Finite Group Databasebeta

← #17949656

Subgroups — #17949656

Order 4352.

690 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 118 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×#10587543C212yesmaximal, complemented
2#14543604C212yesmaximal, complemented
3#14543601C212yesmaximal, complemented
4#14531273C211yesmaximal, characteristic
5C2×#10587543—22no—
6C2×C4×#3624—22no—
7#11763675C2214yes—
8C22×#63256C411yescharacteristic
9C2×#10587578C411yescharacteristic
10C2×C42×D17C2211yescharacteristic
11#11762716C2211yescharacteristic
12#11762711C411yescharacteristic
13#11762710C2211yescharacteristic
14#11762687—22no—
15#11762656—22no—
16C17×#391C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C2×#63256—22no—
19C2×#63256—42no—
20C2×#63256—42no—
21C2×#63256C814yescomplemented
22C22×D68—21no—
23C22×D68C2×C411yescharacteristic
24C22×C4×D17—21no—
25C22×C4×D17C2×C411yescharacteristic
26C22×C4×D17D412yes—
27C22×#3624—24no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C42×C17C2×C411yescharacteristic
30C2×#63227—21no—
31C2×#63227—22no—
32C2×#63227—22no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4×Dic17—21no—
35C2×C4×Dic17D412yes—
36C2×C4×Dic17C2311yescharacteristic
37#10587578—42no—
38C42×D17—42no—
39#10587543—28no—
40C4×#3624—44no—
41C17×#95—22no—
42#10587502—22no—
43C23×D17—21no—
44C23×D17C22⋊C411yescharacteristic
45C2×Q8×C17—21no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17C2×C812yes—
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—21no—
54C22×Dic17C22⋊C411yescharacteristic
55C22×Dic17C2×D412yes—
56#63256—48no—
57#63256—48no—
58C2×D68—24no—
59C2×D68—21no—
60C2×D68—41no—
61C2×D68—41no—
62C2×D68—42no—
63C2×D68—42no—
64C2×D68C2×C812yes—
65C2×C4×D17—22no—
66C2×C4×D17—22no—
67C2×C4×D17—24no—
68C2×C4×D17—24no—
69C2×C4×D17—24no—
70C2×C4×D17—42no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C2×C4×D17C2×C812yes—
74C2×C4×D17M1612yes—
75C2×#3624—28no—
76C2×#3624—44no—
77C2×#3624—44no—
78C4⋊C4×C17—41no—
79C42×C17—41no—
80#63227—42no—
81#63227—44no—
82#63227—44no—
83C4⋊Dic17—41no—
84C4×Dic17—24no—
85C4×Dic17—41no—
86C4×Dic17—42no—
87#7563—171nomaximal, Sylow
88C23×C17C2×C22⋊C411yescharacteristic
89C22×D17—21no—
90C22×D17—21no—
91C22×D17—22no—
92C22×D17—22no—
93C22×D17—22no—
94C22×D17—41no—
95C22×D17—41no—
96C22×D17—42no—
97C22×D17—42no—
98C22×D17C22⋊C812yes—
99Q8×C17—42no—
100Q8×C17—42no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—42no—
110C2×C4×C17#10414yes—
111C2×C4×C17C22×C811yescharacteristic
112C2×C4×C17C2×M1611yescharacteristic
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—41no—
116C2×Dic17—42no—
117C2×Dic17C22⋊C812yes—
118C2×Dic17#9912yes—
119C2×Dic17#10012yes—
120C2×Dic17#13514yes—
121D68—42no—
122D68—42no—
123D68—48no—
124C4×D17—24no—
125C4×D17—24no—
126C4×D17—48no—
127C4×D17—48no—
128#3624—48no—
129#2297—171no—
130C4×C22⋊C8—172no—
131#1272—172no—
132#1270—172no—
133C22×C17C2×C22⋊C811yescharacteristic
134C22×C17C2×#9911yescharacteristic
135C22×C17C2×#10011yescharacteristic
136C22×C17#41612yes—
137C22×C17C2×#10412yes—
138C2×D17—22no—
139C2×D17—24no—
140C2×D17—41no—
141C2×D17—42no—
142C2×D17—41no—
143C2×D17—42no—
144C2×D17—44no—
145C2×D17—81no—
146C2×D17—82no—
147C2×D17—81no—
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—24no—
151C4×C17—42no—
152C4×C17—81no—
153Dic17—24no—
154Dic17—81no—
155Dic17#32314yes—
156C22×#41—171no—
157C2×#117—171no—
158C22×C42—171no—
159C2×C22⋊C8—342no—
160C2×C4×C8—342no—
161#391—171no—
162#391#59511yescharacteristic
163#390—171no—
164#390—171no—
165#386—342no—
166C4×C22⋊C4—342no—
167#336—174no—
168C2×C17C2×#32311yescharacteristic
169C2×C17#147612yes—
170C2×C17#150112yes—
171C2×C17#152511yescharacteristic
172C2×C17#154311yescharacteristic
173D17—42no—
174D17—82no—
175C2×#41—174no—
176C2×#41—342no—
177C2×#41—682no—
178C2×#41—682no—
179C22×Q8C2×#59511yescharacteristic
180C22×D4—171no—
181C22×D4—341no—
182C23×C4—171no—
183C23×C4—172no—
184C23×C4—341no—
185C22×C8—344no—
186#117—682no—
187C2×C4⋊C4—171no—
188C2×C4⋊C4C2×#59511yescharacteristic
189C2×C22⋊C4—342no—
190C2×C22⋊C4—341no—
191C2×C22⋊C4—342no—
192C2×C42—172no—
193C2×C42—171no—
194C2×C42—341no—
195C2×C42—682no—
196C2×C42C2×#59511yescharacteristic
197C22⋊C8—348no—
198C4×C8—684no—
199#95—22no—
200#95—342no—
201C17#756311yescharacteristic, complemented, Sylow
202C24—171no—
203C24—341no—
204#41—688no—
205#41—688no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8#6324212yes—
210C2×D4—172no—
211C2×D4—344no—
212C2×D4—341no—
213C2×D4—681no—
214C2×D4—681no—
215C2×D4—682no—
216C2×D4—682no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—171no—
220C22×C4—172no—
221C22×C4—172no—
222C22×C4—172no—
223C22×C4—342no—
224C22×C4—342no—
225C22×C4—344no—
226C22×C4—344no—
227C22×C4—344no—
228C22×C4—341no—
229C22×C4—682no—
230C22×C4—682no—
231C22×C4—682no—
232C22×C4C22⋊#59512yes—
233C22×C4C22×#59511yescharacteristic
234C2×C8—348no—
235C2×C8—684no—
236C2×C8—684no—
237C4⋊C4—41no—
238C4⋊C4—681no—
239C22⋊C4—684no—
240C22⋊C4—682no—
241C22⋊C4—684no—
242C42—41no—
243C42—344no—
244C42—681no—
245C42—682no—
246C23—172no—
247C23—341no—
248C23—341no—
249C23—342no—
250C23—342no—
251C23—342no—
252C23—681no—
253C23—681no—
254C23—682no—
255C23—682no—
256C23C2×C22⋊#59511yescharacteristic
257Q8—42no—
258Q8—42no—
259D4—682no—
260D4—682no—
261D4—688no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—172no—
272C2×C4—172no—
273C2×C4—172no—
274C2×C4—174no—
275C2×C4—344no—
276C2×C4—344no—
277C2×C4—342no—
278C2×C4—342no—
279C2×C4—688no—
280C2×C4—688no—
281C2×C4—681no—
282C2×C4—682no—
283C2×C4#1058755014yes—
284C2×C4C2×#6324211yescharacteristic
285C2×C4D17⋊M1611yescharacteristic
286C8—688no—
287C22—342no—
288C22—344no—
289C22—682no—
290C22—681no—
291C22—681no—
292C22—682no—
293C22—684no—
294C22—1362no—
295C22—1361no—
296C22—1361no—
297C22#1176377711yescharacteristic
298C22D17⋊#9911yescharacteristic
299C22D17⋊#10011yescharacteristic
300C22#1176379912yes—
301C22C2×#1058755012yes—
302C4—24no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—81no—
307C4—174no—
308C4—344no—
309C4—1361no—
310C2—682no—
311C2—1362no—
312C2#1454401011yescharacteristic
313C2#1454407812yes—
314C2#1454416212yes—
315C2#1454424011yescharacteristic
316C2#1454428811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC23×C17C2×C22⋊C4
prime radicalC22×#63256C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#7563
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C4×#10587543C2semidirect
2. #14543604C2semidirect
3. #14543601C2semidirect
21. C2×#63256C8semidirect
201. C17#7563semidirect

Maximal subgroups

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

subgroupindex
C4×#105875432
#145436042
#145436012
#145312732
#756317