17C2×#4112yes—
143C2×C4×C17H(5,𝔽2)12yes—
144C2×Dic17—24no—
145C2×Dic17—24no—
146C2×Dic17—42no—
147C2×Dic17—42no—
148C2×Dic17—44no—
149Dic34—28no—
150Dic34—44no—
151#3622—44no—
152C2×#541—171no—
153C2×#451—172no—
154#2310D1711yescharacteristic
155#2026—172no—
156#2016—172no—
157#1878—174no—
158#1873—172no—
159#1818—171no—
160C22×C17—22no—
161C22×C17—24no—
162C22×C17—42no—
163C22×C17—42no—
164C22×C17C2×#12011yescharacteristic
165C22×C17#52511yescharacteristic
166C22×C17#53511yescharacteristic
167C22×C17#57814yes—
168C4×C17—22no—
169C4×C17—44no—
170C4×C17—42no—
171C4×C17—42no—
172C4×C17—82no—
173C4×C17#54614yes—
174Dic17—44no—
175Dic17—44no—
176C23×Q8C2×D1711yescharacteristic
177C22×Q16—174no—
178#541—348no—
179C2×C4⋊Q8—171no—
180C2×#126—342no—
181C2×#125—171no—
182C2×#125—342no—
183C2×C22⋊Q8—172no—
184C2×C4×Q8—172no—
185C2×#117C2×D1711yescharacteristic
186C2×#117C2×D1711yescharacteristic
187#451—348no—
188C2×#103—174no—
189C2×#103—174no—
190C2×C22⋊C8—172no—
191#391—22no—
192#391C2×D1712yes—
193#391C2×D1712yes—
194#381—22no—
195#340—174no—
196#336—172no—
197#336—172no—
198C2×C17—42no—
199C2×C17#240411yescharacteristic
200C2×C17#301112yes—
201C2×C17#312712yes—
202C2×C17#322312yes—
203C22×Q8—24no—
204C22×Q8—21no—
205C22×Q8—24no—
206C22×Q8—42no—
207C22×Q8—172no—
208C22×Q8C22×D1712yes—
209C23×C4—21no—
210C23×C4C22×D1711yescharacteristic
211C2×Q16—348no—
212C2×Q16—348no—
213C2×Q16—684no—
214C22×C8—174no—
215C4⋊Q8—344no—
216#126—3416no—
217#125—344no—
218#125—3416no—
219C22⋊Q8—178no—
220C22⋊Q8—344no—
221C4×Q8—344no—
222C4×Q8—344no—
223#117—24no—
224#117—42no—
225C2×C4⋊C4—172no—
226C2×C4⋊C4—342no—
227C2×C4⋊C4—342no—
228C2×C4⋊C4—342no—
229C2×C4⋊C4—342no—
230C2×C4⋊C4C22⋊D1712yes—
231C2×C4⋊C4C22×D1712yes—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—342no—
234C2×C22⋊C4C22⋊D1712yes—
235C2×C42—21no—
236C2×C42—342no—
237C2×C42C22×D1711yescharacteristic
238C2×C42C22×D1711yescharacteristic
239#103—348no—
240#103—684no—
241C22⋊C8—344no—
242#95—24no—
243#95—24no—
244C17#2394311yescharacteristic, complemented, Sylow
245C24C2×C22⋊D1711yescharacteristic
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—416no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—174no—
254C2×Q8—342no—
255C2×Q8—344no—
256C2×Q8—344no—
257C2×Q8#6325214yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—342no—
266C22×C4—342no—
267C22×C4D4×D1714yes—
268C22×C4C2×C22⋊D1712yes—
269C22×C4C2×C22⋊D1712yes—
270C22×C4C2×C22⋊D1711yescharacteristic
271C22×C4C23×D1711yescharacteristic
272Q16—6816no—
273C2×C8—344no—
274C2×C8—344no—
275C2×C8—344no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—174no—
279C4⋊C4—348no—
280C4⋊C4—348no—
281C4⋊C4—348no—
282C4⋊C4—342no—
283C4⋊C4—348no—
284C22⋊C4—24no—
285C22⋊C4—42no—
286C22⋊C4—348no—
287C42—22no—
288C42—22no—
289C42—42no—
290C42—348no—
291C23—21no—
292C23—24no—
293C23—21no—
294C23#1058762312yes—
295C23C22×C22⋊D1711yescharacteristic
296Q8—28no—
297Q8—48no—
298Q8—44no—
299Q8—348no—
300Q8—684no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—344no—
315C2×C4—344no—
316C2×C4—682no—
317C2×C4—684no—
318C2×C4—682no—
319C2×C4#1058762312yes—
320C2×C4C2×#6325212yes—
321C2×C4C2×D4×D1712yes—
322C2×C4#1058769112yes—
323C8—684no—
324C22—22no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22C2×#1058762311yescharacteristic
329C22#1176332011yescharacteristic
330C22#1176337611yescharacteristic
331C22#1176362414yes—
332C4—22no—
333C4—44no—
334C4—42no—
335C4—42no—
336C4—82no—
337C4—684no—
338C4—684no—
339C4#1176344914yes—
340C2—42no—
341C2#1453210311yescharacteristic
342C2#1453807712yes—
343C2#1453908412yes—
344C2#1453986712yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2310C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2310D17
hypocenterC17#23943
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. #14529935C2semidirect
4. #14529883C2semidirect
5. #14529128C2semidirect
244. C17#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#117634212
C2×#117629652
#145299352
#145298832
#145291282
#145290962
#145288552
C17×#23102
#2394317