Finite Group Databasebeta

← #17019620

Subgroups — #17019620

Order 4352.

740 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#14539699C211yesmaximal, characteristic
2C2×#11763009C211yesmaximal, characteristic, complemented
3#14537865C211yesmaximal, characteristic, complemented
4#14528230C214yesmaximal, complemented
5#14528142C214yesmaximal, complemented
6#14527560C211yesmaximal, characteristic, complemented
7#14527453C211yesmaximal, characteristic, complemented
8#14527433C211yesmaximal, characteristic, complemented
9C17×#3037C211yesmaximal, characteristic, complemented
10C22×#63230—21no—
11C2×#10587560C2211yescharacteristic
12#11763185C2212yes—
13#11763157C2212yes—
14#11763009—22no—
15#11763009C2214yes—
16#11763004—22no—
17#11762942—22no—
18Q8×#3622—22no—
19Q8×#3622C2214yescomplemented
20#11762862—22no—
21#11762862C2211yescharacteristic
22#11762862C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587480—21no—
25#11762561C2214yescomplemented
26#11762555C2212yescomplemented
27#11762553C2212yescomplemented
28#11762552C2214yes—
29C2×C17×#137—21no—
30C17×#532C2211yescharacteristic
31C2×C4×Q8×C17C2211yescharacteristic
32C17×#464—22no—
33C17×#462C2212yescomplemented
34C17×#460C2212yescomplemented
35C17×#423C2211yescharacteristic
36C17×#417—21no—
37C2×#41×C17—21no—
38C22×Q8×C17D411yescharacteristic
39C22×D68—21no—
40C22×D68—21no—
41C22×D68—21no—
42C17×#137—44no—
43C2×Q16×C17—22no—
44C2×SD16×C17—22no—
45C2×M16×C17—21no—
46C17×C4⋊Q8C2311yescharacteristic
47C17×#127C2311yescharacteristic
48C17×#124—21no—
49C17×#121—21no—
50C17×#121—21no—
51C4×Q8×C17—21no—
52C4×Q8×C17—21no—
53C4×Q8×C17D412yes—
54C4×Q8×C17C2312yes—
55C4×D4×C17—21no—
56C4×D4×C17—21no—
57C2×C4⋊C4×C17—21no—
58C2×C4⋊C4×C17D411yescharacteristic
59C2×C42×C17—21no—
60C2×C42×C17C2311yescharacteristic
61C2×#63230—24no—
62C2×#63230—44no—
63C2×#63228—24no—
64C22×#3622—21no—
65#10587560—21no—
66#10587560—21no—
67#10587560—21no—
68#10587560C2312yes—
69C17×C4⋊C8—21no—
70C17×C4⋊C8C2312yes—
71C17×#103—22no—
72C17×#102—22no—
73C22⋊#3622—21no—
74C22⋊#3622D412yes—
75#10587480—24no—
76#10587480—24no—
77C4⋊#3622—24no—
78C4⋊#3622—21no—
79C4⋊#3622D412yes—
80C4⋊#3622C2312yes—
81C4⋊#3622C2312yes—
82C4×#3622—24no—
83C4×#3622—21no—
84C4×#3622C2312yes—
85C23×D17—41no—
86#41×C17—42no—
87#41×C17—42no—
88C2×Q8×C17—21no—
89C2×Q8×C17—22no—
90C2×Q8×C17—21no—
91C2×Q8×C17D814yescomplemented
92C2×Q8×C17C2×D412yes—
93C2×D4×C17—21no—
94C2×D4×C17—21no—
95C2×D4×C17—21no—
96C2×D4×C17—41no—
97C22×C4×C17—21no—
98C22×C4×C17—21no—
99C22×C4×C17—41no—
100C22×C4×C17C2×D411yescharacteristic
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17#4111yescharacteristic
103C2×D68—22no—
104C2×D68—22no—
105C2×D68—22no—
106C2×D68—42no—
107C2×D68—42no—
108C2×D68—41no—
109C2×D68—41no—
110C2×D68—41no—
111C2×D68—82no—
112C2×D68—82no—
113Q16×C17—44no—
114SD16×C17—44no—
115M16×C17—42no—
116C2×C8×C17—22no—
117C4⋊C4×C17—22no—
118C4⋊C4×C17—21no—
119C4⋊C4×C17—22no—
120C4⋊C4×C17—21no—
121C4⋊C4×C17—21no—
122C4⋊C4×C17C2×D412yes—
123C22⋊C4×C17—41no—
124C22⋊C4×C17—41no—
125C42×C17—22no—
126C42×C17—22no—
127C42×C17C2×D411yescharacteristic
128C42×C17C2×D411yescharacteristic
129C42×C17#4111yescharacteristic
130C42×C17C2411yescharacteristic
131#63230—48no—
132#63228—84no—
133C2×#3622—22no—
134C2×#3622—22no—
135C2×#3622—42no—
136C2×#3622C2×D414yes—
137#19398—171nomaximal, Sylow
138C23×C17—41no—
139C23×C17#12111yescharacteristic
140C22×D17—42no—
141C22×D17—82no—
142C22×D17—82no—
143C22×D17—82no—
144Q8×C17—22no—
145Q8×C17—24no—
146Q8×C17—41no—
147Q8×C17—41no—
148Q8×C17C2×D814yes—
149D4×C17—41no—
150D4×C17—41no—
151D4×C17—42no—
152D4×C17—81no—
153D4×C17—82no—
154C2×C4×C17—22no—
155C2×C4×C17—24no—
156C2×C4×C17—22no—
157C2×C4×C17—22no—
158C2×C4×C17—41no—
159C2×C4×C17—41no—
160C2×C4×C17—81no—
161C2×C4×C17—82no—
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C2×C4×C17#12111yescharacteristic
165C2×C4×C17C2×D812yes—
166C2×C4×C17#13512yes—
167C2×C4×C17C22×D411yescharacteristic
168C2×C4×C17C22×D411yescharacteristic
169C2×C4×C17C2×#4111yescharacteristic
170D68—42no—
171D68—42no—
172D68—44no—
173D68—82no—
174D68—82no—
175D68—82no—
176D68—82no—
177C8×C17—42no—
178#3622—24no—
179#3622—42no—
180#3622—42no—
181#3205—171no—
182#3037D1711yescharacteristic, complemented
183C2×#460—171no—
184#2982—171no—
185#1682—174no—
186#1668—174no—
187#1571—171no—
188#1545—171no—
189#1540—171no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—81no—
193C22×C17—81no—
194C22×C17—81no—
195C22×C17#46612yes—
196C22×C17C2×#12111yescharacteristic
197C2×D17—82no—
198C2×D17—82no—
199C2×D17—82no—
200C2×D17—164no—
201C4×C17—21no—
202C4×C17—24no—
203C4×C17—21no—
204C4×C17—21no—
205C4×C17—42no—
206C4×C17—81no—
207C4×C17#46612yes—
208C4×C17C22×D811yescharacteristic
209C4×C17C2×#13511yescharacteristic
210C4×C17#57712yes—
211C2×#137—21no—
212C22×SD16—341no—
213#532C2×D1711yescharacteristic
214C2×#127—171no—
215C2×C4×Q8C2×D1711yescharacteristic
216#493—172no—
217#486—172no—
218#464—22no—
219#462C2×D1712yescomplemented
220#460—174no—
221#460—342no—
222#460C2×D1712yescomplemented
223#459—342no—
224#447—342no—
225C8×Q8—174no—
226C8×Q8—342no—
227#431—171no—
228#431—171no—
229#431—342no—
230#423C2×D1711yescharacteristic
231C2×C4⋊C8—171no—
232#417—21no—
233C2×#102—341no—
234#335—174no—
235#332—172no—
236#331—172no—
237#331—174no—
238C2×C17—22no—
239C2×C17—161no—
240C2×C17#304311yescharacteristic
241C2×C17C2×#46611yescharacteristic
242C2×C17#306811yescharacteristic
243D17—162no—
244C2×#41—21no—
245C22×Q8C22⋊D1711yescharacteristic
246C22×D4—341no—
247C22×D4—341no—
248C22×D4—341no—
249#137—44no—
250C2×Q16—22no—
251C2×SD16—22no—
252C2×SD16—344no—
253C2×SD16—684no—
254C2×D8—344no—
255C2×M16—21no—
256C22×C8—341no—
257C4⋊Q8C22×D1711yescharacteristic
258#127—172no—
259#127—341no—
260#127—341no—
261#127—341no—
262#127C22×D1711yescharacteristic
263#124—21no—
264#121—21no—
265#121—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8C22⋊D1712yes—
269C4×Q8C22×D1712yes—
270C4×D4—21no—
271C4×D4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C22⋊D1711yescharacteristic
274C2×C42—21no—
275C2×C42C22×D1711yescharacteristic
276C4⋊C8—21no—
277C4⋊C8—172no—
278C4⋊C8—172no—
279C4⋊C8—172no—
280C4⋊C8—344no—
281C4⋊C8—341no—
282C4⋊C8C22×D1712yes—
283#103—22no—
284#102—22no—
285#102—344no—
286#102—344no—
287C22⋊C8—172no—
288C22⋊C8—341no—
289C4×C8—172no—
290C4×C8—344no—
291C4×C8—341no—
292C17#1939811yescharacteristic, complemented, Sylow
293C24—681no—
294#41—42no—
295#41—42no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—21no—
299C2×Q8#6322814yescomplemented
300C2×Q8C2×C22⋊D1712yes—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—342no—
306C2×D4—342no—
307C2×D4—342no—
308C2×D4—682no—
309C2×D4—681no—
310C2×D4—681no—
311C2×D4—681no—
312C2×D4—682no—
313C2×D4—1362no—
314C2×D4—1362no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4D4×D1711yescharacteristic
319C22×C4#6325411yescharacteristic
320C22×C4C2×C22⋊D1711yescharacteristic
321Q16—44no—
322SD16—44no—
323SD16—688no—
324D8—1364no—
325M16—42no—
326C2×C8—22no—
327C2×C8—174no—
328C2×C8—342no—
329C2×C8—342no—
330C2×C8—682no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4C2×C22⋊D1712yes—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—22no—
340C42—22no—
341C42D4×D1711yescharacteristic
342C42#6325411yescharacteristic
343C42C2×C22⋊D1711yescharacteristic
344C42C23×D1711yescharacteristic
345C23—41no—
346C23—682no—
347C23—1362no—
348C23—1362no—
349C23—1362no—
350C23#1058762511yescharacteristic
351Q8—24no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355Q8C2×#6322814yes—
356D4—41no—
357D4—41no—
358D4—42no—
359D4—81no—
360D4—82no—
361D4—682no—
362D4—682no—
363D4—684no—
364D4—1362no—
365D4—1362no—
366D4—1362no—
367D4—1362no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4C2×#6322812yes—
377C2×C4#1058762511yescharacteristic
378C2×C4#1058762511yescharacteristic
379C2×C4#1058762511yescharacteristic
380C2×C4#1058763612yes—
381C2×C4C2×D4×D1711yescharacteristic
382C2×C4C2×#6325411yescharacteristic
383C2×C4C22×C22⋊D1711yescharacteristic
384C8—42no—
385C8—344no—
386C8—682no—
387C8—682no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—2724no—
394C22—1362no—
395C22—1362no—
396C22—1362no—
397C22#1176304312yes—
398C22C2×#1058762511yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—24no—
403C4—42no—
404C4—81no—
405C4#1176304312yes—
406C4C22×#6322811yescharacteristic
407C4C2×#1058763611yescharacteristic
408C4#1176361812yes—
409C2—22no—
410C2—161no—
411C2—2722no—
412C2#1453830711yescharacteristic
413C2C2×#1176304311yescharacteristic
414C2#1453851411yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3037C2
socleC22×C17C2×#121
prime radical#14539699C2
hypercenter#3037D17
hypocenterC17#19398
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. C2×#11763009C2semidirect
3. #14537865C2semidirect
4. #14528230C2semidirect
5. #14528142C2semidirect
6. #14527560C2semidirect
7. #14527453C2semidirect
8. #14527433C2semidirect
9. C17×#3037C2semidirect
19. Q8×#3622C22semidirect
25. #11762561C22semidirect
26. #11762555C22semidirect
27. #11762553C22semidirect
33. C17×#462C22semidirect
34. C17×#460C22semidirect
91. C2×Q8×C17D8semidirect
182. #3037D17semidirect
219. #462C2×D17semidirect
222. #460C2×D17semidirect
292. C17#19398semidirect
299. C2×Q8#63228semidirect

Maximal subgroups

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

subgroupindex
#145396992
C2×#117630092
#145378652
#145282302
#145281422
#145275602
#145274532
#145274332
C17×#30372
#1939817