Finite Group Databasebeta

← #17961424

Subgroups — #17961424

Order 4352.

524 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C2×#11763715C212yesmaximal
2#14544647C212yesmaximal
3#14543878C212yesmaximal
4#14531498C211yesmaximal, characteristic
5C2×#10587539C2214yes—
6#11763751—22no—
7#11763715—24no—
8#11763687—22no—
9C2×#10587633C2211yescharacteristic
10C2×Q8×Dic17C411yescharacteristic, complemented
11C2×#10587579C2211yescharacteristic
12#11762711C2211yescharacteristic
13#11762698—22no—
14#11762683—22no—
15#11762678C411yescharacteristic, complemented
16#11762655C411yescharacteristic, complemented
17C17×#391C411yescharacteristic, complemented
18C22×Q8×C17C2×C411yescharacteristic
19C22×D68C2311yescharacteristic
20C22×C4×D17D412yes—
21C2×C22⋊#595—24no—
22C2×C4⋊#595—42no—
23C2×C4⋊C4×C17C2×C411yescharacteristic
24C2×C42×C17C2×C411yescharacteristic
25#10587633—22no—
26#10587633D412yes—
27#10587633Q812yes—
28Q8×Dic17—42no—
29C2×#63227—22no—
30C2×#63227—22no—
31C2×#63227D412yes—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×#63225—22no—
35C2×C4×Dic17—21no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17C2×C411yescharacteristic
38#10587579—42no—
39(C2×#595)⋊C4—42no—
40#10587539—24no—
41#10587539—24no—
42C17×#95—22no—
43#10587502—21no—
44#10587502—21no—
45C23×D17C2×D412yes—
46C22×#595—44no—
47C2×Q8×C17—21no—
48C2×Q8×C17—41no—
49C2×Q8×C17—41no—
50C2×Q8×C17C4⋊C412yescomplemented
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17C22⋊C412yes—
54C22×C4×C17C22×C411yescharacteristic
55C22×Dic17—21no—
56C22×Dic17—21no—
57C22×Dic17C22⋊C411yescharacteristic
58C22×Dic17C22⋊C411yescharacteristic
59C2×D68—41no—
60C2×D68—41no—
61C2×D68C2×D411yescharacteristic
62C2×D68C2×Q811yescharacteristic
63C2×D68#4112yes—
64C2×C4×D17—22no—
65C2×C4×D17—22no—
66C2×C4×D17—82no—
67C22⋊#595—44no—
68C22⋊#595—44no—
69C4⋊#595—84no—
70C4⋊C4×C17—41no—
71C42×C17—41no—
72#63227—24no—
73#63227—24no—
74#63227—42no—
75#63227—82no—
76C4⋊Dic17—42no—
77C4⋊Dic17—41no—
78#63225—82no—
79C4×Dic17—21no—
80C4×Dic17—22no—
81C4×Dic17—42no—
82C4×Dic17C4⋊C412yescomplemented
83#12800—171nomaximal, Sylow
84C23×C17C2×C22⋊C411yescharacteristic
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—42no—
88C22×D17—42no—
89C22×D17#12112yes—
90C22×D17C22⋊Q812yes—
91C2×#595—84no—
92C2×#595—84no—
93C2×#595—84no—
94Q8×C17—42no—
95Q8×C17—42no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17C2×C4⋊C411yescharacteristic
107C2×C4×C17#11711yescharacteristic
108C2×Dic17—22no—
109C2×Dic17—21no—
110C2×Dic17—41no—
111C2×Dic17—41no—
112C2×Dic17—41no—
113C2×Dic17—41no—
114C2×Dic17—42no—
115C2×Dic17—41no—
116C2×Dic17#9912yes—
117D68—41no—
118D68—41no—
119D68—42no—
120C4×D17—44no—
121C4×D17—82no—
122#2326—171no—
123C2×#354—172no—
124#1799—172no—
125#1393—172no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17#38011yescharacteristic
129C22×C17C2×#9911yescharacteristic
130C22×C17#41011yescharacteristic
131C2×D17—22no—
132C2×D17—22no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—82no—
138#595—164no—
139C4×C17—21no—
140C4×C17—21no—
141C4×C17—42no—
142C4×C17—42no—
143C4×C17—81no—
144Dic17—22no—
145Dic17—42no—
146Dic17—41no—
147Dic17—81no—
148C2×#124—171no—
149C2×C4×Q8—171no—
150C2×C4×D4—171no—
151C2×#99—174no—
152#391#59511yescharacteristic, complemented
153#390—171no—
154#388—342no—
155#385—342no—
156#384—171no—
157#380—342no—
158#376—171no—
159#354—344no—
160#342—342no—
161C2×C17—21no—
162C2×C17—21no—
163C2×C17#179911yescharacteristic
164C2×C17#212711yescharacteristic
165C2×C17#212911yescharacteristic
166D17—42no—
167D17—42no—
168C22×Q8C2×#59511yescharacteristic
169C22×D4—171no—
170C23×C4—172no—
171#124—172no—
172#124—172no—
173#124—342no—
174C4×Q8—682no—
175C4×D4—682no—
176C2×C4⋊C4—171no—
177C2×C4⋊C4—341no—
178C2×C4⋊C4—342no—
179C2×C4⋊C4—682no—
180C2×C4⋊C4C2×#59511yescharacteristic
181C2×C22⋊C4—172no—
182C2×C22⋊C4—342no—
183C2×C22⋊C4—344no—
184C2×C22⋊C4—342no—
185C2×C42—171no—
186C2×C42—171no—
187C2×C42—341no—
188C2×C42C2×#59511yescharacteristic
189#99—344no—
190#99—344no—
191#95—22no—
192#95—341no—
193#95—341no—
194#95—682no—
195C17#1280011yescharacteristic, complemented, Sylow
196C24—172no—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C4⋊#59512yescomplemented
201C2×D4—171no—
202C2×D4—171no—
203C2×D4—172no—
204C2×D4—681no—
205C2×D4—681no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—171no—
209C22×C4—171no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4—341no—
213C22×C4—341no—
214C22×C4—684no—
215C22×C4—1362no—
216C22×C4C22⋊#59512yes—
217C22×C4C22×#59511yescharacteristic
218C4⋊C4—41no—
219C4⋊C4—681no—
220C4⋊C4—682no—
221C4⋊C4—1362no—
222C4⋊C4—1364no—
223C22⋊C4—344no—
224C22⋊C4—344no—
225C22⋊C4—682no—
226C22⋊C4—684no—
227C22⋊C4—684no—
228C22⋊C4—1362no—
229C42—41no—
230C42—172no—
231C42—342no—
232C42—341no—
233C42—682no—
234C23—172no—
235C23—172no—
236C23—342no—
237C23—342no—
238C23—682no—
239C23—682no—
240C23C2×C22⋊#59511yescharacteristic
241Q8—42no—
242Q8—42no—
243D4—681no—
244D4—681no—
245D4—682no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—172no—
257C2×C4—342no—
258C2×C4—341no—
259C2×C4—684no—
260C2×C4—681no—
261C2×C4—681no—
262C2×C4—681no—
263C2×C4—681no—
264C2×C4—682no—
265C2×C4—681no—
266C2×C4—1362no—
267C2×C4—1364no—
268C2×C4—1364no—
269C2×C4—1364no—
270C2×C4C2×C4⋊#59511yescharacteristic
271C2×C4#1058767711yescharacteristic
272C22—21no—
273C22—21no—
274C22—342no—
275C22—342no—
276C22—682no—
277C22—682no—
278C22—682no—
279C22—682no—
280C22—1362no—
281C22#1176375111yescharacteristic
282C22D17⋊#9911yescharacteristic
283C22#1176378911yescharacteristic
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—81no—
289C4—2724no—
290C4—342no—
291C4—682no—
292C4—681no—
293C4—1361no—
294C2—21no—
295C2—21no—
296C2—682no—
297C2—682no—
298C2#1454464611yescharacteristic
299C2#1454506211yescharacteristic
300C2#1454506711yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC22×C17C2×#99
prime radical, cosocleC22×D68C23
hypercenter#391#595
hypocenterC17#12800
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #391

Complements

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

NH ≅ G/Ntype
10. C2×Q8×Dic17C4semidirect
15. #11762678C4semidirect
16. #11762655C4semidirect
17. C17×#391C4semidirect
50. C2×Q8×C17C4⋊C4semidirect
82. C4×Dic17C4⋊C4semidirect
152. #391#595semidirect
195. C17#12800semidirect
200. C2×Q8C4⋊#595semidirect

Maximal subgroups

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

subgroupindex
C2×#117637152
#145446472
#145438782
#145314982
#1280017