Finite Group Databasebeta

← #16949817

Subgroups — #16949817

Order 4352.

1098 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#11762604C212yesmaximal, complemented
2C2×#11762602C212yesmaximal, complemented
3#14528702C212yesmaximal, complemented
4C2×C17×#532C211yesmaximal, characteristic
5C2×#10587505—22no—
6C2×#10587505C2214yes—
7#11762672—22no—
8#11762604—24no—
9#11762602—24no—
10C22×#41×C17C411yescharacteristic, complemented
11C22×#41×C17C2211yescharacteristic
12C17×#532—24no—
13C17×#532—24no—
14C2×C17×C4⋊Q8C411yescharacteristic, complemented
15C2×C17×#127C2211yescharacteristic
16C2×C17×#124C411yescharacteristic, complemented
17C2×C17×#124C2211yescharacteristic
18C2×C17×#121—22no—
19C2×C4×D4×C17—22no—
20C22×C42×C17C411yescharacteristic, complemented
21C2×#41×C17—22no—
22C2×#41×C17—22no—
23C2×#41×C17—22no—
24C2×#41×C17—22no—
25C2×#41×C17—42no—
26C2×#41×C17—42no—
27C2×#41×C17—42no—
28C2×#41×C17—42no—
29C22×Q8×C17C2×C411yescharacteristic
30C22×Q8×C17C2×C411yescharacteristic
31C22×D4×C17—21no—
32C22×D4×C17—22no—
33C22×D4×C17C2×C411yescharacteristic
34C22×D4×C17D412yes—
35C22×D4×C17C2311yescharacteristic
36C23×C4×C17—21no—
37C23×C4×C17—21no—
38C23×C4×C17C2×C411yescharacteristic
39C23×C4×C17D412yes—
40C17×C4⋊Q8—22no—
41C17×C4⋊Q8—22no—
42C17×#127—22no—
43C17×#127D414yes—
44C17×#124—22no—
45C17×#124—22no—
46C17×#124—22no—
47C17×#124D414yes—
48C17×#121—216no—
49C4×D4×C17—216no—
50C2×C4⋊C4×C17—22no—
51C2×C22⋊C4×C17—22no—
52C2×C22⋊C4×C17—22no—
53C2×C42×C17—21no—
54C2×C42×C17—22no—
55C2×C42×C17—22no—
56C2×C42×C17C2×C411yescharacteristic
57C2×C42×C17C2×C411yescharacteristic
58C2×C22⋊Dic17—24no—
59#10587505—24no—
60#10587505—24no—
61#10587505—44no—
62#10587502—44no—
63C24×C17—21no—
64C24×C17C22⋊C411yescharacteristic
65C24×C17C2×D412yes—
66#41×C17—48no—
67#41×C17—48no—
68#41×C17—48no—
69#41×C17—48no—
70C2×Q8×C17—21no—
71C2×Q8×C17—21no—
72C2×Q8×C17—21no—
73C2×Q8×C17—41no—
74C2×Q8×C17—41no—
75C2×Q8×C17—41no—
76C2×Q8×C17—41no—
77C2×Q8×C17C22⋊C412yes—
78C2×D4×C17—24no—
79C2×D4×C17—24no—
80C2×D4×C17—28no—
81C2×D4×C17—24no—
82C2×D4×C17—21no—
83C2×D4×C17—42no—
84C2×D4×C17—41no—
85C2×D4×C17—41no—
86C2×D4×C17—41no—
87C2×D4×C17—41no—
88C2×D4×C17—48no—
89C2×D4×C17—42no—
90C2×D4×C17—42no—
91C2×D4×C17C22⋊C412yes—
92C2×D4×C17C2×D412yes—
93C2×D4×C17C2×D412yes—
94C22×C4×C17—22no—
95C22×C4×C17—21no—
96C22×C4×C17—22no—
97C22×C4×C17—21no—
98C22×C4×C17—21no—
99C22×C4×C17—24no—
100C22×C4×C17—24no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—24no—
104C22×C4×C17—24no—
105C22×C4×C17—42no—
106C22×C4×C17—44no—
107C22×C4×C17—42no—
108C22×C4×C17—42no—
109C22×C4×C17C22⋊C412yes—
110C22×C4×C17C22⋊C411yescharacteristic
111C22×C4×C17C22×C411yescharacteristic
112C22×Dic17—44no—
113C4⋊C4×C17—28no—
114C22⋊C4×C17—28no—
115C22⋊C4×C17—28no—
116C42×C17—24no—
117C42×C17—21no—
118C42×C17—21no—
119C42×C17C22⋊C412yes—
120C42×C17C22⋊C412yes—
121C22⋊Dic17—44no—
122C22⋊Dic17—44no—
123#12695—171nomaximal, Sylow
124C23×C17—21no—
125C23×C17—22no—
126C23×C17—21no—
127C23×C17—22no—
128C23×C17—22no—
129C23×C17—42no—
130C23×C17—44no—
131C23×C17—44no—
132C23×C17—41no—
133C23×C17—41no—
134C23×C17#9912yes—
135C23×C17C2×C22⋊C411yescharacteristic
136C23×C17#12014yes—
137Q8×C17—42no—
138Q8×C17—42no—
139Q8×C17—42no—
140Q8×C17—42no—
141D4×C17—42no—
142D4×C17—42no—
143D4×C17—42no—
144D4×C17—416no—
145D4×C17—42no—
146D4×C17—48no—
147D4×C17—48no—
148C2×C4×C17—24no—
149C2×C4×C17—24no—
150C2×C4×C17—24no—
151C2×C4×C17—21no—
152C2×C4×C17—22no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—22no—
157C2×C4×C17—22no—
158C2×C4×C17—22no—
159C2×C4×C17—22no—
160C2×C4×C17—22no—
161C2×C4×C17—48no—
162C2×C4×C17—48no—
163C2×C4×C17—48no—
164C2×C4×C17—42no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17#9912yes—
168C2×C4×C17C2×C22⋊C411yescharacteristic
169C2×C4×C17C2×C22⋊C411yescharacteristic
170C2×Dic17—84no—
171C2×Dic17—88no—
172C2×#532D1711yescharacteristic
173C2×#354—172no—
174C2×#353—172no—
175#1782—172no—
176C22×C17—22no—
177C22×C17—24no—
178C22×C17—22no—
179C22×C17—44no—
180C22×C17—42no—
181C22×C17—42no—
182C22×C17—44no—
183C22×C17—44no—
184C22×C17—42no—
185C22×C17—42no—
186C22×C17—82no—
187C22×C17—82no—
188C22×C17—82no—
189C22×C17#37911yescharacteristic
190C22×C17C2×#9911yescharacteristic
191C22×C17C2×#9911yescharacteristic
192C4×C17—24no—
193C4×C17—22no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—42no—
197C4×C17—42no—
198C4×C17—42no—
199Dic17—164no—
200C22×#41Dic1711yescharacteristic, complemented
201C22×#41C2×D1711yescharacteristic
202#532—24no—
203#532—24no—
204C2×C4⋊Q8Dic1711yescharacteristic, complemented
205C2×#127C2×D1711yescharacteristic
206C2×#124Dic1711yescharacteristic, complemented
207C2×#124C2×D1711yescharacteristic
208C2×#121—22no—
209C2×C4×D4—22no—
210C22×C42Dic1711yescharacteristic, complemented
211C2×#99—174no—
212C2×#99—342no—
213#381—342no—
214#354—344no—
215#353—344no—
216C2×C17—21no—
217C2×C17—21no—
218C2×C17—44no—
219C2×C17—42no—
220C2×C17—82no—
221C2×C17#178111yescharacteristic
222C2×C17#212612yes—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—42no—
228C2×#41—42no—
229C2×#41—42no—
230C2×#41—42no—
231C22×Q8C2×Dic1711yescharacteristic
232C22×Q8C2×Dic1711yescharacteristic
233C22×D4—21no—
234C22×D4—22no—
235C22×D4C2×Dic1711yescharacteristic
236C22×D4C22⋊D1712yes—
237C22×D4C22×D1711yescharacteristic
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×Dic1711yescharacteristic
241C23×C4C22⋊D1712yes—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244#127—22no—
245#127C22⋊D1714yes—
246#124—22no—
247#124—22no—
248#124—22no—
249#124C22⋊D1714yes—
250#121—216no—
251C4×D4—216no—
252C2×C4⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—344no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42C2×Dic1711yescharacteristic
260C2×C42C2×Dic1711yescharacteristic
261#99—344no—
262#99—344no—
263#99—684no—
264#95—684no—
265C17#1269511yescharacteristic, complemented, Sylow
266C24—21no—
267C24C22⋊Dic1711yescharacteristic
268C24C2×C22⋊D1712yes—
269#41—48no—
270#41—48no—
271#41—48no—
272#41—48no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8C22⋊Dic1712yes—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—24no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—48no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4C22⋊Dic1712yes—
295C2×D4C2×C22⋊D1712yes—
296C2×D4C2×C22⋊D1712yes—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—684no—
313C22×C4C22⋊Dic1712yes—
314C22×C4C22⋊Dic1711yescharacteristic
315C22×C4C22×Dic1711yescharacteristic
316C4⋊C4—28no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—684no—
320C22⋊C4—684no—
321C42—24no—
322C42—21no—
323C42—21no—
324C42C22⋊Dic1712yes—
325C42C22⋊Dic1712yes—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23—41no—
335C23—41no—
336C23#1058750512yes—
337C23C2×C22⋊Dic1711yescharacteristic
338C23#1058763914yes—
339Q8—42no—
340Q8—42no—
341Q8—42no—
342Q8—42no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—416no—
348D4—48no—
349D4—48no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—1364no—
370C2×C4—1368no—
371C2×C4#1058750512yes—
372C2×C4C2×C22⋊Dic1711yescharacteristic
373C2×C4C2×C22⋊Dic1711yescharacteristic
374C22—22no—
375C22—22no—
376C22—24no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22#1176266511yescharacteristic
388C22C2×#1058750511yescharacteristic
389C22C2×#1058750511yescharacteristic
390C4—24no—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—2724no—
398C2—21no—
399C2—21no—
400C2—44no—
401C2—42no—
402C2—82no—
403C2#1452869911yescharacteristic
404C2#1453044912yes—
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
Fitting, prime radicalC2×C17×#532C2
socleC22×C17C2×#99
cosocleC22×D4×C17C23
hypercenterC2×#532D17
hypocenterC17#12695
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#11762604C2semidirect
2. C2×#11762602C2semidirect
3. #14528702C2semidirect
10. C22×#41×C17C4semidirect
14. C2×C17×C4⋊Q8C4semidirect
16. C2×C17×#124C4semidirect
20. C22×C42×C17C4semidirect
200. C22×#41Dic17semidirect
204. C2×C4⋊Q8Dic17semidirect
206. C2×#124Dic17semidirect
210. C22×C42Dic17semidirect
265. C17#12695semidirect

Maximal subgroups

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

subgroupindex
C2×#117626042
C2×#117626022
#145287022
C2×C17×#5322
#1269517