Finite Group Databasebeta

← #17962508

Subgroups — #17962508

Order 4352.

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

normal subgroups only

This group has 488 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
1#14544648C211yesmaximal, characteristic
2C4×#10587551C211yesmaximal, characteristic
3#14543869C214yesmaximal
4#14533102C211yesmaximal, characteristic
5#11763812—21no—
6C2×#10587553C2212yes—
7C2×#10587551C2212yes—
8C4×C22⋊#595—21no—
9#11763689—21no—
10#11763687—21no—
11C2×C4×C22⋊D17C2211yescharacteristic
12C22×C4⋊Dic17C411yescharacteristic, complemented
13#11762749—21no—
14#11762746—21no—
15#11762708—21no—
16#11762686C2212yes—
17#11762684C412yescomplemented
18#11762666—21no—
19C17×#380C411yescharacteristic, complemented
20C23×C4×C17C2×C411yescharacteristic
21C22×C22⋊D17C2311yescharacteristic
22C23×Dic17C2×C412yes—
23C22×C4×D17D411yescharacteristic
24C2×C22⋊#595—22no—
25C2×#63247—22no—
26C2×C4×#595—41no—
27C2×C4⋊C4×C17—21no—
28C2×C22⋊C4×C17C2×C412yes—
29C2×C22⋊Dic17C2×C411yescharacteristic
30C4×C22⋊D17—41no—
31C4×C22⋊D17—41no—
32C2×#63227—22no—
33C2×#63227D411yescharacteristic
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17—41no—
38C2×C4⋊Dic17—41no—
39C2×#63225D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41#10587553—24no—
42#10587551—22no—
43#10587551Q814yes—
44(C2×#595)⋊C4—41no—
45#10587544—41no—
46C4⋊#3624—41no—
47C17×#95—21no—
48#10587502—22no—
49#10587502—22no—
50C24×C17C22×C411yescharacteristic
51C23×D17C2×D411yescharacteristic
52C22×#595—42no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—22no—
56C22×C4×C17—42no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C2×C22⋊D17—41no—
60C2×C22⋊D17—41no—
61C2×C22⋊D17C2×Q812yes—
62C2×C22⋊D17#4112yes—
63C22×Dic17—22no—
64C22×Dic17—42no—
65C22×Dic17—82no—
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C2×D411yescharacteristic
69C2×C4×D17—21no—
70C2×C4×D17—21no—
71C2×C4×D17—42no—
72C22⋊#595—24no—
73C22⋊#595—42no—
74#63247—42no—
75#63247—42no—
76C2×#3624—42no—
77C4×#595—82no—
78C4⋊C4×C17—42no—
79C22⋊C4×C17—42no—
80C22⋊Dic17—41no—
81#63227—41no—
82#63227—82no—
83C4⋊Dic17—42no—
84C4⋊Dic17—41no—
85C4⋊Dic17—41no—
86C4⋊Dic17—82no—
87#63225—41no—
88C4×Dic17—21no—
89C4×Dic17—21no—
90#13295—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17C2×C22⋊C411yescharacteristic
96C23×C17C4×Q812yes—
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17C22⋊Q812yes—
102C2×#595—44no—
103C2×#595—82no—
104C2×#595—82no—
105C2×C4×C17—22no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—41no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17#9911yescharacteristic
114C2×C4×C17#9911yescharacteristic
115C22⋊D17—42no—
116C22⋊D17—42no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—84no—
127C2×Dic17#12411yescharacteristic
128C2×Dic17C4⋊Q811yescharacteristic
129C4×D17—42no—
130C4×D17—81no—
131#3624—82no—
132#2487—171no—
133#1800—171no—
134C4×#99—171no—
135#1390—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17#39111yescharacteristic
144C22×C17C2×#9911yescharacteristic
145C22×C17#41011yescharacteristic
146C2×D17—22no—
147C2×D17—41no—
148C2×D17—41no—
149C2×D17—42no—
150C2×D17—81no—
151#595—84no—
152C4×C17—21no—
153C4×C17—41no—
154C4×C17—82no—
155C4×C17#35212yes—
156Dic17—21no—
157Dic17—21no—
158Dic17—41no—
159Dic17—41no—
160Dic17—82no—
161C2×C4×D4—171no—
162C22×C4⋊C4—171no—
163#423—341no—
164C2×#100—172no—
165C2×#99—172no—
166#400—341no—
167#399—341no—
168#389—341no—
169#386—172no—
170#385—172no—
171#380—341no—
172#380#59511yescharacteristic, complemented
173C4×C22⋊C4—341no—
174#343—341no—
175#342—341no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17—42no—
179C2×C17#199611yescharacteristic
180C2×C17C2×#35211yescharacteristic
181C2×C17#212211yescharacteristic
182D17—42no—
183C22×D4—171no—
184C23×C4—171no—
185C23×C4—172no—
186C23×C4C2×#59511yescharacteristic
187C2×M16—342no—
188C4×D4—681no—
189C4×D4—681no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4—341no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4—681no—
196C2×C4⋊C4—681no—
197C2×C22⋊C4—171no—
198C2×C22⋊C4—171no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4—342no—
201C2×C22⋊C4C2×#59512yes—
202C2×C42—171no—
203C2×C42—681no—
204C4⋊C8—681no—
205C4⋊C8—681no—
206#100—344no—
207#99—174no—
208#99—342no—
209#95—21no—
210#95—342no—
211#95—342no—
212#95—681no—
213C17#1329511yescharacteristic, complemented, Sylow
214C24—171no—
215C24C22×#59511yescharacteristic
216C2×D4—172no—
217C2×D4—172no—
218C2×D4—681no—
219C2×D4—681no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—171no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4—342no—
230C22×C4—682no—
231C22×C4—682no—
232C22×C4—682no—
233C22×C4—1362no—
234C22×C4C22⋊#59511yescharacteristic
235C22×C4C22⋊#59511yescharacteristic
236M16—682no—
237M16—682no—
238C2×C8—682no—
239C4⋊C4—42no—
240C4⋊C4—681no—
241C4⋊C4—682no—
242C4⋊C4—681no—
243C4⋊C4—681no—
244C4⋊C4—1362no—
245C22⋊C4—42no—
246C22⋊C4—344no—
247C22⋊C4—681no—
248C22⋊C4—681no—
249C22⋊C4—682no—
250C22⋊C4—1362no—
251C42—341no—
252C42—341no—
253C42—1362no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—172no—
259C23—341no—
260C23—341no—
261C23—681no—
262C23—681no—
263C23Q8×#59512yes—
264C23C2×C22⋊#59511yescharacteristic
265D4—682no—
266D4—682no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—171no—
276C2×C4—171no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—682no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—682no—
285C2×C4—682no—
286C2×C4—684no—
287C2×C4—682no—
288C2×C4—1361no—
289C2×C4—1362no—
290C2×C4—1362no—
291C2×C4—1364no—
292C2×C4#1058753911yescharacteristic
293C2×C4#1058753911yescharacteristic
294C8—1362no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—342no—
303C22—682no—
304C22—681no—
305C22—681no—
306C22—1361no—
307C22(C2×#595)⋊Q811yescharacteristic
308C22C2×#1058753911yescharacteristic
309C22#1176378911yescharacteristic
310C4—21no—
311C4—41no—
312C4—82no—
313C4—341no—
314C4—341no—
315C4—681no—
316C4—681no—
317C4—1364no—
318C4—1362no—
319C4#1176371212yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—682no—
324C2#1454497411yescharacteristic
325C2C2×#1176371211yescharacteristic
326C2#1454504511yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC17×#380C4
socleC22×C17C2×#99
prime radical, cosocleC22×C22⋊D17C23
hypercenter#380#595
hypocenterC17#13295
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
12. C22×C4⋊Dic17C4semidirect
17. #11762684C4semidirect
19. C17×#380C4semidirect
172. #380#595semidirect
213. C17#13295semidirect

Maximal subgroups

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

subgroupindex
#145446482
C4×#105875512
#145438692
#145331022
#1329517