Finite Group Databasebeta

← #16949824

Subgroups — #16949824

Order 4352.

1182 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#11762602C214yesmaximal, complemented
2#14528702C212yesmaximal, complemented
3C2×C17×#535C211yesmaximal, characteristic
4C2×#10587505—22no—
5C2×#10587505C2214yescomplemented
6#11762672—22no—
7#11762602—28no—
8C22×#41×C17C2211yescharacteristic
9C23×D4×C17C411yescharacteristic, complemented
10C17×#535—28no—
11C2×C17×#127C2212yes—
12C2×C17×#124C412yescomplemented
13C2×C17×#121—22no—
14C2×C17×#120—22no—
15C2×C17×#117C411yescharacteristic, complemented
16C25×C17—21no—
17C2×#41×C17—24no—
18C2×#41×C17—42no—
19C2×#41×C17—42no—
20C22×Q8×C17C2×C411yescharacteristic
21C22×D4×C17—24no—
22C22×D4×C17—24no—
23C22×D4×C17—21no—
24C22×D4×C17—42no—
25C22×D4×C17—42no—
26C22×D4×C17C2×C412yes—
27C22×D4×C17D412yes—
28C22×D4×C17C2311yescharacteristic
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17D412yes—
31C17×#127—24no—
32C17×#127D418yes—
33C17×#124—24no—
34C17×#124—24no—
35C17×#121—216no—
36C17×#120—216no—
37C17×#117—24no—
38C2×C4⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—24no—
41C2×C42×C17C2×C412yes—
42C2×C22⋊Dic17—24no—
43#10587505—28no—
44#10587505—44no—
45#10587502—44no—
46C24×C17—24no—
47C24×C17—22no—
48C24×C17—42no—
49C24×C17—42no—
50C24×C17—84no—
51C24×C17C22⋊C411yescharacteristic
52C24×C17C2×D412yes—
53#41×C17—416no—
54C2×Q8×C17—22no—
55C2×Q8×C17—41no—
56C2×Q8×C17—41no—
57C2×D4×C17—24no—
58C2×D4×C17—24no—
59C2×D4×C17—24no—
60C2×D4×C17—216no—
61C2×D4×C17—22no—
62C2×D4×C17—42no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C2×D4×C17—42no—
66C2×D4×C17—42no—
67C2×D4×C17—416no—
68C2×D4×C17—42no—
69C2×D4×C17—42no—
70C2×D4×C17—88no—
71C2×D4×C17C22⋊C414yes—
72C2×D4×C17C2×D414yes—
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—22no—
76C22×C4×C17—22no—
77C22×C4×C17—24no—
78C22×C4×C17—24no—
79C22×C4×C17—42no—
80C22×C4×C17C22⋊C412yes—
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22×C411yescharacteristic
83C22×Dic17—44no—
84C4⋊C4×C17—24no—
85C22⋊C4×C17—216no—
86C22⋊C4×C17—24no—
87C42×C17—22no—
88C42×C17C22⋊C414yes—
89C22⋊Dic17—44no—
90C22⋊Dic17—44no—
91#12697—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—22no—
94C23×C17—21no—
95C23×C17—22no—
96C23×C17—24no—
97C23×C17—42no—
98C23×C17—44no—
99C23×C17—48no—
100C23×C17—48no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—88no—
104C23×C17—88no—
105C23×C17—88no—
106C23×C17#9912yes—
107C23×C17C2×C22⋊C411yescharacteristic
108C23×C17#12014yes—
109Q8×C17—44no—
110D4×C17—44no—
111D4×C17—44no—
112D4×C17—48no—
113D4×C17—44no—
114D4×C17—48no—
115D4×C17—816no—
116C2×C4×C17—28no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—24no—
122C2×C4×C17—22no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—48no—
126C2×C4×C17—44no—
127C2×C4×C17#9912yes—
128C2×C4×C17C2×C22⋊C412yes—
129C2×Dic17—84no—
130C2×Dic17—88no—
131C2×#535D1711yescharacteristic
132C2×#353—174no—
133#1782—172no—
134C22×C17—22no—
135C22×C17—24no—
136C22×C17—22no—
137C22×C17—44no—
138C22×C17—42no—
139C22×C17—42no—
140C22×C17—48no—
141C22×C17—44no—
142C22×C17—42no—
143C22×C17—44no—
144C22×C17—82no—
145C22×C17—84no—
146C22×C17—816no—
147C22×C17—84no—
148C22×C17#37911yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17C2×#9911yescharacteristic
151C4×C17—22no—
152C4×C17—22no—
153C4×C17—44no—
154C4×C17—42no—
155Dic17—164no—
156C22×#41C2×D1711yescharacteristic
157C23×D4Dic1711yescharacteristic, complemented
158#535—28no—
159C2×#127C2×D1712yes—
160C2×#124Dic1712yescomplemented
161C2×#121—22no—
162C2×#120—22no—
163C2×#117Dic1711yescharacteristic, complemented
164C2×#99—174no—
165C2×#99—342no—
166#381—342no—
167#353—348no—
168C2×C17—21no—
169C2×C17—21no—
170C2×C17—44no—
171C2×C17—42no—
172C2×C17—84no—
173C2×C17#178111yescharacteristic
174C2×C17#212812yes—
175C25—21no—
176C2×#41—24no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8C2×Dic1711yescharacteristic
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—21no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4C2×Dic1712yes—
186C22×D4C22⋊D1712yes—
187C22×D4C22×D1711yescharacteristic
188C23×C4C2×Dic1711yescharacteristic
189C23×C4C22⋊D1712yes—
190#127—24no—
191#127C22⋊D1718yes—
192#124—24no—
193#124—24no—
194#121—216no—
195#120—216no—
196#117—24no—
197C2×C4⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—344no—
201C2×C42C2×Dic1712yes—
202#99—348no—
203#99—684no—
204#95—684no—
205C17#1269711yescharacteristic, complemented, Sylow
206C24—24no—
207C24—22no—
208C24—42no—
209C24—42no—
210C24—84no—
211C24C22⋊Dic1711yescharacteristic
212C24C2×C22⋊D1712yes—
213#41—416no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—216no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—416no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—88no—
231C2×D4C22⋊Dic1714yes—
232C2×D4C2×C22⋊D1714yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—684no—
241C22×C4C22⋊Dic1712yes—
242C22×C4C22⋊Dic1711yescharacteristic
243C22×C4C22×Dic1711yescharacteristic
244C4⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—216no—
247C22⋊C4—684no—
248C22⋊C4—684no—
249C42—22no—
250C42C22⋊Dic1714yes—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—24no—
256C23—42no—
257C23—44no—
258C23—48no—
259C23—48no—
260C23—42no—
261C23—42no—
262C23—88no—
263C23—88no—
264C23—88no—
265C23#1058750512yes—
266C23C2×C22⋊Dic1711yescharacteristic
267C23#1058763914yes—
268Q8—44no—
269D4—44no—
270D4—44no—
271D4—48no—
272D4—44no—
273D4—48no—
274D4—816no—
275C2×C4—28no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—1364no—
287C2×C4—1368no—
288C2×C4#1058750512yes—
289C2×C4C2×C22⋊Dic1712yes—
290C22—22no—
291C22—22no—
292C22—24no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22—48no—
297C22—44no—
298C22—42no—
299C22—44no—
300C22—82no—
301C22—84no—
302C22—816no—
303C22—84no—
304C22#1176266511yescharacteristic
305C22C2×#1058750511yescharacteristic
306C22C2×#1058750511yescharacteristic
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—2724no—
312C2—21no—
313C2—21no—
314C2—44no—
315C2—42no—
316C2—84no—
317C2#1452869911yescharacteristic
318C2#1453045612yes—
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
Fitting, prime radicalC2×C17×#535C2
socleC22×C17C2×#99
cosocleC22×D4×C17C23
hypercenterC2×#535D17
hypocenterC17#12697
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#11762602C2semidirect
2. #14528702C2semidirect
5. C2×#10587505C22semidirect
9. C23×D4×C17C4semidirect
12. C2×C17×#124C4semidirect
15. C2×C17×#117C4semidirect
157. C23×D4Dic17semidirect
160. C2×#124Dic17semidirect
163. C2×#117Dic17semidirect
205. C17#12697semidirect

Maximal subgroups

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

subgroupindex
C2×#117626022
#145287022
C2×C17×#5352
#1269717