Finite Group Databasebeta

← #16950318

Subgroups — #16950318

Order 4352.

524 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 93 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#14531494C211yesmaximal, characteristic
2C2×#11762603C212yesmaximal
3#14528770C212yesmaximal
4#14526909C211yesmaximal, characteristic
5C17×#1393C211yesmaximal, characteristic
6C2×#10587622C2211yescharacteristic
7C2×#10587573C411yescharacteristic, complemented
8C2×D4×Dic17C2211yescharacteristic
9C2×#10587505C2212yes—
10#11762715C411yescharacteristic, complemented
11#11762703—22no—
12#11762681—22no—
13#11762678C411yescharacteristic, complemented
14#11762667—22no—
15#11762655C411yescharacteristic, complemented
16#11762603—24no—
17#11762575—22no—
18C2×C17×#99C2212yes—
19C17×#390C2211yescharacteristic
20C22×D4×C17C2311yescharacteristic
21C23×Dic17D412yes—
22C22×Dic34C2×C411yescharacteristic
23C2×C4⋊C4×C17C2×C411yescharacteristic
24C2×C22⋊C4×C17—22no—
25C2×C22⋊C4×C17—22no—
26C2×C42×C17C2×C411yescharacteristic
27C2×C22⋊Dic17—22no—
28C2×C22⋊Dic17—22no—
29C2×C22⋊Dic17D412yes—
30#10587622—22no—
31#10587622D412yes—
32#10587622Q812yes—
33D4×Dic17—42no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×#63225—21no—
36C2×#63225—22no—
37C2×#63225—42no—
38C2×C4×Dic17—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17C2×C411yescharacteristic
41#10587573—42no—
42C17×#99—22no—
43C17×#99—22no—
44#10587505—22no—
45#10587505—22no—
46#10587502—21no—
47#10587502—22no—
48#10587502—21no—
49#10587502—42no—
50C24×C17C2×D412yes—
51C2×D4×C17—41no—
52C2×D4×C17—41no—
53C2×D4×C17C2×D411yescharacteristic
54C2×D4×C17C2×Q811yescharacteristic
55C2×D4×C17#4112yes—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—42no—
59C22×C4×C17C22×C411yescharacteristic
60C22×Dic17—21no—
61C22×Dic17—21no—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—42no—
65C22×Dic17—82no—
66C22×Dic17C22⋊C412yes—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C2×Dic34—21no—
70C2×Dic34—41no—
71C2×Dic34—41no—
72C2×Dic34C4⋊C412yescomplemented
73C4⋊C4×C17—41no—
74C22⋊C4×C17—42no—
75C22⋊C4×C17—42no—
76C22⋊C4×C17—82no—
77C42×C17—41no—
78C22⋊Dic17—24no—
79C22⋊Dic17—24no—
80C22⋊Dic17—42no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C4⋊Dic17—41no—
84#63225—42no—
85#63225—82no—
86#63225—84no—
87C4×Dic17—21no—
88C4×Dic17—22no—
89C4×Dic17—42no—
90C4×Dic17C4⋊C412yescomplemented
91#12800—171nomaximal, Sylow
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17C2×C22⋊C411yescharacteristic
97C23×C17#12112yes—
98C23×C17C22⋊Q812yes—
99D4×C17—41no—
100D4×C17—41no—
101D4×C17—42no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—82no—
110C2×C4×C17—82no—
111C2×C4×C17—82no—
112C2×C4×C17C2×C4⋊C411yescharacteristic
113C2×C4×C17#11711yescharacteristic
114C2×Dic17—22no—
115C2×Dic17—22no—
116C2×Dic17—21no—
117C2×Dic17—22no—
118C2×Dic17—41no—
119C2×Dic17—44no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—42no—
125C2×Dic17—41no—
126C2×Dic17—82no—
127C2×Dic17—82no—
128C2×Dic17—82no—
129C2×Dic17—82no—
130C2×Dic17#9912yes—
131Dic34—42no—
132Dic34—42no—
133#2326—171no—
134C2×#354—172no—
135#1799—172no—
136#1393—171no—
137#1393D1711yescharacteristic
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—42no—
146C22×C17—82no—
147C22×C17#38011yescharacteristic
148C22×C17C2×#9911yescharacteristic
149C22×C17#41011yescharacteristic
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—81no—
153C4×C17—81no—
154C4×C17—162no—
155Dic17—22no—
156Dic17—42no—
157Dic17—42no—
158Dic17—42no—
159Dic17—41no—
160Dic17—162no—
161C2×#124—171no—
162C2×C4×Q8—171no—
163C2×C4×D4—171no—
164C2×#99—172no—
165C2×#99C2×D1712yes—
166#391—171no—
167#390C2×D1711yescharacteristic
168#388—342no—
169#385—342no—
170#384—171no—
171#380—342no—
172#376—171no—
173#354—344no—
174#342—342no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17—42no—
178C2×C17—42no—
179C2×C17#179911yescharacteristic
180C2×C17#212711yescharacteristic
181C2×C17#212911yescharacteristic
182C22×Q8—171no—
183C22×D4C22×D1711yescharacteristic
184C23×C4—172no—
185#124—172no—
186#124—172no—
187#124—342no—
188C4×Q8—682no—
189C4×D4—682no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4—342no—
193C2×C4⋊C4—682no—
194C2×C4⋊C4C4×D1711yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—172no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4—342no—
200C2×C42—171no—
201C2×C42—171no—
202C2×C42—341no—
203C2×C42C4×D1711yescharacteristic
204#99—22no—
205#99—22no—
206#99—342no—
207#99—342no—
208#95—341no—
209#95—341no—
210#95—342no—
211#95—682no—
212C17#1280011yescharacteristic, complemented, Sylow
213C24D4×D1712yes—
214C2×Q8—172no—
215C2×Q8—341no—
216C2×Q8—681no—
217C2×Q8—681no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#6325212yes—
221C2×D4D4×D1711yescharacteristic
222C2×D4Q8×D1711yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—172no—
229C22×C4—342no—
230C22×C4—342no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4—682no—
234C22×C4—1362no—
235C22×C4C2×C4×D1711yescharacteristic
236C4⋊C4—41no—
237C4⋊C4—681no—
238C4⋊C4—682no—
239C4⋊C4—1362no—
240C4⋊C4—1364no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—344no—
245C22⋊C4—344no—
246C22⋊C4—682no—
247C22⋊C4—682no—
248C22⋊C4—682no—
249C42—41no—
250C42—172no—
251C42—342no—
252C42—341no—
253C42—682no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23C22⋊C4×D1711yescharacteristic
259C23#1058757012yes—
260C23#1058758212yes—
261Q8—682no—
262Q8—682no—
263D4—42no—
264D4—41no—
265D4—41no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—172no—
277C2×C4—341no—
278C2×C4—342no—
279C2×C4—342no—
280C2×C4—342no—
281C2×C4—681no—
282C2×C4—684no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—682no—
288C2×C4—681no—
289C2×C4—1362no—
290C2×C4—1362no—
291C2×C4—1362no—
292C2×C4—1362no—
293C2×C4#1058755811yescharacteristic
294C2×C4C4⋊C4×D1711yescharacteristic
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22#1176266811yescharacteristic
305C22#99×D1711yescharacteristic
306C22#1176278111yescharacteristic
307C4—21no—
308C4—21no—
309C4—81no—
310C4—81no—
311C4—162no—
312C4—2722no—
313C4—342no—
314C4—682no—
315C4—682no—
316C4—682no—
317C4—681no—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—42no—
322C2#1452877111yescharacteristic
323C2#1453045011yescharacteristic
324C2#1453045711yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1393C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1393D17
hypocenterC17#12800
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
7. C2×#10587573C4semidirect
10. #11762715C4semidirect
13. #11762678C4semidirect
15. #11762655C4semidirect
72. C2×Dic34C4⋊C4semidirect
90. C4×Dic17C4⋊C4semidirect
212. C17#12800semidirect

Maximal subgroups

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

subgroupindex
#145314942
C2×#117626032
#145287702
#145269092
C17×#13932
#1280017