Finite Group Databasebeta

← #17951549

Subgroups — #17951549

Order 4352.

994 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 156 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#14545228C212yesmaximal
2#14543683C214yesmaximal
3#14531163C211yesmaximal, characteristic
4C2×#10587677—22no—
5#11763752—22no—
6#11763751—22no—
7#11763751C414yescomplemented
8#11763750C2214yes—
9C4×C22⋊#595—22no—
10C4×C22⋊#595C414yescomplemented
11C23×C4×D17C2211yescharacteristic
12C2×#10587612C411yescharacteristic, complemented
13#11762670C412yescomplemented
14#11762666—22no—
15#11762664C2212yes—
16C4×#63227—22no—
17C2×C17×#117C411yescharacteristic, complemented
18C24×D17C2311yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C23×Dic17C2×C411yescharacteristic
21C22×C4×D17—42no—
22C22×C4×D17—42no—
23C22×C4×D17C2×C412yes—
24C22×C4×D17C2×C412yes—
25C22×C4×D17D412yes—
26C22×C4×D17D412yes—
27C22×C4×D17D412yes—
28C22×C4×D17Q812yes—
29C2×C22⋊#595—24no—
30C2×C22⋊#595—24no—
31C2×C22⋊#595C2×C418yes—
32#10587677—44no—
33#10587677—44no—
34C2×C4⋊#595—24no—
35C2×C4×#595—24no—
36C17×#117—42no—
37C2×C4⋊C4×C17—21no—
38C2×C22⋊C4×C17C2×C412yes—
39C2×C42×C17—21no—
40C2×C22⋊Dic17C2×C412yes—
41C2×#63227—24no—
42#10587612—42no—
43C2×C4⋊Dic17—21no—
44C2×C4×Dic17—21no—
45(C2×#595)⋊C4—24no—
46(C2×#595)⋊C4—24no—
47#10587502—24no—
48C24×C17C22×C411yescharacteristic
49C23×D17—44no—
50C23×D17—41no—
51C23×D17—41no—
52C23×D17C22×C412yes—
53C23×D17C2×D411yescharacteristic
54C23×D17C2×D412yes—
55C23×D17C2×Q811yescharacteristic
56C22×#595—28no—
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17C4212yes—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C4⋊C412yes—
64C22×Dic17—22no—
65C22×Dic17—41no—
66C22×Dic17—41no—
67C22×Dic17C4211yescharacteristic
68C22×Dic17C4211yescharacteristic
69C22×Dic17C22⋊C412yes—
70C22×Dic17C4⋊C412yes—
71C2×C4×D17—24no—
72C2×C4×D17—24no—
73C2×C4×D17—44no—
74C2×C4×D17—44no—
75C2×C4×D17—44no—
76C2×C4×D17—44no—
77C2×C4×D17—42no—
78C2×C4×D17—42no—
79C2×C4×D17C22⋊C412yes—
80C2×C4×D17C22⋊C412yes—
81C2×C4×D17C22⋊C412yes—
82C2×C4×D17C22⋊C412yes—
83C2×C4×D17C4⋊C412yes—
84C2×C4×D17C4⋊C412yes—
85C2×C4×D17C4⋊C412yes—
86C2×C4×D17C4⋊C412yes—
87C22⋊#595—44no—
88C22⋊#595—44no—
89C22⋊#595—48no—
90C22⋊#595—48no—
91C4⋊#595—48no—
92C4×#595—48no—
93C4⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C42×C17—42no—
96C22⋊Dic17—42no—
97#63227—48no—
98C4⋊Dic17—42no—
99C4×Dic17—42no—
100#8102—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—42no—
104C23×C17C2×C4211yescharacteristic
105C23×C17C2×C22⋊C411yescharacteristic
106C23×C17C2×C4⋊C411yescharacteristic
107C22×D17—22no—
108C22×D17—22no—
109C22×D17—24no—
110C22×D17—24no—
111C22×D17—24no—
112C22×D17—42no—
113C22×D17—41no—
114C22×D17—41no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—48no—
119C22×D17—48no—
120C22×D17C2×C22⋊C412yes—
121C22×D17C2×C4⋊C412yes—
122C2×#595—48no—
123C2×#595—48no—
124C2×#595—48no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—24no—
128C2×C4×C17—24no—
129C2×C4×C17—44no—
130C2×C4×C17—42no—
131C2×C4×C17#9512yes—
132C2×C4×C17#9512yes—
133C2×C4×C17#9912yes—
134C2×C4×C17#9912yes—
135C2×Dic17—22no—
136C2×Dic17—22no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—42no—
140C2×Dic17—44no—
141C2×Dic17—42no—
142C2×Dic17#9512yes—
143C2×Dic17#9512yes—
144C4×D17—24no—
145C4×D17—24no—
146C4×D17—24no—
147C4×D17—24no—
148C4×D17—42no—
149C4×D17—42no—
150C4×D17—42no—
151C4×D17—42no—
152#2280—171no—
153#2280—172no—
154#1304—174no—
155C22×C17—22no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—42no—
161C22×C17—42no—
162C22×C17C2×#9511yescharacteristic
163C22×C17C2×#9911yescharacteristic
164C22×C17C2×#9911yescharacteristic
165C2×D17—22no—
166C2×D17—22no—
167C2×D17—22no—
168C2×D17—22no—
169C2×D17—22no—
170C2×D17—44no—
171C2×D17—44no—
172C2×D17—44no—
173C2×D17—48no—
174C2×D17—44no—
175C2×D17#41014yes—
176#595—88no—
177C4×C17—22no—
178C4×C17—82no—
179C4×C17#34214yes—
180Dic17—22no—
181Dic17—22no—
182Dic17—82no—
183C24×C4—171no—
184C2×#117—171no—
185C2×#117—342no—
186C2×#117#59511yescharacteristic, complemented
187#381—172no—
188#381—342no—
189#380—174no—
190#380—342no—
191#380—342no—
192#379—174no—
193#379—172no—
194C4×C22⋊C4—174no—
195C4×C22⋊C4—342no—
196C4×C22⋊C4—342no—
197C2×C17—21no—
198C2×C17—21no—
199C2×C17—41no—
200C2×C17—41no—
201C2×C17C2×#34211yescharacteristic
202C2×C17#174012yes—
203D17—24no—
204D17—42no—
205C25—171no—
206C23×C4—171no—
207C23×C4—172no—
208C23×C4—172no—
209C23×C4—172no—
210C23×C4—172no—
211C23×C4—172no—
212C23×C4—172no—
213C23×C4—682no—
214C23×C4—682no—
215C23×C4C2×#59511yescharacteristic
216#117—42no—
217#117—684no—
218#117—684no—
219#117—682no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—344no—
222C2×C4⋊C4—341no—
223C2×C22⋊C4—178no—
224C2×C22⋊C4—172no—
225C2×C22⋊C4—344no—
226C2×C22⋊C4—344no—
227C2×C22⋊C4—344no—
228C2×C22⋊C4C2×#59512yes—
229C2×C42—21no—
230C2×C42—344no—
231C2×C42—341no—
232#95—344no—
233#95—344no—
234#95—344no—
235C17#810211yescharacteristic, complemented, Sylow
236C24—171no—
237C24—172no—
238C24—172no—
239C24—171no—
240C24—681no—
241C24—681no—
242C24—684no—
243C24C22×#59511yescharacteristic
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—172no—
248C22×C4—171no—
249C22×C4—171no—
250C22×C4—172no—
251C22×C4—172no—
252C22×C4—172no—
253C22×C4—172no—
254C22×C4—172no—
255C22×C4—172no—
256C22×C4—172no—
257C22×C4—172no—
258C22×C4—172no—
259C22×C4—344no—
260C22×C4—344no—
261C22×C4—348no—
262C22×C4—342no—
263C22×C4—684no—
264C22×C4—684no—
265C22×C4—684no—
266C22×C4—684no—
267C22×C4—682no—
268C22×C4—682no—
269C22×C4—681no—
270C22×C4—681no—
271C22×C4C4×#59512yes—
272C22×C4C4⋊#59512yes—
273C22×C4C22⋊#59511yescharacteristic
274C22×C4C22⋊#59511yescharacteristic
275C4⋊C4—42no—
276C4⋊C4—688no—
277C4⋊C4—682no—
278C22⋊C4—42no—
279C22⋊C4—688no—
280C22⋊C4—688no—
281C22⋊C4—684no—
282C22⋊C4—684no—
283C22⋊C4—682no—
284C22⋊C4—688no—
285C42—42no—
286C42—682no—
287C42—688no—
288C23—21no—
289C23—21no—
290C23—42no—
291C23—172no—
292C23—172no—
293C23—344no—
294C23—344no—
295C23—344no—
296C23—342no—
297C23—342no—
298C23—682no—
299C23—681no—
300C23—681no—
301C23—682no—
302C23—682no—
303C23—682no—
304C23—688no—
305C23—688no—
306C23C2×C4×#59511yescharacteristic
307C23C2×C4⋊#59511yescharacteristic
308C23C2×C22⋊#59511yescharacteristic
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—172no—
316C2×C4—172no—
317C2×C4—342no—
318C2×C4—342no—
319C2×C4—344no—
320C2×C4—344no—
321C2×C4—344no—
322C2×C4—344no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—682no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—682no—
330C2×C4—684no—
331C2×C4—688no—
332C2×C4—688no—
333C2×C4—688no—
334C2×C4—682no—
335C2×C4#1058753912yes—
336C2×C4(C2×#595)⋊C412yes—
337C2×C4(C2×#595)⋊C412yes—
338C2×C4#1058755112yes—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—174no—
347C22—342no—
348C22—342no—
349C22—342no—
350C22—342no—
351C22—342no—
352C22—684no—
353C22—684no—
354C22—684no—
355C22—688no—
356C22—684no—
357C22C2×(C2×#595)⋊C411yescharacteristic
358C22C2×#1058755111yescharacteristic
359C22C2×#1058753911yescharacteristic
360C4—22no—
361C4—82no—
362C4—342no—
363C4—342no—
364C4—1362no—
365C4—1368no—
366C4#1176368714yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—344no—
372C2—682no—
373C2C2×#1176368711yescharacteristic
374C2#1454448212yes—
subgroupG/H
centerC22C2×(C2×#595)⋊C4
commutatorC23×C17C2×C42
FrattiniC24C22×#595
FittingC2×C17×#117C4
socleC22×C17C2×#95
prime radical, cosocleC24×D17C23
hypercenterC2×#117#595
hypocenterC17#8102
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
7. #11763751C4semidirect
10. C4×C22⋊#595C4semidirect
12. C2×#10587612C4semidirect
13. #11762670C4semidirect
17. C2×C17×#117C4semidirect
186. C2×#117#595semidirect
235. C17#8102semidirect

Maximal subgroups

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

subgroupindex
#145452282
#145436832
#145311632
#810217