Finite Group Databasebeta

← #17009223

Subgroups — #17009223

Order 4352.

906 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 191 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#14542594C211yesmaximal, characteristic
2#14539928C212yesmaximal
3#14527791C218yesmaximal
4#14527282C212yesmaximal
5#14527108C211yesmaximal, characteristic
6C17×#2546C211yesmaximal, characteristic
7Q8×Dic34—22no—
8#11763538—22no—
9#11763535—24no—
10#11763440—21no—
11#11763440C2214yes—
12#11763366—22no—
13#11763366C2212yes—
14#11763287—21no—
15#11763205—22no—
16#11763161—22no—
17#11763139C2218yes—
18#11763091—24no—
19#11762862C2212yes—
20#11762545C2218yes—
21#11762535C2214yes—
22C2×C17×C4⋊Q8—21no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C4×Q8×C17C2211yescharacteristic
25C17×#398C2214yes—
26C17×#391—21no—
27C17×#389—22no—
28C4×C4⋊C4×C17C2211yescharacteristic
29C22×Q8×C17—21no—
30C22×Q8×C17D411yescharacteristic
31C17×C4⋊Q8—22no—
32C17×C4⋊Q8—22no—
33C17×C4⋊Q8—44no—
34C4×Q8×C17—22no—
35C4×Q8×C17—22no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—24no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17D411yescharacteristic
40C2×C4⋊C4×C17C2312yes—
41C2×C4⋊C4×C17C2312yes—
42C2×C42×C17—21no—
43C2×C42×C17D412yes—
44C2×C42×C17C2311yescharacteristic
45Q8×Dic17—24no—
46Q8×Dic17—24no—
47#10587612—24no—
48#10587612—22no—
49#10587610—22no—
50#10587610—24no—
51#10587610C2314yes—
52#10587576—24no—
53#10587576—28no—
54#10587574—24no—
55#10587574—28no—
56#10587556—24no—
57#10587556—22no—
58#10587555—22no—
59#10587555—22no—
60#10587555D412yes—
61#10587555D412yes—
62C4×Dic34—24no—
63C4×Dic34—24no—
64C4×Dic34—22no—
65C4×Dic34D414yes—
66C17×#95—24no—
67C17×#95C2312yes—
68C22⋊#3622C2314yes—
69#10587481—28no—
70#10587479—28no—
71C4⋊#3622—22no—
72C4×#3622—22no—
73C2×Q8×C17—22no—
74C2×Q8×C17—22no—
75C2×Q8×C17—42no—
76C2×Q8×C17—42no—
77C2×Q8×C17—42no—
78C2×Q8×C17SD1614yes—
79C22×C4×C17—22no—
80C22×C4×C17—22no—
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D412yes—
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17C2411yescharacteristic
86C2×Dic34—22no—
87C2×Dic34—24no—
88C2×Dic34C2×D414yes—
89C4⋊C4×C17—24no—
90C4⋊C4×C17—22no—
91C4⋊C4×C17—24no—
92C4⋊C4×C17—24no—
93C4⋊C4×C17—41no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17—48no—
96C42×C17—22no—
97C42×C17—22no—
98C42×C17—42no—
99C42×C17C2×D412yes—
100C42×C17C2×D412yes—
101C22⋊Dic17—24no—
102C4⋊Dic17—24no—
103C4⋊Dic17—24no—
104C4⋊Dic17—24no—
105C4⋊Dic17—22no—
106C4⋊Dic17—22no—
107C4⋊Dic17—22no—
108C4⋊Dic17C2×D414yes—
109#63225—28no—
110#63225—24no—
111#63225—24no—
112C4×Dic17—28no—
113C4×Dic17—24no—
114C2×#3622—24no—
115#18410—171nomaximal, Sylow
116C23×C17C22×D411yescharacteristic
117Q8×C17—28no—
118Q8×C17—84no—
119C2×C4×C17—22no—
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—24no—
123C2×C4×C17—22no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17#12011yescharacteristic
132C2×C4×C17#12012yes—
133C2×C4×C17#12011yescharacteristic
134C2×C4×C17C2×SD1614yes—
135C2×C4×C17C2×SD1612yes—
136C2×C4×C17#13712yes—
137C2×C4×C17C22×D412yes—
138C2×Dic17—24no—
139C2×Dic17—24no—
140Dic34—44no—
141Dic34—44no—
142Dic34—84no—
143#3622—44no—
144#3502—171no—
145#3229—172no—
146#2546D1711yescharacteristic
147#1611—178no—
148#1503—172no—
149#1461—171no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17C2×#12011yescharacteristic
153C22×C17C22×SD1611yescharacteristic
154C22×C17C2×#13711yescharacteristic
155C4×C17—24no—
156C4×C17—21no—
157C4×C17—22no—
158C4×C17—44no—
159C4×C17—42no—
160C4×C17—82no—
161C4×C17#44814yes—
162C4×C17#45612yes—
163Dic17—44no—
164Dic17—44no—
165Q82—342no—
166#557—344no—
167#557—342no—
168#544—174no—
169#544—341no—
170#533—172no—
171#533—342no—
172C2×C4⋊Q8—21no—
173C2×C4⋊Q8C2×D1711yescharacteristic
174#519—341no—
175C2×C4×Q8C2×D1711yescharacteristic
176#498—342no—
177#487—342no—
178#483—178no—
179#475—344no—
180#431—172no—
181#398C2×D1714yes—
182#391—21no—
183#389—22no—
184C4×C4⋊C4C2×D1711yescharacteristic
185#328—178no—
186#324—174no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17C2×#44811yescharacteristic
190C2×C17C2×#45611yescharacteristic
191C2×C17#302811yescharacteristic
192C22×Q8—21no—
193C22×Q8C22⋊D1711yescharacteristic
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—44no—
197C4⋊Q8—172no—
198C4⋊Q8—172no—
199C4⋊Q8—342no—
200C4⋊Q8—342no—
201C4⋊Q8—348no—
202C4⋊Q8—344no—
203#125—344no—
204#125—348no—
205#125—344no—
206#125—342no—
207C22⋊Q8—174no—
208C22⋊Q8—342no—
209C22⋊Q8—344no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—174no—
213C4×Q8—344no—
214C4×Q8—344no—
215C4×Q8—344no—
216C4×Q8—344no—
217C4×Q8—342no—
218#117—342no—
219#117—344no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22⋊D1711yescharacteristic
224C2×C4⋊C4C22×D1712yes—
225C2×C4⋊C4C22×D1712yes—
226C2×C42—21no—
227C2×C42C22⋊D1712yes—
228C2×C42C22×D1711yescharacteristic
229#106—348no—
230C4⋊C8—342no—
231#103—348no—
232C22⋊C8—174no—
233C4×C8—342no—
234#95—24no—
235#95C22×D1712yes—
236C17#1841011yescharacteristic, complemented, Sylow
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—174no—
243C2×Q8—344no—
244C2×Q8—342no—
245C2×Q8#6322914yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4C2×C22⋊D1712yes—
249C22×C4C2×C22⋊D1711yescharacteristic
250C22×C4C2×C22⋊D1712yes—
251C22×C4C2×C22⋊D1711yescharacteristic
252C22×C4C23×D1711yescharacteristic
253C2×C8—344no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—24no—
258C4⋊C4—48no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—174no—
262C4⋊C4—344no—
263C4⋊C4—344no—
264C4⋊C4—344no—
265C4⋊C4—348no—
266C4⋊C4—344no—
267C4⋊C4—342no—
268C4⋊C4—342no—
269C4⋊C4—344no—
270C4⋊C4—342no—
271C22⋊C4—344no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—348no—
276C42—344no—
277C42D4×D1712yes—
278C42D4×D1712yes—
279C23C22×C22⋊D1711yescharacteristic
280Q8—28no—
281Q8—84no—
282Q8—684no—
283Q8—684no—
284Q8—1364no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—344no—
298C2×C4—344no—
299C2×C4C2×#6322914yes—
300C2×C4C2×#6322912yes—
301C2×C4#1058762311yescharacteristic
302C2×C4#1058762312yes—
303C2×C4#1058762311yescharacteristic
304C2×C4#1058763712yes—
305C2×C4C2×D4×D1712yes—
306C8—684no—
307C22—21no—
308C22—21no—
309C22C2×#1058762311yescharacteristic
310C22C22×#6322911yescharacteristic
311C22C2×#1058763711yescharacteristic
312C4—21no—
313C4—22no—
314C4—24no—
315C4—44no—
316C4—42no—
317C4—82no—
318C4—684no—
319C4—684no—
320C4#1176294714yes—
321C4#1176299112yes—
322C2—21no—
323C2—21no—
324C2C2×#1176294711yescharacteristic
325C2C2×#1176299111yescharacteristic
326C2#1453818911yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2546C2
socleC22×C17C2×#120
prime radicalC23×C17C22×D4
hypercenter#2546D17
hypocenterC17#18410
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
236. C17#18410semidirect

Maximal subgroups

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

subgroupindex
#145425942
#145399282
#145277912
#145272822
#145271082
C17×#25462
#1841017