Finite Group Databasebeta

← #17962520

Subgroups — #17962520

Order 4352.

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

normal subgroups only

This group has 500 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#14533958C211yesmaximal, characteristic
5C2×#10587539C2212yes—
6C2×#10587539C2212yes—
7#11763751—21no—
8C4×C22⋊#595—21no—
9#11763687—21no—
10#11763687—21no—
11C2×C4×D68C2211yescharacteristic
12#11762720—21no—
13#11762720—21no—
14#11762710C2212yes—
15#11762674C412yescomplemented
16#11762666—21no—
17#11762666—21no—
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
33C2×#63227—22no—
34C2×#63227—22no—
35C2×#63227D411yescharacteristic
36C2×#63227D411yescharacteristic
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2×C412yes—
40C4×D68—41no—
41C4×D68—41no—
42(C2×#595)⋊C4—41no—
43(C2×#595)⋊C4—41no—
44#10587539—22no—
45#10587539—24no—
46#10587539Q814yescomplemented
47#10587502—21no—
48#10587502—22no—
49#10587502—22no—
50C23×D17C2×D411yescharacteristic
51C23×D17C2×D411yescharacteristic
52C22×#595—42no—
53C22×#595—42no—
54C2×Q8×C17—42no—
55C2×Q8×C17—82no—
56C22×C4×C17—22no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22×C411yescharacteristic
60C22×Dic17—22no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C22⋊C411yescharacteristic
63C2×D68—41no—
64C2×D68—41no—
65C2×D68C2×Q812yes—
66C2×D68#4112yes—
67C2×C4×D17—21no—
68C2×C4×D17—21no—
69C2×C4×D17—21no—
70C2×C4×D17—21no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C22⋊#595—24no—
74C22⋊#595—42no—
75C22⋊#595—42no—
76C22⋊#595—42no—
77C4⋊#595—82no—
78C4×#595—82no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C42×C17—21no—
83C42×C17—21no—
84#63227—41no—
85#63227—41no—
86#63227—82no—
87#63227—82no—
88C4⋊Dic17—41no—
89C4⋊Dic17—42no—
90C4×Dic17—42no—
91#13299—171nomaximal, Sylow
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17—41no—
98C22×D17—41no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17C22⋊Q812yes—
102C22×D17#12411yescharacteristic
103C22×D17C4⋊Q811yescharacteristic
104C2×#595—44no—
105C2×#595—82no—
106C2×#595—82no—
107C2×#595—82no—
108C2×#595—82no—
109C2×#595—82no—
110Q8×C17—84no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—22no—
114C2×C4×C17—21no—
115C2×C4×C17—41no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17#9911yescharacteristic
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17C4×Q812yes—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129D68—42no—
130D68—42no—
131C4×D17—42no—
132C4×D17—42no—
133C4×D17—81no—
134C4×D17—81no—
135#2563—171no—
136#1799—171no—
137C4×#99—171no—
138#1393—174no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17#39111yescharacteristic
142C22×C17C2×#9911yescharacteristic
143C22×C17#41011yescharacteristic
144C2×D17—22no—
145C2×D17—21no—
146C2×D17—21no—
147C2×D17—41no—
148C2×D17—41no—
149C2×D17—41no—
150C2×D17—41no—
151C2×D17—42no—
152C2×D17—41no—
153C2×D17—41no—
154C2×D17—81no—
155C2×D17—81no—
156#595—84no—
157#595—162no—
158C4×C17—21no—
159C4×C17—22no—
160C4×C17—41no—
161C4×C17—82no—
162C4×C17#35412yes—
163Dic17—41no—
164Dic17—41no—
165Dic17—82no—
166C2×C4⋊Q8#59511yescharacteristic, complemented
167C2×C4×D4—171no—
168C2×#99—172no—
169C2×#99—172no—
170#393—341no—
171#393—341no—
172#390—172no—
173#382—172no—
174#380—341no—
175#380—341no—
176#380—341no—
177C4×C4⋊C4—171no—
178C4×C22⋊C4—341no—
179#342—341no—
180#342—341no—
181C2×C17—21no—
182C2×C17—21no—
183C2×C17#199611yescharacteristic
184C2×C17C2×#35411yescharacteristic
185C2×C17#212811yescharacteristic
186D17—42no—
187D17—41no—
188D17—41no—
189C22×Q8—21no—
190C22×D4—171no—
191C23×C4—171no—
192C23×C4—171no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4×D4—681no—
196C4×D4—681no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4—681no—
201C2×C4⋊C4C2×#59512yes—
202C2×C22⋊C4—171no—
203C2×C22⋊C4—171no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4—342no—
207C2×C22⋊C4—342no—
208C2×C42—172no—
209C2×C42—681no—
210C2×C42C2×#59511yescharacteristic
211#99—174no—
212#99—342no—
213#99—344no—
214#95—342no—
215#95—342no—
216#95—341no—
217#95—681no—
218#95—681no—
219C17#1329911yescharacteristic, complemented, Sylow
220C24—171no—
221C24—171no—
222C2×Q8—42no—
223C2×Q8—82no—
224C2×D4—172no—
225C2×D4—172no—
226C2×D4—681no—
227C2×D4—681no—
228C22×C4—22no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4—342no—
236C22×C4—682no—
237C22×C4—682no—
238C22×C4—682no—
239C22×C4—682no—
240C22×C4C22⋊#59511yescharacteristic
241C22×C4C22⋊#59511yescharacteristic
242C22×C4C22×#59511yescharacteristic
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—681no—
247C4⋊C4—682no—
248C4⋊C4—1362no—
249C22⋊C4—344no—
250C22⋊C4—681no—
251C22⋊C4—681no—
252C22⋊C4—682no—
253C22⋊C4—682no—
254C22⋊C4—682no—
255C22⋊C4—1362no—
256C22⋊C4—1362no—
257C42—21no—
258C42—21no—
259C42—682no—
260C42—1362no—
261C23—172no—
262C23—171no—
263C23—171no—
264C23—341no—
265C23—341no—
266C23—341no—
267C23—341no—
268C23—681no—
269C23—681no—
270C23—681no—
271C23—681no—
272C23C2×C22⋊#59511yescharacteristic
273Q8—84no—
274D4—682no—
275D4—682no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—684no—
293C2×C4—682no—
294C2×C4—1361no—
295C2×C4—1361no—
296C2×C4—1362no—
297C2×C4—1362no—
298C2×C4—1362no—
299C2×C4—1362no—
300C2×C4—1362no—
301C2×C4#1058755111yescharacteristic
302C2×C4#1058755111yescharacteristic
303C2×C4Q8×#59512yes—
304C22—21no—
305C22—21no—
306C22—342no—
307C22—341no—
308C22—341no—
309C22—681no—
310C22—682no—
311C22—681no—
312C22—681no—
313C22—681no—
314C22—681no—
315C22—681no—
316C22—1361no—
317C22—1361no—
318C22(C2×#595)⋊Q811yescharacteristic
319C22C2×#1058755111yescharacteristic
320C22#1176379011yescharacteristic
321C4—22no—
322C4—21no—
323C4—41no—
324C4—82no—
325C4—2722no—
326C4—681no—
327C4—681no—
328C4—1362no—
329C4—1364no—
330C4#1176371512yes—
331C2—21no—
332C2—21no—
333C2—681no—
334C2—682no—
335C2—681no—
336C2#1454497311yescharacteristic
337C2C2×#1176371511yescharacteristic
338C2#1454506311yescharacteristic
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#13299
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
15. #11762674C4semidirect
18. C4×C4⋊Dic17C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
46. #10587539Q8semidirect
166. C2×C4⋊Q8#595semidirect
219. C17#13299semidirect

Maximal subgroups

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

subgroupindex
#145446452
C4×#105875392
#145438792
#145339582
#1329917