Finite Group Databasebeta

← #16925937

Subgroups — #16925937

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14528883C212yesmaximal, complemented
2C4×#10587505C211yesmaximal, characteristic
3#14527304C212yesmaximal, complemented
4#14527122C211yesmaximal, characteristic
5C2×C17×#556C211yesmaximal, characteristic
6C2×#10587505—21no—
7#11762773C2212yes—
8C4×C22⋊Dic17C2212yes—
9#11762576—21no—
10#11762568—24no—
11#11762546—24no—
12#11762535—21no—
13#11762533—21no—
14C17×#556—24no—
15C17×#556—24no—
16C2×C17×C4⋊Q8C411yescharacteristic, complemented
17C2×C17×#126—21no—
18C2×C17×#126—21no—
19C2×C17×#124—21no—
20C2×C17×C22⋊Q8C411yescharacteristic, complemented
21C2×C17×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8×C17—21no—
23C2×C4×Q8×C17C411yescharacteristic, complemented
24C2×C4×D4×C17C2211yescharacteristic
25C2×C17×#117C411yescharacteristic, complemented
26C2×C17×#117C2211yescharacteristic
27C22×Q8×C17—21no—
28C22×Q8×C17C2×C411yescharacteristic
29C22×D4×C17D411yescharacteristic
30C23×C4×C17C2×C411yescharacteristic
31C23×C4×C17C2311yescharacteristic
32C17×C4⋊Q8—22no—
33C17×C4⋊Q8—22no—
34C17×#126—28no—
35C17×#126—28no—
36C17×#124—28no—
37C17×C22⋊Q8—22no—
38C17×C22⋊Q8—22no—
39C17×C22⋊Q8—22no—
40C17×C22⋊Q8D414yes—
41C4×Q8×C17—28no—
42C4×Q8×C17—22no—
43C4×Q8×C17—22no—
44C4×D4×C17—22no—
45C4×D4×C17—22no—
46C17×#117—22no—
47C17×#117—22no—
48C17×#117—22no—
49C17×#117D414yes—
50C2×C4⋊C4×C17—21no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—21no—
53C2×C4⋊C4×C17—21no—
54C2×C4⋊C4×C17C2×C411yescharacteristic
55C2×C4⋊C4×C17C2×C411yescharacteristic
56C2×C4⋊C4×C17C2×C411yescharacteristic
57C2×C4⋊C4×C17D411yescharacteristic
58C2×C22⋊C4×C17—21no—
59C2×C22⋊C4×C17—21no—
60C2×C22⋊C4×C17—21no—
61C2×C22⋊C4×C17—21no—
62C2×C22⋊C4×C17D411yescharacteristic
63C2×C22⋊C4×C17D411yescharacteristic
64C2×C42×C17—21no—
65C2×C42×C17—21no—
66C2×C42×C17—21no—
67C2×C42×C17C2×C411yescharacteristic
68C2×C22⋊Dic17—22no—
69C2×C4×Dic17—22no—
70C2×#63223—22no—
71#10587505—24no—
72#10587502—22no—
73C22⋊#3622—22no—
74C22⋊#3622—22no—
75C24×C17C22⋊C411yescharacteristic
76C24×C17C2×D411yescharacteristic
77C2×Q8×C17—24no—
78C2×Q8×C17—21no—
79C2×Q8×C17—42no—
80C2×Q8×C17—42no—
81C2×Q8×C17—42no—
82C2×Q8×C17C22⋊C412yes—
83C2×D4×C17—22no—
84C2×D4×C17—22no—
85C2×D4×C17—41no—
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—21no—
92C22×C4×C17—42no—
93C22×C4×C17—42no—
94C22×C4×C17C22⋊C411yescharacteristic
95C22×C4×C17C22⋊C411yescharacteristic
96C22×C4×C17C22⋊C411yescharacteristic
97C22×C4×C17C22×C411yescharacteristic
98C22×C4×C17C2×D412yes—
99C22×C4×C17C2×D412yes—
100C22×C4×C17C2×D411yescharacteristic
101C22×Dic17—42no—
102C4⋊C4×C17—24no—
103C4⋊C4×C17—24no—
104C4⋊C4×C17—24no—
105C4⋊C4×C17—21no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—21no—
108C4⋊C4×C17—24no—
109C4⋊C4×C17—22no—
110C4⋊C4×C17C22⋊C412yes—
111C4⋊C4×C17C22⋊C412yes—
112C4⋊C4×C17C22⋊C412yes—
113C22⋊C4×C17—24no—
114C22⋊C4×C17—24no—
115C22⋊C4×C17—24no—
116C22⋊C4×C17—24no—
117C22⋊C4×C17—22no—
118C22⋊C4×C17—22no—
119C42×C17—24no—
120C42×C17—24no—
121C42×C17—21no—
122C42×C17—21no—
123C42×C17—24no—
124C22⋊Dic17—24no—
125C22⋊Dic17—42no—
126C4×Dic17—44no—
127#63223—44no—
128C2×#3622—42no—
129#8365—171nomaximal, Sylow
130C23×C17—21no—
131C23×C17—21no—
132C23×C17—21no—
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—41no—
136C23×C17—41no—
137C23×C17—41no—
138C23×C17—41no—
139C23×C17C2×C22⋊C411yescharacteristic
140C23×C17#12012yes—
141Q8×C17—42no—
142Q8×C17—48no—
143Q8×C17—42no—
144D4×C17—44no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—22no—
148C2×C4×C17—24no—
149C2×C4×C17—22no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—22no—
153C2×C4×C17—22no—
154C2×C4×C17—22no—
155C2×C4×C17—41no—
156C2×C4×C17—42no—
157C2×C4×C17—42no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—41no—
164C2×C4×C17—42no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17#10412yes—
168C2×C4×C17C2×C22⋊C411yescharacteristic
169C2×C4×C17C2×C22⋊C411yescharacteristic
170C2×C4×C17#12012yes—
171C2×Dic17—44no—
172C2×Dic17—82no—
173C2×Dic17—82no—
174#3622—82no—
175C2×#556D1711yescharacteristic
176#1826—172no—
177C4×#99—171no—
178#1508—172no—
179#1465—171no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—22no—
183C22×C17—22no—
184C22×C17—22no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—81no—
189C22×C17—81no—
190C22×C17—81no—
191C22×C17—81no—
192C22×C17#37911yescharacteristic
193C22×C17C2×#10411yescharacteristic
194C22×C17#42111yescharacteristic
195C4×C17—22no—
196C4×C17—21no—
197C4×C17—22no—
198C4×C17—41no—
199C4×C17—42no—
200C4×C17—41no—
201C4×C17—42no—
202C4×C17—42no—
203C4×C17—42no—
204C4×C17#41012yes—
205Dic17—84no—
206#556—24no—
207#556—24no—
208C2×C4⋊Q8Dic1711yescharacteristic, complemented
209C2×#126—21no—
210C2×#126—21no—
211C2×#124—21no—
212C2×C22⋊Q8Dic1711yescharacteristic, complemented
213C2×C22⋊Q8C2×D1711yescharacteristic
214C2×C4×Q8—21no—
215C2×C4×Q8Dic1711yescharacteristic, complemented
216C2×C4×D4C2×D1711yescharacteristic
217C2×#117Dic1711yescharacteristic, complemented
218C2×#117C2×D1711yescharacteristic
219C2×#99—341no—
220#407—172no—
221C4×C22⋊C4—172no—
222#342—341no—
223#339—344no—
224#328—344no—
225#324—341no—
226#323—341no—
227C2×C17—21no—
228C2×C17—21no—
229C2×C17—42no—
230C2×C17—81no—
231C2×C17#178211yescharacteristic
232C2×C17#178911yescharacteristic
233C2×C17#179611yescharacteristic
234C22×Q8—21no—
235C22×Q8C2×Dic1711yescharacteristic
236C22×D4C22⋊D1711yescharacteristic
237C23×C4C2×Dic1711yescharacteristic
238C23×C4C22×D1711yescharacteristic
239C2×M16—342no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242#126—28no—
243#126—28no—
244#124—28no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8C22⋊D1714yes—
249C4×Q8—28no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×D4—22no—
253C4×D4—22no—
254#117—22no—
255#117—22no—
256#117—22no—
257#117C22⋊D1714yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C2×Dic1711yescharacteristic
263C2×C4⋊C4C2×Dic1711yescharacteristic
264C2×C4⋊C4C2×Dic1711yescharacteristic
265C2×C4⋊C4C22⋊D1711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—342no—
271C2×C22⋊C4C22⋊D1711yescharacteristic
272C2×C22⋊C4C22⋊D1711yescharacteristic
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—342no—
277C2×C42C2×Dic1711yescharacteristic
278#99—344no—
279C22⋊C8—342no—
280C22⋊C8—342no—
281#95—342no—
282C17#836511yescharacteristic, complemented, Sylow
283C24C22⋊Dic1711yescharacteristic
284C24C2×C22⋊D1711yescharacteristic
285C2×Q8—24no—
286C2×Q8—21no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8C22⋊Dic1712yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—682no—
303C22×C4C22⋊Dic1711yescharacteristic
304C22×C4C22⋊Dic1711yescharacteristic
305C22×C4C22⋊Dic1711yescharacteristic
306C22×C4C22×Dic1711yescharacteristic
307C22×C4C2×C22⋊D1712yes—
308C22×C4C2×C22⋊D1712yes—
309C22×C4C2×C22⋊D1711yescharacteristic
310M16—684no—
311C2×C8—682no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4C22⋊Dic1712yes—
321C4⋊C4C22⋊Dic1712yes—
322C4⋊C4C22⋊Dic1712yes—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—344no—
330C22⋊C4—682no—
331C42—24no—
332C42—24no—
333C42—21no—
334C42—21no—
335C42—24no—
336C42—684no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C2×C22⋊Dic1711yescharacteristic
347C23#1058763912yes—
348Q8—42no—
349Q8—48no—
350Q8—42no—
351D4—44no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—684no—
375C2×C4—1362no—
376C2×C4—1362no—
377C2×C4#1058750812yes—
378C2×C4C2×C22⋊Dic1711yescharacteristic
379C2×C4C2×C22⋊Dic1711yescharacteristic
380C2×C4#1058763912yes—
381C8—1362no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22#1176266511yescharacteristic
395C22C2×#1058750811yescharacteristic
396C22#1176282611yescharacteristic
397C4—22no—
398C4—21no—
399C4—22no—
400C4—41no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—1364no—
407C4#1176278312yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—81no—
412C2#1452870211yescharacteristic
413C2#1452872811yescharacteristic
414C2#1452875911yescharacteristic
subgroupG/H
centerC22#11762665
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#556C2
socleC22×C17#379
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenterC2×#556D17
hypocenterC17#8365
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #14528883C2semidirect
3. #14527304C2semidirect
16. C2×C17×C4⋊Q8C4semidirect
20. C2×C17×C22⋊Q8C4semidirect
23. C2×C4×Q8×C17C4semidirect
25. C2×C17×#117C4semidirect
208. C2×C4⋊Q8Dic17semidirect
212. C2×C22⋊Q8Dic17semidirect
215. C2×C4×Q8Dic17semidirect
217. C2×#117Dic17semidirect
282. C17#8365semidirect

Maximal subgroups

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

subgroupindex
#145288832
C4×#105875052
#145273042
#145271222
C2×C17×#5562
#836517