Finite Group Databasebeta

← #16932279

Subgroups — #16932279

Order 4352.

682 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 111 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#14537825C211yesmaximal, characteristic
2#14528708C212yesmaximal, complemented
3#14527122C211yesmaximal, characteristic
4#14526690C212yesmaximal, complemented
5C17×#2409C211yesmaximal, characteristic
6#11762887—24no—
7#11762862—21no—
8C2×#10587507—24no—
9#11762773—21no—
10#11762773C2212yes—
11#11762773C2212yes—
12#11762535—21no—
13#11762533—21no—
14C23×Q8×C17C2211yescharacteristic
15C2×C17×C22⋊Q8—21no—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#395—21no—
18C17×#393—21no—
19C17×#391—21no—
20C17×#386C412yes—
21C17×#385C412yes—
22C17×#380C2211yescharacteristic
23C22×Q8×C17—21no—
24C22×Q8×C17—22no—
25C22×Q8×C17—21no—
26C22×Q8×C17—22no—
27C22×Q8×C17—44no—
28C22×D4×C17C2×C411yescharacteristic
29C23×C4×C17C2×C412yes—
30C23×C4×C17D411yescharacteristic
31C23×C4×C17C2311yescharacteristic
32C17×C22⋊Q8—44no—
33C4×D4×C17—22no—
34C4×D4×C17—22no—
35C2×C4⋊C4×C17—22no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17C2×C411yescharacteristic
40C2×C22⋊C4×C17C2×C412yes—
41C2×C22⋊C4×C17D411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×#63223—22no—
44C2×#63223—22no—
45C2×#63223D414yes—
46C17×#95—22no—
47C17×#95—22no—
48C17×#95—21no—
49#10587507—48no—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C22⋊#3622—22no—
54C22⋊#3622—22no—
55C4⋊#3622—22no—
56C4×#3622—22no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C22×C411yescharacteristic
59C2×Q8×C17—24no—
60C2×Q8×C17—28no—
61C2×Q8×C17—42no—
62C2×Q8×C17—44no—
63C2×Q8×C17—44no—
64C2×Q8×C17—48no—
65C2×Q8×C17—41no—
66C2×Q8×C17—42no—
67C2×Q8×C17—41no—
68C2×D4×C17—22no—
69C2×D4×C17—22no—
70C2×D4×C17—41no—
71C22×C4×C17—22no—
72C22×C4×C17—24no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—22no—
76C22×C4×C17—44no—
77C22×C4×C17—42no—
78C22×C4×C17—42no—
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17#4112yes—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17—44no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17—22no—
91C22⋊C4×C17—44no—
92C22⋊C4×C17—42no—
93C42×C17—21no—
94C42×C17—21no—
95#63223—44no—
96#63223—44no—
97#63223—82no—
98C2×#3622—24no—
99C2×#3622—42no—
100#9506—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17C2×C22⋊C411yescharacteristic
111C23×C17C4×D412yes—
112Q8×C17—48no—
113Q8×C17—44no—
114Q8×C17—44no—
115D4×C17—44no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—22no—
119C2×C4×C17—24no—
120C2×C4×C17—24no—
121C2×C4×C17—28no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—44no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17—42no—
135C2×C4×C17—81no—
136C2×C4×C17#10112yescomplemented
137C2×C4×C17#12011yescharacteristic
138C2×C4×C17#12112yes—
139C2×C4×C17#12311yescharacteristic
140#3622—44no—
141#3622—82no—
142#2975—171no—
143#2409D1711yescharacteristic
144#1784—172no—
145#1465—171no—
146#1305—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—22no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—42no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—81no—
159C22×C17#38611yescharacteristic
160C22×C17C2×#10111yescharacteristic
161C22×C17#41311yescharacteristic
162C4×C17—22no—
163C4×C17—21no—
164C4×C17—22no—
165C4×C17—44no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—82no—
169C4×C17B(4,𝔽2)12yes—
170C23×Q8C2×D1711yescharacteristic
171C2×C22⋊Q8—21no—
172C2×C4×D4C2×D1711yescharacteristic
173#436—344no—
174#431—341no—
175C2×#101—344no—
176#407—172no—
177#407—172no—
178#407—341no—
179#395—21no—
180#393—21no—
181#391—21no—
182#386Dic1712yes—
183#385Dic1712yes—
184#380C2×D1711yescharacteristic
185#324—341no—
186#323—341no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—42no—
190C2×C17—81no—
191C2×C17#188911yescharacteristic
192C2×C17#190111yescharacteristic
193C2×C17#191511yescharacteristic
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—44no—
199C22×D4C2×Dic1711yescharacteristic
200C23×C4C2×Dic1712yes—
201C23×C4C22⋊D1711yescharacteristic
202C23×C4C22×D1711yescharacteristic
203C2×M16—174no—
204C2×M16—342no—
205C2×M16—342no—
206C22⋊Q8—44no—
207C4×D4—22no—
208C4×D4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22⋊D1711yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4C2×Dic1711yescharacteristic
214C2×C22⋊C4C2×Dic1712yes—
215C2×C22⋊C4C22⋊D1711yescharacteristic
216C2×C42C22⋊D1711yescharacteristic
217C4⋊C8—342no—
218#101—688no—
219C22⋊C8—342no—
220C22⋊C8—342no—
221C22⋊C8—342no—
222C22⋊C8—342no—
223C22⋊C8—342no—
224C4×C8—342no—
225#95—22no—
226#95—22no—
227#95—21no—
228C17#950611yescharacteristic, complemented, Sylow
229C24C22⋊Dic1711yescharacteristic
230C24C22×Dic1711yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊Dic1711yescharacteristic
252C22×C4C22⋊Dic1711yescharacteristic
253C22×C4C22⋊Dic1711yescharacteristic
254C22×C4D4×D1712yes—
255C22×C4#6325412yes—
256C22×C4C2×C22⋊D1711yescharacteristic
257C22×C4C2×C22⋊D1711yescharacteristic
258M16—684no—
259M16—684no—
260M16—1362no—
261C2×C8—344no—
262C2×C8—682no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23D4×Dic1712yes—
282C23C2×C22⋊Dic1711yescharacteristic
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4#1058750712yescomplemented
308C2×C4#1058762111yescharacteristic
309C2×C4#1058762311yescharacteristic
310C2×C4#1058762512yes—
311C8—684no—
312C8—1362no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1176268911yescharacteristic
326C22C2×#1058750711yescharacteristic
327C22#1176279311yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—41no—
332C4—44no—
333C4—41no—
334C4—82no—
335C4#1176299512yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#1452919311yescharacteristic
341C2#1452925311yescharacteristic
342C2#1452933711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2409C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC23×C4×C17C23
hypercenter#2409D17
hypocenterC17#9506
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. #14528708C2semidirect
4. #14526690C2semidirect
136. C2×C4×C17#101semidirect
228. C17#9506semidirect
307. C2×C4#10587507semidirect

Maximal subgroups

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

subgroupindex
#145378252
#145287082
#145271222
#145266902
C17×#24092
#950617