Finite Group Databasebeta

← #16944906

Subgroups — #16944906

Order 4352.

768 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 87 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#14532140C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic, complemented
3#14526786C214yesmaximal, complemented
4C4×C17×#100C211yesmaximal, characteristic, complemented
5C23×D68C411yescharacteristic
6C2×#10587616—21no—
7C2×#10587623—21no—
8#11762773—21no—
9#11762726C2212yes—
10#11762718C412yes—
11#11762711—21no—
12#11762669C411yescharacteristic
13#11762664—21no—
14#11762535C2212yescomplemented
15#11762533—21no—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×#100—21no—
18C4×M16×C17—21no—
19C17×#343C2212yescomplemented
20C24×D17—21no—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17C2×C411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C22×C22⋊D17—22no—
25C22×D68—21no—
26C22×D68—44no—
27C22×D68—41no—
28C22×D68—41no—
29C22×D68C2×C412yes—
30C2×M16×C17—22no—
31C4×D4×C17—41no—
32C4×D4×C17—41no—
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C22⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—21no—
38#10587623—44no—
39#10587616—44no—
40C2×#63227—22no—
41C2×#63227—22no—
42C2×#63227—22no—
43C2×C4⋊Dic17—21no—
44C2×#63223—41no—
45C17×#100—24no—
46C17×#97—41no—
47C4×C8×C17—41no—
48#10587502C2×C412yes—
49C22⋊#3622—22no—
50C22⋊#3622—41no—
51C24×C17C22⋊C411yescharacteristic
52C24×C17C22⋊C411yescharacteristic
53C23×D17—22no—
54C23×D17—44no—
55C23×D17—42no—
56C23×D17—42no—
57C23×D17—42no—
58C23×D17—82no—
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—41no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C2×C22⋊D17—44no—
74C2×C22⋊D17—44no—
75C2×C22⋊D17—82no—
76C22×Dic17—22no—
77C2×D68—24no—
78C2×D68—42no—
79C2×D68—42no—
80C2×D68—48no—
81C2×D68—44no—
82C2×D68—41no—
83C2×D68—41no—
84C2×D68—44no—
85C2×D68—44no—
86M16×C17—24no—
87M16×C17—42no—
88C2×C8×C17—42no—
89C4⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C42×C17—21no—
93C42×C17—21no—
94C22⋊Dic17—42no—
95#63227—44no—
96#63227—44no—
97#63227—82no—
98C4⋊Dic17—42no—
99#63223—82no—
100C2×#3622—42no—
101#11779—171nomaximal, Sylow
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17#10412yes—
112C23×C17C2×C22⋊C411yescharacteristic
113C22×D17—24no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—48no—
118C22×D17—42no—
119C22×D17—42no—
120C22×D17—42no—
121C22×D17—88no—
122C22×D17—88no—
123C22×D17—84no—
124C22×D17—84no—
125D4×C17—44no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—22no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17—42no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—81no—
139C2×C4×C17#9911yescharacteristic
140C2×C4×C17#9911yescharacteristic
141C2×C4×C17AGL(1,ℤ/8)12yes—
142C22⋊D17—88no—
143C2×Dic17—42no—
144C2×Dic17—42no—
145C2×Dic17—42no—
146D68—44no—
147D68—44no—
148D68—44no—
149D68—44no—
150C8×C17—44no—
151#3622—82no—
152#2408—171no—
153C4×#100D1711yescharacteristic, complemented
154#1465—171no—
155#1345—174no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—22no—
159C22×C17—42no—
160C22×C17—42no—
161C22×C17—42no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—81no—
168C22×C17C2×#9911yescharacteristic
169C22×C17#41811yescharacteristic
170C22×C17C2×#10411yescharacteristic
171C2×D17—44no—
172C2×D17—44no—
173C2×D17—44no—
174C2×D17—82no—
175C2×D17—82no—
176C2×D17—816no—
177C2×D17—82no—
178C2×D17—82no—
179C4×C17—21no—
180C4×C17—22no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17AΣL(1,𝔽16)12yes—
185Dic17—82no—
186C23×D4—171no—
187C2×#121—341no—
188C2×#120—341no—
189C2×C4×D4C2×D1711yescharacteristic
190C2×#100—21no—
191#407—341no—
192C4×M16—21no—
193#394—172no—
194#392—172no—
195#390—341no—
196#380—171no—
197#379—341no—
198#343C2×D1712yescomplemented
199#324—172no—
200#323—341no—
201C2×C17—21no—
202C2×C17—21no—
203C2×C17—42no—
204C2×C17—81no—
205C2×C17#150711yescharacteristic
206C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
207C2×C17#213411yescharacteristic
208D17—84no—
209C25—341no—
210C22×D4—172no—
211C22×D4—342no—
212C22×D4—341no—
213C22×D4—681no—
214C22×D4—681no—
215C22×D4—684no—
216C22×D4C22⋊D1711yescharacteristic
217C23×C4C4×D1711yescharacteristic
218C23×C4D6811yescharacteristic
219C2×M16—22no—
220C2×M16—681no—
221#121—684no—
222#120—684no—
223C4×D4—41no—
224C4×D4—41no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4C22×D1711yescharacteristic
227C2×C22⋊C4—341no—
228C2×C22⋊C4—342no—
229C2×C22⋊C4—342no—
230C2×C22⋊C4—342no—
231C2×C22⋊C4C4×D1711yescharacteristic
232C2×C22⋊C4D6811yescharacteristic
233C2×C42C22⋊D1711yescharacteristic
234#100—24no—
235C22⋊C8—342no—
236C22⋊C8—681no—
237#97—41no—
238C4×C8—41no—
239#95—172no—
240C17#1177911yescharacteristic, complemented, Sylow
241C24—342no—
242C24—682no—
243C24—682no—
244C24—684no—
245C24—682no—
246C24—1362no—
247C24#6322711yescharacteristic
248C24#6322711yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—344no—
253C2×D4—682no—
254C2×D4—682no—
255C2×D4—688no—
256C2×D4—684no—
257C2×D4—684no—
258C2×D4—684no—
259C2×D4—684no—
260C2×D4—684no—
261C2×D4—681no—
262C2×D4—681no—
263C2×D4—1362no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—342no—
271C22×C4#6322711yescharacteristic
272C22×C4#6322711yescharacteristic
273C22×C4C2×C4×D1711yescharacteristic
274C22×C4C2×D6811yescharacteristic
275C22×C4C2×C22⋊D1711yescharacteristic
276M16—24no—
277M16—42no—
278M16—1362no—
279C2×C8—42no—
280C2×C8—682no—
281C4⋊C4—41no—
282C4⋊C4—682no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—684no—
286C22⋊C4—682no—
287C22⋊C4—684no—
288C22⋊C4—1362no—
289C42—21no—
290C42—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—344no—
301C23—682no—
302C23—682no—
303C23—682no—
304C23—688no—
305C23—682no—
306C23—682no—
307C23—682no—
308C23—1364no—
309C23—1368no—
310C23—1368no—
311C23—1364no—
312C23#1058747612yes—
313C23C2×#6322711yescharacteristic
314D4—44no—
315D4—684no—
316D4—684no—
317D4—684no—
318D4—684no—
319D4—1368no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—682no—
334C2×C4—682no—
335C2×C4—682no—
336C2×C4#1058747711yescharacteristic
337C2×C4#1058747711yescharacteristic
338C2×C4#1058763512yes—
339C8—44no—
340C8—1362no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—81no—
353C22—684no—
354C22—684no—
355C22—684no—
356C22—1362no—
357C22—1362no—
358C22—13616no—
359C22—1362no—
360C22—1362no—
361C22C2×#1058747711yescharacteristic
362C22#1176281311yescharacteristic
363C22C2×#1058747611yescharacteristic
364C4—22no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—1362no—
370C4#1176259712yes—
371C2—21no—
372C2—21no—
373C2—42no—
374C2—81no—
375C2—1364no—
376C2#1452729911yescharacteristic
377C2C2×#1176259711yescharacteristic
378C2#1453047511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC4×C17×#100C2
socleC22×C17C2×#99
prime radical#14532140C2
cosocleC2×C4⋊C4×C17C23
hypercenterC4×#100D17
hypocenterC17#11779
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. #14527122C2semidirect
3. #14526786C2semidirect
4. C4×C17×#100C2semidirect
14. #11762535C22semidirect
19. C17×#343C22semidirect
153. C4×#100D17semidirect
198. #343C2×D17semidirect
240. C17#11779semidirect

Maximal subgroups

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

subgroupindex
#145321402
#145271222
#145267862
C4×C17×#1002
#1177917