Finite Group Databasebeta

← #16945261

Subgroups — #16945261

Order 4352.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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#14530580C211yesmaximal, characteristic, complemented
3#14528913C211yesmaximal, 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#11762674—21no—
19#11762627—21no—
20C22⋊#63214C2211yescharacteristic
21C16×Dic17—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×C4×Dic17D411yescharacteristic
37C22×#3622D411yescharacteristic
38C2×D136C2×C411yescharacteristic
39#10587584—41no—
40#10587579—21no—
41#10587578—41no—
42#10587576—21no—
43#10587572—41no—
44#10587567—41no—
45C2×C16×C17—21no—
46C17×#107C2×C411yescharacteristic
47C17×#103—21no—
48#10587506—21no—
49#10587502—41no—
50#10587502—41no—
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—
89C4×Dic17—41no—
90C4×Dic17C22⋊C411yescharacteristic
91C4×Dic17C22⋊C411yescharacteristic
92C4×Dic17SD1611yescharacteristic
93C4×Dic17SD1611yescharacteristic
94C2×#3622—41no—
95C2×#3622C22⋊C411yescharacteristic
96C2×#3622C22⋊C411yescharacteristic
97D136—41no—
98C16×C17—41no—
99#63214—41no—
100#11855—171nomaximal, Sylow
101C23×C17C2×D811yescharacteristic
102C22×D17—41no—
103Q8×C17—41no—
104Q8×C17—41no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—81no—
110C2×C4×C17—81no—
111C2×C4×C17—81no—
112C2×C4×C17C2×C22⋊C411yescharacteristic
113C2×C4×C17C2×SD1611yescharacteristic
114C22⋊D17—81no—
115C22⋊D17—81no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—81no—
121C2×Dic17—81no—
122C2×Dic17—81no—
123C2×Dic17#10211yescharacteristic
124C2×Dic17#10211yescharacteristic
125C2×Dic17#10211yescharacteristic
126C2×Dic17#10211yescharacteristic
127D68—41no—
128D68—41no—
129C4×D17—82no—
130C8×C17—21no—
131C8×C17—21no—
132#3622—21no—
133#3622—21no—
134#3168—171no—
135#2162—171no—
136#1835—171no—
137#1348—171no—
138#1348D1711yescharacteristic, complemented
139#1347—171no—
140#1347—171no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17C2×#10211yescharacteristic
144C22×C17#50811yescharacteristic
145C22×C17#50811yescharacteristic
146C2×D17—81no—
147C2×D17—81no—
148C2×D17—81no—
149C4×C17—21no—
150C4×C17—81no—
151C4×C17—81no—
152C4×C17—161no—
153C4×C17#41011yescharacteristic
154C4×C17#42111yescharacteristic
155Dic17—21no—
156Dic17—21no—
157Dic17—21no—
158Dic17—21no—
159Dic17—81no—
160Dic17—161no—
161#529—341no—
162#499—171no—
163#495—171no—
164#488—341no—
165#467C2×D1711yescharacteristic
166#466—171no—
167#465—341no—
168C4×Q16—171no—
169C4×SD16—341no—
170C4×D8—171no—
171C2×C4×C8—171no—
172#382—341no—
173C4⋊C16—341no—
174C22⋊C16—171no—
175C22⋊C16C2×D1711yescharacteristic
176C4×C16—341no—
177#340—171no—
178#340—341no—
179#340C2×D1711yescharacteristic
180C2×C17—21no—
181C2×C17—21no—
182C2×C17#151711yescharacteristic
183C2×C17#213811yescharacteristic
184C2×C17#214511yescharacteristic
185D17—161no—
186C2×Q16C4×D1711yescharacteristic
187C2×SD16—341no—
188C2×D8—171no—
189C22×C8—171no—
190C22×C8C22×D1711yescharacteristic
191#126—681no—
192#125—341no—
193#124—341no—
194#123—681no—
195C22⋊Q8—21no—
196#121—341no—
197C4×Q8—341no—
198C4×D4—341no—
199C4×D4—681no—
200#117—681no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—341no—
203C2×C42—171no—
204C2×C16—21no—
205C2×C16—341no—
206#107—171no—
207#107C4×D1711yescharacteristic
208#106—341no—
209#103—21no—
210#103—341no—
211#102—341no—
212#102—341no—
213C4×C8—171no—
214C4×C8—171no—
215C4×C8—171no—
216C4×C8—171no—
217#95—681no—
218#95—681no—
219C17#1185511yescharacteristic, complemented, Sylow
220C2×Q8—21no—
221C2×D4—341no—
222C2×D4—681no—
223C22×C4—41no—
224C22×C4—171no—
225C22×C4—171no—
226C22×C4—681no—
227C22×C4—681no—
228C22×C4D4×D1711yescharacteristic
229Q16—41no—
230SD16—682no—
231D8—681no—
232C2×C8—41no—
233C2×C8—171no—
234C2×C8—171no—
235C2×C8—681no—
236C2×C8C2×C4×D1711yescharacteristic
237C2×C8D4×D1711yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—341no—
243C4⋊C4—681no—
244C4⋊C4—681no—
245C4⋊C4—681no—
246C4⋊C4—681no—
247C4⋊C4—681no—
248C22⋊C4—41no—
249C22⋊C4—681no—
250C22⋊C4—681no—
251C22⋊C4—681no—
252C22⋊C4—681no—
253C42—171no—
254C42—171no—
255C42—171no—
256C42—171no—
257C42—681no—
258C16—41no—
259C16—681no—
260C23—681no—
261C23D8×D1711yescharacteristic
262Q8—41no—
263Q8—41no—
264D4—681no—
265D4—681no—
266D4—1361no—
267D4—1361no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—171no—
276C2×C4—171no—
277C2×C4—171no—
278C2×C4—171no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—1362no—
284C2×C4—1361no—
285C2×C4—1361no—
286C2×C4—1361no—
287C2×C4C22⋊C4×D1711yescharacteristic
288C2×C4SD16×D1711yescharacteristic
289C8—21no—
290C8—21no—
291C8—341no—
292C8—341no—
293C22—21no—
294C22—21no—
295C22—1361no—
296C22—1361no—
297C22—1361no—
298C22#102×D1711yescharacteristic
299C22#1176324011yescharacteristic
300C22#1176324211yescharacteristic
301C4—21no—
302C4—81no—
303C4—81no—
304C4—161no—
305C4—2721no—
306C4—341no—
307C4—341no—
308C4—341no—
309C4—341no—
310C4—1361no—
311C4#1176278111yescharacteristic
312C4#1176282411yescharacteristic
313C2—21no—
314C2—21no—
315C2—2721no—
316C2#1452734211yescharacteristic
317C2#1453049011yescharacteristic
318C2#1453051811yescharacteristic
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#11855
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. #14530580C2semidirect
3. #14528913C2semidirect
4. #14526796C2semidirect
7. C17×#1348C2semidirect
9. #11763210C4semidirect
10. #11763194C4semidirect
14. Q16×Dic17C4semidirect
16. #11762891C4semidirect
138. #1348D17semidirect
219. C17#11855semidirect

Maximal subgroups

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

subgroupindex
#145393552
#145305802
#145289132
#145267962
#145267942
#145267922
C17×#13482
#1185517