Finite Group Databasebeta

← #17009178

Subgroups — #17009178

Order 4352.

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

normal subgroups only

This group has 910 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#14539732C212yesmaximal
3#14527757C218yesmaximal
4#14527282C212yesmaximal
5#14527108C211yesmaximal, characteristic
6C17×#2567C211yesmaximal, characteristic
7Q8×Dic34—22no—
8#11763538—22no—
9#11763535—24no—
10#11763440—21no—
11#11763440C2214yes—
12#11763366—22no—
13#11763366C2212yes—
14#11763287—21no—
15#11763188—22no—
16#11763159—22no—
17#11763016—24no—
18#11762965C2218yes—
19#11762862C2212yes—
20#11762545C2218yes—
21#11762535C2214yes—
22C2×C17×C4⋊Q8C2211yescharacteristic
23C2×C4×Q8×C17C2211yescharacteristic
24C17×#395C2214yes—
25C17×#391—21no—
26C17×#389—22no—
27C17×#384—21no—
28C4×C4⋊C4×C17C2211yescharacteristic
29C22×Q8×C17D411yescharacteristic
30C22×Q8×C17C2312yes—
31C17×C4⋊Q8—22no—
32C17×C4⋊Q8—22no—
33C4×Q8×C17—22no—
34C4×Q8×C17—22no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—24no—
39C2×C4⋊C4×C17D411yescharacteristic
40C2×C4⋊C4×C17C2312yes—
41C2×C42×C17—21no—
42C2×C42×C17D412yes—
43C2×C42×C17C2311yescharacteristic
44Q8×Dic17—24no—
45Q8×Dic17—24no—
46C2×#63231—28no—
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—
68#10587506—28no—
69C22⋊#3622C2314yes—
70C4⋊#3622—22no—
71C4×#3622—22no—
72C2×Q8×C17—22no—
73C2×Q8×C17—22no—
74C2×Q8×C17—24no—
75C2×Q8×C17—82no—
76C2×Q8×C17Q1614yescomplemented
77C22×C4×C17—22no—
78C22×C4×C17—22no—
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D412yes—
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×D411yescharacteristic
83C22×C4×C17C2411yescharacteristic
84C2×Dic34—22no—
85C2×Dic34—24no—
86C2×Dic34C2×D414yes—
87C4⋊C4×C17—24no—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—24no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—42no—
92C4⋊C4×C17—41no—
93C4⋊C4×C17—84no—
94C42×C17—22no—
95C42×C17—22no—
96C42×C17—42no—
97C42×C17C2×D412yes—
98C42×C17C2×D412yes—
99C22⋊Dic17—24no—
100#63231—88no—
101C4⋊Dic17—24no—
102C4⋊Dic17—24no—
103C4⋊Dic17—24no—
104C4⋊Dic17—22no—
105C4⋊Dic17—22no—
106C4⋊Dic17—22no—
107C4⋊Dic17C2×D414yes—
108#63225—28no—
109#63225—24no—
110#63225—24no—
111C4×Dic17—28no—
112C4×Dic17—24no—
113C2×#3622—24no—
114#18404—171nomaximal, Sylow
115C23×C17C22×D411yescharacteristic
116Q8×C17—28no—
117Q8×C17—48no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—22no—
121C2×C4×C17—24no—
122C2×C4×C17—22no—
123C2×C4×C17—41no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—44no—
128C2×C4×C17—42no—
129C2×C4×C17#12011yescharacteristic
130C2×C4×C17#12012yes—
131C2×C4×C17#12011yescharacteristic
132C2×C4×C17C2×Q1614yes—
133C2×C4×C17C2×Q1612yes—
134C2×C4×C17#13712yes—
135C2×C4×C17C22×D412yes—
136C2×Dic17—24no—
137C2×Dic17—24no—
138Dic34—44no—
139Dic34—44no—
140Dic34—84no—
141#3622—44no—
142#3502—171no—
143#3209—172no—
144#2567D1711yescharacteristic
145#1605—178no—
146#1503—172no—
147#1461—171no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17C2×#12011yescharacteristic
151C22×C17C22×Q1611yescharacteristic
152C22×C17C2×#13711yescharacteristic
153C4×C17—24no—
154C4×C17—21no—
155C4×C17—22no—
156C4×C17—44no—
157C4×C17—42no—
158C4×C17—82no—
159C4×C17#45114yes—
160C4×C17#45612yes—
161Dic17—44no—
162Dic17—44no—
163Q82—342no—
164#557—344no—
165#557—342no—
166#544—174no—
167#544—341no—
168#533—172no—
169#533—342no—
170C2×C4⋊Q8C2×D1711yescharacteristic
171#519—341no—
172C2×C4×Q8C2×D1711yescharacteristic
173#494—342no—
174#487—342no—
175#462—344no—
176#451—178no—
177#431—172no—
178#395C2×D1714yes—
179#391—21no—
180#389—22no—
181#384—21no—
182C4×C4⋊C4C2×D1711yescharacteristic
183#328—178no—
184#324—174no—
185C2×C17—21no—
186C2×C17—21no—
187C2×C17C2×#45111yescharacteristic
188C2×C17C2×#45611yescharacteristic
189C2×C17#302711yescharacteristic
190C22×Q8C22⋊D1711yescharacteristic
191C22×Q8C22×D1712yes—
192C2×Q16—348no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—172no—
196C4⋊Q8—172no—
197C4⋊Q8—342no—
198C4⋊Q8—342no—
199C4⋊Q8—348no—
200C4⋊Q8—344no—
201#125—344no—
202#125—348no—
203#125—344no—
204#125—342no—
205C22⋊Q8—174no—
206C22⋊Q8—342no—
207C22⋊Q8—344no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—174no—
211C4×Q8—344no—
212C4×Q8—344no—
213C4×Q8—344no—
214C4×Q8—344no—
215C4×Q8—342no—
216#117—342no—
217#117—344no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4C22⋊D1711yescharacteristic
223C2×C4⋊C4C22×D1712yes—
224C2×C42—21no—
225C2×C42C22⋊D1712yes—
226C2×C42C22×D1711yescharacteristic
227C4⋊C8—342no—
228#103—348no—
229C22⋊C8—174no—
230C4×C8—342no—
231#95—24no—
232#95C22×D1712yes—
233C17#1840411yescharacteristic, complemented, Sylow
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—82no—
238C2×Q8—174no—
239C2×Q8—344no—
240C2×Q8—342no—
241C2×Q8#6323114yescomplemented
242C22×C4—22no—
243C22×C4—22no—
244C22×C4C2×C22⋊D1711yescharacteristic
245C22×C4C2×C22⋊D1712yes—
246C22×C4C2×C22⋊D1712yes—
247C22×C4C2×C22⋊D1711yescharacteristic
248C22×C4C23×D1711yescharacteristic
249Q16—1368no—
250C2×C8—344no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—84no—
258C4⋊C4—174no—
259C4⋊C4—344no—
260C4⋊C4—344no—
261C4⋊C4—344no—
262C4⋊C4—348no—
263C4⋊C4—344no—
264C4⋊C4—342no—
265C4⋊C4—342no—
266C4⋊C4—344no—
267C4⋊C4—342no—
268C22⋊C4—344no—
269C42—22no—
270C42—22no—
271C42—42no—
272C42—348no—
273C42—344no—
274C42D4×D1712yes—
275C42D4×D1712yes—
276C23C22×C22⋊D1711yescharacteristic
277Q8—28no—
278Q8—48no—
279Q8—684no—
280Q8—684no—
281Q8—1364no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—344no—
294C2×C4—344no—
295C2×C4#1058762311yescharacteristic
296C2×C4#1058762312yes—
297C2×C4#1058762311yescharacteristic
298C2×C4C2×#6323114yes—
299C2×C4C2×#6323112yes—
300C2×C4#1058763712yes—
301C2×C4C2×D4×D1712yes—
302C8—684no—
303C22—21no—
304C22—21no—
305C22C2×#1058762311yescharacteristic
306C22C22×#6323111yescharacteristic
307C22C2×#1058763711yescharacteristic
308C4—21no—
309C4—22no—
310C4—24no—
311C4—44no—
312C4—42no—
313C4—82no—
314C4—684no—
315C4—684no—
316C4#1176296414yes—
317C4#1176299112yes—
318C2—21no—
319C2—21no—
320C2C2×#1176296411yescharacteristic
321C2C2×#1176299111yescharacteristic
322C2#1453818511yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2567C2
socleC22×C17C2×#120
prime radicalC23×C17C22×D4
hypercenter#2567D17
hypocenterC17#18404
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
76. C2×Q8×C17Q16semidirect
233. C17#18404semidirect
241. C2×Q8#63231semidirect

Maximal subgroups

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

subgroupindex
#145425942
#145397322
#145277572
#145272822
#145271082
C17×#25672
#1840417