Finite Group Databasebeta

← #17038709

Subgroups — #17038709

Order 4352.

770 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587636C211yesmaximal, characteristic
2#14537426C211yesmaximal, characteristic, complemented
3#14529671C212yesmaximal
4#14529529C212yesmaximal, complemented
5#14529491C212yesmaximal, complemented
6#14529487C212yesmaximal, complemented
7C4×#10587493C211yesmaximal, characteristic, complemented
8C4×#10587496C211yesmaximal, characteristic, complemented
9C4×#10587508C211yesmaximal, characteristic, complemented
10C4×C8×Dic17C211yesmaximal, characteristic, complemented
11C17×#2922C211yesmaximal, characteristic, complemented
12C2×#10587636C2211yescharacteristic
13C4×#63252C2211yescharacteristic
14C4×#63231C412yescomplemented
15C4×#63230C412yescomplemented
16C4×#63229C412yescomplemented
17C4×#63228C412yescomplemented
18#11762868—22no—
19#11762860—22no—
20C2×#10587493C2212yes—
21#11762845C2211yescharacteristic
22#11762835C2212yes—
23C2×#10587508C2212yes—
24C2×#10587496C2212yes—
25#11762805C2211yescharacteristic
26#11762803C2211yescharacteristic
27#11762775C2212yes—
28C4×#63223C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30C2×C8×Dic17C2212yes—
31C2×C8×Dic17C2212yes—
32C42×Dic17C2211yescharacteristic
33#11762573C2212yes—
34#11762570C2212yes—
35#11762568C2212yes—
36C4×#41×C17C2211yescharacteristic
37C17×#433—22no—
38C17×#431—22no—
39C17×#424C2212yes—
40C17×#408C2212yes—
41C4×M16×C17C2211yescharacteristic
42C2×C4×C8×C17C2211yescharacteristic
43C2×#41×C17C2311yescharacteristic
44C2×#63252C2311yescharacteristic
45C2×M16×C17C2312yes—
46C22×C8×C17C2312yes—
47C4×Q8×C17C2×C412yes—
48C4×D4×C17—22no—
49C4×D4×C17C2×C412yes—
50C17×#117—21no—
51C17×#117C2311yescharacteristic
52C2×C42×C17—21no—
53C2×C42×C17C2311yescharacteristic
54#10587636—24no—
55C2×#63231C2×C411yescharacteristic
56C2×#63230C2×C411yescharacteristic
57C2×#63229C2×C411yescharacteristic
58C2×#63228C2×C411yescharacteristic
59C4×C22⋊D17—22no—
60#10587612C2311yescharacteristic
61C2×C4×Dic17—22no—
62C2×C4×Dic17C2312yes—
63C2×#63223C2312yes—
64C22×#3622C2311yescharacteristic
65C22×#3622C2311yescharacteristic
66C4×D68C2×C412yes—
67#10587558—21no—
68C42×D17—21no—
69C4×Dic34C2×C412yes—
70C17×C4⋊C8—24no—
71C17×C22⋊C8—24no—
72C17×#97—21no—
73C4×C8×C17—21no—
74C4×C8×C17D412yes—
75C4×C8×C17D412yes—
76#10587508—24no—
77#10587506C2×C412yes—
78#10587503C2×C412yes—
79#10587496—24no—
80#10587493—24no—
81#10587491—24no—
82#10587485—24no—
83C8×Dic17—24no—
84C8×Dic17—24no—
85#10587481C2×C412yes—
86#10587480C2×C412yes—
87#10587479C2×C412yes—
88#10587478C2×C412yes—
89#10587474—21no—
90C4×#3622—21no—
91C4×#3622C2×C412yes—
92C4×#3622C2×C412yes—
93#41×C17—22no—
94#41×C17—22no—
95C2×Q8×C17C22×C411yescharacteristic
96C2×D4×C17—21no—
97C2×D4×C17C22×C411yescharacteristic
98C22×C4×C17—22no—
99C22×C4×C17—21no—
100C22×C4×C17#4112yes—
101C22×C4×C17C2411yescharacteristic
102C2×C22⋊D17—21no—
103C22×Dic17—22no—
104#63252—22no—
105#63252—22no—
106C2×D68C22×C411yescharacteristic
107C2×C4×D17—22no—
108C2×C4×D17—21no—
109C2×Dic34C22×C411yescharacteristic
110M16×C17—22no—
111M16×C17—22no—
112C2×C8×C17—22no—
113C2×C8×C17—22no—
114C2×C8×C17—22no—
115C2×C8×C17C2×D414yes—
116C4⋊C4×C17—22no—
117C4⋊C4×C17C22×C412yes—
118C22⋊C4×C17—21no—
119C22⋊C4×C17—22no—
120C42×C17—22no—
121C42×C17—21no—
122C42×C17C22×C412yes—
123C42×C17C2×D411yescharacteristic
124C42×C17C2×D411yescharacteristic
125C22⋊Dic17—21no—
126#63231—22no—
127#63230—22no—
128#63229—22no—
129#63228—22no—
130#63227—22no—
131C4⋊Dic17C22×C412yes—
132#63225—22no—
133C4×Dic17—28no—
134C4×Dic17—22no—
135C4×Dic17—21no—
136C4×Dic17—22no—
137C4×Dic17#4114yes—
138#63223—22no—
139#63223—22no—
140C2×#3622—22no—
141C2×#3622—22no—
142C2×#3622C22×C412yes—
143C2×#3622C22×C411yescharacteristic
144C2×#3622C22×C411yescharacteristic
145C2×#3622#4114yes—
146#21292—171nomaximal, Sylow
147C23×C17—21no—
148C23×C17C2×#4111yescharacteristic
149C22×D17—21no—
150Q8×C17—21no—
151Q8×C17—21no—
152D4×C17—21no—
153D4×C17—21no—
154D4×C17—42no—
155C2×C4×C17—24no—
156C2×C4×C17—22no—
157C2×C4×C17—21no—
158C2×C4×C17—42no—
159C2×C4×C17—44no—
160C2×C4×C17C4×D412yes—
161C2×C4×C17C4×D412yes—
162C2×C4×C17C23×C411yescharacteristic
163C2×C4×C17C22×D411yescharacteristic
164C2×C4×C17C2×#4112yes—
165C2×C4×C17C2×#4112yes—
166C22⋊D17—42no—
167C2×Dic17—28no—
168C2×Dic17—24no—
169C2×Dic17—22no—
170C2×Dic17—21no—
171D68—21no—
172D68—21no—
173C4×D17—42no—
174C4×D17—44no—
175Dic34—21no—
176Dic34—21no—
177C8×C17—24no—
178C8×C17—42no—
179#3622—22no—
180#3622—42no—
181#3622#11712yes—
182#3622#11712yes—
183C4×#135—171no—
184#2922—171no—
185#2922D1711yescharacteristic, complemented
186#1974—172no—
187#1945—172no—
188#1937—172no—
189#1937—172no—
190C4×#108—171no—
191C4×#104—171no—
192C4×#104—171no—
193C42×C8—171no—
194C22×C17—22no—
195C22×C17—42no—
196C22×C17—41no—
197C22×C17C2×C4×D411yescharacteristic
198C22×C17#52912yes—
199C2×D17—42no—
200C2×D17—41no—
201C4×C17—22no—
202C4×C17—42no—
203C4×C17—41no—
204C4×C17#44314yes—
205C4×C17C2×#11711yescharacteristic
206C4×C17C4×#4111yescharacteristic
207C4×C17#53212yes—
208Dic17—28no—
209Dic17—42no—
210Dic17—41no—
211C2×#135—171no—
212C4×#41—171no—
213C4×#41C2×D1711yescharacteristic
214C4×Q16—172no—
215C4×SD16—172no—
216C4×SD16—172no—
217C4×D8—172no—
218#433—22no—
219#433—342no—
220#431—22no—
221#431—342no—
222C2×#108—172no—
223#427—171no—
224#424—172no—
225#424C2×D1712yes—
226C2×#104—172no—
227C2×#104—172no—
228#416—171no—
229#416—171no—
230#408—172no—
231#408C2×D1712yes—
232C4×M16—171no—
233C4×M16C2×D1711yescharacteristic
234C2×C4×C8—171no—
235C2×C4×C8—172no—
236C2×C4×C8—172no—
237C2×C4×C8C2×D1711yescharacteristic
238C43—171no—
239#341—172no—
240#339—172no—
241#339—172no—
242C2×C17—22no—
243C2×C17—81no—
244C2×C17#229311yescharacteristic
245C2×C17C2×#44312yes—
246D17—81no—
247C2×#41—171no—
248C2×#41C22×D1711yescharacteristic
249#135—344no—
250C2×Q16—171no—
251C2×SD16—171no—
252C2×SD16—171no—
253C2×D8—171no—
254C2×M16—172no—
255C2×M16C22×D1712yes—
256C22×C8—171no—
257C22×C8—171no—
258C22×C8C22×D1712yes—
259C4×Q8—172no—
260C4×Q8C4×D1712yes—
261C4×D4—22no—
262C4×D4—172no—
263C4×D4—342no—
264C4×D4C4×D1712yes—
265#117—21no—
266#117—171no—
267#117—341no—
268#117C22×D1711yescharacteristic
269C2×C42—21no—
270C2×C42—172no—
271C2×C42—342no—
272C2×C42—341no—
273C2×C42C22×D1711yescharacteristic
274#108—344no—
275#107—172no—
276#106—172no—
277C4⋊C8—24no—
278C4⋊C8—344no—
279#104—344no—
280#104—344no—
281#103—172no—
282#103—172no—
283#102—172no—
284#102—172no—
285C22⋊C8—24no—
286C22⋊C8—344no—
287#97—21no—
288#97—341no—
289C4×C8—21no—
290C4×C8—172no—
291C4×C8—172no—
292C4×C8—344no—
293C4×C8—344no—
294C4×C8—341no—
295C4×C8C22⋊D1712yes—
296C4×C8C22⋊D1712yes—
297C17#2129211yescharacteristic, complemented, Sylow
298#41—22no—
299#41—22no—
300#41—342no—
301#41—342no—
302C2×Q8—171no—
303C2×Q8C2×C4×D1711yescharacteristic
304C2×D4—21no—
305C2×D4—171no—
306C2×D4—341no—
307C2×D4C2×C4×D1711yescharacteristic
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—342no—
311C22×C4—341no—
312C22×C4—342no—
313C22×C4#6325212yes—
314C22×C4C23×D1711yescharacteristic
315Q16—342no—
316SD16—342no—
317SD16—342no—
318D8—342no—
319M16—22no—
320M16—22no—
321M16—342no—
322M16—342no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—171no—
327C2×C8—171no—
328C2×C8—174no—
329C2×C8—172no—
330C2×C8—342no—
331C2×C8—342no—
332C2×C8C2×C22⋊D1714yes—
333C4⋊C4—22no—
334C4⋊C4—172no—
335C4⋊C4—342no—
336C4⋊C4C2×C4×D1712yes—
337C22⋊C4—21no—
338C22⋊C4—22no—
339C22⋊C4—341no—
340C22⋊C4—342no—
341C42—21no—
342C42—22no—
343C42—174no—
344C42—348no—
345C42—342no—
346C42—341no—
347C42—342no—
348C42C2×C4×D1712yes—
349C42C2×C22⋊D1711yescharacteristic
350C42C2×C22⋊D1711yescharacteristic
351C23—21no—
352C23—341no—
353C23C2×#6325211yescharacteristic
354Q8—21no—
355Q8—21no—
356Q8—341no—
357Q8—341no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—341no—
362D4—341no—
363D4—682no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—348no—
370C2×C4—344no—
371C2×C4—342no—
372C2×C4—341no—
373C2×C4—682no—
374C2×C4—684no—
375C2×C4C4×C22⋊D1712yes—
376C2×C4C4×C22⋊D1712yes—
377C2×C4C22×C4×D1711yescharacteristic
378C2×C4#41×D1712yes—
379C2×C4#41×D1712yes—
380C2×C4C22×C22⋊D1711yescharacteristic
381C8—24no—
382C8—42no—
383C8—172no—
384C8—172no—
385C8—342no—
386C8—682no—
387C22—22no—
388C22—41no—
389C22—42no—
390C22—681no—
391C22—682no—
392C22C2×C4×C22⋊D1711yescharacteristic
393C22#1176333712yes—
394C4—22no—
395C4—41no—
396C4—42no—
397C4—348no—
398C4—681no—
399C4—682no—
400C4#1176292414yes—
401C4#117×D1711yescharacteristic
402C4#1176327711yescharacteristic
403C4#1176335512yes—
404C2—22no—
405C2—81no—
406C2—1361no—
407C2#1453124311yescharacteristic
408C2C2×#1176292412yes—
subgroupG/H
centerC42C2×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2922C2
socleC22×C17C2×C4×D4
prime radicalC2×#10587636C22
cosocleC22×C4×C17C24
hypercenter#2922D17
hypocenterC17#21292
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C42 ◁ C2×C4×C8 ◁ #2922

Complements

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

NH ≅ G/Ntype
2. #14537426C2semidirect
4. #14529529C2semidirect
5. #14529491C2semidirect
6. #14529487C2semidirect
7. C4×#10587493C2semidirect
8. C4×#10587496C2semidirect
9. C4×#10587508C2semidirect
10. C4×C8×Dic17C2semidirect
11. C17×#2922C2semidirect
14. C4×#63231C4semidirect
15. C4×#63230C4semidirect
16. C4×#63229C4semidirect
17. C4×#63228C4semidirect
185. #2922D17semidirect
297. C17#21292semidirect

Maximal subgroups

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

subgroupindex
C4×#105876362
#145374262
#145296712
#145295292
#145294912
#145294872
C4×#105874932
C4×#105874962
C4×#105875082
C4×C8×Dic172
C17×#29222
#2129217