Finite Group Databasebeta

← #16945062

Subgroups — #16945062

Order 4352.

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

normal subgroups only

This group has 586 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#14528864C211yesmaximal, characteristic
2#14527114C211yesmaximal, characteristic
3#14526814C214yesmaximal, complemented
4C2×C17×#395C211yesmaximal, characteristic
5#11762773—21no—
6#11762773—21no—
7#11762578—21no—
8#11762578C2212yes—
9#11762535—21no—
10#11762535C2212yes—
11C23×Q8×C17C411yescharacteristic
12C22×C4⋊C4×C17—21no—
13C22×C4⋊C4×C17C2211yescharacteristic
14C17×#395—21no—
15C17×#395—21no—
16C17×#395C412yes—
17C17×#395C2212yes—
18C2×C17×#95—21no—
19C2×C17×#95C411yescharacteristic
20C22×Q8×C17—21no—
21C22×Q8×C17—44no—
22C22×Q8×C17—41no—
23C22×Q8×C17—41no—
24C22×Q8×C17C2×C412yes—
25C23×C4×C17—21no—
26C23×C4×C17—21no—
27C23×C4×C17C2×C411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17—41no—
33C2×C4⋊C4×C17—41no—
34C2×C4⋊C4×C17—44no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C4⋊C4×C17C2311yescharacteristic
39C2×#63223—22no—
40C2×#63223—41no—
41C17×#95—22no—
42C17×#95—21no—
43C17×#95—22no—
44C17×#95C2×C412yes—
45C22⋊#3622—22no—
46C22⋊#3622—41no—
47C22⋊#3622—41no—
48C22⋊#3622—41no—
49C24×C17C22⋊C411yescharacteristic
50C2×Q8×C17—44no—
51C2×Q8×C17—44no—
52C2×Q8×C17—42no—
53C2×Q8×C17—42no—
54C2×Q8×C17—48no—
55C2×Q8×C17—44no—
56C2×Q8×C17—44no—
57C22×C4×C17—24no—
58C22×C4×C17—24no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—22no—
65C22×C4×C17—22no—
66C22×C4×C17—21no—
67C22×C4×C17—22no—
68C22×C4×C17—41no—
69C22×C4×C17—42no—
70C22×C4×C17—42no—
71C22×C4×C17—42no—
72C22×C4×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—42no—
75C22×C4×C17—42no—
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22×C411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—44no—
87C4⋊C4×C17—44no—
88#63223—42no—
89#63223—42no—
90#63223—82no—
91C2×#3622—42no—
92C2×#3622—42no—
93#11815—171nomaximal, Sylow
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17#9911yescharacteristic
99C23×C17#9911yescharacteristic
100C23×C17C2×C22⋊C411yescharacteristic
101Q8×C17—44no—
102Q8×C17—44no—
103Q8×C17—48no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—24no—
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—21no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—48no—
119C2×C4×C17—42no—
120C2×C4×C17—44no—
121C2×C4×C17—42no—
122C2×C4×C17—44no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—81no—
132C2×C4×C17#13712yes—
133#3622—82no—
134#3622—82no—
135C2×#395D1711yescharacteristic
136#1821—171no—
137#1463—171no—
138#1354—174no—
139C22×C17—24no—
140C22×C17—24no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17AΣL(1,𝔽16)12yes—
148C22×C17#35612yes—
149C22×C17C2×#9911yescharacteristic
150C22×C17#41911yescharacteristic
151C22×C17#42111yescharacteristic
152C4×C17—22no—
153C4×C17—44no—
154C4×C17—41no—
155C4×C17—44no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17—41no—
160C23×Q8Dic1711yescharacteristic
161C22×C4⋊C4—21no—
162C22×C4⋊C4C2×D1711yescharacteristic
163#407—341no—
164#407—341no—
165#395—21no—
166#395—21no—
167#395Dic1712yes—
168#395C2×D1712yes—
169C2×#95—21no—
170C2×#95Dic1711yescharacteristic
171#343—172no—
172#343—341no—
173#324—172no—
174#324—341no—
175C2×C17—21no—
176C2×C17—22no—
177C2×C17—22no—
178C2×C17—21no—
179C2×C17#151111yescharacteristic
180C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
181C2×C17C2×#35611yescharacteristic
182C22×Q8—21no—
183C22×Q8—44no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8C2×Dic1712yes—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C2×Dic1711yescharacteristic
190C23×C4C22⋊D1711yescharacteristic
191C23×C4C22⋊D1711yescharacteristic
192C2×M16—342no—
193C2×M16—681no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C2×Dic1711yescharacteristic
200C2×C4⋊C4C22⋊D1711yescharacteristic
201C2×C4⋊C4C22⋊D1711yescharacteristic
202C2×C4⋊C4C22×D1711yescharacteristic
203C22⋊C8—342no—
204C22⋊C8—681no—
205C22⋊C8—681no—
206C22⋊C8—681no—
207#95—22no—
208#95—21no—
209#95—22no—
210#95C2×Dic1712yes—
211C17#1181511yescharacteristic, complemented, Sylow
212C24C22⋊Dic1711yescharacteristic
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—48no—
218C2×Q8—44no—
219C2×Q8—44no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊Dic1711yescharacteristic
240C22×C4C22⋊Dic1711yescharacteristic
241C22×C4C22⋊Dic1711yescharacteristic
242C22×C4C22×Dic1711yescharacteristic
243C22×C4C2×C22⋊D1711yescharacteristic
244C22×C4C2×C22⋊D1711yescharacteristic
245M16—682no—
246M16—682no—
247M16—1362no—
248C2×C8—682no—
249C2×C8—682no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1058750511yescharacteristic
261C23#1058750511yescharacteristic
262C23C2×C22⋊Dic1711yescharacteristic
263Q8—44no—
264Q8—44no—
265Q8—48no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4#1058762812yes—
295C8—1362no—
296C8—1362no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1176259612yes—
306C22#1176260812yes—
307C22C2×#1058750511yescharacteristic
308C22#1176281811yescharacteristic
309C22#1176282611yescharacteristic
310C4—22no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—44no—
315C4—41no—
316C4—41no—
317C4—41no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#1452731711yescharacteristic
323C2C2×#1176259611yescharacteristic
324C2C2×#1176260811yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC2×C17×#395C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenterC2×#395D17
hypocenterC17#11815
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
3. #14526814C2semidirect
211. C17#11815semidirect

Maximal subgroups

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

subgroupindex
#145288642
#145271142
#145268142
C2×C17×#3952
#1181517