Finite Group Databasebeta

← #16932056

Subgroups — #16932056

Order 4352.

700 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 107 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#14532424C211yesmaximal, characteristic
2#14531370C211yesmaximal, characteristic
3#14527318C212yesmaximal
4#14526638C212yesmaximal
5C17×#1784C211yesmaximal, characteristic
6C2×#10587630C412yescomplemented
7C2×#10587610C2211yescharacteristic
8C2×#10587612—21no—
9#11762728—22no—
10#11762716—24no—
11#11762716C412yescomplemented
12#11762682—21no—
13#11762682—21no—
14#11762672—21no—
15#11762672C2212yes—
16C4×C22⋊Dic17C2211yescharacteristic
17#11762547—24no—
18C23×Q8×C17C2211yescharacteristic
19C2×C17×#101—22no—
20C17×#407C2212yes—
21C22×Q8×C17—22no—
22C22×Q8×C17—22no—
23C22×Q8×C17—44no—
24C22×Q8×C17C2×C412yes—
25C22×Q8×C17D412yes—
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C23×C4×C17C2311yescharacteristic
29C22×Dic34C2×C411yescharacteristic
30C2×M16×C17—22no—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—21no—
34#10587630—44no—
35#10587612—44no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38#10587610—22no—
39#10587610—22no—
40C2×#63225—22no—
41C2×#63225—22no—
42C2×#63225—21no—
43C2×C4×Dic17—22no—
44C2×C4×Dic17C2×C412yes—
45C17×#101—44no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—22no—
48#10587502—24no—
49#10587502—22no—
50#10587502—22no—
51#10587502—22no—
52#10587502—21no—
53#10587502D414yes—
54C24×C17C2×D411yescharacteristic
55C2×Q8×C17—24no—
56C2×Q8×C17—24no—
57C2×Q8×C17—24no—
58C2×Q8×C17—24no—
59C2×Q8×C17—42no—
60C2×Q8×C17—42no—
61C2×Q8×C17—48no—
62C2×Q8×C17—44no—
63C2×Q8×C17—44no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—24no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—42no—
70C22×C4×C17—42no—
71C22×C4×C17C22⋊C412yes—
72C22×C4×C17C22⋊C412yes—
73C22×C4×C17C22×C411yescharacteristic
74C22×C4×C17C2×D412yes—
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17#4112yes—
77C22×Dic17—22no—
78C22×Dic17—21no—
79C22×Dic17—21no—
80C22×Dic17—24no—
81C2×Dic34—22no—
82C2×Dic34—42no—
83M16×C17—44no—
84C2×C8×C17—42no—
85C22⋊Dic17—24no—
86C22⋊Dic17—44no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89C4⋊Dic17—22no—
90C4⋊Dic17—44no—
91#63225—44no—
92#63225—44no—
93#63225—42no—
94C4×Dic17—44no—
95C4×Dic17—42no—
96#9462—171nomaximal, Sylow
97C23×C17—22no—
98C23×C17—22no—
99C23×C17—41no—
100C23×C17C2×C22⋊C411yescharacteristic
101C23×C17#12011yescharacteristic
102C23×C17#12311yescharacteristic
103Q8×C17—48no—
104Q8×C17—48no—
105C2×C4×C17—24no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17#9914yes—
120C2×C4×C17C4×D412yes—
121C2×C4×C17#12112yes—
122C2×Dic17—24no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—44no—
126C2×Dic17—44no—
127C2×Dic17—42no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—42no—
131C2×Dic17—44no—
132Dic34—82no—
133C8×C17—82no—
134#2434—171no—
135#2310—171no—
136#1784D1711yescharacteristic
137#1511—172no—
138#1284—172no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—44no—
147C22×C17#38611yescharacteristic
148C22×C17C2×#9912yes—
149C22×C17#45612yes—
150C4×C17—22no—
151C4×C17—22no—
152C4×C17—24no—
153C4×C17—42no—
154Dic17—44no—
155Dic17—84no—
156Dic17—81no—
157Dic17—81no—
158C23×Q8C2×D1711yescharacteristic
159C2×C4⋊Q8—172no—
160C2×C22⋊Q8—171no—
161C2×#117—341no—
162C2×#101—22no—
163#407C2×D1712yes—
164#395—342no—
165#391—172no—
166#391—344no—
167#385—341no—
168#385—341no—
169#381—172no—
170#381—341no—
171C4×C22⋊C4—171no—
172#328—344no—
173C2×C17—22no—
174C2×C17—22no—
175C2×C17—41no—
176C2×C17#188611yescharacteristic
177C2×C17#189811yescharacteristic
178C2×C17#191211yescharacteristic
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—44no—
182C22×Q8—171no—
183C22×Q8C4×D1712yes—
184C22×Q8C22⋊D1712yes—
185C23×C4D6811yescharacteristic
186C23×C4D6811yescharacteristic
187C23×C4C22×D1711yescharacteristic
188C2×M16—22no—
189C4⋊Q8—684no—
190C22⋊Q8—342no—
191C22⋊Q8—342no—
192#117—684no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—342no—
198C2×C22⋊C4—341no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4—341no—
201C2×C42—172no—
202C2×C42—342no—
203#101—44no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206#95—174no—
207#95—344no—
208#95—342no—
209#95—341no—
210#95—342no—
211#95—342no—
212C17#946211yescharacteristic, complemented, Sylow
213C24C2×D6811yescharacteristic
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—342no—
224C2×Q8—682no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—342no—
233C22×C4—344no—
234C22×C4—341no—
235C22×C4—341no—
236C22×C4#6322712yes—
237C22×C4#6322712yes—
238C22×C4C2×C4×D1711yescharacteristic
239C22×C4D4×D1712yes—
240C22×C4#6325412yes—
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—44no—
243C2×C8—42no—
244C4⋊C4—342no—
245C4⋊C4—684no—
246C4⋊C4—684no—
247C4⋊C4—684no—
248C4⋊C4—682no—
249C22⋊C4—344no—
250C22⋊C4—684no—
251C22⋊C4—681no—
252C22⋊C4—681no—
253C42—682no—
254C42—684no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23#1058756811yescharacteristic
259C23#1058757211yescharacteristic
260C23C2×#6322711yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—1362no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—344no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—684no—
282C2×C4—684no—
283C2×C4—682no—
284C2×C4—684no—
285C2×C4—682no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4#1058747714yes—
289C2×C4#1058756712yes—
290C2×C4#1058762512yes—
291C8—82no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#1176268611yescharacteristic
301C22C2×#1058747712yes—
302C22#1176299412yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—42no—
307C4—684no—
308C4—1364no—
309C4—1361no—
310C4—1361no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#1452917511yescharacteristic
315C2#1452923511yescharacteristic
316C2#1452931911yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1784C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1784D17
hypocenterC17#9462
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
6. C2×#10587630C4semidirect
11. #11762716C4semidirect
212. C17#9462semidirect

Maximal subgroups

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

subgroupindex
#145324242
#145313702
#145273182
#145266382
C17×#17842
#946217