Finite Group Databasebeta

← #17009179

Subgroups — #17009179

Order 4352.

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

normal subgroups only

This group has 910 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#14539729C212yesmaximal
2#14533989C211yesmaximal, characteristic
3#14527758C218yesmaximal
4#14527283C212yesmaximal
5#14527108C211yesmaximal, characteristic
6C17×#3502C211yesmaximal, characteristic
7C2×#10587555C2211yescharacteristic
8#11763188—22no—
9#11763161—22no—
10#11763019—24no—
11#11762964C2218yes—
12#11762862C2212yes—
13#11762729C2214yes—
14#11762716—21no—
15#11762707—22no—
16#11762678—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×Dic34C2312yes—
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×#63231—28no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17—21no—
61C2×C4⋊Dic17—21no—
62C2×#63225—24no—
63C2×C4×Dic17—21no—
64#10587555—22no—
65#10587555—22no—
66#10587502—24no—
67#10587502C2312yes—
68C22⋊#3622C2314yes—
69#10587481—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×C17Q1614yescomplemented
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—24no—
86C2×Dic34—82no—
87C4⋊C4×C17—24no—
88C4⋊C4×C17—28no—
89C4⋊C4×C17—24no—
90C4⋊C4×C17—24no—
91C4⋊C4×C17—24no—
92C4⋊C4×C17—22no—
93C4⋊C4×C17—22no—
94C4⋊C4×C17—22no—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—24no—
97C4⋊C4×C17—24no—
98C4⋊C4×C17C2×D414yes—
99C22⋊C4×C17—24no—
100C42×C17—28no—
101C42×C17—22no—
102C42×C17—24no—
103C42×C17—22no—
104C42×C17C2×D412yes—
105C42×C17C2×D412yes—
106#63231—88no—
107C4⋊Dic17—24no—
108C4⋊Dic17—41no—
109C4⋊Dic17—42no—
110C4⋊Dic17—41no—
111#63225—84no—
112C4×Dic17—42no—
113C2×#3622—24no—
114#18404—171nomaximal, Sylow
115C23×C17C22×D411yescharacteristic
116Q8×C17—28no—
117Q8×C17—44no—
118Q8×C17—44no—
119Q8×C17—84no—
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—24no—
124C2×C4×C17—24no—
125C2×C4×C17—22no—
126C2×C4×C17—41no—
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—42no—
137C2×Dic17—42no—
138C2×Dic17—44no—
139C2×Dic17—42no—
140Dic34—48no—
141#3622—44no—
142#3502D1711yescharacteristic
143#3209—172no—
144#2567—171no—
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—44no—
159C4×C17#45114yes—
160C4×C17#45612yes—
161Dic17—44no—
162Dic17—82no—
163Q82—22no—
164#557—22no—
165#557—24no—
166#544—21no—
167#544C2×D1714yes—
168#533—22no—
169#533C2×D1712yes—
170C2×C4⋊Q8—171no—
171#519—21no—
172C2×C4×Q8C2×D1711yescharacteristic
173#494—342no—
174#487—342no—
175#462—344no—
176#451—178no—
177#431—172no—
178#395—174no—
179#391—341no—
180#389—342no—
181#384—341no—
182C4×C4⋊C4—171no—
183#328—178no—
184#324—174no—
185C2×C17—21no—
186C2×C17—21no—
187C2×C17C2×#45111yescharacteristic
188C2×C17C2×#45611yescharacteristic
189C2×C17#302711yescharacteristic
190C22×Q8—172no—
191C22×Q8C22⋊D1711yescharacteristic
192C2×Q16—348no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—24no—
196C4⋊Q8—28no—
197C4⋊Q8—342no—
198C4⋊Q8—342no—
199C4⋊Q8C22⋊D1712yes—
200C4⋊Q8C22⋊D1712yes—
201#125—24no—
202#125—24no—
203#125—28no—
204#125—22no—
205C22⋊Q8—22no—
206C22⋊Q8—24no—
207C22⋊Q8C22×D1714yes—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8C22⋊D1714yes—
216#117—24no—
217#117—22no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—344no—
222C2×C4⋊C4C22⋊D1711yescharacteristic
223C2×C4⋊C4C22×D1712yes—
224C2×C42—341no—
225C2×C42C22⋊D1712yes—
226C2×C42C22×D1711yescharacteristic
227C4⋊C8—342no—
228#103—348no—
229C22⋊C8—174no—
230C4×C8—342no—
231#95—172no—
232#95—344no—
233C17#1840411yescharacteristic, complemented, Sylow
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—344no—
239C2×Q8—1362no—
240C2×Q8#6323114yescomplemented
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
249Q16—1368no—
250C2×C8—344no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—344no—
263C4⋊C4—681no—
264C4⋊C4—682no—
265C4⋊C4—681no—
266C4⋊C4—1364no—
267C4⋊C4C2×C22⋊D1714yes—
268C22⋊C4—24no—
269C42—28no—
270C42—22no—
271C42—22no—
272C42—24no—
273C42—682no—
274C42C2×C22⋊D1712yes—
275C42C2×C22⋊D1712yes—
276C23C22×C22⋊D1711yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—44no—
280Q8—84no—
281Q8—688no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—344no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—684no—
294C2×C4—682no—
295C2×C4#1058762812yes—
296C2×C4C2×#6323114yes—
297C2×C4C2×#6323112yes—
298C2×C4#1058763911yescharacteristic
299C2×C4#1058763912yes—
300C2×C4#1058763911yescharacteristic
301C2×C4C22×C22⋊D1712yes—
302C8—684no—
303C22—21no—
304C22—21no—
305C22C2×#1058763911yescharacteristic
306C22C22×#6323111yescharacteristic
307C22C2×#1058762811yescharacteristic
308C4—21no—
309C4—22no—
310C4—24no—
311C4—44no—
312C4—42no—
313C4—44no—
314C4—684no—
315C4—1362no—
316C4#1176296514yes—
317C4#1176299212yes—
318C2—21no—
319C2—21no—
320C2C2×#1176296511yescharacteristic
321C2C2×#1176299211yescharacteristic
322C2#1453818611yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#3502C2
socleC22×C17C2×#120
prime radicalC23×C17C22×D4
hypercenter#3502D17
hypocenterC17#18404
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
76. C2×Q8×C17Q16semidirect
233. C17#18404semidirect
240. C2×Q8#63231semidirect

Maximal subgroups

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

subgroupindex
#145397292
#145339892
#145277582
#145272832
#145271082
C17×#35022
#1840417