Finite Group Databasebeta

← #17959785

Subgroups — #17959785

Order 4352.

452 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 80 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
2#14544021C211yesmaximal, characteristic
3#14543779C214yesmaximal, complemented
4#14533958C211yesmaximal, characteristic
5C2×#10587539—21no—
6#11763777—21no—
7#11763751—21no—
8#11763687C2212yes—
9#11763654—21no—
10#11763653C2212yes—
11C2×C4×D68C2211yescharacteristic
12#11762720C412yescomplemented
13#11762720C2212yes—
14#11762710—21no—
15#11762674—21no—
16#11762666—21no—
17#11762666—21no—
18C4×C4⋊Dic17C411yescharacteristic, complemented
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C22×Q8×C17C2×C412yes—
21C22×D68D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17D411yescharacteristic
24C2×C22⋊#595—22no—
25C2×C4⋊#595—41no—
26C2×#63242—41no—
27C17×C4⋊Q8—41no—
28C17×C4⋊Q8—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C42×C17C2×C411yescharacteristic
32C2×#63227—22no—
33C2×#63227—22no—
34C2×#63227C2×C411yescharacteristic
35C2×#63227C2311yescharacteristic
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17D411yescharacteristic
38C2×C4×Dic17—21no—
39C4×D68—41no—
40C4×D68—41no—
41C22⋊#3624—22no—
42(C2×#595)⋊C4—41no—
43#10587543—41no—
44#10587539—42no—
45#10587502—22no—
46#10587502—22no—
47#10587502C2×C412yes—
48C23×D17C22⋊C411yescharacteristic
49C23×D17C2×D411yescharacteristic
50C22×#595—42no—
51C2×Q8×C17—42no—
52C2×Q8×C17—82no—
53C22×C4×C17—22no—
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22×C411yescharacteristic
57C22×Dic17—22no—
58C22×Dic17C22⋊C411yescharacteristic
59C22×Dic17C2×D411yescharacteristic
60C2×D68—41no—
61C2×D68—41no—
62C2×D68SD1612yes—
63C2×D68Q1612yes—
64C2×C4×D17—21no—
65C2×C4×D17—21no—
66C2×C4×D17—21no—
67C2×C4×D17—21no—
68C2×C4×D17—42no—
69C2×C4×D17—42no—
70C22⋊#595—42no—
71C22⋊#595—42no—
72C2×#3624—42no—
73C4⋊#595—82no—
74#63242—82no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C42×C17—21no—
79C42×C17—21no—
80#63227—41no—
81#63227—41no—
82#63227—82no—
83#63227—82no—
84C4⋊Dic17—41no—
85C4⋊Dic17—42no—
86C4×Dic17—42no—
87#11900—171nomaximal, Sylow
88C23×C17C2×C22⋊C411yescharacteristic
89C22×D17—21no—
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—41no—
94C22×D17—41no—
95C22×D17—41no—
96C22×D17—41no—
97C22×D17#10312yes—
98C22×D17C2×SD1611yescharacteristic
99C22×D17C2×Q1611yescharacteristic
100C2×#595—82no—
101C2×#595—82no—
102C2×#595—82no—
103Q8×C17—84no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—41no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17#9911yescharacteristic
113C2×C4×C17#9911yescharacteristic
114C2×C4×C17#10312yes—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122D68—42no—
123D68—42no—
124C4×D17—42no—
125C4×D17—42no—
126C4×D17—81no—
127C4×D17—81no—
128#3624—82no—
129#2563—171no—
130#1799—171no—
131#1459—171no—
132#1351—174no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17C2×#9911yescharacteristic
136C22×C17C2×#10311yescharacteristic
137C22×C17#42111yescharacteristic
138C2×D17—21no—
139C2×D17—22no—
140C2×D17—21no—
141C2×D17—41no—
142C2×D17—41no—
143C2×D17—41no—
144C2×D17—41no—
145C2×D17—42no—
146C2×D17—41no—
147C2×D17—41no—
148C2×D17—81no—
149C2×D17—81no—
150#595—162no—
151C4×C17—21no—
152C4×C17—22no—
153C4×C17—41no—
154C4×C17—82no—
155C4×C17#35412yes—
156Dic17—41no—
157Dic17—41no—
158Dic17—82no—
159C2×C4⋊Q8#59511yescharacteristic, complemented
160C2×C4×D4—171no—
161C2×#99—341no—
162C2×C22⋊C8—341no—
163#393—172no—
164#393—172no—
165#390—341no—
166#382—341no—
167#380—341no—
168#380—341no—
169#380—341no—
170C4×C4⋊C4—171no—
171#342—172no—
172#324—341no—
173#323—172no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17#152011yescharacteristic
177C2×C17#212211yescharacteristic
178C2×C17C2×#35411yescharacteristic
179D17—41no—
180D17—42no—
181D17—41no—
182C22×Q8C2×#59512yes—
183C22×D4—171no—
184C23×C4—171no—
185C23×C4—171no—
186C22×C8—681no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4×D4—681no—
190C4×D4—681no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—681no—
196C2×C22⋊C4—171no—
197C2×C22⋊C4—171no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4—342no—
201C2×C42—341no—
202C2×C42C2×#59511yescharacteristic
203#99—682no—
204C22⋊C8—342no—
205C22⋊C8—681no—
206#95—172no—
207#95—342no—
208#95—342no—
209#95—681no—
210C17#1190011yescharacteristic, complemented, Sylow
211C24—171no—
212C24—171no—
213C2×Q8—42no—
214C2×Q8—82no—
215C2×D4—172no—
216C2×D4—172no—
217C2×D4—681no—
218C2×D4—681no—
219C22×C4—22no—
220C22×C4—171no—
221C22×C4—171no—
222C22×C4—341no—
223C22×C4—341no—
224C22×C4—341no—
225C22×C4—341no—
226C22×C4—342no—
227C22×C4—682no—
228C22×C4—682no—
229C22×C4—682no—
230C22×C4C22⋊#59511yescharacteristic
231C22×C4C22⋊#59511yescharacteristic
232C22×C4C22×#59511yescharacteristic
233C2×C8—682no—
234C2×C8—1362no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—681no—
239C4⋊C4—682no—
240C4⋊C4—1362no—
241C22⋊C4—681no—
242C22⋊C4—681no—
243C22⋊C4—682no—
244C22⋊C4—682no—
245C22⋊C4—1362no—
246C22⋊C4—1362no—
247C42—21no—
248C42—21no—
249C42—682no—
250C23—171no—
251C23—172no—
252C23—171no—
253C23—341no—
254C23—341no—
255C23—341no—
256C23—341no—
257C23—681no—
258C23—681no—
259C23—681no—
260C23—681no—
261C23C2×C22⋊#59511yescharacteristic
262Q8—84no—
263D4—682no—
264D4—682no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—682no—
276C2×C4—682no—
277C2×C4—681no—
278C2×C4—681no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—1361no—
283C2×C4—1361no—
284C2×C4—1362no—
285C2×C4—1362no—
286C2×C4—1362no—
287C2×C4#1058754912yes—
288C2×C4#1058755111yescharacteristic
289C2×C4#1058755111yescharacteristic
290C8—1362no—
291C22—21no—
292C22—21no—
293C22—341no—
294C22—342no—
295C22—341no—
296C22—681no—
297C22—682no—
298C22—681no—
299C22—681no—
300C22—681no—
301C22—681no—
302C22—681no—
303C22—1361no—
304C22—1361no—
305C22C2×#1058755111yescharacteristic
306C22C2×#1058754911yescharacteristic
307C22#1176380811yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—82no—
312C4—2722no—
313C4—681no—
314C4—681no—
315C4—1362no—
316C4#1176371512yes—
317C2—21no—
318C2—21no—
319C2—681no—
320C2—681no—
321C2—682no—
322C2#1454422411yescharacteristic
323C2#1454504311yescharacteristic
324C2C2×#1176371511yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC22×D68D4
cosocleC2×#63227C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11900
LCSG ▷ C22×C4×C17 ▷ C23×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
3. #14543779C2semidirect
12. #11762720C4semidirect
18. C4×C4⋊Dic17C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
159. C2×C4⋊Q8#595semidirect
210. C17#11900semidirect

Maximal subgroups

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

subgroupindex
#145446452
#145440212
#145437792
#145339582
#1190017