Finite Group Databasebeta

← #17951616

Subgroups — #17951616

Order 4352.

942 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 148 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
1D17⋊#407C212yesmaximal
2#14543687C214yesmaximal, complemented
3D17×#381C211yesmaximal, characteristic
4C2×D17⋊M16—22no—
5#11763781—22no—
6#11763780—22no—
7#11763780C414yescomplemented
8#11763779C2214yes—
9#11763777—22no—
10#11763777C414yescomplemented
11C23×C4×D17C2211yescharacteristic
12C2×C22⋊C4×D17C412yes—
13#11762672C411yescharacteristic
14#11762671—22no—
15#11762670C2212yes—
16#95×D17—22no—
17C17×#381C411yescharacteristic
18C24×D17C2×C411yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C23×Dic17C2311yescharacteristic
21C22×C4×D17—22no—
22C22×C4×D17—44no—
23C22×C4×D17C2×C412yes—
24C22×C4×D17C2×C412yes—
25C22×C4×D17D412yes—
26C22×C4×D17D412yes—
27C22×C4×D17D412yes—
28C22×C4×D17Q812yes—
29C2×#63247—24no—
30D17⋊M16—48no—
31C2×#63243—24no—
32C2×#63242—24no—
33C2×C22⋊C4×C17C2×C412yes—
34C2×C22⋊Dic17C2×C412yes—
35C2×#63227—22no—
36C22⋊C4×D17—44no—
37C22⋊#3624—24no—
38C22⋊#3624C2×C418yescomplemented
39#10587543—24no—
40#10587543—24no—
41C17×#95—22no—
42#10587502—22no—
43#10587502—24no—
44C24×C17C22×C411yescharacteristic
45C23×D17—24no—
46C23×D17—44no—
47C23×D17C4212yes—
48C23×D17C22⋊C411yescharacteristic
49C23×D17C22⋊C411yescharacteristic
50C23×D17C4⋊C412yes—
51C22×C4×C17—22no—
52C22×C4×C17—42no—
53C22×C4×C17C4211yescharacteristic
54C22×C4×C17C4211yescharacteristic
55C22×C4×C17C22⋊C412yes—
56C22×C4×C17C4⋊C412yes—
57C22×Dic17—22no—
58C22×Dic17—42no—
59C22×Dic17C22×C412yes—
60C22×Dic17C2×D412yes—
61C22×Dic17C2×D411yescharacteristic
62C22×Dic17C2×Q811yescharacteristic
63C2×C4×D17—28no—
64C2×C4×D17—44no—
65C2×C4×D17—44no—
66C2×C4×D17—44no—
67C2×C4×D17—44no—
68C2×C4×D17—44no—
69C2×C4×D17—48no—
70C2×C4×D17—44no—
71C2×C4×D17C22⋊C412yes—
72C2×C4×D17C22⋊C412yes—
73C2×C4×D17C22⋊C412yes—
74C2×C4×D17C22⋊C412yes—
75C2×C4×D17C4⋊C412yes—
76C2×C4×D17C4⋊C412yes—
77C2×C4×D17C4⋊C412yes—
78C2×C4×D17C4⋊C412yes—
79#63247—48no—
80C2×#3624—28no—
81#63243—48no—
82#63242—48no—
83C22⋊C4×C17—42no—
84C22⋊Dic17—42no—
85#63227—44no—
86#8114—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—42no—
89C23×C17C2×C4211yescharacteristic
90C23×C17C2×C22⋊C411yescharacteristic
91C23×C17C2×C4⋊C411yescharacteristic
92C22×D17—22no—
93C22×D17—24no—
94C22×D17—24no—
95C22×D17—24no—
96C22×D17—24no—
97C22×D17—24no—
98C22×D17—42no—
99C22×D17—42no—
100C22×D17—42no—
101C22×D17—416no—
102C22×D17#9512yes—
103C22×D17#9512yes—
104C22×D17#9914yes—
105C2×C4×C17—24no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—44no—
110C2×C4×C17—42no—
111C2×C4×C17#9512yes—
112C2×C4×C17#9512yes—
113C2×Dic17—24no—
114C2×Dic17—42no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—44no—
118C2×Dic17—42no—
119C2×Dic17C2×C22⋊C412yes—
120C2×Dic17C2×C4⋊C412yes—
121C4×D17—24no—
122C4×D17—24no—
123C4×D17—24no—
124C4×D17—24no—
125C4×D17—42no—
126C4×D17—42no—
127C4×D17—42no—
128C4×D17—42no—
129C4×D17—88no—
130#3624—48no—
131C2×#407—172no—
132C2×#381—171no—
133#1306—174no—
134C22×C17—22no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—44no—
140C22×C17C2×#9511yescharacteristic
141C22×C17C2×#9912yes—
142C2×D17—28no—
143C2×D17—22no—
144C2×D17—22no—
145C2×D17—22no—
146C2×D17—22no—
147C2×D17—44no—
148C2×D17—48no—
149C2×D17—48no—
150C2×D17#34214yes—
151C2×D17#34414yes—
152C4×C17—22no—
153C4×C17—22no—
154C4×C17—82no—
155Dic17—22no—
156Dic17—22no—
157Dic17—82no—
158C24×C4—171no—
159C22×M16—342no—
160C22×C22⋊C4—172no—
161#407—174no—
162#407—174no—
163#407—342no—
164#407—342no—
165C2×C22⋊C8—174no—
166C2×C22⋊C8—342no—
167#381—171no—
168#381—172no—
169#381—342no—
170#381#59511yescharacteristic
171C2×#95—342no—
172C2×C17—22no—
173C2×C17—41no—
174C2×C17—41no—
175C2×C17C2×#34211yescharacteristic
176C2×C17C2×#34412yes—
177D17—24no—
178D17—24no—
179C25—171no—
180C23×C4—171no—
181C23×C4—172no—
182C23×C4—172no—
183C23×C4—172no—
184C23×C4—172no—
185C23×C4—172no—
186C23×C4—172no—
187C23×C4—342no—
188C23×C4—684no—
189C23×C4C2×#59511yescharacteristic
190C2×M16—344no—
191C2×M16—344no—
192C2×M16—688no—
193C22×C8—344no—
194C2×C22⋊C4—172no—
195C2×C22⋊C4—342no—
196C2×C22⋊C4—684no—
197C2×C22⋊C4C2×#59512yes—
198C22⋊C8—178no—
199C22⋊C8—344no—
200C22⋊C8—344no—
201C22⋊C8—344no—
202#95—22no—
203#95—344no—
204#95—342no—
205C17#811411yescharacteristic, complemented, Sylow
206C24—171no—
207C24—171no—
208C24—172no—
209C24—172no—
210C24—344no—
211C24—684no—
212C24C22×#59511yescharacteristic
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—172no—
216C22×C4—172no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—172no—
220C22×C4—172no—
221C22×C4—172no—
222C22×C4—172no—
223C22×C4—172no—
224C22×C4—172no—
225C22×C4—172no—
226C22×C4—172no—
227C22×C4—348no—
228C22×C4—342no—
229C22×C4—684no—
230C22×C4—684no—
231C22×C4—684no—
232C22×C4—684no—
233C22×C4—684no—
234C22×C4—684no—
235C22×C4—688no—
236C22×C4—682no—
237C22×C4C4×#59511yescharacteristic
238C22×C4C4×#59511yescharacteristic
239C22×C4C4⋊#59512yes—
240C22×C4C22⋊#59512yes—
241M16—688no—
242M16—688no—
243C2×C8—348no—
244C2×C8—688no—
245C22⋊C4—42no—
246C22⋊C4—684no—
247C22⋊C4—682no—
248C23—22no—
249C23—42no—
250C23—172no—
251C23—172no—
252C23—174no—
253C23—344no—
254C23—344no—
255C23—344no—
256C23—344no—
257C23—344no—
258C23—342no—
259C23—682no—
260C23—682no—
261C23—682no—
262C23—6816no—
263C23C2×C4×#59511yescharacteristic
264C23C2×C4⋊#59511yescharacteristic
265C23C2×C22⋊#59511yescharacteristic
266C2×C4—24no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—172no—
273C2×C4—172no—
274C2×C4—344no—
275C2×C4—344no—
276C2×C4—344no—
277C2×C4—344no—
278C2×C4—344no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—681no—
284C2×C4—682no—
285C2×C4—681no—
286C2×C4—684no—
287C2×C4—682no—
288C2×C4—1368no—
289C2×C4(C2×#595)⋊C412yes—
290C2×C4(C2×#595)⋊C412yes—
291C8—688no—
292C22—22no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—44no—
298C22—174no—
299C22—174no—
300C22—348no—
301C22—342no—
302C22—342no—
303C22—342no—
304C22—342no—
305C22—684no—
306C22—688no—
307C22—688no—
308C22C2×(C2×#595)⋊C411yescharacteristic
309C22D17⋊#9912yes—
310C4—22no—
311C4—22no—
312C4—82no—
313C4—342no—
314C4—342no—
315C4—1362no—
316C2—22no—
317C2—41no—
318C2—41no—
319C2—344no—
320C2—344no—
321C2#1454448111yescharacteristic
322C2#1454449212yes—
subgroupG/H
centerC22C2×(C2×#595)⋊C4
commutatorC23×C17C2×C42
FrattiniC24C22×#595
FittingC17×#381C4
socleC22×C17C2×#95
prime radicalC24×D17C2×C4
cosocleC23×Dic17C23
hypercenter#381#595
hypocenterC17#8114
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #14543687C2semidirect
7. #11763780C4semidirect
10. #11763777C4semidirect
38. C22⋊#3624C2×C4semidirect
205. C17#8114semidirect

Maximal subgroups

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

subgroupindex
D17⋊#4072
#145436872
D17×#3812
#811417