Finite Group Databasebeta

← #17034890

Subgroups — #17034890

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762765C211yesmaximal, characteristic
2#14531393C211yesmaximal, characteristic
3#14528946C211yesmaximal, characteristic, complemented
4#14528943C211yesmaximal, characteristic, complemented
5#14528919C211yesmaximal, characteristic, complemented
6#14528917C211yesmaximal, characteristic, complemented
7#14528815C212yesmaximal
8#14528811C212yesmaximal
9#14528682C212yesmaximal, complemented
10C4×#10587478C212yesmaximal, complemented
11C17×#2314C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C23×#3622C2211yescharacteristic
14C2×#10587478C2212yes—
15C2×#10587478C2212yes—
16C2×#10587478C2212yes—
17C2×#10587479C2212yes—
18#11762765C418yescomplemented
19C2×#10587474C2211yescharacteristic
20C2×#10587474C2211yescharacteristic
21C2×C4×#3622C2211yescharacteristic
22C2×C4×#3622C2211yescharacteristic
23#11762702—22no—
24#11762695—22no—
25#11762679C2211yescharacteristic
26#11762675C2211yescharacteristic
27#11762669C2212yes—
28C4×C4⋊Dic17C2212yes—
29#11762571C2212yes—
30#11762571C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33C2×C17×#117C2211yescharacteristic
34C17×#388—22no—
35C17×#387—22no—
36C17×#384C2211yescharacteristic
37C17×#382C2211yescharacteristic
38C17×#380C2212yes—
39C4×C4⋊C4×C17C2212yes—
40C23×C4×C17C2311yescharacteristic
41C17×#117C2×C414yes—
42C17×#117Q814yes—
43C2×C4⋊C4×C17—22no—
44C2×C4⋊C4×C17D412yes—
45C2×C4⋊C4×C17C2312yes—
46C2×C4⋊C4×C17C2312yes—
47C2×C22⋊C4×C17—22no—
48C2×C22⋊C4×C17D412yes—
49C2×C42×C17—22no—
50C2×C42×C17C2311yescharacteristic
51C2×C42×C17C2311yescharacteristic
52C2×C22⋊Dic17—22no—
53C2×C4⋊Dic17—22no—
54C2×C4⋊Dic17C2312yes—
55C2×C4⋊Dic17C2312yes—
56C2×C4×Dic17—22no—
57C2×#63223—22no—
58C2×#63223—22no—
59C2×#63223C2×C414yes—
60C22×#3622—22no—
61C22×#3622—22no—
62C22×#3622C2×C414yes—
63C22×#3622C2311yescharacteristic
64C22×#3622C2311yescharacteristic
65C22×#3622C2311yescharacteristic
66C22×#3622C2311yescharacteristic
67C17×#95—22no—
68C17×#95—22no—
69C17×#95—22no—
70#10587502—22no—
71#10587502—22no—
72#10587502—22no—
73#10587479—24no—
74#10587478—24no—
75#10587478—24no—
76#10587478—24no—
77#10587474C2×C414yes—
78#10587474C2×C414yes—
79C4×#3622C2×C414yes—
80C4×#3622C2×C414yes—
81C24×C17C2×D411yescharacteristic
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—22no—
87C22×C4×C17—22no—
88C22×C4×C17C22×C412yes—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D412yes—
91C22×C4×C17C2×D412yes—
92C22×C4×C17C2×Q812yes—
93C22×C4×C17C2411yescharacteristic
94C22×Dic17—22no—
95C22×Dic17—22no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—22no—
99C4⋊C4×C17—22no—
100C4⋊C4×C17—82no—
101C4⋊C4×C17C4⋊C418yes—
102C22⋊C4×C17—82no—
103C22⋊C4×C17C4⋊C418yes—
104C42×C17—82no—
105C42×C17C22×C412yes—
106C42×C17C22×C412yes—
107C42×C17C2×Q812yes—
108C42×C17C2×Q812yes—
109C22⋊Dic17—82no—
110C4⋊Dic17—22no—
111C4⋊Dic17—22no—
112C4⋊Dic17—22no—
113C4⋊Dic17—22no—
114C4⋊Dic17—82no—
115C4×Dic17—82no—
116#63223—28no—
117C2×#3622—21no—
118C2×#3622—21no—
119C2×#3622—21no—
120C2×#3622—21no—
121C2×#3622—28no—
122C2×#3622C22×C412yes—
123C2×#3622C22×C412yes—
124C2×#3622C22×C412yes—
125C2×#3622C22×C412yes—
126C2×#3622#4112yes—
127C2×#3622#4112yes—
128C2×#3622#4112yes—
129C2×#3622#4112yes—
130#20821—171nomaximal, Sylow
131C23×C17—21no—
132C23×C17—21no—
133C23×C17—22no—
134C23×C17—22no—
135C23×C17C2×C4⋊C412yes—
136C23×C17C22×D411yescharacteristic
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—22no—
140C2×C4×C17—22no—
141C2×C4×C17—22no—
142C2×C4×C17—22no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—42no—
148C2×C4×C17—42no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17C2×C4⋊C414yes—
154C2×C4×C17C2×C4⋊C414yes—
155C2×C4×C17C2×C4⋊C412yes—
156C2×C4×C17C23×C411yescharacteristic
157C2×C4×C17C22×Q811yescharacteristic
158C2×C4×C17C2×#4111yescharacteristic
159C2×C4×C17C2×#4111yescharacteristic
160C2×C4×C17C2×#4111yescharacteristic
161C2×C4×C17C2×#4111yescharacteristic
162C2×Dic17—42no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—42no—
168#3622—22no—
169#3622—22no—
170#3622#11714yes—
171#3622#11714yes—
172C2×#405—171no—
173#2314—171no—
174#2314D1711yescharacteristic
175#1846—171no—
176#1845—171no—
177#1836—171no—
178#1836—171no—
179#1809—172no—
180#1808—172no—
181#1776—172no—
182C4×#107—172no—
183C22×C17—24no—
184C22×C17—22no—
185C22×C17—42no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17C22×C4⋊C411yescharacteristic
189C22×C17#53211yescharacteristic
190C22×C17#53211yescharacteristic
191C22×C17#57612yes—
192C22×C17#57812yes—
193C4×C17—24no—
194C4×C17—24no—
195C4×C17—22no—
196C4×C17—82no—
197C4×C17—82no—
198C4×C17C2×#11711yescharacteristic
199C4×C17C2×#11711yescharacteristic
200C4×C17#53311yescharacteristic
201C4×C17#53311yescharacteristic
202Dic17—82no—
203Dic17—82no—
204C22×M16—171no—
205C23×C8—171no—
206C2×#117C2×D1711yescharacteristic
207C2×#107—172no—
208C2×#107—172no—
209C2×#107—172no—
210C2×#106—172no—
211#405—178no—
212C2×#97—171no—
213C2×#97—171no—
214C2×C4×C8—171no—
215C2×C4×C8—171no—
216#388—22no—
217#388—342no—
218#387—22no—
219#387—342no—
220#384—171no—
221#384C2×D1711yescharacteristic
222#382—171no—
223#382C2×D1711yescharacteristic
224#380—172no—
225#380C2×D1712yes—
226C4×C4⋊C4—172no—
227C4×C4⋊C4C2×D1712yes—
228#340—172no—
229#340—172no—
230#340—172no—
231#340—172no—
232C2×C17—42no—
233C2×C17#228511yescharacteristic
234C2×C17#291312yes—
235C2×C17#317512yes—
236C2×C17#317611yescharacteristic
237C2×C17#317711yescharacteristic
238C23×C4C22×D1711yescharacteristic
239C2×M16—174no—
240C2×M16—342no—
241C2×M16—342no—
242C22×C8—171no—
243C22×C8—171no—
244C22×C8—174no—
245C22×C8—171no—
246C22×C8—171no—
247C22×C8—342no—
248C22×C8—342no—
249#117Dic3414yes—
250#117C4×D1714yes—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—172no—
253C2×C4⋊C4—172no—
254C2×C4⋊C4—342no—
255C2×C4⋊C4C22⋊D1712yes—
256C2×C4⋊C4C22×D1712yes—
257C2×C4⋊C4C22×D1712yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—342no—
260C2×C22⋊C4C22⋊D1712yes—
261C2×C42—22no—
262C2×C42—342no—
263C2×C42C22×D1711yescharacteristic
264C2×C42C22×D1711yescharacteristic
265#107—344no—
266#107—344no—
267#107—344no—
268#106—344no—
269#97—174no—
270#97—174no—
271C4×C8—174no—
272C4×C8—174no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—342no—
277#95—342no—
278#95—342no—
279C17#2082111yescharacteristic, complemented, Sylow
280C24C2×C22⋊D1711yescharacteristic
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—342no—
288C22×C4—342no—
289C22×C4C2×Dic3412yes—
290C22×C4C2×C4×D1712yes—
291C22×C4C2×C22⋊D1711yescharacteristic
292C22×C4C2×C22⋊D1712yes—
293C22×C4C2×C22⋊D1712yes—
294C22×C4C23×D1711yescharacteristic
295M16—348no—
296C2×C8—172no—
297C2×C8—172no—
298C2×C8—172no—
299C2×C8—172no—
300C2×C8—172no—
301C2×C8—172no—
302C2×C8—172no—
303C2×C8—172no—
304C2×C8—348no—
305C2×C8—341no—
306C2×C8—341no—
307C2×C8—341no—
308C2×C8—341no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—82no—
314C4⋊C4—342no—
315C4⋊C4—342no—
316C4⋊C4—342no—
317C4⋊C4—342no—
318C4⋊C4—1362no—
319C4⋊C4#6322518yes—
320C22⋊C4—82no—
321C22⋊C4—1362no—
322C22⋊C4#6322518yes—
323C42—82no—
324C42—1362no—
325C42C2×Dic3412yes—
326C42C2×Dic3412yes—
327C42C2×C4×D1712yes—
328C42C2×C4×D1712yes—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—22no—
333C23C2×#6322512yes—
334C23C22×C22⋊D1711yescharacteristic
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—682no—
352C2×C4—682no—
353C2×C4—682no—
354C2×C4—682no—
355C2×C4—682no—
356C2×C4—682no—
357C2×C4C2×#6322514yes—
358C2×C4C2×#6322514yes—
359C2×C4C2×#6322512yes—
360C2×C4C22×Dic3411yescharacteristic
361C2×C4C22×C4×D1711yescharacteristic
362C2×C4#41×D1711yescharacteristic
363C2×C4#41×D1711yescharacteristic
364C2×C4#41×D1711yescharacteristic
365C2×C4#41×D1711yescharacteristic
366C8—174no—
367C8—174no—
368C8—342no—
369C8—342no—
370C22—22no—
371C22—24no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22C22×#6322511yescharacteristic
376C22#1176335511yescharacteristic
377C22#1176335511yescharacteristic
378C22#1176361212yes—
379C22#1176362412yes—
380C4—22no—
381C4—24no—
382C4—24no—
383C4—82no—
384C4—82no—
385C4—1362no—
386C4—1362no—
387C4#117×D1711yescharacteristic
388C4#117×D1711yescharacteristic
389C4#1176336211yescharacteristic
390C4#1176336211yescharacteristic
391C2—42no—
392C2#1453118711yescharacteristic
393C2#1453738212yes—
394C2#1453941712yes—
395C2#1453942811yescharacteristic
396C2#1453943711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2314C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#2314D17
hypocenterC17#20821
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #14528946C2semidirect
4. #14528943C2semidirect
5. #14528919C2semidirect
6. #14528917C2semidirect
9. #14528682C2semidirect
10. C4×#10587478C2semidirect
18. #11762765C4semidirect
279. C17#20821semidirect

Maximal subgroups

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

subgroupindex
C2×#117627652
#145313932
#145289462
#145289432
#145289192
#145289172
#145288152
#145288112
#145286822
C4×#105874782
C17×#23142
#2082117