Finite Group Databasebeta

← #16945197

Subgroups — #16945197

Order 4352.

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

normal subgroups only

This group has 390 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#14528861C211yesmaximal, characteristic
2#14527098C211yesmaximal, characteristic
3#14526805C212yesmaximal
4#14526784C212yesmaximal
5C17×#2599C211yesmaximal, characteristic
6#11762862—21no—
7#11762672—21no—
8#11762576—21no—
9#11762576C2212yes—
10#11762535—21no—
11#11762533C2212yes—
12C2×C4×Q8×C17C411yescharacteristic, complemented
13C2×C17×#117C2211yescharacteristic
14C17×#398C411yescharacteristic, complemented
15C17×#397C2211yescharacteristic
16C17×#395C411yescharacteristic, complemented
17C17×#393C2211yescharacteristic
18C17×#388—21no—
19C17×#387—21no—
20C17×#387—21no—
21C17×#385—21no—
22C17×#383C411yescharacteristic, complemented
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C4×Q8×C17—41no—
26C4×Q8×C17—41no—
27C17×#117—41no—
28C17×#117—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17C2311yescharacteristic
39C2×C42×C17—21no—
40C2×C42×C17C2×C411yescharacteristic
41C2×C42×C17D411yescharacteristic
42C2×C22⋊Dic17—22no—
43C17×#95—21no—
44C17×#95—21no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51#10587502—41no—
52#10587502—41no—
53C22⋊#3622—22no—
54C4⋊#3622—41no—
55C4×#3622—41no—
56C24×C17C2×D411yescharacteristic
57C2×Q8×C17—42no—
58C2×Q8×C17—41no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—41no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—42no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—42no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17—21no—
89C42×C17—42no—
90C22⋊Dic17—42no—
91C22⋊Dic17—42no—
92C2×#3622—42no—
93#11842—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17C2×C22⋊C411yescharacteristic
99C23×C17#13512yes—
100Q8×C17—42no—
101Q8×C17—42no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17—21no—
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—42no—
113C2×C4×C17—42no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17#9911yescharacteristic
130C2×C4×C17#9911yescharacteristic
131C2×Dic17—82no—
132C2×Dic17—82no—
133C2×Dic17—82no—
134#3622—82no—
135#2599D1711yescharacteristic
136#1820—171no—
137#1458—171no—
138#1351—172no—
139#1344—172no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—22no—
143C22×C17—42no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—81no—
147C22×C17C2×#9911yescharacteristic
148C22×C17#41611yescharacteristic
149C22×C17#42111yescharacteristic
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—41no—
153C4×C17—42no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—81no—
159C4×C17—81no—
160C4×C17—81no—
161Dic17—162no—
162C2×C4×Q8Dic1711yescharacteristic, complemented
163C2×#117C2×D1711yescharacteristic
164#431—341no—
165#398Dic1711yescharacteristic, complemented
166#397C2×D1711yescharacteristic
167#395Dic1711yescharacteristic, complemented
168#393C2×D1711yescharacteristic
169#388—21no—
170#387—21no—
171#387—21no—
172#385—21no—
173#383Dic1711yescharacteristic, complemented
174#381—341no—
175#342—172no—
176#342—341no—
177#324—341no—
178#323—172no—
179C2×C17—21no—
180C2×C17—21no—
181C2×C17—42no—
182C2×C17#151611yescharacteristic
183C2×C17#212011yescharacteristic
184C2×C17#212911yescharacteristic
185C22×Q8C2×Dic1711yescharacteristic
186C23×C4C22⋊D1711yescharacteristic
187C4×Q8—41no—
188C4×Q8—41no—
189#117—41no—
190#117—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×Dic1711yescharacteristic
195C2×C4⋊C4C2×Dic1711yescharacteristic
196C2×C4⋊C4C22⋊D1711yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4C22⋊D1711yescharacteristic
201C2×C22⋊C4C22×D1711yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic1711yescharacteristic
204C2×C42C22⋊D1711yescharacteristic
205C4⋊C8—681no—
206C22⋊C8—342no—
207C4×C8—681no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—681no—
215#95—681no—
216#95C2×Dic1711yescharacteristic
217#95C2×Dic1711yescharacteristic
218C17#1184211yescharacteristic, complemented, Sylow
219C24C2×C22⋊D1711yescharacteristic
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—682no—
231C22×C4C22⋊Dic1711yescharacteristic
232C22×C4C22⋊Dic1711yescharacteristic
233C22×C4C22⋊Dic1711yescharacteristic
234C22×C4C22⋊Dic1711yescharacteristic
235C22×C4C22×Dic1711yescharacteristic
236C22×C4C2×C22⋊D1711yescharacteristic
237C2×C8—682no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—682no—
250C22⋊C4—682no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—42no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23#1058763612yes—
261C23C2×C22⋊Dic1711yescharacteristic
262Q8—42no—
263Q8—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—1362no—
292C2×C4—1362no—
293C2×C4—1362no—
294C2×C4#1058750511yescharacteristic
295C2×C4#1058750511yescharacteristic
296C8—1362no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1058750511yescharacteristic
305C22#1176280511yescharacteristic
306C22#1176282611yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—2722no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#1452734011yescharacteristic
323C2#1453042511yescharacteristic
324C2#1453045911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2599C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2599D17
hypocenterC17#11842
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C17C4semidirect
14. C17×#398C4semidirect
16. C17×#395C4semidirect
22. C17×#383C4semidirect
162. C2×C4×Q8Dic17semidirect
165. #398Dic17semidirect
167. #395Dic17semidirect
173. #383Dic17semidirect
218. C17#11842semidirect

Maximal subgroups

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

subgroupindex
#145288612
#145270982
#145268052
#145267842
C17×#25992
#1184217