Finite Group Databasebeta

← #17961560

Subgroups — #17961560

Order 4352.

670 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 76 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#14544615C212yesmaximal
2#14543879C212yesmaximal, complemented
3#14543878C212yesmaximal, complemented
4#14531273C211yesmaximal, characteristic
5C2×#10587539—22no—
6C2×#10587539C2214yes—
7#11763752—22no—
8C22×#63256C2211yescharacteristic
9C2×#10587578C411yescharacteristic, complemented
10C2×C42×D17C411yescharacteristic, complemented
11#11762716C411yescharacteristic, complemented
12#11762711C2211yescharacteristic
13#11762710C2211yescharacteristic
14#11762687—22no—
15#11762656—22no—
16C17×#391C411yescharacteristic, complemented
17C22×Q8×C17C2×C411yescharacteristic
18C2×#63256—24no—
19C2×#63256—42no—
20C2×#63256—42no—
21C22×D68D412yes—
22C22×D68C2311yescharacteristic
23C22×C4×D17—21no—
24C22×C4×D17C2×C411yescharacteristic
25C22×C4×D17D412yes—
26C2×C22⋊#595—24no—
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C42×C17C2×C411yescharacteristic
29C2×#63227—21no—
30C2×#63227—22no—
31C2×#63227—22no—
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×C4×Dic17—21no—
34C2×C4×Dic17—21no—
35C2×C4×Dic17C2×C411yescharacteristic
36#10587578—42no—
37C42×D17—42no—
38(C2×#595)⋊C4—44no—
39#10587539—28no—
40#10587539—44no—
41C17×#95—22no—
42#10587502—22no—
43C23×D17C22⋊C411yescharacteristic
44C23×D17C2×D412yes—
45C22×#595—44no—
46C2×Q8×C17—22no—
47C2×Q8×C17—41no—
48C2×Q8×C17—41no—
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—21no—
54C22×Dic17—21no—
55C22×Dic17C22⋊C411yescharacteristic
56#63256—416no—
57C2×D68—24no—
58C2×D68—24no—
59C2×D68—42no—
60C2×D68—41no—
61C2×D68—41no—
62C2×D68#4114yes—
63C2×C4×D17—22no—
64C2×C4×D17—22no—
65C2×C4×D17—22no—
66C2×C4×D17—24no—
67C2×C4×D17—24no—
68C2×C4×D17—42no—
69C2×C4×D17—42no—
70C2×C4×D17—42no—
71C2×C4×D17—42no—
72C22⋊#595—44no—
73C22⋊#595—44no—
74C4⋊C4×C17—41no—
75C42×C17—41no—
76#63227—44no—
77#63227—44no—
78#63227—42no—
79C4⋊Dic17—41no—
80C4×Dic17—42no—
81C4×Dic17—42no—
82C4×Dic17—41no—
83#12857—171nomaximal, Sylow
84C23×C17C2×C22⋊C411yescharacteristic
85C22×D17—21no—
86C22×D17—22no—
87C22×D17—21no—
88C22×D17—22no—
89C22×D17—41no—
90C22×D17—41no—
91C22×D17—42no—
92C22×D17—42no—
93C22×D17#9912yes—
94C22×D17#12314yes—
95C2×#595—84no—
96C2×#595—84no—
97C2×#595—84no—
98Q8×C17—44no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17—41no—
109C2×C4×C17#11712yes—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—22no—
113C2×Dic17—41no—
114C2×Dic17—41no—
115C2×Dic17—42no—
116C2×Dic17—41no—
117C2×Dic17#9912yes—
118D68—44no—
119D68—48no—
120C4×D17—28no—
121C4×D17—48no—
122C4×D17—48no—
123#2297—171no—
124#1782—172no—
125#1393—172no—
126#1393—172no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17#38111yescharacteristic
130C22×C17C2×#9911yescharacteristic
131C22×C17C2×#9911yescharacteristic
132C2×D17—22no—
133C2×D17—24no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—41no—
138C2×D17—41no—
139C2×D17—42no—
140C2×D17—44no—
141C2×D17—82no—
142#595—164no—
143C4×C17—22no—
144C4×C17—42no—
145C4×C17—42no—
146C4×C17—81no—
147Dic17—22no—
148Dic17—42no—
149Dic17—81no—
150C22×#41—171no—
151C2×#117—171no—
152C22×C42—171no—
153C2×#99—174no—
154C2×#99—342no—
155#391—171no—
156#391#59511yescharacteristic, complemented
157#390—171no—
158#390—171no—
159#386—342no—
160#381—342no—
161C4×C22⋊C4—342no—
162C2×C17—21no—
163C2×C17—21no—
164C2×C17#181911yescharacteristic
165C2×C17#212611yescharacteristic
166C2×C17#212611yescharacteristic
167D17—44no—
168D17—42no—
169C2×#41—344no—
170C2×#41—682no—
171C2×#41—682no—
172C22×Q8C2×#59511yescharacteristic
173C22×D4—172no—
174C22×D4—171no—
175C23×C4—171no—
176C23×C4—172no—
177C23×C4—341no—
178#117—682no—
179C2×C4⋊C4—171no—
180C2×C4⋊C4C2×#59511yescharacteristic
181C2×C22⋊C4—341no—
182C2×C22⋊C4—342no—
183C2×C22⋊C4—342no—
184C2×C22⋊C4—344no—
185C2×C42—171no—
186C2×C42—341no—
187C2×C42—341no—
188C2×C42—682no—
189C2×C42C2×#59511yescharacteristic
190#99—348no—
191#99—684no—
192#95—22no—
193#95—342no—
194#95—684no—
195C17#1285711yescharacteristic, complemented, Sylow
196C24—171no—
197C24—172no—
198#41—6816no—
199C2×Q8—22no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×D4—174no—
203C2×D4—344no—
204C2×D4—344no—
205C2×D4—681no—
206C2×D4—681no—
207C2×D4—682no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—171no—
211C22×C4—342no—
212C22×C4—344no—
213C22×C4—342no—
214C22×C4—344no—
215C22×C4—342no—
216C22×C4—341no—
217C22×C4—341no—
218C22×C4—682no—
219C22×C4—682no—
220C22×C4—682no—
221C22×C4—682no—
222C22×C4—684no—
223C22×C4C22⋊#59512yes—
224C22×C4C22×#59511yescharacteristic
225C4⋊C4—41no—
226C4⋊C4—681no—
227C22⋊C4—684no—
228C22⋊C4—684no—
229C22⋊C4—682no—
230C22⋊C4—684no—
231C22⋊C4—684no—
232C42—41no—
233C42—681no—
234C42—682no—
235C42—682no—
236C23—174no—
237C23—172no—
238C23—341no—
239C23—341no—
240C23—342no—
241C23—342no—
242C23—681no—
243C23—681no—
244C23—682no—
245C23—682no—
246C23C2×C22⋊#59511yescharacteristic
247Q8—44no—
248D4—684no—
249D4—688no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—172no—
261C2×C4—348no—
262C2×C4—341no—
263C2×C4—341no—
264C2×C4—342no—
265C2×C4—688no—
266C2×C4—688no—
267C2×C4—681no—
268C2×C4—681no—
269C2×C4—682no—
270C2×C4—681no—
271C2×C4—1364no—
272C2×C4—1364no—
273C2×C4—1364no—
274C2×C4#1058767712yes—
275C22—21no—
276C22—21no—
277C22—342no—
278C22—344no—
279C22—682no—
280C22—684no—
281C22—682no—
282C22—682no—
283C22—681no—
284C22—682no—
285C22—681no—
286C22—1362no—
287C22#1176375211yescharacteristic
288C22D17⋊#9911yescharacteristic
289C22D17⋊#9911yescharacteristic
290C4—22no—
291C4—42no—
292C4—42no—
293C4—81no—
294C4—2724no—
295C4—342no—
296C4—682no—
297C4—1361no—
298C2—21no—
299C2—21no—
300C2—684no—
301C2—682no—
302C2#1454468011yescharacteristic
303C2#1454505611yescharacteristic
304C2#1454505811yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC22×C17C2×#99
prime radicalC22×#63256C22
cosocleC22×D68C23
hypercenter#391#595
hypocenterC17#12857
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #14543879C2semidirect
3. #14543878C2semidirect
9. C2×#10587578C4semidirect
10. C2×C42×D17C4semidirect
11. #11762716C4semidirect
16. C17×#391C4semidirect
156. #391#595semidirect
195. C17#12857semidirect

Maximal subgroups

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

subgroupindex
#145446152
#145438792
#145438782
#145312732
#1285717