Finite Group Databasebeta

← #17066287

Subgroups — #17066287

Order 4352.

1158 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 207 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×#11762964C212yesmaximal
2#14531368C211yesmaximal, characteristic
3#14529932C212yesmaximal, complemented
4#14529884C212yesmaximal, complemented
5#14529125C214yesmaximal, complemented
6#14529097C212yesmaximal
7#14528855C211yesmaximal, characteristic
8C2×C17×#541C211yesmaximal, characteristic
9C23×Dic34C2211yescharacteristic
10C22×#63231C2214yes—
11C2×#10587612C2211yescharacteristic
12#11762964—28no—
13C2×#10587506C2214yes—
14C2×#10587481C2214yes—
15C2×C22⋊#3622C2212yes—
16#11762715C2212yes—
17#11762715C2212yes—
18#11762714—22no—
19#11762670—22no—
20#11762571C2214yes—
21#11762564C2212yes—
22#11762564C2212yes—
23C17×#541—28no—
24C2×C17×C4⋊Q8C2211yescharacteristic
25C2×C17×#126—22no—
26C2×C17×#125—22no—
27C2×C17×#125C2211yescharacteristic
28C2×C17×C22⋊Q8C2212yes—
29C2×C4×Q8×C17C2212yes—
30C2×C17×#117C2211yescharacteristic
31C22×Q8×C17C2312yes—
32C23×C4×C17C2311yescharacteristic
33C23×Dic17—21no—
34C22×Dic34—21no—
35C22×Dic34—24no—
36C22×Dic34—24no—
37C22×Dic34—42no—
38C22×Dic34C2312yes—
39C17×C4⋊Q8—24no—
40C17×#126—216no—
41C17×#125—216no—
42C17×#125—24no—
43C17×C22⋊Q8—24no—
44C17×C22⋊Q8D418yes—
45C4×Q8×C17—24no—
46C4×Q8×C17—24no—
47C17×#117—24no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17—22no—
50C2×C4⋊C4×C17—22no—
51C2×C4⋊C4×C17—22no—
52C2×C4⋊C4×C17D412yes—
53C2×C4⋊C4×C17C2312yes—
54C2×C22⋊C4×C17—22no—
55C2×C22⋊C4×C17D412yes—
56C2×C42×C17—22no—
57C2×C42×C17C2311yescharacteristic
58C2×C42×C17C2311yescharacteristic
59C2×C22⋊Dic17—21no—
60C2×#63231—28no—
61C2×#63231—28no—
62C2×#63231—44no—
63#10587612—42no—
64C2×C4⋊Dic17C2312yes—
65C2×C4×Dic17—21no—
66C22×#3622C2314yes—
67#10587506—44no—
68#10587502—24no—
69#10587502—24no—
70C22⋊#3622—24no—
71#10587481—28no—
72C24×C17C2×D411yescharacteristic
73C2×Q8×C17—22no—
74C2×Q8×C17—24no—
75C2×Q8×C17—24no—
76C2×Q8×C17C2×D414yes—
77C22×C4×C17—22no—
78C22×C4×C17—22no—
79C22×C4×C17—22no—
80C22×C4×C17—22no—
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D414yes—
85C22×C4×C17C2411yescharacteristic
86C22×Dic17—24no—
87C22×Dic17—22no—
88C22×Dic17—22no—
89C22×Dic17—42no—
90C22×Dic17—42no—
91C2×Dic34—24no—
92C2×Dic34—24no—
93C2×Dic34—24no—
94C2×Dic34—22no—
95C2×Dic34—416no—
96C2×Dic34—42no—
97C2×Dic34—42no—
98C2×Dic34#4114yes—
99C4⋊C4×C17—28no—
100C4⋊C4×C17—28no—
101C4⋊C4×C17—28no—
102C4⋊C4×C17—28no—
103C4⋊C4×C17—24no—
104C4⋊C4×C17—22no—
105C4⋊C4×C17C2×D414yes—
106C22⋊C4×C17—28no—
107C22⋊C4×C17—24no—
108C42×C17—28no—
109C42×C17—22no—
110C42×C17—22no—
111C22⋊Dic17—42no—
112#63231—416no—
113C4⋊Dic17—24no—
114C4×Dic17—42no—
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—44no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—28no—
129C2×C4×C17—22no—
130C2×C4×C17—22no—
131C2×C4×C17—24no—
132C2×C4×C17—24no—
133C2×C4×C17—24no—
134C2×C4×C17—24no—
135C2×C4×C17—42no—
136C2×C4×C17—42no—
137C2×C4×C17—44no—
138C2×C4×C17#12012yes—
139C2×C4×C17C22×D412yes—
140C2×C4×C17C2×#4112yes—
141C2×C4×C17H(5,𝔽2)12yes—
142C2×Dic17—24no—
143C2×Dic17—48no—
144C2×Dic17—42no—
145C2×Dic17—42no—
146C2×Dic17—42no—
147C2×Dic17—44no—
148Dic34—28no—
149Dic34—48no—
150Dic34—44no—
151#3622—44no—
152C2×#541D1711yescharacteristic
153C2×#451—172no—
154#2310—171no—
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—44no—
173C4×C17#54614yes—
174Dic17—44no—
175Dic17—82no—
176C23×Q8—171no—
177C22×Q16—174no—
178#541—28no—
179C2×C4⋊Q8C2×D1711yescharacteristic
180C2×#126—22no—
181C2×#125—22no—
182C2×#125C2×D1711yescharacteristic
183C2×C22⋊Q8C2×D1712yes—
184C2×C4×Q8C2×D1712yes—
185C2×#117—171no—
186C2×#117C2×D1711yescharacteristic
187#451—348no—
188C2×#103—174no—
189C2×#103—174no—
190C2×C22⋊C8—172no—
191#391—172no—
192#391—172no—
193#391—342no—
194#381—342no—
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—172no—
204C22×Q8—344no—
205C22×Q8—341no—
206C22×Q8—344no—
207C22×Q8—682no—
208C22×Q8C22×D1712yes—
209C23×C4—341no—
210C23×C4C22×D1711yescharacteristic
211C2×Q16—348no—
212C2×Q16—348no—
213C2×Q16—684no—
214C22×C8—174no—
215C4⋊Q8—24no—
216#126—216no—
217#125—216no—
218#125—24no—
219C22⋊Q8—24no—
220C22⋊Q8C22⋊D1718yes—
221C4×Q8—24no—
222C4×Q8—24no—
223#117—24no—
224#117—682no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—172no—
230C2×C4⋊C4C22⋊D1712yes—
231C2×C4⋊C4C22×D1712yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—341no—
234C2×C22⋊C4C22⋊D1712yes—
235C2×C42—22no—
236C2×C42—341no—
237C2×C42C22×D1711yescharacteristic
238C2×C42C22×D1711yescharacteristic
239#103—348no—
240#103—684no—
241C22⋊C8—344no—
242#95—344no—
243#95—344no—
244C17#2394311yescharacteristic, complemented, Sylow
245C24C2×C22⋊D1711yescharacteristic
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—174no—
250C2×Q8—344no—
251C2×Q8—342no—
252C2×Q8—344no—
253C2×Q8—344no—
254C2×Q8—6816no—
255C2×Q8—682no—
256C2×Q8—682no—
257C2×Q8C2×C22⋊D1714yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—344no—
263C22×C4—342no—
264C22×C4—342no—
265C22×C4—682no—
266C22×C4—682no—
267C22×C4C2×C22⋊D1712yes—
268C22×C4C2×C22⋊D1712yes—
269C22×C4C2×C22⋊D1714yes—
270C22×C4C2×C22⋊D1711yescharacteristic
271C22×C4C23×D1711yescharacteristic
272Q16—6816no—
273C2×C8—344no—
274C2×C8—344no—
275C2×C8—344no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—22no—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—344no—
283C4⋊C4C2×C22⋊D1714yes—
284C22⋊C4—28no—
285C22⋊C4—24no—
286C22⋊C4—682no—
287C42—28no—
288C42—22no—
289C42—22no—
290C42—682no—
291C23—21no—
292C23—24no—
293C23—21no—
294C23#1058763912yes—
295C23C22×C22⋊D1711yescharacteristic
296Q8—28no—
297Q8—44no—
298Q8—348no—
299Q8—688no—
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—24no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—344no—
314C2×C4—688no—
315C2×C4—682no—
316C2×C4—682no—
317C2×C4—684no—
318C2×C4—682no—
319C2×C4#1058763912yes—
320C2×C4#41×D1712yes—
321C2×C4#1058769612yes—
322C2×C4C22×C22⋊D1712yes—
323C8—684no—
324C22—22no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22C2×#1058763911yescharacteristic
329C22#1176331611yescharacteristic
330C22#1176337311yescharacteristic
331C22#1176362414yes—
332C4—22no—
333C4—44no—
334C4—42no—
335C4—42no—
336C4—44no—
337C4—684no—
338C4—1362no—
339C4#1176344714yes—
340C2—42no—
341C2#1453209911yescharacteristic
342C2#1453807912yes—
343C2#1453907912yes—
344C2#1453986112yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#541C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterC2×#541D17
hypocenterC17#23943
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
3. #14529932C2semidirect
4. #14529884C2semidirect
5. #14529125C2semidirect
244. C17#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#117629642
#145313682
#145299322
#145298842
#145291252
#145290972
#145288552
C2×C17×#5412
#2394317