Finite Group Databasebeta

← #16944959

Subgroups — #16944959

Order 4352.

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

normal subgroups only

This group has 572 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#14532149C211yesmaximal, characteristic
2C4×#10587504C211yesmaximal, characteristic
3#14526815C214yesmaximal, complemented
4C17×#1465C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic
6C2×#10587610—21no—
7C2×#10587504—21no—
8C4×#63223—21no—
9#11762729C2212yes—
10#11762721C412yes—
11#11762715—21no—
12#11762689—21no—
13#11762681—21no—
14#11762669C411yescharacteristic
15#11762578C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#407—21no—
18C17×#324C2212yes—
19C17×#323—21no—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17—21no—
24C22×Dic34—21no—
25C22×Dic34—44no—
26C22×Dic34—41no—
27C22×Dic34—41no—
28C22×Dic34C2×C412yes—
29C2×M16×C17—41no—
30C4×D4×C17—41no—
31C4×D4×C17—41no—
32C2×C4⋊C4×C17C2311yescharacteristic
33C2×C22⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C22⋊Dic17—22no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39#10587610—44no—
40C2×#63225—22no—
41C2×#63223—22no—
42C17×C22⋊C8—22no—
43C17×C22⋊C8—41no—
44#10587504—24no—
45#10587502—22no—
46#10587502—22no—
47#10587502C2×C412yes—
48#10587474—41no—
49C4×#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C2×D4×C17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—41no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—42no—
60C22×C4×C17—42no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—44no—
69C22×Dic17—42no—
70C2×Dic34—24no—
71C2×Dic34—42no—
72C2×Dic34—48no—
73C2×Dic34—44no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C2×Dic34—42no—
77C2×Dic34—44no—
78C2×Dic34—44no—
79M16×C17—82no—
80C2×C8×C17—42no—
81C4⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C42×C17—21no—
85C42×C17—21no—
86C22⋊Dic17—44no—
87C22⋊Dic17—42no—
88C4⋊Dic17—42no—
89#63225—44no—
90#63223—24no—
91#63223—42no—
92C2×#3622—42no—
93#11791—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17#10412yes—
104C23×C17C2×C22⋊C411yescharacteristic
105D4×C17—44no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—81no—
119C2×C4×C17#9911yescharacteristic
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#13712yes—
122C2×Dic17—24no—
123C2×Dic17—42no—
124C2×Dic17—48no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129Dic34—44no—
130Dic34—44no—
131Dic34—44no—
132Dic34—44no—
133C8×C17—82no—
134#3622—44no—
135#2409—171no—
136C4×#100—171no—
137#1465D1711yescharacteristic
138#1354—174no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—22no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—81no—
151C22×C17C2×#9911yescharacteristic
152C22×C17#41911yescharacteristic
153C22×C17C2×#10411yescharacteristic
154C4×C17—21no—
155C4×C17—22no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17AΣL(1,𝔽16)12yes—
160Dic17—44no—
161Dic17—82no—
162C23×Q8—171no—
163C2×C22⋊Q8—341no—
164C2×C4×D4C2×D1711yescharacteristic
165C2×#100—341no—
166#407—21no—
167C4×M16—341no—
168#395—172no—
169#393—172no—
170#391—341no—
171#386—341no—
172#385—341no—
173#380—171no—
174#343—172no—
175#324C2×D1712yes—
176#323—21no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—42no—
180C2×C17—81no—
181C2×C17#150811yescharacteristic
182C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
183C2×C17#213511yescharacteristic
184C22×Q8—172no—
185C22×Q8—341no—
186C22×Q8—681no—
187C22×Q8—681no—
188C22×Q8—684no—
189C22×D4D6811yescharacteristic
190C23×C4—341no—
191C23×C4C4×D1711yescharacteristic
192C23×C4C22⋊D1711yescharacteristic
193C2×M16—41no—
194C2×M16—342no—
195C22⋊Q8—684no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—342no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4C22×D1711yescharacteristic
201C2×C22⋊C4—342no—
202C2×C22⋊C4—341no—
203C2×C22⋊C4C4×D1711yescharacteristic
204C2×C22⋊C4C22⋊D1711yescharacteristic
205C2×C42D6811yescharacteristic
206#100—344no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209#97—681no—
210C4×C8—681no—
211#95—172no—
212#95—342no—
213#95—342no—
214C17#1179111yescharacteristic, complemented, Sylow
215C24#6322711yescharacteristic
216C24#6322711yescharacteristic
217C2×Q8—344no—
218C2×Q8—682no—
219C2×Q8—682no—
220C2×Q8—684no—
221C2×Q8—684no—
222C2×Q8—688no—
223C2×Q8—684no—
224C2×Q8—681no—
225C2×Q8—681no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—342no—
236C22×C4—342no—
237C22×C4—684no—
238C22×C4—682no—
239C22×C4#6322711yescharacteristic
240C22×C4#6322711yescharacteristic
241C22×C4C2×C4×D1711yescharacteristic
242C22×C4C2×D6811yescharacteristic
243C22×C4C2×C22⋊D1711yescharacteristic
244M16—82no—
245M16—344no—
246M16—682no—
247C2×C8—42no—
248C2×C8—682no—
249C4⋊C4—41no—
250C4⋊C4—682no—
251C4⋊C4—684no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—684no—
255C22⋊C4—682no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#1058749612yes—
268C23C2×#6322711yescharacteristic
269Q8—684no—
270Q8—684no—
271Q8—684no—
272Q8—684no—
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—344no—
288C2×C4—682no—
289C2×C4—688no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4#1058747711yescharacteristic
295C2×C4#1058747711yescharacteristic
296C2×C4#1058759512yes—
297C8—82no—
298C8—684no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1058747711yescharacteristic
312C22#1176281911yescharacteristic
313C22C2×#1058749611yescharacteristic
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—684no—
320C4—1362no—
321C4#1176259512yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2#1452730511yescharacteristic
327C2C2×#1176259511yescharacteristic
328C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11791
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14526815C2semidirect
214. C17#11791semidirect

Maximal subgroups

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

subgroupindex
#145321492
C4×#105875042
#145268152
C17×#14652
#1179117