Finite Group Databasebeta

← #16945282

Subgroups — #16945282

Order 4352.

336 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 81 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#14539355C211yesmaximal, characteristic
2#14530584C211yesmaximal, characteristic, complemented
3#14528910C211yesmaximal, characteristic, complemented
4#14526796C211yesmaximal, characteristic, complemented
5#14526794C211yesmaximal, characteristic
6#14526792C211yesmaximal, characteristic
7C17×#1348C211yesmaximal, characteristic, complemented
8#11763345—21no—
9#11763210C411yescharacteristic, complemented
10#11763194C411yescharacteristic, complemented
11#11763165—21no—
12#11763045C2211yescharacteristic
13#11763040—21no—
14Q16×Dic17C411yescharacteristic, complemented
15#11762894—21no—
16#11762891C411yescharacteristic, complemented
17C2×C8×Dic17C2211yescharacteristic
18#11762678—21no—
19#11762627—21no—
20C22⋊#63214C2211yescharacteristic
21#11762584—21no—
22#11762572C2211yescharacteristic
23#11762571—21no—
24C17×#467C2211yescharacteristic
25C17×C22⋊C16C2211yescharacteristic
26C17×#340C2211yescharacteristic
27C2×Q16×C17C2×C411yescharacteristic
28C22×C8×C17C2311yescharacteristic
29C17×C22⋊Q8—21no—
30C2×C4⋊C4×C17—21no—
31#10587633—21no—
32Q8×Dic17—21no—
33C2×#63230—21no—
34#10587616—21no—
35C2×C4⋊Dic17—21no—
36C2×#63225—41no—
37C2×#63225—41no—
38C2×C4×Dic17D411yescharacteristic
39C22×#3622D411yescharacteristic
40C2×D136C2×C411yescharacteristic
41#10587584—41no—
42#10587579—21no—
43#10587578—41no—
44#10587576—21no—
45#10587572—41no—
46#10587567—41no—
47C2×C16×C17—21no—
48C17×#107C2×C411yescharacteristic
49C17×#103—21no—
50#10587506—21no—
51#10587492—21no—
52#10587489C2×C411yescharacteristic
53C8×Dic17C2×C411yescharacteristic
54C8×Dic17C2×C411yescharacteristic
55C8×Dic17D411yescharacteristic
56C8×Dic17D411yescharacteristic
57C2×#63214—21no—
58#10587480—21no—
59#10587479—21no—
60C2×Q8×C17—21no—
61C22×C4×C17—41no—
62C22×C4×C17C2×D411yescharacteristic
63C2×C22⋊D17—41no—
64C22×Dic17—41no—
65C22×Dic17D811yescharacteristic
66C22×Dic17D811yescharacteristic
67C2×D68—21no—
68C2×C4×D17—41no—
69Q16×C17—41no—
70C2×C8×C17—41no—
71C2×C8×C17C22×C411yescharacteristic
72C2×C8×C17C2×D411yescharacteristic
73C4⋊C4×C17—21no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78#63230—42no—
79#63227—41no—
80#63227—41no—
81#63227—41no—
82#63227—41no—
83C4⋊Dic17—21no—
84C4⋊Dic17—41no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87C4⋊Dic17—41no—
88#63225—41no—
89#63225—82no—
90#63225—82no—
91C4×Dic17—41no—
92C4×Dic17C22⋊C411yescharacteristic
93C4×Dic17C22⋊C411yescharacteristic
94C4×Dic17SD1611yescharacteristic
95C4×Dic17SD1611yescharacteristic
96C2×#3622—41no—
97C2×#3622C22⋊C411yescharacteristic
98C2×#3622C22⋊C411yescharacteristic
99D136—41no—
100C16×C17—41no—
101#63214—41no—
102#11860—171nomaximal, Sylow
103C23×C17C2×D811yescharacteristic
104C22×D17—41no—
105Q8×C17—41no—
106Q8×C17—41no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—81no—
112C2×C4×C17—81no—
113C2×C4×C17—81no—
114C2×C4×C17C2×C22⋊C411yescharacteristic
115C2×C4×C17C2×SD1611yescharacteristic
116C22⋊D17—81no—
117C22⋊D17—81no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—22no—
121C2×Dic17—41no—
122C2×Dic17—81no—
123C2×Dic17—81no—
124C2×Dic17—81no—
125C2×Dic17#10211yescharacteristic
126C2×Dic17#10211yescharacteristic
127C2×Dic17#10211yescharacteristic
128C2×Dic17#10211yescharacteristic
129D68—41no—
130D68—41no—
131C4×D17—82no—
132C8×C17—21no—
133C8×C17—21no—
134#3622—21no—
135#3622—21no—
136#3168—171no—
137#2163—171no—
138#1834—171no—
139#1348—171no—
140#1348D1711yescharacteristic, complemented
141#1347—171no—
142#1347—171no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17C2×#10211yescharacteristic
146C22×C17#50911yescharacteristic
147C22×C17#51011yescharacteristic
148C2×D17—81no—
149C2×D17—81no—
150C2×D17—81no—
151C4×C17—21no—
152C4×C17—81no—
153C4×C17—81no—
154C4×C17—161no—
155C4×C17#41011yescharacteristic
156C4×C17#42111yescharacteristic
157Dic17—21no—
158Dic17—21no—
159Dic17—21no—
160Dic17—81no—
161Dic17—161no—
162Dic17#36012yes—
163#529—341no—
164#499—171no—
165#495—171no—
166#488—341no—
167#467C2×D1711yescharacteristic
168#466—171no—
169#465—341no—
170C4×Q16—171no—
171C4×SD16—341no—
172C4×D8—171no—
173C2×C4×C8—171no—
174#384—341no—
175C4⋊C16—341no—
176C22⋊C16—171no—
177C22⋊C16C2×D1711yescharacteristic
178#346—341no—
179#340—171no—
180#340—341no—
181#340C2×D1711yescharacteristic
182C2×C17—21no—
183C2×C17—21no—
184C2×C17#151711yescharacteristic
185C2×C17#213911yescharacteristic
186C2×C17C2×#36011yescharacteristic
187D17—161no—
188C2×Q16C4×D1711yescharacteristic
189C2×SD16—341no—
190C2×D8—171no—
191C22×C8—171no—
192C22×C8C22×D1711yescharacteristic
193#126—681no—
194#125—341no—
195#124—341no—
196#123—681no—
197C22⋊Q8—21no—
198#121—341no—
199C4×Q8—341no—
200C4×D4—341no—
201C4×D4—681no—
202#117—681no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—681no—
206C2×C4⋊C4—681no—
207C2×C42—171no—
208C2×C16—21no—
209C2×C16—341no—
210#107—171no—
211#107C4×D1711yescharacteristic
212#106—341no—
213#103—21no—
214#103—341no—
215#102—341no—
216#102—341no—
217C4×C8—171no—
218C4×C8—171no—
219C4×C8—171no—
220C4×C8—171no—
221C17#1186011yescharacteristic, complemented, Sylow
222C2×Q8—21no—
223C2×D4—341no—
224C2×D4—681no—
225C22×C4—41no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—681no—
229C22×C4—681no—
230C22×C4D4×D1711yescharacteristic
231Q16—41no—
232SD16—682no—
233D8—681no—
234C2×C8—41no—
235C2×C8—171no—
236C2×C8—171no—
237C2×C8—681no—
238C2×C8C2×C4×D1711yescharacteristic
239C2×C8D4×D1711yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—341no—
245C4⋊C4—681no—
246C4⋊C4—681no—
247C4⋊C4—681no—
248C4⋊C4—681no—
249C4⋊C4—681no—
250C4⋊C4—1362no—
251C4⋊C4—1362no—
252C22⋊C4—41no—
253C22⋊C4—681no—
254C22⋊C4—681no—
255C22⋊C4—681no—
256C22⋊C4—681no—
257C42—171no—
258C42—171no—
259C42—171no—
260C42—171no—
261C42—681no—
262C16—41no—
263C16—681no—
264C23—681no—
265C23D8×D1711yescharacteristic
266Q8—41no—
267Q8—41no—
268D4—681no—
269D4—681no—
270D4—1361no—
271D4—1361no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—171no—
280C2×C4—171no—
281C2×C4—171no—
282C2×C4—171no—
283C2×C4—342no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—681no—
287C2×C4—1362no—
288C2×C4—1361no—
289C2×C4—1361no—
290C2×C4—1361no—
291C2×C4C22⋊C4×D1711yescharacteristic
292C2×C4SD16×D1711yescharacteristic
293C8—21no—
294C8—21no—
295C8—341no—
296C8—341no—
297C22—21no—
298C22—21no—
299C22—1361no—
300C22—1361no—
301C22—1361no—
302C22#102×D1711yescharacteristic
303C22#1176324611yescharacteristic
304C22#1176325111yescharacteristic
305C4—21no—
306C4—81no—
307C4—81no—
308C4—161no—
309C4—2721no—
310C4—172no—
311C4—341no—
312C4—341no—
313C4—341no—
314C4—1361no—
315C4#1176278111yescharacteristic
316C4#1176282411yescharacteristic
317C2—21no—
318C2—21no—
319C2—2721no—
320C2#1452734211yescharacteristic
321C2#1453049511yescharacteristic
322C2D17×#36011yescharacteristic
subgroupG/H
centerC22#102×D17
commutatorC2×C8×C17C22×C4
FrattiniC22×C8C22×D17
FittingC17×#1348C2
socleC22×C17C2×#102
prime radical#11763045C22
cosocleC22×C8×C17C23
hypercenter#1348D17
hypocenterC17#11860
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #14530584C2semidirect
3. #14528910C2semidirect
4. #14526796C2semidirect
7. C17×#1348C2semidirect
9. #11763210C4semidirect
10. #11763194C4semidirect
14. Q16×Dic17C4semidirect
16. #11762891C4semidirect
140. #1348D17semidirect
221. C17#11860semidirect

Maximal subgroups

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

subgroupindex
#145393552
#145305842
#145289102
#145267962
#145267942
#145267922
C17×#13482
#1186017