Finite Group Databasebeta

← #16943997

Subgroups — #16943997

Order 4352.

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

normal subgroups only

This group has 554 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
1#14527125C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526732C214yesmaximal, complemented
4C17×#2409C211yesmaximal, characteristic
5#11762866—21no—
6C2×C22⋊#3622—21no—
7#11762535—21no—
8#11762533—21no—
9#11762533C2212yes—
10#11762533C2212yes—
11C23×Q8×C17C411yescharacteristic
12C2×C17×C22⋊Q8—21no—
13C2×C4×D4×C17C2211yescharacteristic
14C17×#395—21no—
15C17×#393—21no—
16C17×#391—21no—
17C17×#386C412yes—
18C17×#385C2212yes—
19C17×#380C411yescharacteristic
20C22×Q8×C17—21no—
21C22×Q8×C17—21no—
22C22×Q8×C17—41no—
23C22×Q8×C17—41no—
24C22×Q8×C17—44no—
25C22×D4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C23×C4×C17C2×C412yes—
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×C17D411yescharacteristic
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17C2×C412yes—
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17C2311yescharacteristic
39C2×C42×C17D411yescharacteristic
40C22×#3622—41no—
41C17×#95—21no—
42C17×#95—22no—
43C17×#95—22no—
44C22⋊#3622—22no—
45C22⋊#3622—22no—
46C22⋊#3622—41no—
47C4⋊#3622—41no—
48#10587474—41no—
49C24×C17C22⋊C411yescharacteristic
50C24×C17C22×C411yescharacteristic
51C2×Q8×C17—24no—
52C2×Q8×C17—42no—
53C2×Q8×C17—44no—
54C2×Q8×C17—42no—
55C2×Q8×C17—44no—
56C2×Q8×C17—48no—
57C2×Q8×C17—44no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—41no—
63C22×C4×C17—22no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—22no—
69C22×C4×C17—42no—
70C22×C4×C17—44no—
71C22×C4×C17—42no—
72C22×C4×C17—42no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C4⋊C4×C17—41no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—44no—
81C22⋊C4×C17—24no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—44no—
85C42×C17—21no—
86C42×C17—21no—
87C2×#3622—42no—
88C2×#3622—42no—
89C2×#3622—82no—
90#11575—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17#10112yes—
100C23×C17C2×C22⋊C411yescharacteristic
101C23×C17#13112yes—
102Q8×C17—44no—
103Q8×C17—48no—
104Q8×C17—44no—
105D4×C17—44no—
106C2×C4×C17—22no—
107C2×C4×C17—24no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—44no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—44no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—81no—
128C2×C4×C17#9911yescharacteristic
129C2×C4×C17#9911yescharacteristic
130#3622—82no—
131#3622—82no—
132#2409D1711yescharacteristic
133#1466—171no—
134#1459—171no—
135#1326—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—81no—
148C22×C17#40811yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17C2×#10111yescharacteristic
151C4×C17—21no—
152C4×C17—41no—
153C4×C17—44no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—82no—
158C4×C17AΣL(1,𝔽16)12yes—
159C23×Q8Dic1711yescharacteristic
160C2×C22⋊Q8—21no—
161C2×C4×D4C2×D1711yescharacteristic
162#432—341no—
163C2×C22⋊C8—341no—
164#395—21no—
165#393—21no—
166#391—21no—
167#386Dic1712yes—
168#385C2×D1712yes—
169#380Dic1711yescharacteristic
170#324—341no—
171#323—172no—
172#323—172no—
173#323—341no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—42no—
177C2×C17—42no—
178C2×C17#147211yescharacteristic
179C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
180C2×C17#212311yescharacteristic
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—44no—
186C22×D4C2×Dic1711yescharacteristic
187C23×C4C2×Dic1711yescharacteristic
188C23×C4C2×Dic1712yes—
189C23×C4C22⋊D1711yescharacteristic
190C22×C8—681no—
191C22⋊Q8—44no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C22⋊D1711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4C2×Dic1712yes—
199C2×C22⋊C4C22⋊D1711yescharacteristic
200C2×C22⋊C4C22×D1711yescharacteristic
201C2×C42C22⋊D1711yescharacteristic
202C4⋊C8—681no—
203C22⋊C8—342no—
204C22⋊C8—342no—
205C22⋊C8—681no—
206#97—681no—
207#95—21no—
208#95—22no—
209#95—22no—
210C17#1157511yescharacteristic, complemented, Sylow
211C24C22⋊Dic1711yescharacteristic
212C24C22×Dic1711yescharacteristic
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4C22⋊Dic1711yescharacteristic
236C22×C4C22⋊Dic1711yescharacteristic
237C22×C4C22⋊Dic1711yescharacteristic
238C22×C4C2×C22⋊D1711yescharacteristic
239C22×C4C2×C22⋊D1711yescharacteristic
240C2×C8—682no—
241C2×C8—682no—
242C2×C8—1362no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C22⋊C4—24no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—44no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#1058750712yes—
261C23#1058763412yes—
262C23C2×C22⋊Dic1711yescharacteristic
263Q8—44no—
264Q8—48no—
265Q8—44no—
266D4—44no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4#1058750511yescharacteristic
290C2×C4#1058750511yescharacteristic
291C8—1362no—
292C8—1362no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#1176277611yescharacteristic
306C22C2×#1058750511yescharacteristic
307C22C2×#1058750711yescharacteristic
308C4—21no—
309C4—41no—
310C4—44no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—82no—
315C4#1176259612yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2#1452714811yescharacteristic
321C2C2×#1176259611yescharacteristic
322C2#1453043811yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2409C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C22⋊C4×C17C23
hypercenter#2409D17
hypocenterC17#11575
LCSG ▷ C24×C17 ▷ C22×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. #14526732C2semidirect
210. C17#11575semidirect

Maximal subgroups

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

subgroupindex
#145271252
#145271022
#145267322
C17×#24092
#1157517