Finite Group Databasebeta

← #17958387

Subgroups — #17958387

Order 4352.

706 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 78 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#14544030C212yesmaximal
2#14543711C212yesmaximal, complemented
3#14543710C212yesmaximal, complemented
4D17×#381C211yesmaximal, characteristic
5#11763780—22no—
6#11763653—22no—
7#11763652C2214yes—
8C23×C4×D17C411yescharacteristic
9C2×C22⋊C4×D17C411yescharacteristic
10C2×C22⋊C4×D17C2211yescharacteristic
11#11762672C2211yescharacteristic
12#11762671—22no—
13#11762670C411yescharacteristic
14#11762670C2211yescharacteristic
15#95×D17—22no—
16C17×#381C411yescharacteristic
17C24×D17C2×C411yescharacteristic
18C23×C4×C17C2×C411yescharacteristic
19C23×Dic17C2×C411yescharacteristic
20C22×C4×D17—21no—
21C22×C4×D17—21no—
22C22×C4×D17—22no—
23C22×C4×D17—22no—
24C22×C4×D17—21no—
25C22×C4×D17—42no—
26C22×C4×D17—42no—
27C22×C4×D17D412yes—
28C2×#63243—42no—
29C2×C22⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊Dic17C2×C411yescharacteristic
32C2×C22⋊Dic17C2311yescharacteristic
33C2×#63227—21no—
34C2×#63227D412yes—
35C22⋊C4×D17—42no—
36C22⋊C4×D17—42no—
37C22⋊#3624—24no—
38#10587543—42no—
39C17×#95—22no—
40#10587502—22no—
41#10587502—22no—
42#10587502—22no—
43C24×C17C22×C411yescharacteristic
44C23×D17—21no—
45C23×D17—21no—
46C23×D17—22no—
47C23×D17—22no—
48C23×D17—44no—
49C23×D17C22⋊C411yescharacteristic
50C23×D17C22⋊C411yescharacteristic
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—41no—
55C22×C4×C17—41no—
56C22×C4×C17C22⋊C412yes—
57C22×C4×C17M1612yescomplemented
58C22×Dic17—21no—
59C22×Dic17—21no—
60C22×Dic17—21no—
61C22×Dic17—41no—
62C22×Dic17—41no—
63C22×Dic17M1612yescomplemented
64C22×Dic17C2×D412yes—
65C2×C4×D17—24no—
66C2×C4×D17—24no—
67C2×C4×D17—22no—
68C2×C4×D17—22no—
69C2×C4×D17—48no—
70C2×C4×D17—42no—
71C2×C4×D17—42no—
72C2×C4×D17—48no—
73C2×C4×D17—42no—
74C2×C4×D17—42no—
75C2×C4×D17—44no—
76C2×C4×D17—44no—
77C2×C4×D17—42no—
78C2×#3624—44no—
79#63243—84no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊Dic17—41no—
83C22⋊Dic17—41no—
84#63227—42no—
85#63227—42no—
86#11500—171nomaximal, Sylow
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—42no—
90C23×C17C2×C22⋊C411yescharacteristic
91C23×C17C2×M1611yescharacteristic
92C23×C17C2×M1611yescharacteristic
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—24no—
96C22×D17—24no—
97C22×D17—24no—
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—48no—
101C22×D17—44no—
102C22×D17—44no—
103C22×D17—44no—
104C22×D17—44no—
105C22×D17—42no—
106C22×D17#9912yes—
107C22×D17#10012yes—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—42no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×Dic17—22no—
118C2×Dic17—22no—
119C2×Dic17—42no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—41no—
126C4×D17—48no—
127C4×D17—48no—
128C4×D17—44no—
129C4×D17—82no—
130C4×D17—84no—
131#3624—84no—
132C2×#381—171no—
133#1462—172no—
134#1317—172no—
135#1317—172no—
136C22×C17—22no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17#40711yescharacteristic
144C22×C17C2×#9911yescharacteristic
145C22×C17C2×#10011yescharacteristic
146C2×D17—24no—
147C2×D17—24no—
148C2×D17—22no—
149C2×D17—22no—
150C2×D17—22no—
151C2×D17—48no—
152C2×D17—48no—
153C2×D17—42no—
154C2×D17—42no—
155C2×D17—42no—
156C2×D17#35214yes—
157C4×C17—42no—
158C4×C17—42no—
159C4×C17—81no—
160Dic17—42no—
161Dic17—42no—
162Dic17—81no—
163C24×C4—171no—
164C22×C22⋊C4—171no—
165C22×C22⋊C4—171no—
166#407—342no—
167#381—171no—
168#381—171no—
169#381—171no—
170#381—342no—
171#381#59511yescharacteristic
172C2×#95—342no—
173#323—174no—
174#323—342no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17—41no—
178C2×C17—41no—
179C2×C17#146211yescharacteristic
180C2×C17C2×#35212yes—
181D17—24no—
182D17—42no—
183C25—171no—
184C23×C4—171no—
185C23×C4—172no—
186C23×C4—342no—
187C23×C4—341no—
188C23×C4—341no—
189C23×C4—341no—
190C23×C4—342no—
191C23×C4—682no—
192C23×C4—682no—
193C23×C4C2×#59511yescharacteristic
194C2×M16—682no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—172no—
197C2×C22⋊C4—171no—
198C2×C22⋊C4—341no—
199C2×C22⋊C4—682no—
200C2×C22⋊C4—682no—
201C2×C22⋊C4C2×#59511yescharacteristic
202C2×C22⋊C4C2×#59511yescharacteristic
203C22⋊C8—344no—
204C22⋊C8—682no—
205#95—22no—
206#95—342no—
207#95—342no—
208#95—342no—
209C17#1150011yescharacteristic, complemented, Sylow
210C24—171no—
211C24—171no—
212C24—342no—
213C24—342no—
214C24—341no—
215C24—341no—
216C24—684no—
217C24C22×#59511yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—172no—
224C22×C4—172no—
225C22×C4—342no—
226C22×C4—344no—
227C22×C4—344no—
228C22×C4—342no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—341no—
232C22×C4—682no—
233C22×C4—682no—
234C22×C4—688no—
235C22×C4—682no—
236C22×C4—682no—
237C22×C4—688no—
238C22×C4—684no—
239C22×C4—682no—
240C22×C4—684no—
241C22×C4—681no—
242C22×C4—681no—
243C22×C4#6324312yescomplemented
244C22×C4C22⋊#59512yes—
245M16—1364no—
246C2×C8—684no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—682no—
250C22⋊C4—682no—
251C22⋊C4—681no—
252C22⋊C4—681no—
253C23—21no—
254C23—21no—
255C23—42no—
256C23—172no—
257C23—172no—
258C23—344no—
259C23—344no—
260C23—344no—
261C23—341no—
262C23—341no—
263C23—341no—
264C23—341no—
265C23—682no—
266C23—684no—
267C23—684no—
268C23—684no—
269C23—688no—
270C23—684no—
271C23C2×#6324311yescharacteristic
272C23D17⋊M1611yescharacteristic
273C23C2×C22⋊#59511yescharacteristic
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—342no—
284C2×C4—342no—
285C2×C4—684no—
286C2×C4—688no—
287C2×C4—688no—
288C2×C4—682no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—681no—
295C2×C4—1362no—
296C2×C4—1364no—
297C8—1364no—
298C22—22no—
299C22—21no—
300C22—21no—
301C22—41no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—174no—
306C22—344no—
307C22—344no—
308C22—342no—
309C22—342no—
310C22—342no—
311C22—682no—
312C22—682no—
313C22—688no—
314C22—688no—
315C22—682no—
316C22#1176378011yescharacteristic
317C22D17⋊#9911yescharacteristic
318C22D17⋊#10011yescharacteristic
319C4—42no—
320C4—42no—
321C4—81no—
322C4—682no—
323C4—682no—
324C4—1361no—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2—344no—
330C2—682no—
331C2#1454403011yescharacteristic
332C2D17⋊#35212yes—
subgroupG/H
centerC22D17⋊#99
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC17×#381C4
socleC22×C17C2×#99
prime radicalC24×D17C2×C4
cosocleC2×C22⋊Dic17C23
hypercenter#381#595
hypocenterC17#11500
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #14543711C2semidirect
3. #14543710C2semidirect
57. C22×C4×C17M16semidirect
63. C22×Dic17M16semidirect
209. C17#11500semidirect
243. C22×C4#63243semidirect

Maximal subgroups

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

subgroupindex
#145440302
#145437112
#145437102
D17×#3812
#1150017