Finite Group Databasebeta

← #17951602

Subgroups — #17951602

Order 4352.

942 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 152 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×D17D411yescharacteristic
27C22×C4×D17D411yescharacteristic
28C22×C4×D17D412yes—
29C22×C4×D17Q811yescharacteristic
30C22×C4×D17Q811yescharacteristic
31C2×#63247—24no—
32D17⋊M16—48no—
33C2×#63243—24no—
34C2×#63242—24no—
35C2×C22⋊C4×C17C2×C412yes—
36C2×C22⋊Dic17C2×C412yes—
37C2×#63227—22no—
38C22⋊C4×D17—44no—
39C22⋊#3624—24no—
40C22⋊#3624C2×C418yescomplemented
41#10587543—24no—
42#10587543—24no—
43C17×#95—22no—
44#10587502—22no—
45#10587502—24no—
46C24×C17C22×C411yescharacteristic
47C23×D17—24no—
48C23×D17—44no—
49C23×D17C4211yescharacteristic
50C23×D17C4211yescharacteristic
51C23×D17C22⋊C411yescharacteristic
52C23×D17C22⋊C411yescharacteristic
53C23×D17C4⋊C411yescharacteristic
54C23×D17C4⋊C411yescharacteristic
55C22×C4×C17—22no—
56C22×C4×C17—42no—
57C22×C4×C17C4212yes—
58C22×C4×C17C22⋊C412yes—
59C22×C4×C17C4⋊C411yescharacteristic
60C22×C4×C17C4⋊C411yescharacteristic
61C22×Dic17—22no—
62C22×Dic17—42no—
63C22×Dic17C22×C412yes—
64C22×Dic17C2×D412yes—
65C22×Dic17C2×D411yescharacteristic
66C22×Dic17C2×Q811yescharacteristic
67C2×C4×D17—28no—
68C2×C4×D17—44no—
69C2×C4×D17—44no—
70C2×C4×D17—42no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C2×C4×D17—42no—
74C2×C4×D17—44no—
75C2×C4×D17—48no—
76C2×C4×D17—44no—
77C2×C4×D17C22⋊C412yes—
78C2×C4×D17C22⋊C412yes—
79C2×C4×D17C22⋊C412yes—
80C2×C4×D17C22⋊C412yes—
81C2×C4×D17C4⋊C412yes—
82C2×C4×D17C4⋊C412yes—
83C2×C4×D17C4⋊C412yes—
84C2×C4×D17C4⋊C412yes—
85#63247—48no—
86C2×#3624—28no—
87#63243—48no—
88#63242—48no—
89C22⋊C4×C17—42no—
90C22⋊Dic17—42no—
91#63227—44no—
92C2×#1306—171nomaximal, Sylow
93C23×C17—22no—
94C23×C17—42no—
95C23×C17C2×C4211yescharacteristic
96C23×C17C2×C22⋊C411yescharacteristic
97C23×C17C2×C4⋊C411yescharacteristic
98C22×D17—24no—
99C22×D17—22no—
100C22×D17—22no—
101C22×D17—22no—
102C22×D17—22no—
103C22×D17—22no—
104C22×D17—24no—
105C22×D17—24no—
106C22×D17—42no—
107C22×D17—42no—
108C22×D17—42no—
109C22×D17—416no—
110C22×D17#9511yescharacteristic
111C22×D17#9511yescharacteristic
112C22×D17#9511yescharacteristic
113C22×D17#9511yescharacteristic
114C22×D17#9914yes—
115C2×C4×C17—24no—
116C2×C4×C17—42no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—44no—
120C2×C4×C17—42no—
121C2×C4×C17#9512yes—
122C2×C4×C17#9512yes—
123C2×Dic17—24no—
124C2×Dic17—42no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—44no—
128C2×Dic17—42no—
129C2×Dic17C2×C22⋊C412yes—
130C2×Dic17C2×C4⋊C412yes—
131C4×D17—24no—
132C4×D17—24no—
133C4×D17—24no—
134C4×D17—24no—
135C4×D17—42no—
136C4×D17—42no—
137C4×D17—42no—
138C4×D17—42no—
139C4×D17—88no—
140#3624—48no—
141C2×#407—172no—
142C2×#381—171no—
143#1306—174no—
144C22×C17—22no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—22no—
148C22×C17—22no—
149C22×C17—44no—
150C22×C17C2×#9511yescharacteristic
151C22×C17C2×#9912yes—
152C2×D17—22no—
153C2×D17—28no—
154C2×D17—21no—
155C2×D17—21no—
156C2×D17—22no—
157C2×D17—22no—
158C2×D17—44no—
159C2×D17—44no—
160C2×D17—44no—
161C2×D17—44no—
162C2×D17—44no—
163C2×D17#34212yes—
164C2×D17#34212yes—
165C2×D17#34414yes—
166C4×C17—22no—
167C4×C17—22no—
168C4×C17—82no—
169Dic17—22no—
170Dic17—22no—
171Dic17—82no—
172C24×C4—171no—
173C22×M16—342no—
174C22×C22⋊C4—172no—
175#407—174no—
176#407—174no—
177#407—342no—
178#407—342no—
179C2×C22⋊C8—174no—
180C2×C22⋊C8—342no—
181#381—171no—
182#381—172no—
183#381—342no—
184#381#59511yescharacteristic
185C2×#95—342no—
186C2×C17—22no—
187C2×C17—41no—
188C2×C17—41no—
189C2×C17C2×#34211yescharacteristic
190C2×C17C2×#34412yes—
191D17—22no—
192D17—41no—
193D17—41no—
194D17#130614yescomplemented
195C25—171no—
196C23×C4—171no—
197C23×C4—172no—
198C23×C4—172no—
199C23×C4—172no—
200C23×C4—171no—
201C23×C4—171no—
202C23×C4—171no—
203C23×C4—171no—
204C23×C4—172no—
205C23×C4—342no—
206C23×C4—684no—
207C23×C4C2×#59511yescharacteristic
208C2×M16—344no—
209C2×M16—344no—
210C2×M16—688no—
211C22×C8—344no—
212C2×C22⋊C4—172no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4—684no—
215C2×C22⋊C4C2×#59512yes—
216C22⋊C8—178no—
217C22⋊C8—344no—
218C22⋊C8—344no—
219C22⋊C8—344no—
220#95—22no—
221#95—344no—
222#95—342no—
223C17C2×#130611yescharacteristic, complemented, Sylow
224C24—171no—
225C24—171no—
226C24—171no—
227C24—171no—
228C24—171no—
229C24—171no—
230C24—344no—
231C24—684no—
232C24C22×#59511yescharacteristic
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—172no—
236C22×C4—172no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—172no—
240C22×C4—172no—
241C22×C4—172no—
242C22×C4—172no—
243C22×C4—172no—
244C22×C4—172no—
245C22×C4—172no—
246C22×C4—172no—
247C22×C4—348no—
248C22×C4—342no—
249C22×C4—684no—
250C22×C4—684no—
251C22×C4—684no—
252C22×C4—682no—
253C22×C4—682no—
254C22×C4—682no—
255C22×C4—682no—
256C22×C4—684no—
257C22×C4—688no—
258C22×C4—682no—
259C22×C4C4×#59512yes—
260C22×C4C4⋊#59511yescharacteristic
261C22×C4C4⋊#59511yescharacteristic
262C22×C4C22⋊#59512yes—
263M16—688no—
264M16—688no—
265C2×C8—348no—
266C2×C8—688no—
267C22⋊C4—42no—
268C22⋊C4—684no—
269C22⋊C4—682no—
270C23—22no—
271C23—42no—
272C23—171no—
273C23—171no—
274C23—174no—
275C23—171no—
276C23—171no—
277C23—344no—
278C23—342no—
279C23—342no—
280C23—342no—
281C23—342no—
282C23—344no—
283C23—344no—
284C23—342no—
285C23—682no—
286C23—682no—
287C23—682no—
288C23—6816no—
289C23C2×C4×#59511yescharacteristic
290C23C2×C4⋊#59511yescharacteristic
291C23C2×C22⋊#59511yescharacteristic
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—172no—
299C2×C4—172no—
300C2×C4—344no—
301C2×C4—344no—
302C2×C4—344no—
303C2×C4—344no—
304C2×C4—344no—
305C2×C4—682no—
306C2×C4—682no—
307C2×C4—682no—
308C2×C4—682no—
309C2×C4—681no—
310C2×C4—682no—
311C2×C4—681no—
312C2×C4—684no—
313C2×C4—682no—
314C2×C4—1368no—
315C2×C4(C2×#595)⋊C412yes—
316C2×C4(C2×#595)⋊C412yes—
317C8—688no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—44no—
324C22—172no—
325C22—174no—
326C22—172no—
327C22—348no—
328C22—342no—
329C22—341no—
330C22—341no—
331C22—342no—
332C22—342no—
333C22—684no—
334C22—684no—
335C22—684no—
336C22—684no—
337C22—684no—
338C22C2×(C2×#595)⋊C411yescharacteristic
339C22D17⋊#9912yes—
340C4—22no—
341C4—22no—
342C4—82no—
343C4—342no—
344C4—342no—
345C4—1362no—
346C2—22no—
347C2—41no—
348C2—41no—
349C2—174no—
350C2—342no—
351C2—681no—
352C2—681no—
353C2#1454448111yescharacteristic
354C2#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
hypocenterC17C2×#1306
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
40. C22⋊#3624C2×C4semidirect
194. D17#1306semidirect
223. C17C2×#1306semidirect

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
C2×#130617