Finite Group Databasebeta

← #16922749

Subgroups — #16922749

Order 4352.

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

normal subgroups only

This group has 550 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#14537115C211yesmaximal, characteristic
2C4×C22⋊#3622C211yesmaximal, characteristic, complemented
3#14526606C212yesmaximal, complemented
4#14526604C212yesmaximal, complemented
5C17×#1749C211yesmaximal, characteristic, complemented
6C2×#10587615—21no—
7C2×#10587616C411yescharacteristic
8C2×C22⋊#3622—21no—
9C2×C4×#3622—21no—
10#11762711C411yescharacteristic
11#11762711C2211yescharacteristic
12#11762707C411yescharacteristic
13#11762707C2211yescharacteristic
14#11762700—21no—
15#11762700—21no—
16#11762688—21no—
17#11762669C411yescharacteristic
18#11762564C2212yes—
19C22×C42×C17C2211yescharacteristic
20C2×C17×C22⋊C8—21no—
21C2×C17×#97—21no—
22C17×#336C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×C22⋊D17—21no—
27C22×D68C2×C411yescharacteristic
28C22×C8×C17—22no—
29C2×C42×C17—42no—
30C2×C42×C17C2×C411yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C42×C17C2311yescharacteristic
34C2×C22⋊Dic17—21no—
35#10587616—42no—
36#10587615—44no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×C4⋊Dic17C2×C411yescharacteristic
44C2×C4⋊Dic17C2×C411yescharacteristic
45C2×#63225—21no—
46C2×#63225—21no—
47C22×#3622—22no—
48C17×C22⋊C8—24no—
49C17×#97—42no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C22⋊#3622—24no—
54C4×#3622—42no—
55C24×C17C22⋊C411yescharacteristic
56C23×D17—21no—
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17—22no—
63C22×C4×C17—41no—
64C22×C4×C17—41no—
65C22×C4×C17—41no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C2×C22⋊D17—42no—
73C2×C22⋊D17—42no—
74C2×C22⋊D17—42no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C2×D68—22no—
79C2×D68—41no—
80C2×D68—41no—
81C2×C8×C17—24no—
82C2×C8×C17—42no—
83C2×C8×C17—42no—
84C42×C17—22no—
85C42×C17—22no—
86C42×C17—41no—
87C42×C17—41no—
88C22⋊Dic17—42no—
89#63227—42no—
90#63227—42no—
91#63227—42no—
92#63227—42no—
93#63227—42no—
94C4⋊Dic17—22no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97#63225—42no—
98#63225—42no—
99C2×#3622—24no—
100C2×#3622—42no—
101C2×#3622—42no—
102#7674—171nomaximal, Sylow
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17#10412yes—
110C23×C17C2×C22⋊C411yescharacteristic
111C22×D17—41no—
112C22×D17—41no—
113C22×D17—41no—
114C22×D17—41no—
115C22×D17—41no—
116C22×D17—41no—
117C22×D17—41no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—44no—
132C2×C4×C17—44no—
133C2×C4×C17—44no—
134C2×C4×C17#9912yes—
135C2×C4×C17#10412yes—
136C2×C4×C17#13112yes—
137C2×C4×C17#13512yes—
138C2×C4×C17AGL(1,ℤ/8)12yes—
139C22⋊D17—84no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149D68—82no—
150C8×C17—44no—
151#3622—44no—
152#2854—171no—
153#1749D1711yescharacteristic, complemented
154C4×C22⋊C8—171no—
155#1271—172no—
156#1270—172no—
157C22×C17—22no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17#40811yescharacteristic
165C22×C17C2×#9911yescharacteristic
166C22×C17#41311yescharacteristic
167C22×C17#41611yescharacteristic
168C22×C17#41811yescharacteristic
169C22×C17C2×#10411yescharacteristic
170C22×C17C2×#10411yescharacteristic
171C2×D17—81no—
172C2×D17—81no—
173C2×D17—81no—
174C2×D17—81no—
175C2×D17—81no—
176C2×D17—81no—
177C2×D17—81no—
178C4×C17—22no—
179C4×C17—22no—
180C4×C17—22no—
181C4×C17—22no—
182C4×C17—22no—
183C4×C17—41no—
184Dic17—81no—
185Dic17—81no—
186Dic17—81no—
187C2×#123—341no—
188C2×#121—171no—
189C22×C42C2×D1711yescharacteristic
190C2×C22⋊C8—21no—
191C2×C22⋊C8—341no—
192C2×#97—21no—
193C2×C4×C8—341no—
194#390—171no—
195#390—171no—
196#389—171no—
197#389—171no—
198#388—341no—
199#388—341no—
200#386—341no—
201#380—171no—
202#336—172no—
203#336C2×D1712yes—
204C2×C17—42no—
205C2×C17#146811yescharacteristic
206C2×C17#148411yescharacteristic
207C2×C17#149411yescharacteristic
208C2×C17#150111yescharacteristic
209C2×C17#150711yescharacteristic
210C2×C17#152411yescharacteristic
211C2×C17#155311yescharacteristic
212D17—161no—
213C22×D4—171no—
214C22×D4—341no—
215C23×C4C4×D1711yescharacteristic
216C23×C4D6811yescharacteristic
217C23×C4C22⋊D1711yescharacteristic
218C22×C8—22no—
219C22×C8—342no—
220#123—684no—
221#121—682no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C22⋊C4—341no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4—341no—
230C2×C22⋊C4—341no—
231C2×C22⋊C4—341no—
232C2×C22⋊C4—341no—
233C2×C42—42no—
234C2×C42C4×D1711yescharacteristic
235C2×C42D6811yescharacteristic
236C2×C42C22⋊D1711yescharacteristic
237C2×C42C22×D1711yescharacteristic
238C22⋊C8—24no—
239C22⋊C8—344no—
240#97—42no—
241C4×C8—682no—
242#95—341no—
243#95—341no—
244#95—341no—
245C17#767411yescharacteristic, complemented, Sylow
246C24—341no—
247C24#6322711yescharacteristic
248C2×D4—342no—
249C2×D4—681no—
250C2×D4—681no—
251C2×D4—682no—
252C2×D4—682no—
253C2×D4—682no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4#6322711yescharacteristic
267C22×C4#6322711yescharacteristic
268C22×C4#6322711yescharacteristic
269C22×C4C2×C4×D1711yescharacteristic
270C22×C4C2×D6811yescharacteristic
271C22×C4C2×C22⋊D1711yescharacteristic
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—344no—
276C2×C8—682no—
277C2×C8—682no—
278C4⋊C4—342no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C4⋊C4—682no—
282C4⋊C4—682no—
283C22⋊C4—682no—
284C22⋊C4—682no—
285C22⋊C4—682no—
286C22⋊C4—682no—
287C22⋊C4—682no—
288C22⋊C4—682no—
289C42—22no—
290C42—22no—
291C42—41no—
292C42—41no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—681no—
300C23—681no—
301C23—681no—
302C23—681no—
303C23—681no—
304C23—681no—
305C23—681no—
306C23#1058749612yes—
307C23C2×#6322711yescharacteristic
308D4—1362no—
309D4—1364no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4#1058747612yes—
336C2×C4#1058747712yes—
337C2×C4#1058759312yes—
338C2×C4#1058759412yes—
339C2×C4#1058763612yes—
340C8—44no—
341C8—684no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—1361no—
350C22—1361no—
351C22—1361no—
352C22—1361no—
353C22—1361no—
354C22—1361no—
355C22—1361no—
356C22#1176277711yescharacteristic
357C22C2×#1058747711yescharacteristic
358C22#1176279111yescharacteristic
359C22#1176280311yescharacteristic
360C22#1176281511yescharacteristic
361C22C2×#1058749611yescharacteristic
362C22C2×#1058747611yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—41no—
369C4—1361no—
370C4—1361no—
371C4—1361no—
372C2—42no—
373C2—2721no—
374C2#1452713311yescharacteristic
375C2#1452720211yescharacteristic
376C2#1452724211yescharacteristic
377C2#1452727311yescharacteristic
378C2#1452730111yescharacteristic
379C2#1452737011yescharacteristic
380C2#1452749011yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1749C2
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587616C4
cosocleC2×C42×C17C23
hypercenter#1749D17
hypocenterC17#7674
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊#3622C2semidirect
3. #14526606C2semidirect
4. #14526604C2semidirect
5. C17×#1749C2semidirect
153. #1749D17semidirect
245. C17#7674semidirect

Maximal subgroups

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

subgroupindex
#145371152
C4×C22⋊#36222
#145266062
#145266042
C17×#17492
#767417