Finite Group Databasebeta

← #16927633

Subgroups — #16927633

Order 4352.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587489C211yesmaximal, characteristic, complemented
2#14527547C211yesmaximal, characteristic, complemented
3#14527309C212yesmaximal
4#14526641C212yesmaximal, complemented
5C17×#2966C211yesmaximal, characteristic
6#11762866C2212yes—
7C2×#10587489—21no—
8C4×C4⋊Dic17C2212yes—
9#11762572—21no—
10#11762562—21no—
11#11762560—21no—
12#11762546—22no—
13#11762543—22no—
14C17×#529C2211yescharacteristic
15C17×#446C411yescharacteristic, complemented
16C8×Q8×C17C411yescharacteristic, complemented
17C17×#436C411yescharacteristic, complemented
18C17×#435—21no—
19C8×D4×C17—21no—
20C8×D4×C17C411yescharacteristic, complemented
21C17×#433—21no—
22C17×#432C2211yescharacteristic
23C17×#405—21no—
24C2×C4×C8×C17C2211yescharacteristic
25C2×M16×C17—21no—
26C22×C8×C17—21no—
27C22×C8×C17D411yescharacteristic
28C22×C8×C17D411yescharacteristic
29C17×#126—21no—
30C17×#125Q811yescharacteristic
31C17×#124Q811yescharacteristic
32C17×#123—21no—
33C17×C22⋊Q8D411yescharacteristic
34C17×#121D411yescharacteristic
35C4×Q8×C17C2×C411yescharacteristic
36C4×D4×C17—21no—
37C4×D4×C17C2×C411yescharacteristic
38C17×#117—21no—
39C2×C42×C17C2311yescharacteristic
40C2×C4⋊Dic17—22no—
41C2×C4×Dic17—22no—
42C17×C4⋊C8—21no—
43C17×C4⋊C8—21no—
44C17×C4⋊C8C2×C411yescharacteristic
45C17×C4⋊C8C2×C411yescharacteristic
46C17×C22⋊C8—21no—
47C17×C22⋊C8—21no—
48C17×C22⋊C8—21no—
49C17×#97—21no—
50C17×#97—21no—
51C4×C8×C17—21no—
52C4×C8×C17C2×C411yescharacteristic
53C4×C8×C17C2×C411yescharacteristic
54C4×C8×C17D411yescharacteristic
55C4×C8×C17D411yescharacteristic
56#10587502—22no—
57C22⋊#3622—22no—
58#10587489—24no—
59C4⋊#3622—22no—
60#10587474—22no—
61C2×Q8×C17C4⋊C411yescharacteristic
62C2×D4×C17—21no—
63C2×D4×C17C4⋊C411yescharacteristic
64C22×C4×C17—21no—
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17#4111yescharacteristic
68C22×Dic17—42no—
69M16×C17—42no—
70C2×C8×C17—22no—
71C2×C8×C17—21no—
72C2×C8×C17—21no—
73C2×C8×C17—41no—
74C2×C8×C17—42no—
75C2×C8×C17C22⋊C411yescharacteristic
76C2×C8×C17C22⋊C411yescharacteristic
77C2×C8×C17C22⋊C411yescharacteristic
78C2×C8×C17C22⋊C411yescharacteristic
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17C4⋊C411yescharacteristic
84C4⋊C4×C17C4⋊C411yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C22⋊C4×C17—21no—
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C42×C17—21no—
91C42×C17C22×C411yescharacteristic
92C42×C17C2×D411yescharacteristic
93C42×C17C2×Q811yescharacteristic
94C42×C17#4111yescharacteristic
95C4⋊Dic17—24no—
96C4⋊Dic17—42no—
97C4×Dic17—82no—
98C2×#3622—42no—
99#8695—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17#12111yescharacteristic
102Q8×C17—41no—
103D4×C17—41no—
104D4×C17—41no—
105D4×C17—41no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17C2×C22⋊C411yescharacteristic
117C2×C4×C17C2×C4⋊C411yescharacteristic
118C2×C4×C17#11711yescharacteristic
119C2×C4×C17#12111yescharacteristic
120C2×C4×C17C22⋊Q811yescharacteristic
121C2×C4×C17C22⋊Q811yescharacteristic
122C2×Dic17—44no—
123C2×Dic17—82no—
124C2×Dic17—82no—
125C8×C17—21no—
126C8×C17—21no—
127C8×C17—21no—
128C8×C17—41no—
129C8×C17—41no—
130C8×C17#10412yes—
131#3622—82no—
132#2966D1711yescharacteristic
133C4×#107—171no—
134#1568—171no—
135#1509—172no—
136#1285—172no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17#38011yescharacteristic
143C22×C17#47211yescharacteristic
144C22×C17#47311yescharacteristic
145C4×C17—21no—
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—41no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17C2×#10411yescharacteristic
154C4×C17#42111yescharacteristic
155Dic17—84no—
156#529C2×D1711yescharacteristic
157#446Dic1711yescharacteristic, complemented
158C8×Q8Dic1711yescharacteristic, complemented
159#436Dic1711yescharacteristic, complemented
160#435—21no—
161C8×D4—21no—
162C8×D4Dic1711yescharacteristic, complemented
163#433—21no—
164#432—172no—
165#432C2×D1711yescharacteristic
166C2×#107—341no—
167#405—21no—
168C2×C4×C8C2×D1711yescharacteristic
169C4×C4⋊C4—172no—
170#340—341no—
171#335—341no—
172#334—341no—
173#328—342no—
174#327—342no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17—81no—
178C2×C17#180611yescharacteristic
179C2×C17#181211yescharacteristic
180C2×C17#181611yescharacteristic
181C2×M16—21no—
182C22×C8—21no—
183C22×C8C22⋊D1711yescharacteristic
184C22×C8C22⋊D1711yescharacteristic
185#126—21no—
186#125Dic3411yescharacteristic
187#124Dic3411yescharacteristic
188#123—21no—
189C22⋊Q8D6811yescharacteristic
190#121D6811yescharacteristic
191C4×Q8C2×Dic1711yescharacteristic
192C4×D4—21no—
193C4×D4C2×Dic1711yescharacteristic
194#117—21no—
195C2×C4⋊C4—342no—
196C2×C42—342no—
197C2×C42C22×D1711yescharacteristic
198#107—344no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—342no—
202C4⋊C8C2×Dic1711yescharacteristic
203C4⋊C8C2×Dic1711yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—342no—
208#97—21no—
209#97—21no—
210#97—342no—
211C4×C8—21no—
212C4×C8C2×Dic1711yescharacteristic
213C4×C8C2×Dic1711yescharacteristic
214C4×C8C22⋊D1711yescharacteristic
215C4×C8C22⋊D1711yescharacteristic
216#95—342no—
217C17#869511yescharacteristic, complemented, Sylow
218C2×Q8C4⋊Dic1711yescharacteristic
219C2×D4—21no—
220C2×D4C4⋊Dic1711yescharacteristic
221C22×C4—21no—
222C22×C4—682no—
223C22×C4C2×D6811yescharacteristic
224C22×C4#6325211yescharacteristic
225C22×C4C2×C22⋊D1711yescharacteristic
226M16—42no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—682no—
233C2×C8C22⋊Dic1711yescharacteristic
234C2×C8C22⋊Dic1711yescharacteristic
235C2×C8C22⋊Dic1711yescharacteristic
236C2×C8C22⋊Dic1711yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—344no—
242C4⋊C4—682no—
243C4⋊C4C4⋊Dic1711yescharacteristic
244C4⋊C4C4⋊Dic1711yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—21no—
247C22⋊C4—21no—
248C22⋊C4—21no—
249C22⋊C4—21no—
250C42—21no—
251C42—1362no—
252C42C2×Dic3411yescharacteristic
253C42#6325211yescharacteristic
254C42C22×Dic1711yescharacteristic
255C42C2×C22⋊D1711yescharacteristic
256C23—21no—
257C23#1058761611yescharacteristic
258Q8—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—684no—
273C2×C4—1362no—
274C2×C4—1362no—
275C2×C4#1058761011yescharacteristic
276C2×C4#1058761011yescharacteristic
277C2×C4C2×C4⋊Dic1711yescharacteristic
278C2×C4#1058761211yescharacteristic
279C2×C4#1058761611yescharacteristic
280C2×C4C2×C22⋊Dic1711yescharacteristic
281C8—21no—
282C8—21no—
283C8—21no—
284C8—41no—
285C8—41no—
286C8—1362no—
287C8#1058750812yes—
288C22—21no—
289C22—21no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22#1176266911yescharacteristic
294C22#1176307911yescharacteristic
295C22#1176308411yescharacteristic
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—1364no—
305C4C2×#1058750811yescharacteristic
306C4#1176282611yescharacteristic
307C2—21no—
308C2—21no—
309C2—81no—
310C2#1452880411yescharacteristic
311C2#1452883011yescharacteristic
312C2#1452884811yescharacteristic
subgroupG/H
centerC22#11762669
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2966C2
socleC22×C17#380
prime radicalC17×#121D4
cosocleC2×C42×C17C23
hypercenter#2966D17
hypocenterC17#8695
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
1. C4×#10587489C2semidirect
2. #14527547C2semidirect
4. #14526641C2semidirect
15. C17×#446C4semidirect
16. C8×Q8×C17C4semidirect
17. C17×#436C4semidirect
20. C8×D4×C17C4semidirect
157. #446Dic17semidirect
158. C8×Q8Dic17semidirect
159. #436Dic17semidirect
162. C8×D4Dic17semidirect
217. C17#8695semidirect

Maximal subgroups

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

subgroupindex
C4×#105874892
#145275472
#145273092
#145266412
C17×#29662
#869517