Finite Group Databasebeta

← #17009230

Subgroups — #17009230

Order 4352.

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

normal subgroups only

This group has 870 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#14539932C212yesmaximal
2#14533996C211yesmaximal, characteristic
3#14527826C218yesmaximal
4#14527283C212yesmaximal
5#14527108C211yesmaximal, characteristic
6C17×#3502C211yesmaximal, characteristic
7C2×#10587555C2211yescharacteristic
8#11763212—22no—
9#11763159—22no—
10#11763142C2218yes—
11#11763104—24no—
12#11762862C2212yes—
13#11762748C2214yes—
14#11762716—21no—
15#11762715—22no—
16#11762674—21no—
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
58C2×C4⋊Dic17—21no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17C2312yes—
61C2×C4×Dic17—21no—
62#10587555—22no—
63#10587555—22no—
64#10587506—28no—
65#10587502—24no—
66#10587502—24no—
67#10587502C2312yes—
68C22⋊#3622C2314yes—
69#10587478—28no—
70C4⋊#3622—22no—
71C4×#3622—22no—
72C2×Q8×C17—24no—
73C2×Q8×C17—22no—
74C2×Q8×C17—22no—
75C2×Q8×C17—22no—
76C2×Q8×C17Q1614yes—
77C2×Q8×C17C2×D414yes—
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D412yes—
80C22×C4×C17C2×D412yes—
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2411yescharacteristic
83C22×Dic17—22no—
84C22×Dic17—22no—
85C2×Dic34—44no—
86C2×Dic34—41no—
87C2×Dic34—41no—
88C4⋊C4×C17—24no—
89C4⋊C4×C17—28no—
90C4⋊C4×C17—24no—
91C4⋊C4×C17—24no—
92C4⋊C4×C17—24no—
93C4⋊C4×C17—22no—
94C4⋊C4×C17—22no—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—24no—
98C4⋊C4×C17—24no—
99C4⋊C4×C17C2×D414yes—
100C22⋊C4×C17—24no—
101C42×C17—28no—
102C42×C17—22no—
103C42×C17—24no—
104C42×C17—22no—
105C42×C17C2×D412yes—
106C42×C17C2×D412yes—
107C4⋊Dic17—24no—
108C4⋊Dic17—42no—
109C4⋊Dic17—42no—
110C4×Dic17—42no—
111C2×#3622—24no—
112#18411—171nomaximal, Sylow
113C23×C17C22×D411yescharacteristic
114Q8×C17—28no—
115Q8×C17—44no—
116Q8×C17—44no—
117Q8×C17—84no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—22no—
121C2×C4×C17—24no—
122C2×C4×C17—24no—
123C2×C4×C17—22no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17#12011yescharacteristic
127C2×C4×C17#12012yes—
128C2×C4×C17#12011yescharacteristic
129C2×C4×C17C2×Q1614yes—
130C2×C4×C17C2×Q1612yes—
131C2×C4×C17#13712yes—
132C2×C4×C17C22×D412yes—
133C2×Dic17—44no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138Dic34—82no—
139Dic34—82no—
140#3622—44no—
141#3502D1711yescharacteristic
142#3230—172no—
143#2568—171no—
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—44no—
156C4×C17—42no—
157C4×C17—44no—
158C4×C17#45114yes—
159C4×C17#45412yes—
160Dic17—82no—
161Dic17—82no—
162Q82—22no—
163#557—22no—
164#557—24no—
165#544—21no—
166#544C2×D1714yes—
167#533—22no—
168#533C2×D1712yes—
169C2×C4⋊Q8—171no—
170#519—21no—
171C2×C4×Q8C2×D1711yescharacteristic
172#500—342no—
173#487—342no—
174#484—178no—
175#477—344no—
176#431—172no—
177#400—174no—
178#391—341no—
179#391—342no—
180#382—341no—
181C4×C4⋊C4—171no—
182#328—178no—
183#324—174no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17C2×#45111yescharacteristic
187C2×C17C2×#45411yescharacteristic
188C2×C17#302811yescharacteristic
189C22×Q8—341no—
190C22×Q8C22⋊D1711yescharacteristic
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—24no—
194C4⋊Q8—28no—
195C4⋊Q8—342no—
196C4⋊Q8—342no—
197C4⋊Q8C22⋊D1712yes—
198C4⋊Q8C22⋊D1712yes—
199#125—24no—
200#125—24no—
201#125—28no—
202#125—22no—
203C22⋊Q8—22no—
204C22⋊Q8—24no—
205C22⋊Q8C22×D1714yes—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8C22⋊D1714yes—
214#117—24no—
215#117—22no—
216C2×C4⋊C4—172no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4C22⋊D1711yescharacteristic
220C2×C4⋊C4C22×D1712yes—
221C2×C42—341no—
222C2×C42C22⋊D1712yes—
223C2×C42C22×D1711yescharacteristic
224#107—348no—
225C4⋊C8—342no—
226#103—348no—
227C22⋊C8—174no—
228C4×C8—342no—
229#95—172no—
230#95—344no—
231#95—344no—
232C17#1841111yescharacteristic, complemented, Sylow
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—684no—
238C2×Q8—681no—
239C2×Q8—681no—
240C2×Q8#6323114yes—
241C2×Q8C2×C22⋊D1714yes—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4C2×C22⋊D1712yes—
245C22×C4C2×C22⋊D1711yescharacteristic
246C22×C4C2×C22⋊D1712yes—
247C22×C4C2×C22⋊D1711yescharacteristic
248C22×C4C23×D1711yescharacteristic
249C2×C8—344no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—28no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—344no—
262C4⋊C4—682no—
263C4⋊C4—682no—
264C4⋊C4C2×C22⋊D1714yes—
265C22⋊C4—24no—
266C42—28no—
267C42—22no—
268C42—22no—
269C42—24no—
270C42—682no—
271C42C2×C22⋊D1712yes—
272C42C2×C22⋊D1712yes—
273C23C22×C22⋊D1711yescharacteristic
274Q8—28no—
275Q8—44no—
276Q8—44no—
277Q8—84no—
278Q8—1362no—
279Q8—1362no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—682no—
289C2×C4—684no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4#1058762812yes—
294C2×C4C2×#6323114yes—
295C2×C4C2×#6323112yes—
296C2×C4#1058763911yescharacteristic
297C2×C4#1058763912yes—
298C2×C4#1058763911yescharacteristic
299C2×C4C22×C22⋊D1712yes—
300C8—684no—
301C22—21no—
302C22—21no—
303C22C2×#1058763911yescharacteristic
304C22C22×#6323111yescharacteristic
305C22C2×#1058762811yescharacteristic
306C4—21no—
307C4—22no—
308C4—24no—
309C4—44no—
310C4—42no—
311C4—44no—
312C4—1362no—
313C4—1362no—
314C4#1176296514yes—
315C4#1176298212yes—
316C2—21no—
317C2—21no—
318C2C2×#1176296511yescharacteristic
319C2C2×#1176298211yescharacteristic
320C2#1453819011yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#3502C2
socleC22×C17C2×#120
prime radicalC23×C17C22×D4
hypercenter#3502D17
hypocenterC17#18411
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
232. C17#18411semidirect

Maximal subgroups

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

subgroupindex
#145399322
#145339962
#145278262
#145272832
#145271082
C17×#35022
#1841117