Finite Group Databasebeta

← #17009224

Subgroups — #17009224

Order 4352.

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

normal subgroups only

This group has 906 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
1#14539925C212yesmaximal
2#14533769C211yesmaximal, characteristic
3#14527792C218yesmaximal
4#14527283C212yesmaximal
5#14527108C211yesmaximal, characteristic
6C17×#3502C211yesmaximal, characteristic
7C2×#10587555C2211yescharacteristic
8C2×#10587630—21no—
9#11763205—22no—
10#11763159—22no—
11#11763137C2218yes—
12#11763094—24no—
13#11762862C2212yes—
14#11762739C2214yes—
15#11762714—21no—
16#11762707—22no—
17C4×C4⋊Dic17C2211yescharacteristic
18#11762546C2218yes—
19#11762535C2214yes—
20Q82×C17—22no—
21C17×#557—24no—
22C17×#557—22no—
23C17×#544—21no—
24C17×#544C2214yes—
25C17×#533—22no—
26C17×#533C2212yes—
27C17×#519—21no—
28C2×C4×Q8×C17C2211yescharacteristic
29C22×Q8×C17D411yescharacteristic
30C22×Dic34—21no—
31C17×C4⋊Q8—22no—
32C17×C4⋊Q8—22no—
33C17×C4⋊Q8—24no—
34C17×C4⋊Q8—28no—
35C17×C4⋊Q8D412yes—
36C17×C4⋊Q8D412yes—
37C17×#125—24no—
38C17×#125—24no—
39C17×#125—28no—
40C17×#125—22no—
41C17×C22⋊Q8—22no—
42C17×C22⋊Q8—24no—
43C17×C22⋊Q8C2314yes—
44C4×Q8×C17—24no—
45C4×Q8×C17—24no—
46C4×Q8×C17—24no—
47C4×Q8×C17—24no—
48C4×Q8×C17—22no—
49C4×Q8×C17—22no—
50C4×Q8×C17—22no—
51C4×Q8×C17D414yes—
52C17×#117—24no—
53C17×#117—22no—
54C2×C4⋊C4×C17D411yescharacteristic
55C2×C4⋊C4×C17C2312yes—
56C2×C42×C17D412yes—
57C2×C42×C17C2311yescharacteristic
58#10587630—44no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17—21no—
61C2×C4⋊Dic17C2312yes—
62C2×#63225—24no—
63C2×C4×Dic17—21no—
64#10587555—22no—
65#10587555—22no—
66#10587506—28no—
67#10587502—24no—
68#10587502C2312yes—
69C22⋊#3622C2314yes—
70#10587479—28no—
71C4⋊#3622—22no—
72C4×#3622—22no—
73C2×Q8×C17—24no—
74C2×Q8×C17—22no—
75C2×Q8×C17—22no—
76C2×Q8×C17—22no—
77C2×Q8×C17SD1614yes—
78C2×Q8×C17C2×D414yes—
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D412yes—
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×D411yescharacteristic
83C22×C4×C17C2411yescharacteristic
84C22×Dic17—22no—
85C22×Dic17—22no—
86C2×Dic34—42no—
87C2×Dic34—42no—
88C2×Dic34—42no—
89C4⋊C4×C17—24no—
90C4⋊C4×C17—28no—
91C4⋊C4×C17—24no—
92C4⋊C4×C17—24no—
93C4⋊C4×C17—24no—
94C4⋊C4×C17—22no—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—24no—
99C4⋊C4×C17—24no—
100C4⋊C4×C17C2×D414yes—
101C22⋊C4×C17—24no—
102C42×C17—28no—
103C42×C17—22no—
104C42×C17—24no—
105C42×C17—22no—
106C42×C17C2×D412yes—
107C42×C17C2×D412yes—
108C4⋊Dic17—24no—
109C4⋊Dic17—24no—
110C4⋊Dic17—41no—
111C4⋊Dic17—41no—
112#63225—48no—
113C4×Dic17—42no—
114C2×#3622—24no—
115#18410—171nomaximal, Sylow
116C23×C17C22×D411yescharacteristic
117Q8×C17—28no—
118Q8×C17—44no—
119Q8×C17—44no—
120Q8×C17—84no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—22no—
124C2×C4×C17—24no—
125C2×C4×C17—24no—
126C2×C4×C17—22no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17#12011yescharacteristic
130C2×C4×C17#12012yes—
131C2×C4×C17#12011yescharacteristic
132C2×C4×C17C2×SD1614yes—
133C2×C4×C17C2×SD1612yes—
134C2×C4×C17#13712yes—
135C2×C4×C17C22×D412yes—
136C2×Dic17—24no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142Dic34—84no—
143#3622—44no—
144#3502D1711yescharacteristic
145#3229—172no—
146#2546—171no—
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—44no—
161C4×C17#44814yes—
162C4×C17#45612yes—
163Dic17—44no—
164Dic17—82no—
165Q82—22no—
166#557—22no—
167#557—24no—
168#544—21no—
169#544C2×D1714yes—
170#533—22no—
171#533C2×D1712yes—
172C2×C4⋊Q8—171no—
173C2×C4⋊Q8—341no—
174#519—21no—
175C2×C4×Q8C2×D1711yescharacteristic
176#498—342no—
177#487—342no—
178#483—178no—
179#475—344no—
180#431—172no—
181#398—174no—
182#391—341no—
183#389—342no—
184C4×C4⋊C4—171no—
185#328—178no—
186#324—174no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17C2×#44811yescharacteristic
190C2×C17C2×#45611yescharacteristic
191C2×C17#302811yescharacteristic
192C22×Q8—341no—
193C22×Q8C22⋊D1711yescharacteristic
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—24no—
197C4⋊Q8—28no—
198C4⋊Q8—342no—
199C4⋊Q8—342no—
200C4⋊Q8—684no—
201C4⋊Q8C22⋊D1712yes—
202C4⋊Q8C22⋊D1712yes—
203#125—24no—
204#125—24no—
205#125—28no—
206#125—22no—
207C22⋊Q8—22no—
208C22⋊Q8—24no—
209C22⋊Q8C22×D1714yes—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8C22⋊D1714yes—
218#117—24no—
219#117—22no—
220C2×C4⋊C4—172no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—344no—
224C2×C4⋊C4C22⋊D1711yescharacteristic
225C2×C4⋊C4C22×D1712yes—
226C2×C42—341no—
227C2×C42C22⋊D1712yes—
228C2×C42C22×D1711yescharacteristic
229#106—348no—
230C4⋊C8—342no—
231#103—348no—
232C22⋊C8—174no—
233C4×C8—342no—
234#95—172no—
235#95—344no—
236C17#1841011yescharacteristic, complemented, Sylow
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—682no—
242C2×Q8—682no—
243C2×Q8—682no—
244C2×Q8#6323014yes—
245C2×Q8C2×C22⋊D1714yes—
246C22×C4—342no—
247C22×C4—342no—
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—24no—
257C4⋊C4—28no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—344no—
266C4⋊C4—344no—
267C4⋊C4—688no—
268C4⋊C4—681no—
269C4⋊C4—681no—
270C4⋊C4C2×C22⋊D1714yes—
271C22⋊C4—24no—
272C42—28no—
273C42—22no—
274C42—22no—
275C42—24no—
276C42—682no—
277C42C2×C22⋊D1712yes—
278C42C2×C22⋊D1712yes—
279C23C22×C22⋊D1711yescharacteristic
280Q8—28no—
281Q8—44no—
282Q8—44no—
283Q8—84no—
284Q8—1364no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—344no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4C2×#6323014yes—
300C2×C4C2×#6323012yes—
301C2×C4#1058762812yes—
302C2×C4#1058763911yescharacteristic
303C2×C4#1058763912yes—
304C2×C4#1058763911yescharacteristic
305C2×C4C22×C22⋊D1712yes—
306C8—684no—
307C22—21no—
308C22—21no—
309C22C2×#1058763911yescharacteristic
310C22C22×#6323011yescharacteristic
311C22C2×#1058762811yescharacteristic
312C4—21no—
313C4—22no—
314C4—24no—
315C4—44no—
316C4—42no—
317C4—44no—
318C4—684no—
319C4—1362no—
320C4#1176294814yes—
321C4#1176299212yes—
322C2—21no—
323C2—21no—
324C2C2×#1176294811yescharacteristic
325C2C2×#1176299211yescharacteristic
326C2#1453819011yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#3502C2
socleC22×C17C2×#120
prime radicalC23×C17C22×D4
hypercenter#3502D17
hypocenterC17#18410
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

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
#145399252
#145337692
#145277922
#145272832
#145271082
C17×#35022
#1841017