Finite Group Databasebeta

← #16944958

Subgroups — #16944958

Order 4352.

572 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#10587504C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526814C214yesmaximal, complemented
4C17×#2409C211yesmaximal, characteristic
5C2×#10587504—21no—
6#11762773—21no—
7C4×#63223—21no—
8#11762578C2212yes—
9#11762535C2212yes—
10#11762533—21no—
11C23×Q8×C17C411yescharacteristic
12C2×C17×C22⋊Q8—21no—
13C2×C4×D4×C17C2211yescharacteristic
14C17×#395C2212yes—
15C17×#393C412yes—
16C17×#391—21no—
17C17×#386—21no—
18C17×#385—21no—
19C17×#380C411yescharacteristic
20C22×Q8×C17—21no—
21C22×Q8×C17—41no—
22C22×Q8×C17—41no—
23C22×Q8×C17—44no—
24C22×Q8×C17C2×C412yes—
25C22×D4×C17D411yescharacteristic
26C23×C4×C17—21no—
27C23×C4×C17C2×C411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C17×C22⋊Q8—44no—
30C4×D4×C17—41no—
31C4×D4×C17—41no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17C2311yescharacteristic
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17C2×C411yescharacteristic
38C2×C22⋊C4×C17D411yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×#63223—22no—
41C2×#63223—41no—
42C17×#95—22no—
43C17×#95—22no—
44C17×#95C2×C412yes—
45#10587504—24no—
46C22⋊#3622—22no—
47C22⋊#3622—41no—
48#10587474—41no—
49C4×#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C2×Q8×C17—24no—
53C2×Q8×C17—44no—
54C2×Q8×C17—42no—
55C2×Q8×C17—42no—
56C2×Q8×C17—44no—
57C2×Q8×C17—48no—
58C2×Q8×C17—44no—
59C2×Q8×C17—41no—
60C2×Q8×C17—41no—
61C2×D4×C17—22no—
62C2×D4×C17—22no—
63C2×D4×C17—41no—
64C22×C4×C17—22no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—22no—
70C22×C4×C17—44no—
71C22×C4×C17—42no—
72C22×C4×C17—42no—
73C22×C4×C17—42no—
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C4⋊C4×C17—41no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—44no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—44no—
85C22⋊C4×C17—42no—
86C42×C17—21no—
87C42×C17—21no—
88#63223—24no—
89#63223—42no—
90#63223—82no—
91C2×#3622—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
105Q8×C17—44no—
106Q8×C17—44no—
107Q8×C17—44no—
108Q8×C17—44no—
109D4×C17—44no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—24no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—48no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—81no—
130C2×C4×C17#9911yescharacteristic
131C2×C4×C17#9911yescharacteristic
132C2×C4×C17#13712yes—
133#3622—44no—
134#3622—82no—
135#2409D1711yescharacteristic
136C4×#100—171no—
137#1465—171no—
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—44no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—82no—
161C4×C17AΣL(1,𝔽16)12yes—
162C23×Q8Dic1711yescharacteristic
163C2×C22⋊Q8—21no—
164C2×C4×D4C2×D1711yescharacteristic
165C2×#100—341no—
166#407—341no—
167C4×M16—341no—
168#395C2×D1712yes—
169#393Dic1712yes—
170#391—21no—
171#386—21no—
172#385—21no—
173#380Dic1711yescharacteristic
174#343—172no—
175#324—172no—
176#323—341no—
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—21no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—44no—
188C22×Q8C2×Dic1712yes—
189C22×D4C22⋊D1711yescharacteristic
190C23×C4—21no—
191C23×C4C2×Dic1711yescharacteristic
192C23×C4C22⋊D1711yescharacteristic
193C2×M16—342no—
194C2×M16—681no—
195C22⋊Q8—44no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C22×D1711yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C2×Dic1711yescharacteristic
204C2×C22⋊C4C22⋊D1711yescharacteristic
205C2×C42C22⋊D1711yescharacteristic
206#100—344no—
207C22⋊C8—342no—
208C22⋊C8—681no—
209#97—681no—
210C4×C8—681no—
211#95—22no—
212#95—22no—
213#95C2×Dic1712yes—
214C17#1179111yescharacteristic, complemented, Sylow
215C24C22⋊Dic1711yescharacteristic
216C24C22⋊Dic1711yescharacteristic
217C2×Q8—24no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—44no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊Dic1711yescharacteristic
240C22×C4C22⋊Dic1711yescharacteristic
241C22×C4C22×Dic1711yescharacteristic
242C22×C4C2×C22⋊D1711yescharacteristic
243C22×C4C2×C22⋊D1711yescharacteristic
244M16—344no—
245M16—682no—
246M16—1362no—
247C2×C8—682no—
248C2×C8—682no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#1058750812yes—
268C23C2×C22⋊Dic1711yescharacteristic
269Q8—44no—
270Q8—44no—
271Q8—44no—
272Q8—44no—
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4#1058750511yescharacteristic
295C2×C4#1058750511yescharacteristic
296C2×C4#1058762812yes—
297C8—684no—
298C8—1362no—
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×#1058750511yescharacteristic
312C22#1176281811yescharacteristic
313C22C2×#1058750811yescharacteristic
314C4—22no—
315C4—21no—
316C4—41no—
317C4—44no—
318C4—41no—
319C4—41no—
320C4—82no—
321C4#1176259612yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2#1452730411yescharacteristic
327C2C2×#1176259611yescharacteristic
328C2#1453047911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2409C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#2409D17
hypocenterC17#11791
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #14526814C2semidirect
214. C17#11791semidirect

Maximal subgroups

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

subgroupindex
C4×#105875042
#145271222
#145268142
C17×#24092
#1179117