Finite Group Databasebeta

← #16932269

Subgroups — #16932269

Order 4352.

878 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 111 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#14537825C211yesmaximal, characteristic, complemented
2#14528705C212yesmaximal, complemented
3#14527122C211yesmaximal, characteristic, complemented
4#14526690C212yesmaximal, complemented
5C17×#2408C211yesmaximal, characteristic
6#11762887—24no—
7#11762862—21no—
8C2×#10587504—24no—
9#11762773—21no—
10#11762773C2212yes—
11#11762773C2212yes—
12#11762535—21no—
13#11762533—21no—
14C23×D4×C17C2211yescharacteristic
15C2×C17×#121—21no—
16C2×C17×#120C412yes—
17C2×C4×D4×C17C2211yescharacteristic
18C17×#394—21no—
19C17×#392—21no—
20C17×#390—21no—
21C17×#380C2211yescharacteristic
22C17×#379C412yes—
23C25×C17C2×C412yes—
24C22×D4×C17—22no—
25C22×D4×C17—21no—
26C22×D4×C17—22no—
27C22×D4×C17—21no—
28C22×D4×C17—22no—
29C22×D4×C17—44no—
30C22×D4×C17C2×C411yescharacteristic
31C23×C4×C17D411yescharacteristic
32C23×C4×C17C2311yescharacteristic
33C17×#121—44no—
34C17×#120—44no—
35C4×D4×C17—22no—
36C4×D4×C17—22no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17D411yescharacteristic
39C2×C22⋊C4×C17—22no—
40C2×C22⋊C4×C17—22no—
41C2×C22⋊C4×C17—22no—
42C2×C22⋊C4×C17C2×C411yescharacteristic
43C2×C22⋊C4×C17C2×C412yes—
44C2×C22⋊C4×C17D411yescharacteristic
45C2×C42×C17D411yescharacteristic
46C2×#63223—22no—
47C2×#63223—22no—
48C2×#63223D414yescomplemented
49C17×#95—21no—
50#10587504—48no—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C22⋊#3622—22no—
54C22⋊#3622—22no—
55C22⋊#3622—22no—
56C4⋊#3622—22no—
57C4×#3622—22no—
58C24×C17—22no—
59C24×C17—24no—
60C24×C17—44no—
61C24×C17—42no—
62C24×C17—42no—
63C24×C17—82no—
64C24×C17C22⋊C411yescharacteristic
65C24×C17C22×C411yescharacteristic
66C2×D4×C17—28no—
67C2×D4×C17—24no—
68C2×D4×C17—22no—
69C2×D4×C17—22no—
70C2×D4×C17—41no—
71C2×D4×C17—42no—
72C2×D4×C17—44no—
73C2×D4×C17—44no—
74C2×D4×C17—48no—
75C2×D4×C17—44no—
76C2×D4×C17—44no—
77C2×D4×C17—41no—
78C2×D4×C17—42no—
79C2×D4×C17—41no—
80C2×D4×C17—82no—
81C22×C4×C17—21no—
82C22×C4×C17—21no—
83C22×C4×C17—22no—
84C22×C4×C17—42no—
85C22×C4×C17—42no—
86C22×C4×C17C22⋊C411yescharacteristic
87C22×C4×C17C22⋊C411yescharacteristic
88C22×C4×C17C22⋊C411yescharacteristic
89C22×C4×C17C2×D412yes—
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17#4112yes—
93C4⋊C4×C17—22no—
94C4⋊C4×C17—42no—
95C22⋊C4×C17—22no—
96C22⋊C4×C17—22no—
97C22⋊C4×C17—44no—
98C22⋊C4×C17—44no—
99C22⋊C4×C17—42no—
100C22⋊C4×C17—82no—
101C42×C17—21no—
102C42×C17—21no—
103#63223—44no—
104#63223—44no—
105#63223—82no—
106C2×#3622—24no—
107C2×#3622—42no—
108#9504—171nomaximal, Sylow
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—24no—
112C23×C17—28no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17—41no—
120C23×C17—42no—
121C23×C17—42no—
122C23×C17—44no—
123C23×C17—42no—
124C23×C17—42no—
125C23×C17—42no—
126C23×C17—42no—
127C23×C17—84no—
128C23×C17—84no—
129C23×C17—88no—
130C23×C17—88no—
131C23×C17C2×C22⋊C411yescharacteristic
132C23×C17C4×D412yes—
133D4×C17—44no—
134D4×C17—48no—
135D4×C17—44no—
136D4×C17—44no—
137D4×C17—88no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—22no—
141C2×C4×C17—24no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—41no—
145C2×C4×C17—42no—
146C2×C4×C17—42no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—81no—
153C2×C4×C17#10012yescomplemented
154C2×C4×C17#12011yescharacteristic
155C2×C4×C17#12112yes—
156C2×C4×C17#12311yescharacteristic
157#3622—44no—
158#3622—82no—
159#2975—171no—
160#2408D1711yescharacteristic
161#1783—172no—
162#1465—171no—
163#1305—172no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—22no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17—42no—
170C22×C17—42no—
171C22×C17—42no—
172C22×C17—44no—
173C22×C17—44no—
174C22×C17—41no—
175C22×C17—44no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—81no—
179C22×C17—82no—
180C22×C17—82no—
181C22×C17—88no—
182C22×C17—88no—
183C22×C17—82no—
184C22×C17—82no—
185C22×C17#38611yescharacteristic
186C22×C17C2×#10011yescharacteristic
187C22×C17#41311yescharacteristic
188C4×C17—22no—
189C4×C17—21no—
190C4×C17—22no—
191C4×C17—41no—
192C4×C17—41no—
193C4×C17—82no—
194C4×C17B(4,𝔽2)12yes—
195C23×D4C2×D1711yescharacteristic
196C2×#121—21no—
197C2×#120Dic1712yes—
198C2×C4×D4C2×D1711yescharacteristic
199#436—344no—
200#431—341no—
201C2×#100—344no—
202#407—172no—
203#407—172no—
204#407—341no—
205#394—21no—
206#392—21no—
207#390—21no—
208#380C2×D1711yescharacteristic
209#379Dic1712yes—
210#324—341no—
211#323—341no—
212C2×C17—21no—
213C2×C17—21no—
214C2×C17—42no—
215C2×C17—81no—
216C2×C17—84no—
217C2×C17#188911yescharacteristic
218C2×C17#190011yescharacteristic
219C2×C17#191411yescharacteristic
220C25C2×Dic1712yes—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—44no—
227C22×D4C2×Dic1711yescharacteristic
228C23×C4C22⋊D1711yescharacteristic
229C23×C4C22×D1711yescharacteristic
230C2×M16—174no—
231C2×M16—342no—
232C2×M16—342no—
233#121—44no—
234#120—44no—
235C4×D4—22no—
236C4×D4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C22⋊D1711yescharacteristic
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4C2×Dic1711yescharacteristic
243C2×C22⋊C4C2×Dic1712yes—
244C2×C22⋊C4C22⋊D1711yescharacteristic
245C2×C42C22⋊D1711yescharacteristic
246C4⋊C8—342no—
247#100—688no—
248C22⋊C8—342no—
249C22⋊C8—342no—
250C22⋊C8—342no—
251C22⋊C8—342no—
252C22⋊C8—342no—
253C4×C8—342no—
254#95—21no—
255C17#950411yescharacteristic, complemented, Sylow
256C24—22no—
257C24—24no—
258C24—44no—
259C24—42no—
260C24—42no—
261C24—82no—
262C24C22⋊Dic1711yescharacteristic
263C24C22×Dic1711yescharacteristic
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—82no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C22⋊Dic1711yescharacteristic
285C22×C4C22⋊Dic1711yescharacteristic
286C22×C4C22⋊Dic1711yescharacteristic
287C22×C4D4×D1712yes—
288C22×C4#6325412yes—
289C22×C4C2×C22⋊D1711yescharacteristic
290C22×C4C2×C22⋊D1711yescharacteristic
291M16—684no—
292M16—684no—
293M16—1362no—
294C2×C8—344no—
295C2×C8—682no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—82no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—24no—
309C23—28no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—44no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—84no—
325C23—84no—
326C23—88no—
327C23—88no—
328C23D4×Dic1712yes—
329C23C2×C22⋊Dic1711yescharacteristic
330D4—44no—
331D4—48no—
332D4—44no—
333D4—44no—
334D4—88no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4#1058750412yescomplemented
351C2×C4#1058762111yescharacteristic
352C2×C4#1058762311yescharacteristic
353C2×C4#1058762512yes—
354C8—684no—
355C8—1362no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—44no—
365C22—44no—
366C22—41no—
367C22—41no—
368C22—44no—
369C22—41no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—88no—
374C22—88no—
375C22—82no—
376C22—82no—
377C22#1176268911yescharacteristic
378C22C2×#1058750411yescharacteristic
379C22#1176279311yescharacteristic
380C4—22no—
381C4—21no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4#1176299512yes—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—81no—
391C2—84no—
392C2#1452919311yescharacteristic
393C2#1452924811yescharacteristic
394C2#1452933211yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
Fitting, prime radicalC17×#2408C2
socleC22×C17#386
cosocleC23×C4×C17C23
hypercenter#2408D17
hypocenterC17#9504
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #14537825C2semidirect
2. #14528705C2semidirect
3. #14527122C2semidirect
4. #14526690C2semidirect
48. C2×#63223D4semidirect
153. C2×C4×C17#100semidirect
255. C17#9504semidirect
350. C2×C4#10587504semidirect

Maximal subgroups

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

subgroupindex
#145378252
#145287052
#145271222
#145266902
C17×#24082
#950417