Finite Group Databasebeta

← #16943922

Subgroups — #16943922

Order 4352.

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

normal subgroups only

This group has 446 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#14533092C211yesmaximal, characteristic
2#14527125C211yesmaximal, characteristic
3#14526713C214yesmaximal, complemented
4C17×#1459C211yesmaximal, characteristic
5C2×#10587621C412yes—
6C22×C4⋊Dic17C411yescharacteristic
7#11762866—21no—
8#11762739—21no—
9#11762734—21no—
10#11762707—21no—
11#11762670C2212yes—
12#11762669—21no—
13#11762669C411yescharacteristic
14#11762533—21no—
15#11762533C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×C22⋊C8—21no—
18C17×#324—21no—
19C17×#323C2212yes—
20C22×D4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17C2×C412yes—
24C22×C8×C17—41no—
25C4×D4×C17—41no—
26C4×D4×C17—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17C2311yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17C2×C412yes—
33#10587621—44no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—41no—
38C2×C4⋊Dic17—41no—
39C2×#63225—22no—
40C17×C22⋊C8—22no—
41C17×C22⋊C8—41no—
42#10587502—21no—
43#10587502—22no—
44#10587502—22no—
45C22⋊#3622—22no—
46C4⋊#3622—41no—
47#10587474—41no—
48C24×C17C22⋊C411yescharacteristic
49C24×C17C22×C411yescharacteristic
50C2×D4×C17—22no—
51C2×D4×C17—22no—
52C2×D4×C17—41no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—42no—
58C22×C4×C17—42no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×Dic17—22no—
65C22×Dic17—22no—
66C22×Dic17—24no—
67C22×Dic17—42no—
68C22×Dic17—42no—
69C2×C8×C17—42no—
70C2×C8×C17—82no—
71C4⋊C4×C17—41no—
72C22⋊C4×C17—41no—
73C22⋊C4×C17—41no—
74C42×C17—21no—
75C42×C17—21no—
76C22⋊Dic17—44no—
77C22⋊Dic17—42no—
78C4⋊Dic17—42no—
79C4⋊Dic17—42no—
80C4⋊Dic17—41no—
81C4⋊Dic17—41no—
82#63225—44no—
83C2×#3622—42no—
84#11558—171nomaximal, Sylow
85C23×C17—21no—
86C23×C17—21no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—41no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17#10012yes—
94C23×C17C2×C22⋊C411yescharacteristic
95C23×C17#13112yes—
96D4×C17—44no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—81no—
111C2×C4×C17#9911yescharacteristic
112C2×C4×C17#9911yescharacteristic
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—84no—
120C8×C17—82no—
121#3622—82no—
122#2486—171no—
123#1466—171no—
124#1459D1711yescharacteristic
125#1317—174no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17—22no—
129C22×C17—22no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17—41no—
136C22×C17—41no—
137C22×C17—81no—
138C22×C17#40811yescharacteristic
139C22×C17C2×#9911yescharacteristic
140C22×C17C2×#10011yescharacteristic
141C4×C17—21no—
142C4×C17—41no—
143C4×C17—41no—
144C4×C17—41no—
145C4×C17—41no—
146C4×C17#35212yes—
147Dic17—82no—
148Dic17—82no—
149C2×#123—172no—
150C2×C4×D4C2×D1711yescharacteristic
151C22×C4⋊C4—171no—
152#432—341no—
153C2×C22⋊C8—21no—
154#398—341no—
155#397—341no—
156#389—341no—
157#381—172no—
158#380—171no—
159#380—341no—
160#324—21no—
161#323—172no—
162#323—341no—
163#323C2×D1712yes—
164C2×C17—21no—
165C2×C17—21no—
166C2×C17—42no—
167C2×C17—42no—
168C2×C17#147111yescharacteristic
169C2×C17C2×#35211yescharacteristic
170C2×C17#212311yescharacteristic
171C22×D4C4×D1711yescharacteristic
172C23×C4—172no—
173C23×C4C4×D1711yescharacteristic
174C23×C4C22⋊D1711yescharacteristic
175C22×C8—41no—
176#123—684no—
177C4×D4—41no—
178C4×D4—41no—
179C2×C4⋊C4—341no—
180C2×C4⋊C4—342no—
181C2×C4⋊C4—341no—
182C2×C4⋊C4—341no—
183C2×C4⋊C4—681no—
184C2×C4⋊C4—681no—
185C2×C4⋊C4C22⋊D1711yescharacteristic
186C2×C22⋊C4—172no—
187C2×C22⋊C4—342no—
188C2×C22⋊C4D6811yescharacteristic
189C2×C22⋊C4C22×D1711yescharacteristic
190C2×C42D6811yescharacteristic
191C4⋊C8—681no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—342no—
195#97—681no—
196#95—342no—
197#95—342no—
198#95—341no—
199C17#1155811yescharacteristic, complemented, Sylow
200C24#6322711yescharacteristic
201C24C2×C4×D1711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—342no—
212C22×C4—342no—
213C22×C4—344no—
214C22×C4—682no—
215C22×C4—682no—
216C22×C4#6322711yescharacteristic
217C22×C4#6322711yescharacteristic
218C22×C4#6322711yescharacteristic
219C22×C4C2×D6811yescharacteristic
220C22×C4C2×C22⋊D1711yescharacteristic
221C2×C8—42no—
222C2×C8—82no—
223C2×C8—682no—
224C4⋊C4—41no—
225C4⋊C4—682no—
226C4⋊C4—684no—
227C4⋊C4—682no—
228C4⋊C4—681no—
229C4⋊C4—681no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—684no—
233C22⋊C4—682no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#1058749412yes—
245C23#1058758712yes—
246C23C2×#6322711yescharacteristic
247D4—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—682no—
263C2×C4—682no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4—682no—
267C2×C4—682no—
268C2×C4—1364no—
269C2×C4#1058747711yescharacteristic
270C2×C4#1058747711yescharacteristic
271C8—82no—
272C8—1362no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#1176277511yescharacteristic
286C22C2×#1058747711yescharacteristic
287C22C2×#1058749411yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—1362no—
294C4—1362no—
295C4#1176259812yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—42no—
300C2#1452714311yescharacteristic
301C2C2×#1176259811yescharacteristic
302C2#1453043911yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1459C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C22⋊C4×C17C23
hypercenter#1459D17
hypocenterC17#11558
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #14526713C2semidirect
199. C17#11558semidirect

Maximal subgroups

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

subgroupindex
#145330922
#145271252
#145267132
C17×#14592
#1155817