Finite Group Databasebeta

← #17953302

Subgroups — #17953302

Order 4352.

550 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 108 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#14544660C212yesmaximal, complemented
2#14544644C212yesmaximal
3#14543647C212yesmaximal, complemented
4C2×C8×C22⋊D17C211yesmaximal, characteristic
5C2×#10587533—22no—
6C2×#10587551—22no—
7#11763751C2214yes—
8#11763687—22no—
9#11763683—22no—
10C22×C8×D17C2211yescharacteristic
11C2×C4×C22⋊D17C2211yescharacteristic
12C8×C22⋊D17—22no—
13C8×C22⋊D17—22no—
14C2×#10587485C411yescharacteristic, complemented
15C2×C22⋊#3622C411yescharacteristic, complemented
16C2×#10587491C2211yescharacteristic
17C2×C8×Dic17C411yescharacteristic, complemented
18C23×C8×C17C411yescharacteristic, complemented
19C23×C4×C17C2×C411yescharacteristic
20C22×C22⋊D17D411yescharacteristic
21C22×C4×D17C2311yescharacteristic
22C2×C22⋊#595—24no—
23C2×C4⋊#595—24no—
24C22×C8×C17—21no—
25C22×C8×C17—21no—
26C22×C8×C17—42no—
27C22×C8×C17C2×C411yescharacteristic
28C22×C8×C17C2×C411yescharacteristic
29C2×C22⋊Dic17Q811yescharacteristic
30C4×C22⋊D17—22no—
31C4×C22⋊D17—22no—
32C2×#63227D411yescharacteristic
33C2×#63225Q811yescharacteristic
34C2×C4×Dic17C2×C411yescharacteristic
35C22×#3622C2×C411yescharacteristic
36C22×#3622C2×C411yescharacteristic
37C2×C8×D17—42no—
38C2×C8×D17D412yes—
39C2×C8×D17D412yes—
40#10587551—44no—
41(C2×#595)⋊C4—24no—
42#10587533—44no—
43C22⋊#3622—22no—
44#10587491—22no—
45#10587485—22no—
46C8×Dic17—21no—
47C8×Dic17—21no—
48C24×C17C4⋊C411yescharacteristic
49C23×D17C2×D411yescharacteristic
50C22×#595—44no—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—41no—
54C22×C4×C17—41no—
55C22×C4×C17C4⋊C411yescharacteristic
56C22×C4×C17C22×C411yescharacteristic
57C2×C22⋊D17—22no—
58C2×C22⋊D17—22no—
59C2×C22⋊D17SD1614yes—
60C22×Dic17C4⋊C411yescharacteristic
61C22×Dic17C4⋊C411yescharacteristic
62C22×Dic17C2×Q811yescharacteristic
63C2×C4×D17—41no—
64C2×C4×D17—41no—
65C2×C4×D17C2×D412yes—
66C2×C4×D17#4112yes—
67C22⋊#595—44no—
68C22⋊#595—44no—
69C4⋊#595—44no—
70C4⋊#595—44no—
71C2×C8×C17—22no—
72C2×C8×C17—21no—
73C2×C8×C17—42no—
74C2×C8×C17—42no—
75C2×C8×C17—41no—
76C2×C8×C17C22⋊C411yescharacteristic
77C2×C8×C17C22⋊C411yescharacteristic
78C22⋊Dic17—22no—
79#63227—22no—
80#63225—22no—
81C4×Dic17—21no—
82C4×Dic17—21no—
83C2×#3622—21no—
84C2×#3622—41no—
85C2×#3622—41no—
86C2×#3622C22⋊C412yes—
87C8×D17—24no—
88C8×D17—42no—
89#8608—171nomaximal, Sylow
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17#10612yescomplemented
95C23×C17C2×C4⋊C411yescharacteristic
96C22×D17—41no—
97C22×D17—41no—
98C22×D17#12112yes—
99C22×D17C2×SD1612yes—
100C22×D17#13512yes—
101C2×#595—84no—
102C2×#595—84no—
103C2×#595—84no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17C2×C22⋊C411yescharacteristic
112C2×C4×C17#11711yescharacteristic
113C22⋊D17—28no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17#10612yescomplemented
120C2×Dic17C22⋊Q812yes—
121C4×D17—22no—
122C4×D17—22no—
123C4×D17—42no—
124C8×C17—21no—
125C8×C17—41no—
126C8×C17#9912yescomplemented
127#3622—22no—
128#3622—41no—
129C2×C8×D4—171no—
130#1807—172no—
131#1799—172no—
132#1289—172no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—22no—
136C22×C17—42no—
137C22×C17—42no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17#38011yescharacteristic
141C22×C17C2×#10611yescharacteristic
142C22×C17#42711yescharacteristic
143C2×D17—22no—
144C2×D17—22no—
145C2×D17—42no—
146C2×D17—42no—
147C2×D17#46514yes—
148#595—164no—
149C4×C17—21no—
150C4×C17—41no—
151C4×C17C2×#9911yescharacteristic
152C4×C17#41011yescharacteristic
153Dic17—22no—
154Dic17—22no—
155Dic17—41no—
156C23×C8—171no—
157C23×C8#59511yescharacteristic, complemented
158C2×C4×D4—171no—
159C8×D4—342no—
160C8×D4—342no—
161C2×#106—342no—
162C2×C4⋊C8—171no—
163C2×#99—342no—
164C2×C22⋊C8—171no—
165C2×C22⋊C8—171no—
166C2×C4×C8—171no—
167#380—174no—
168#342—342no—
169#340—342no—
170C2×C17—21no—
171C2×C17—21no—
172C2×C17—42no—
173C2×C17#179911yescharacteristic
174C2×C17#180711yescharacteristic
175C2×C17#181211yescharacteristic
176D17—24no—
177C22×D4—171no—
178C23×C4—171no—
179C23×C4C2×#59511yescharacteristic
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—171no—
184C22×C8—172no—
185C22×C8—172no—
186C22×C8—171no—
187C22×C8—682no—
188C22×C8C2×#59511yescharacteristic
189C22×C8C2×#59511yescharacteristic
190C4×D4—342no—
191C4×D4—342no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—344no—
194C2×C22⋊C4—171no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—344no—
197C2×C42—171no—
198#106—684no—
199C4⋊C8—342no—
200#99—684no—
201C22⋊C8—342no—
202C22⋊C8—342no—
203C4×C8—341no—
204C4×C8—341no—
205#95—344no—
206C17#860811yescharacteristic, complemented, Sylow
207C24—171no—
208C24C4⋊#59511yescharacteristic
209C2×D4—174no—
210C2×D4—342no—
211C2×D4—342no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—172no—
219C22×C4—172no—
220C22×C4—171no—
221C22×C4—681no—
222C22×C4—681no—
223C22×C4—684no—
224C22×C4C4⋊#59511yescharacteristic
225C22×C4C22×#59511yescharacteristic
226C2×C8—22no—
227C2×C8—21no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—172no—
232C2×C8—344no—
233C2×C8—341no—
234C2×C8—682no—
235C2×C8—681no—
236C2×C8—681no—
237C2×C8C22⋊#59511yescharacteristic
238C2×C8C22⋊#59511yescharacteristic
239C4⋊C4—342no—
240C4⋊C4—684no—
241C4⋊C4—684no—
242C22⋊C4—342no—
243C22⋊C4—342no—
244C22⋊C4—684no—
245C22⋊C4—684no—
246C42—341no—
247C42—341no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—172no—
253C23—172no—
254C23—172no—
255C23—681no—
256C23—681no—
257C23#1058753312yescomplemented
258C23C2×C4⋊#59511yescharacteristic
259D4—348no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—172no—
268C2×C4—172no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—341no—
274C2×C4—682no—
275C2×C4—681no—
276C2×C4—681no—
277C2×C4—1364no—
278C2×C4—1364no—
279C2×C4—1364no—
280C2×C4#1058767711yescharacteristic
281C2×C4C2×C22⋊#59511yescharacteristic
282C8—21no—
283C8—41no—
284C8—342no—
285C8—681no—
286C8#1058755112yescomplemented
287C22—21no—
288C22—21no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—174no—
295C22—342no—
296C22—342no—
297C22—682no—
298C22—682no—
299C22#1176375111yescharacteristic
300C22C2×#1058753311yescharacteristic
301C22#1176381911yescharacteristic
302C4—21no—
303C4—41no—
304C4—2724no—
305C4—342no—
306C4—342no—
307C4—681no—
308C4C2×#1058755111yescharacteristic
309C4#1176378811yescharacteristic
310C2—21no—
311C2—21no—
312C2—42no—
313C2—344no—
314C2#1454464411yescharacteristic
315C2#1454466011yescharacteristic
316C2#1454466611yescharacteristic
subgroupG/H
centerC22#11763751
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC23×C8×C17C4
socleC22×C17#380
prime radicalC22×C22⋊D17D4
cosocleC22×C4×D17C23
hypercenterC23×C8#595
hypocenterC17#8608
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. #14544660C2semidirect
3. #14543647C2semidirect
14. C2×#10587485C4semidirect
15. C2×C22⋊#3622C4semidirect
17. C2×C8×Dic17C4semidirect
18. C23×C8×C17C4semidirect
94. C23×C17#106semidirect
119. C2×Dic17#106semidirect
126. C8×C17#99semidirect
157. C23×C8#595semidirect
206. C17#8608semidirect
257. C23#10587533semidirect
286. C8#10587551semidirect

Maximal subgroups

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

subgroupindex
#145446602
#145446442
#145436472
C2×C8×C22⋊D172
#860817