Finite Group Databasebeta

← #17034926

Subgroups — #17034926

Order 4352.

842 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762761C211yesmaximal, characteristic
2#14531180C212yesmaximal
3#14528947C211yesmaximal, characteristic, complemented
4#14528945C211yesmaximal, characteristic, complemented
5#14528913C212yesmaximal, complemented
6#14528818C211yesmaximal, characteristic
7#14528816C212yesmaximal
8#14528814C211yesmaximal, characteristic
9#14528682C212yesmaximal, complemented
10#14528681C211yesmaximal, characteristic, complemented
11C17×#1776C211yesmaximal, characteristic, complemented
12C22×#63218C2211yescharacteristic
13C22×#63218C2211yescharacteristic
14C2×C42×D17C2211yescharacteristic
15C2×#10587489C2211yescharacteristic
16C2×#10587478C2212yes—
17C2×#10587488C2211yescharacteristic
18C2×#10587479C2212yes—
19#11762761C418yescomplemented
20C2×#10587474C2211yescharacteristic
21C2×C8×Dic17C2211yescharacteristic
22C2×C8×Dic17C2211yescharacteristic
23#11762676—22no—
24#11762674C2212yes—
25#11762674C2212yes—
26#11762671—22no—
27#11762666C2212yes—
28#11762666C2212yes—
29C4×C4⋊Dic17C2212yes—
30C4×#63227—22no—
31#11762655—22no—
32#11762572C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35C2×C17×#107C2211yescharacteristic
36C2×C17×#106C2211yescharacteristic
37C2×C17×#97C2211yescharacteristic
38C4×C4⋊C4×C17C2212yes—
39C17×#340C2212yes—
40C22×C4×D17D412yes—
41C22×C4×D17C2311yescharacteristic
42C22×C8×C17C2311yescharacteristic
43C22×C8×C17C2311yescharacteristic
44C2×C4⋊C4×C17C2312yes—
45C2×C4⋊C4×C17C2312yes—
46C2×C42×C17—22no—
47C2×C42×C17C2311yescharacteristic
48C2×#63227—22no—
49C2×#63227—22no—
50C2×C4⋊Dic17C2312yes—
51C2×C4⋊Dic17C2312yes—
52C2×C4×Dic17—22no—
53C2×C4×Dic17D412yes—
54C2×C4×Dic17C2311yescharacteristic
55C22×#3622C2311yescharacteristic
56C22×#3622C2311yescharacteristic
57C2×#63218—22no—
58C2×#63218—22no—
59C2×#63218—22no—
60C2×#63218—22no—
61C2×#63218C2×C414yes—
62C2×#63218C2×C414yes—
63C42×D17C2×C414yes—
64C42×D17Q814yes—
65C17×#107—22no—
66C17×#106—22no—
67C17×#97C2×C414yes—
68C17×#95—22no—
69#10587502—22no—
70#10587502—22no—
71#10587502—22no—
72#10587502—22no—
73#10587502—22no—
74#10587502—22no—
75#10587502—22no—
76#10587489—22no—
77#10587488—22no—
78C8×Dic17C2×C414yes—
79C8×Dic17C2×C414yes—
80#10587479—24no—
81#10587478—24no—
82#10587474C2×C414yes—
83C23×D17C2×D411yescharacteristic
84C22×C4×C17—22no—
85C22×C4×C17—22no—
86C22×C4×C17C2×D412yes—
87C22×C4×C17C2411yescharacteristic
88C22×Dic17—22no—
89C22×Dic17—22no—
90C22×Dic17C2×D412yes—
91C22×Dic17C2×D411yescharacteristic
92C2×C4×D17—21no—
93C2×C4×D17—21no—
94C2×C4×D17—21no—
95C2×C4×D17—21no—
96C2×C4×D17—42no—
97C2×C4×D17—42no—
98C2×C4×D17C4⋊C418yes—
99C2×C4×D17C22×C412yes—
100C2×C4×D17C2×Q812yes—
101C2×C8×C17—21no—
102C2×C8×C17—21no—
103C2×C8×C17C22×C412yes—
104C2×C8×C17C22×C412yes—
105C2×C8×C17#4112yes—
106C2×C8×C17#4112yes—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—22no—
109C4⋊C4×C17—22no—
110C4⋊C4×C17—22no—
111C42×C17—44no—
112C42×C17C22×C412yes—
113C42×C17C2×Q812yes—
114#63227—44no—
115#63227—44no—
116C4⋊Dic17—22no—
117C4⋊Dic17—22no—
118C4⋊Dic17—22no—
119C4⋊Dic17—22no—
120C4×Dic17—44no—
121C4×Dic17C4⋊C418yes—
122C4×Dic17C22×C412yes—
123C4×Dic17C2×Q812yes—
124C2×#3622—21no—
125C2×#3622—21no—
126C2×#3622C22×C412yes—
127C2×#3622C22×C412yes—
128C2×#3622#4112yes—
129C2×#3622#4112yes—
130#63218—28no—
131#63218—28no—
132#20825—171nomaximal, Sylow
133C23×C17C22×D411yescharacteristic
134C22×D17—21no—
135C22×D17—21no—
136C22×D17—22no—
137C22×D17—22no—
138C22×D17C2×C4⋊C412yes—
139C2×C4×C17—22no—
140C2×C4×C17—22no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—42no—
147C2×C4×C17C2×C4⋊C414yes—
148C2×C4×C17C23×C411yescharacteristic
149C2×C4×C17C22×Q811yescharacteristic
150C2×C4×C17C2×#4111yescharacteristic
151C2×C4×C17C2×#4111yescharacteristic
152C2×C4×C17C2×#4111yescharacteristic
153C2×C4×C17C2×#4111yescharacteristic
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—22no—
157C2×Dic17—22no—
158C2×Dic17—42no—
159C2×Dic17—42no—
160C2×Dic17—42no—
161C2×Dic17—42no—
162C2×Dic17—42no—
163C2×Dic17—42no—
164C2×Dic17C2×C4⋊C414yes—
165C2×Dic17C2×C4⋊C412yes—
166C4×D17—22no—
167C4×D17—22no—
168C4×D17—22no—
169C4×D17—22no—
170C4×D17—48no—
171C8×C17—22no—
172C8×C17#11714yes—
173#3622—22no—
174#3622#11714yes—
175C2×C4×M16—171no—
176#2283—172no—
177#1846—171no—
178#1846—171no—
179#1835—172no—
180#1809—171no—
181#1809—171no—
182#1809—172no—
183#1776—171no—
184#1776—172no—
185#1776D1711yescharacteristic, complemented
186C22×C17C22×C4⋊C411yescharacteristic
187C22×C17#53211yescharacteristic
188C22×C17#53211yescharacteristic
189C22×C17#57512yes—
190C22×C17#57512yes—
191C2×D17—24no—
192C2×D17—22no—
193C2×D17—42no—
194C2×D17—41no—
195C2×D17—41no—
196C4×C17—24no—
197C4×C17—82no—
198C4×C17—82no—
199C4×C17C2×#11711yescharacteristic
200C4×C17C2×#11711yescharacteristic
201C4×C17#53311yescharacteristic
202C4×C17#53311yescharacteristic
203Dic17—24no—
204Dic17—22no—
205Dic17—82no—
206Dic17—82no—
207C22×M16—171no—
208C22×M16—171no—
209C22×C42—171no—
210C2×#107—171no—
211C2×#107—172no—
212C2×#107C2×D1711yescharacteristic
213C2×#106—171no—
214C2×#106—172no—
215C2×#106C2×D1711yescharacteristic
216C4×M16—178no—
217C2×#97—171no—
218C2×#97C2×D1711yescharacteristic
219C2×C4×C8—171no—
220C2×C4×C8—171no—
221#383—342no—
222#382—172no—
223#382—172no—
224#381—342no—
225#380—172no—
226#380—172no—
227C4×C4⋊C4—172no—
228C4×C4⋊C4C2×D1712yes—
229C4×C22⋊C4—342no—
230#376—342no—
231#340—172no—
232#340—172no—
233#340—172no—
234#340C2×D1712yes—
235C2×C17#228511yescharacteristic
236C2×C17#291112yes—
237C2×C17#318312yes—
238C2×C17#318311yescharacteristic
239C2×C17#318311yescharacteristic
240D17—42no—
241C23×C4—171no—
242C23×C4—172no—
243C2×M16—174no—
244C2×M16—174no—
245C2×M16—342no—
246C2×M16—342no—
247C2×M16—342no—
248C2×M16—342no—
249C22×C8—171no—
250C22×C8—171no—
251C22×C8C22×D1711yescharacteristic
252C22×C8C22×D1711yescharacteristic
253C2×C4⋊C4—172no—
254C2×C4⋊C4—172no—
255C2×C4⋊C4C22×D1712yes—
256C2×C4⋊C4C22×D1712yes—
257C2×C22⋊C4—342no—
258C2×C22⋊C4—342no—
259C2×C42—22no—
260C2×C42—172no—
261C2×C42—171no—
262C2×C42—174no—
263C2×C42—174no—
264C2×C42—342no—
265C2×C42C22×D1711yescharacteristic
266#107—22no—
267#107—344no—
268#107—342no—
269#106—22no—
270#106—344no—
271#106—342no—
272#97—174no—
273#97C4×D1714yes—
274C4×C8—174no—
275C4×C8—174no—
276#95—22no—
277#95—342no—
278#95—342no—
279#95—342no—
280#95—342no—
281#95—342no—
282#95—342no—
283#95—342no—
284C17#2082511yescharacteristic, complemented, Sylow
285C24—171no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—172no—
289C22×C4—171no—
290C22×C4—172no—
291C22×C4—172no—
292C22×C4—178no—
293C22×C4—341no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4—341no—
297C22×C4—342no—
298C22×C4—342no—
299C22×C4—682no—
300C22×C4—682no—
301C22×C4D4×D1712yes—
302C22×C4C23×D1711yescharacteristic
303M16—348no—
304M16—348no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—172no—
308C2×C8—172no—
309C2×C8—172no—
310C2×C8—172no—
311C2×C8—341no—
312C2×C8—341no—
313C2×C8C2×C4×D1712yes—
314C2×C8C2×C4×D1712yes—
315C2×C8#6325212yes—
316C2×C8#6325212yes—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—342no—
322C4⋊C4—342no—
323C4⋊C4—342no—
324C4⋊C4—342no—
325C22⋊C4—684no—
326C22⋊C4—684no—
327C42—44no—
328C42—172no—
329C42—172no—
330C42—178no—
331C42—684no—
332C42C2×C4×D1712yes—
333C42Q8×D1712yes—
334C23—172no—
335C23—342no—
336C23—342no—
337C23—341no—
338C23—341no—
339C23C2×D4×D1711yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—172no—
349C2×C4—174no—
350C2×C4—342no—
351C2×C4—342no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—342no—
355C2×C4—342no—
356C2×C4—342no—
357C2×C4—342no—
358C2×C4—688no—
359C2×C4—682no—
360C2×C4—682no—
361C2×C4—682no—
362C2×C4—682no—
363C2×C4—682no—
364C2×C4—682no—
365C2×C4C4⋊C4×D1714yes—
366C2×C4C22×C4×D1711yescharacteristic
367C2×C4C2×#6325211yescharacteristic
368C2×C4C2×#6325211yescharacteristic
369C2×C4C2×Q8×D1711yescharacteristic
370C2×C4#41×D1711yescharacteristic
371C2×C4#41×D1711yescharacteristic
372C8—22no—
373C8—174no—
374C8—342no—
375C8#1058755814yes—
376C22—342no—
377C22—344no—
378C22—682no—
379C22—681no—
380C22—681no—
381C22C2×C4⋊C4×D1711yescharacteristic
382C22#1176335611yescharacteristic
383C22#1176335611yescharacteristic
384C22#1176360512yes—
385C22#1176360812yes—
386C4—24no—
387C4—82no—
388C4—82no—
389C4—342no—
390C4—344no—
391C4—1362no—
392C4—1362no—
393C4#117×D1711yescharacteristic
394C4C2×#1058755811yescharacteristic
395C4#1176336311yescharacteristic
396C4#1176336311yescharacteristic
397C2—682no—
398C2#1453118811yescharacteristic
399C2#1453737112yes—
400C2#1453950912yes—
401C2#1453951111yescharacteristic
402C2#1453951611yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1776C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1776D17
hypocenterC17#20825
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
3. #14528947C2semidirect
4. #14528945C2semidirect
5. #14528913C2semidirect
9. #14528682C2semidirect
10. #14528681C2semidirect
11. C17×#1776C2semidirect
19. #11762761C4semidirect
185. #1776D17semidirect
284. C17#20825semidirect

Maximal subgroups

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

subgroupindex
C2×#117627612
#145311802
#145289472
#145289452
#145289132
#145288182
#145288162
#145288142
#145286822
#145286812
C17×#17762
#2082517