Finite Group Databasebeta

← #17959353

Subgroups — #17959353

Order 4352.

480 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#10587539C211yesmaximal, characteristic
2#14544041C211yesmaximal, characteristic
3#14543764C214yesmaximal, complemented
4#14533780C211yesmaximal, characteristic
5C2×#10587539—21no—
6#11763780—21no—
7C4×C22⋊#595—21no—
8#11763687C2212yes—
9#11763654—21no—
10#11763653C2212yes—
11C2×#10587633—21no—
12C2×C4×D68C2211yescharacteristic
13#11762735C412yescomplemented
14#11762735C2212yes—
15#11762716—21no—
16#11762666—21no—
17#11762666—21no—
18C4×C4⋊Dic17C411yescharacteristic, complemented
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×D68D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17D411yescharacteristic
24C2×C22⋊#595—22no—
25C2×C4×#595—41no—
26C2×#63243—41no—
27C17×C4⋊Q8—41no—
28C17×C4⋊Q8—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C42×C17C2×C411yescharacteristic
32#10587633—44no—
33C2×#63227—22no—
34C2×#63227—22no—
35C2×#63227C2×C411yescharacteristic
36C2×#63227C2311yescharacteristic
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17D411yescharacteristic
39C2×C4×Dic17—21no—
40C4×D68—41no—
41C4×D68—41no—
42C22⋊#3624—22no—
43(C2×#595)⋊C4—41no—
44#10587543—41no—
45#10587539—24no—
46#10587502—22no—
47#10587502—22no—
48#10587502C2×C412yes—
49C23×D17C22⋊C411yescharacteristic
50C23×D17C2×D411yescharacteristic
51C22×#595—42no—
52C2×Q8×C17—42no—
53C2×Q8×C17—42no—
54C2×Q8×C17—42no—
55C22×C4×C17—22no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C22×Dic17—22no—
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C2×D411yescharacteristic
62C2×D68—22no—
63C2×D68—22no—
64C2×D68—41no—
65C2×C4×D17—21no—
66C2×C4×D17—21no—
67C2×C4×D17—21no—
68C2×C4×D17—21no—
69C2×C4×D17—42no—
70C2×C4×D17—42no—
71C22⋊#595—24no—
72C22⋊#595—42no—
73C2×#3624—42no—
74C4×#595—82no—
75#63243—82no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C42×C17—21no—
80C42×C17—21no—
81#63227—41no—
82#63227—41no—
83#63227—44no—
84#63227—44no—
85C4⋊Dic17—41no—
86C4⋊Dic17—42no—
87C4×Dic17—42no—
88#11777—171nomaximal, Sylow
89C23×C17C2×C22⋊C411yescharacteristic
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—41no—
96C22×D17—41no—
97C22×D17—41no—
98C22×D17—41no—
99C22×D17AGL(1,ℤ/8)12yes—
100C2×#595—44no—
101C2×#595—82no—
102C2×#595—82no—
103Q8×C17—84no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—41no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17#9911yescharacteristic
113C2×C4×C17#9911yescharacteristic
114C2×C4×C17#10412yes—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122D68—44no—
123C4×D17—42no—
124C4×D17—42no—
125C4×D17—81no—
126C4×D17—81no—
127#3624—82no—
128#2547—171no—
129C4×#99—171no—
130#1465—171no—
131#1344—174no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17C2×#9911yescharacteristic
135C22×C17#41811yescharacteristic
136C22×C17C2×#10411yescharacteristic
137C2×D17—22no—
138C2×D17—41no—
139C2×D17—41no—
140C2×D17—42no—
141C2×D17—42no—
142C2×D17—41no—
143C2×D17—41no—
144C2×D17—41no—
145C2×D17—81no—
146C2×D17—81no—
147#595—84no—
148C4×C17—21no—
149C4×C17—22no—
150C4×C17—41no—
151C4×C17—82no—
152C4×C17#35412yes—
153Dic17—41no—
154Dic17—41no—
155Dic17—82no—
156C2×C4⋊Q8#59511yescharacteristic, complemented
157C2×#124—341no—
158C2×C4×D4—171no—
159C2×#99—341no—
160#407—341no—
161#397—172no—
162#397—172no—
163#391—341no—
164#380—341no—
165#380—341no—
166C4×C4⋊C4—171no—
167C4×C22⋊C4—341no—
168#342—172no—
169#324—341no—
170#323—172no—
171C2×C17—21no—
172C2×C17—21no—
173C2×C17#150711yescharacteristic
174C2×C17#212311yescharacteristic
175C2×C17C2×#35411yescharacteristic
176D17—42no—
177D17—81no—
178C22×Q8—21no—
179C22×D4—171no—
180C23×C4—171no—
181C23×C4—171no—
182C2×M16—681no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185#124—684no—
186C4×D4—681no—
187C4×D4—681no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4C2×#59512yes—
192C2×C22⋊C4—171no—
193C2×C22⋊C4—171no—
194C2×C22⋊C4—342no—
195C2×C22⋊C4—342no—
196C2×C22⋊C4—342no—
197C2×C42—341no—
198C2×C42—681no—
199C2×C42C2×#59511yescharacteristic
200#99—344no—
201C22⋊C8—342no—
202C22⋊C8—681no—
203#95—172no—
204#95—342no—
205#95—342no—
206#95—681no—
207C17#1177711yescharacteristic, complemented, Sylow
208C24—171no—
209C24—171no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×D4—342no—
214C2×D4—342no—
215C2×D4—681no—
216C22×C4—22no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—341no—
220C22×C4—341no—
221C22×C4—341no—
222C22×C4—341no—
223C22×C4—342no—
224C22×C4—682no—
225C22×C4—682no—
226C22×C4—682no—
227C22×C4C22⋊#59511yescharacteristic
228C22×C4C22⋊#59511yescharacteristic
229C22×C4C22×#59511yescharacteristic
230M16—1362no—
231C2×C8—682no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—681no—
236C4⋊C4—682no—
237C22⋊C4—344no—
238C22⋊C4—681no—
239C22⋊C4—681no—
240C22⋊C4—684no—
241C22⋊C4—684no—
242C22⋊C4—682no—
243C42—21no—
244C42—21no—
245C42—682no—
246C42—1362no—
247C23—172no—
248C23—341no—
249C23—341no—
250C23—341no—
251C23—341no—
252C23—341no—
253C23—681no—
254C23—681no—
255C23—681no—
256C23—681no—
257C23C2×C22⋊#59511yescharacteristic
258Q8—84no—
259D4—684no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—341no—
269C2×C4—341no—
270C2×C4—682no—
271C2×C4—682no—
272C2×C4—681no—
273C2×C4—681no—
274C2×C4—682no—
275C2×C4—682no—
276C2×C4—684no—
277C2×C4—682no—
278C2×C4—1361no—
279C2×C4—1361no—
280C2×C4—1362no—
281C2×C4—1362no—
282C2×C4#1058754812yes—
283C2×C4#1058755111yescharacteristic
284C2×C4#1058755111yescharacteristic
285C8—1362no—
286C22—21no—
287C22—21no—
288C22—342no—
289C22—682no—
290C22—682no—
291C22—681no—
292C22—681no—
293C22—681no—
294C22—681no—
295C22—681no—
296C22—1361no—
297C22—1361no—
298C22C2×#1058755111yescharacteristic
299C22#1176380111yescharacteristic
300C22C2×#1058754811yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—82no—
305C4—681no—
306C4—681no—
307C4—1362no—
308C4—1364no—
309C4#1176371512yes—
310C2—21no—
311C2—21no—
312C2—682no—
313C2—1361no—
314C2#1454418011yescharacteristic
315C2#1454504611yescharacteristic
316C2C2×#1176371511yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC22×D68D4
cosocleC2×#63227C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11777
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543764C2semidirect
13. #11762735C4semidirect
18. C4×C4⋊Dic17C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
156. C2×C4⋊Q8#595semidirect
207. C17#11777semidirect

Maximal subgroups

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

subgroupindex
C4×#105875392
#145440412
#145437642
#145337802
#1177717