Finite Group Databasebeta

← #17962517

Subgroups — #17962517

Order 4352.

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

normal subgroups only

This group has 556 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#14544645C211yesmaximal, characteristic
2C4×#10587539C211yesmaximal, characteristic
3#14543879C214yesmaximal
4#14533738C211yesmaximal, characteristic
5C2×#10587539C2212yes—
6C2×#10587539C2212yes—
7#11763751—21no—
8C4×C22⋊#595—21no—
9#11763687—21no—
10#11763687—21no—
11C2×#10587632—21no—
12C2×#10587632—21no—
13C2×C4×D68C2211yescharacteristic
14#11762744—21no—
15#11762744—21no—
16#11762710C2212yes—
17#11762678C412yescomplemented
18C4×C4⋊Dic17C411yescharacteristic, complemented
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×D68C2311yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17D411yescharacteristic
24C2×C22⋊#595—22no—
25C2×C22⋊#595—22no—
26C2×C4⋊#595—41no—
27C2×C4×#595—41no—
28C17×C4⋊Q8—41no—
29C17×C4⋊Q8—41no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C42×C17C2×C411yescharacteristic
33#10587632—44no—
34#10587632—44no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227D411yescharacteristic
38C2×#63227D411yescharacteristic
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—22no—
42C2×#63225—22no—
43C2×C4×Dic17C2×C412yes—
44C4×D68—41no—
45C4×D68—41no—
46(C2×#595)⋊C4—41no—
47(C2×#595)⋊C4—41no—
48#10587539—22no—
49#10587539—24no—
50#10587539Q814yescomplemented
51#10587502—21no—
52C23×D17C2×D411yescharacteristic
53C23×D17C2×D411yescharacteristic
54C22×#595—42no—
55C22×#595—42no—
56C2×Q8×C17—42no—
57C2×Q8×C17—42no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C22×C4×C17—22no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×Dic17—22no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C22⋊C411yescharacteristic
67C2×D68—41no—
68C2×D68—41no—
69C2×D68C2×Q812yes—
70C2×D68#4112yes—
71C2×C4×D17—21no—
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17—42no—
76C2×C4×D17—42no—
77C22⋊#595—24no—
78C22⋊#595—42no—
79C22⋊#595—42no—
80C22⋊#595—42no—
81C4⋊#595—82no—
82C4×#595—82no—
83C4⋊C4×C17—42no—
84C4⋊C4×C17—42no—
85C42×C17—21no—
86C42×C17—21no—
87#63227—41no—
88#63227—41no—
89#63227—44no—
90#63227—44no—
91C4⋊Dic17—24no—
92C4⋊Dic17—41no—
93#63225—44no—
94#63225—44no—
95C4×Dic17—42no—
96#13298—171nomaximal, Sylow
97C23×C17C2×C22⋊C411yescharacteristic
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—41no—
103C22×D17—41no—
104C22×D17—41no—
105C22×D17—41no—
106C22×D17C22⋊Q812yes—
107C22×D17#12411yescharacteristic
108C22×D17C4⋊Q811yescharacteristic
109C2×#595—44no—
110C2×#595—82no—
111C2×#595—82no—
112C2×#595—82no—
113C2×#595—82no—
114C2×#595—82no—
115Q8×C17—82no—
116Q8×C17—82no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—22no—
120C2×C4×C17—21no—
121C2×C4×C17—41no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#9911yescharacteristic
127C2×C4×C17C4×Q812yes—
128C2×Dic17—24no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135D68—42no—
136D68—42no—
137C4×D17—42no—
138C4×D17—42no—
139C4×D17—81no—
140C4×D17—81no—
141#2542—171no—
142#1799—171no—
143C4×#99—171no—
144#1393—174no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17#39111yescharacteristic
148C22×C17C2×#9911yescharacteristic
149C22×C17#41011yescharacteristic
150C2×D17—22no—
151C2×D17—21no—
152C2×D17—21no—
153C2×D17—41no—
154C2×D17—41no—
155C2×D17—41no—
156C2×D17—41no—
157C2×D17—42no—
158C2×D17—41no—
159C2×D17—41no—
160C2×D17—81no—
161C2×D17—81no—
162#595—84no—
163#595—162no—
164C4×C17—21no—
165C4×C17—22no—
166C4×C17—41no—
167C4×C17—82no—
168C4×C17#35312yes—
169Dic17—44no—
170Dic17—41no—
171Dic17—41no—
172C2×C4⋊Q8#59511yescharacteristic, complemented
173C2×C22⋊Q8—341no—
174C2×C22⋊Q8—341no—
175C2×C4×D4—171no—
176C2×#99—172no—
177C2×#99—172no—
178#399—341no—
179#399—341no—
180#390—172no—
181#384—172no—
182#380—341no—
183C4×C4⋊C4—171no—
184C4×C22⋊C4—341no—
185#342—341no—
186#342—341no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17#199611yescharacteristic
190C2×C17C2×#35311yescharacteristic
191C2×C17#212911yescharacteristic
192D17—42no—
193D17—41no—
194D17—41no—
195C22×Q8—21no—
196C22×D4—171no—
197C23×C4—171no—
198C23×C4—171no—
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201C22⋊Q8—684no—
202C22⋊Q8—684no—
203C4×D4—681no—
204C4×D4—681no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—342no—
209C2×C4⋊C4—342no—
210C2×C4⋊C4—681no—
211C2×C4⋊C4C2×#59512yes—
212C2×C22⋊C4—171no—
213C2×C22⋊C4—171no—
214C2×C22⋊C4—342no—
215C2×C22⋊C4—342no—
216C2×C22⋊C4—342no—
217C2×C22⋊C4—342no—
218C2×C42—172no—
219C2×C42—681no—
220C2×C42C2×#59511yescharacteristic
221#99—174no—
222#99—342no—
223#99—344no—
224#95—341no—
225#95—681no—
226#95—681no—
227C17#1329811yescharacteristic, complemented, Sylow
228C24—171no—
229C24—171no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×D4—172no—
235C2×D4—172no—
236C2×D4—681no—
237C2×D4—681no—
238C22×C4—22no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—342no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—682no—
247C22×C4—682no—
248C22×C4—682no—
249C22×C4—682no—
250C22×C4C22⋊#59511yescharacteristic
251C22×C4C22⋊#59511yescharacteristic
252C22×C4C22×#59511yescharacteristic
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—344no—
256C4⋊C4—681no—
257C4⋊C4—684no—
258C4⋊C4—684no—
259C4⋊C4—1362no—
260C22⋊C4—344no—
261C22⋊C4—681no—
262C22⋊C4—681no—
263C22⋊C4—684no—
264C22⋊C4—684no—
265C22⋊C4—682no—
266C22⋊C4—682no—
267C22⋊C4—682no—
268C42—21no—
269C42—21no—
270C42—682no—
271C42—1362no—
272C23—172no—
273C23—171no—
274C23—171no—
275C23—341no—
276C23—341no—
277C23—341no—
278C23—341no—
279C23—681no—
280C23—681no—
281C23—681no—
282C23—681no—
283C23C2×C22⋊#59511yescharacteristic
284Q8—82no—
285Q8—82no—
286D4—682no—
287D4—682no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—344no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4—682no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—684no—
306C2×C4—1361no—
307C2×C4—1361no—
308C2×C4—1362no—
309C2×C4—1362no—
310C2×C4—1362no—
311C2×C4—1362no—
312C2×C4—1362no—
313C2×C4#1058755111yescharacteristic
314C2×C4#1058755111yescharacteristic
315C2×C4Q8×#59512yes—
316C22—21no—
317C22—21no—
318C22—342no—
319C22—341no—
320C22—341no—
321C22—681no—
322C22—682no—
323C22—681no—
324C22—681no—
325C22—681no—
326C22—681no—
327C22—681no—
328C22—1361no—
329C22—1361no—
330C22(C2×#595)⋊Q811yescharacteristic
331C22C2×#1058755111yescharacteristic
332C22#1176379011yescharacteristic
333C4—22no—
334C4—21no—
335C4—41no—
336C4—82no—
337C4—2722no—
338C4—684no—
339C4—681no—
340C4—681no—
341C4—1364no—
342C4#1176371312yes—
343C2—21no—
344C2—21no—
345C2—681no—
346C2—682no—
347C2—681no—
348C2#1454497311yescharacteristic
349C2C2×#1176371311yescharacteristic
350C2#1454506611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radical, cosocleC22×D68C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#13298
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
17. #11762678C4semidirect
18. C4×C4⋊Dic17C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
50. #10587539Q8semidirect
172. C2×C4⋊Q8#595semidirect
227. C17#13298semidirect

Maximal subgroups

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

subgroupindex
#145446452
C4×#105875392
#145438792
#145337382
#1329817