Finite Group Databasebeta

← #17009229

Subgroups — #17009229

Order 4352.

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

normal subgroups only

This group has 870 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#14539935C212yesmaximal
3#14527825C218yesmaximal
4#14527282C212yesmaximal
5#14527108C211yesmaximal, characteristic
6C17×#2568C211yesmaximal, characteristic
7Q8×Dic34—22no—
8#11763538—22no—
9#11763535—24no—
10#11763440—21no—
11#11763440C2214yes—
12#11763366—22no—
13#11763366C2212yes—
14#11763287—21no—
15#11763212—22no—
16#11763161—22no—
17#11763144C2218yes—
18#11763101—24no—
19#11762862C2212yes—
20#11762545C2218yes—
21#11762535C2214yes—
22C2×C17×C4⋊Q8C2211yescharacteristic
23C2×C4×Q8×C17C2211yescharacteristic
24C17×#400C2214yes—
25C17×#391—22no—
26C17×#391—21no—
27C17×#382—21no—
28C4×C4⋊C4×C17C2211yescharacteristic
29C22×Q8×C17—21no—
30C22×Q8×C17D411yescharacteristic
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×C17D411yescharacteristic
38C2×C4⋊C4×C17C2312yes—
39C2×C4⋊C4×C17C2312yes—
40C2×C42×C17—21no—
41C2×C42×C17D412yes—
42C2×C42×C17C2311yescharacteristic
43Q8×Dic17—24no—
44Q8×Dic17—24no—
45#10587612—24no—
46#10587612—22no—
47#10587610—22no—
48#10587610—24no—
49#10587610C2314yes—
50#10587576—24no—
51#10587576—28no—
52#10587574—24no—
53#10587574—28no—
54#10587556—24no—
55#10587556—22no—
56#10587555—22no—
57#10587555—22no—
58#10587555D412yes—
59#10587555D412yes—
60C4×Dic34—24no—
61C4×Dic34—24no—
62C4×Dic34—22no—
63C4×Dic34D414yes—
64C17×#95—24no—
65C17×#95—24no—
66C17×#95C2312yes—
67C22⋊#3622C2314yes—
68#10587481—28no—
69#10587478—28no—
70C4⋊#3622—22no—
71C4×#3622—22no—
72C2×Q8×C17—22no—
73C2×Q8×C17—22no—
74C2×Q8×C17—44no—
75C2×Q8×C17—41no—
76C2×Q8×C17—41no—
77C2×Q8×C17Q1614yes—
78C22×C4×C17—22no—
79C22×C4×C17—22no—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2411yescharacteristic
85C2×Dic34—22no—
86C2×Dic34—24no—
87C2×Dic34C2×D414yes—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—24no—
90C4⋊C4×C17—24no—
91C4⋊C4×C17—42no—
92C4⋊C4×C17—42no—
93C42×C17—22no—
94C42×C17—22no—
95C42×C17—42no—
96C42×C17C2×D412yes—
97C42×C17C2×D412yes—
98C22⋊Dic17—24no—
99C4⋊Dic17—24no—
100C4⋊Dic17—24no—
101C4⋊Dic17—24no—
102C4⋊Dic17—22no—
103C4⋊Dic17—22no—
104C4⋊Dic17—22no—
105C4⋊Dic17C2×D414yes—
106#63225—28no—
107#63225—24no—
108#63225—24no—
109C4×Dic17—28no—
110C4×Dic17—24no—
111C2×#3622—24no—
112#18411—171nomaximal, Sylow
113C23×C17C22×D411yescharacteristic
114Q8×C17—28no—
115Q8×C17—82no—
116Q8×C17—82no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—22no—
121C2×C4×C17—41no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—44no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17#12011yescharacteristic
129C2×C4×C17#12012yes—
130C2×C4×C17#12011yescharacteristic
131C2×C4×C17C2×Q1614yes—
132C2×C4×C17C2×Q1612yes—
133C2×C4×C17#13712yes—
134C2×C4×C17C22×D412yes—
135C2×Dic17—24no—
136C2×Dic17—24no—
137Dic34—44no—
138Dic34—44no—
139Dic34—84no—
140#3622—44no—
141#3502—171no—
142#3230—172no—
143#2568D1711yescharacteristic
144#1617—178no—
145#1503—172no—
146#1461—171no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17C2×#12011yescharacteristic
150C22×C17C22×Q1611yescharacteristic
151C22×C17C2×#13711yescharacteristic
152C4×C17—24no—
153C4×C17—21no—
154C4×C17—22no—
155C4×C17—42no—
156C4×C17—82no—
157C4×C17—82no—
158C4×C17#45114yes—
159C4×C17#45412yes—
160Dic17—44no—
161Dic17—44no—
162Q82—342no—
163#557—344no—
164#557—342no—
165#544—174no—
166#544—341no—
167#533—172no—
168#533—342no—
169C2×C4⋊Q8C2×D1711yescharacteristic
170#519—341no—
171C2×C4×Q8C2×D1711yescharacteristic
172#500—342no—
173#487—342no—
174#484—178no—
175#477—344no—
176#431—172no—
177#400C2×D1714yes—
178#391—21no—
179#391—22no—
180#382—21no—
181C4×C4⋊C4C2×D1711yescharacteristic
182#328—178no—
183#324—174no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17C2×#45111yescharacteristic
187C2×C17C2×#45411yescharacteristic
188C2×C17#302811yescharacteristic
189C22×Q8—21no—
190C22×Q8C22⋊D1711yescharacteristic
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—172no—
194C4⋊Q8—172no—
195C4⋊Q8—342no—
196C4⋊Q8—342no—
197C4⋊Q8—348no—
198C4⋊Q8—344no—
199#125—344no—
200#125—348no—
201#125—344no—
202#125—342no—
203C22⋊Q8—174no—
204C22⋊Q8—342no—
205C22⋊Q8—344no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—174no—
209C4×Q8—344no—
210C4×Q8—344no—
211C4×Q8—344no—
212C4×Q8—344no—
213C4×Q8—342no—
214#117—342no—
215#117—344no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22⋊D1711yescharacteristic
219C2×C4⋊C4C22×D1712yes—
220C2×C4⋊C4C22×D1712yes—
221C2×C42—21no—
222C2×C42C22⋊D1712yes—
223C2×C42C22×D1711yescharacteristic
224#107—348no—
225C4⋊C8—342no—
226#103—348no—
227C22⋊C8—174no—
228C4×C8—342no—
229#95—24no—
230#95—24no—
231#95C22×D1712yes—
232C17#1841111yescharacteristic, complemented, Sylow
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—174no—
239C2×Q8—344no—
240C2×Q8—342no—
241C2×Q8#6323114yes—
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
249C2×C8—344no—
250C4⋊C4—24no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—174no—
256C4⋊C4—344no—
257C4⋊C4—344no—
258C4⋊C4—344no—
259C4⋊C4—348no—
260C4⋊C4—344no—
261C4⋊C4—342no—
262C4⋊C4—342no—
263C4⋊C4—344no—
264C4⋊C4—342no—
265C22⋊C4—344no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—348no—
270C42—344no—
271C42D4×D1712yes—
272C42D4×D1712yes—
273C23C22×C22⋊D1711yescharacteristic
274Q8—28no—
275Q8—82no—
276Q8—82no—
277Q8—684no—
278Q8—684no—
279Q8—1364no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—344no—
292C2×C4—344no—
293C2×C4#1058762311yescharacteristic
294C2×C4#1058762312yes—
295C2×C4#1058762311yescharacteristic
296C2×C4C2×#6323114yes—
297C2×C4C2×#6323112yes—
298C2×C4#1058763712yes—
299C2×C4C2×D4×D1712yes—
300C8—684no—
301C22—21no—
302C22—21no—
303C22C2×#1058762311yescharacteristic
304C22C22×#6323111yescharacteristic
305C22C2×#1058763711yescharacteristic
306C4—21no—
307C4—22no—
308C4—24no—
309C4—42no—
310C4—82no—
311C4—82no—
312C4—684no—
313C4—684no—
314C4#1176296414yes—
315C4#1176298112yes—
316C2—21no—
317C2—21no—
318C2C2×#1176296411yescharacteristic
319C2C2×#1176298111yescharacteristic
320C2#1453818911yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2568C2
socleC22×C17C2×#120
prime radicalC23×C17C22×D4
hypercenter#2568D17
hypocenterC17#18411
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
232. C17#18411semidirect

Maximal subgroups

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

subgroupindex
#145425942
#145399352
#145278252
#145272822
#145271082
C17×#25682
#1841117