Finite Group Databasebeta

← #17050530

Subgroups — #17050530

Order 4352.

1062 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×#11763532C211yesmaximal, characteristic
2M16×D68C212yesmaximal
3#14529773C212yesmaximal, complemented
4#14529249C214yesmaximal, complemented
5#14528866C212yesmaximal, complemented
6#14528780C211yesmaximal, characteristic, complemented
7#14528521C212yesmaximal, complemented
8C4×C17×#101C211yesmaximal, characteristic, complemented
9C2×M16×D17C2214yes—
10#11763532—22no—
11#11763532—24no—
12#11763532—22no—
13C2×#10587555C2211yescharacteristic
14C2×#10587584C414yes—
15C2×#10587561C412yes—
16C2×#10587632C2212yes—
17C2×C4×Dic34C412yes—
18C2×C4×D68C2211yescharacteristic
19C2×#10587578C2212yes—
20#11762888—22no—
21#11762882—24no—
22C8×D68—22no—
23#11762833C2212yes—
24C2×#10587507C2212yes—
25#11762772C2214yes—
26#11762578C2212yes—
27#11762577C2214yes—
28#11762577C2214yes—
29C2×C4×Q8×C17C2211yescharacteristic
30C2×C17×#101C2212yes—
31C4×M16×C17C2212yes—
32C17×#343C2211yescharacteristic
33C17×#343C2211yescharacteristic
34C22×Q8×C17C2311yescharacteristic
35C22×D68C2×C411yescharacteristic
36C22×C4×D17C2312yes—
37C22×Dic34C2×C412yes—
38C2×M16×C17C2312yes—
39C2×M16×C17C2312yes—
40C4×Q8×C17—22no—
41C4×Q8×C17—22no—
42C2×C4⋊C4×C17C2×C412yes—
43C2×C4⋊C4×C17C2311yescharacteristic
44C2×C42×C17C2×C412yes—
45C2×C42×C17C2311yescharacteristic
46#10587632—24no—
47#10587632—24no—
48C2×#63227C2×C414yes—
49C2×#63227C2×C414yes—
50C2×#63227C2×C412yes—
51C2×C4⋊Dic17C2×C412yes—
52C2×C4⋊Dic17C2×C411yescharacteristic
53C2×C4⋊Dic17C2312yes—
54C2×#63225C2×C414yes—
55C2×C4×Dic17C2×C414yes—
56C2×#63223C2314yes—
57M16×D17—28no—
58M16×D17—44no—
59C2×#63218—24no—
60C2×C8×D17—24no—
61#10587584—28no—
62#10587584—28no—
63#10587578—24no—
64#10587578—24no—
65#10587561—24no—
66#10587561—24no—
67C4×D68D412yes—
68C4×D68D414yes—
69C4×D68D412yes—
70#10587555—22no—
71#10587555—22no—
72C4×Dic34—24no—
73C4×Dic34—22no—
74C4×Dic34—22no—
75C17×#101—24no—
76C17×#97—22no—
77C4×C8×C17—22no—
78#10587507—42no—
79#10587491—24no—
80#10587491—24no—
81C4⋊#3622—22no—
82C4⋊#3622—22no—
83C23×D17C22×C412yes—
84C2×Q8×C17—22no—
85C2×Q8×C17—22no—
86C2×Q8×C17—21no—
87C2×Q8×C17—21no—
88C22×C4×C17C22×C412yes—
89C22×C4×C17C22×C411yescharacteristic
90C22×C4×C17C22×C412yes—
91C22×C4×C17C22×C411yescharacteristic
92C22×C4×C17C2411yescharacteristic
93C22×Dic17C22×C412yes—
94C22×Dic17C22×C414yes—
95C2×D68—42no—
96C2×D68C22⋊C412yes—
97C2×D68C22⋊C412yes—
98C2×C4×D17—24no—
99C2×C4×D17—22no—
100C2×C4×D17—22no—
101C2×C4×D17C2×D414yes—
102C2×C4×D17C2×D414yes—
103C2×Dic34—22no—
104C2×Dic34—22no—
105C2×Dic34—44no—
106M16×C17—22no—
107M16×C17—22no—
108M16×C17—22no—
109M16×C17#4114yes—
110C2×C8×C17—22no—
111C2×C8×C17—22no—
112C4⋊C4×C17—24no—
113C4⋊C4×C17—22no—
114C42×C17—22no—
115C42×C17—22no—
116C42×C17C2×D412yes—
117C42×C17C2×D412yes—
118#63227—28no—
119#63227—28no—
120#63227C22⋊C414yes—
121#63227C22⋊C414yes—
122C4⋊Dic17—24no—
123C4⋊Dic17—24no—
124C4⋊Dic17C22⋊C412yes—
125C4⋊Dic17C22⋊C411yescharacteristic
126C4⋊Dic17C22⋊C411yescharacteristic
127#63225—24no—
128#63225—24no—
129C4×Dic17—24no—
130C4×Dic17—24no—
131#63223—24no—
132#63223—24no—
133C2×#3622—24no—
134#63218—48no—
135C8×D17—48no—
136#22484—171nomaximal, Sylow
137C23×C17C23×C411yescharacteristic
138C22×D17—22no—
139C22×D17—22no—
140C22×D17—42no—
141C22×D17—42no—
142C22×D17C2×C22⋊C414yes—
143Q8×C17—42no—
144Q8×C17—42no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—41no—
151C2×C4×C17—42no—
152C2×C4×C17C2×C22⋊C411yescharacteristic
153C2×C4×C17C2×C22⋊C412yes—
154C2×C4×C17C2×C22⋊C411yescharacteristic
155C2×C4×C17C22×D412yes—
156C2×C4×C17C2×#4112yes—
157C2×C4×C17H(5,𝔽2)11yescharacteristic
158C2×C4×C172−1+411yescharacteristic
159C2×Dic17—22no—
160C2×Dic17—22no—
161C2×Dic17—24no—
162C2×Dic17—44no—
163C2×Dic17C2×C22⋊C412yes—
164C2×Dic17C2×C22⋊C412yes—
165D68—24no—
166D68—42no—
167C4×D17—24no—
168C4×D17—24no—
169C4×D17—24no—
170C4×D17—24no—
171Dic34—44no—
172Dic34—44no—
173C8×C17—24no—
174C8×C17—42no—
175#3622—44no—
176C2×#556—171no—
177D4×M16—172no—
178#1997—172no—
179#1901—174no—
180#1822—172no—
181#1801—171no—
182C4×#101D1711yescharacteristic, complemented
183#1742—172no—
184C22×C17—21no—
185C22×C17—21no—
186C22×C17C22×C22⋊C411yescharacteristic
187C22×C17C4×#4111yescharacteristic
188C22×C17#52011yescharacteristic
189C2×D17—24no—
190C2×D17—24no—
191C2×D17—24no—
192C2×D17—82no—
193C2×D17—82no—
194C4×C17—21no—
195C4×C17—22no—
196C4×C17—42no—
197C4×C17—42no—
198C4×C17#41312yes—
199C4×C17#54612yes—
200C4×C17#54712yes—
201Dic17—22no—
202Dic17—22no—
203Dic17—44no—
204C22×M16—174no—
205#556—342no—
206#556—344no—
207#556—342no—
208C2×C4⋊Q8—171no—
209C2×#126—174no—
210C2×#124—172no—
211C2×C22⋊Q8—172no—
212C2×C4×Q8—172no—
213C2×C4×Q8C2×D1711yescharacteristic
214C2×C4×D4—171no—
215C2×#117—172no—
216#436—342no—
217#435—344no—
218C8×D4—342no—
219#423—172no—
220C2×#101—172no—
221C2×#101C2×D1712yes—
222#407—174no—
223C4×M16C2×D1712yes—
224#343—174no—
225#343—174no—
226#343—172no—
227#343C2×D1711yescharacteristic
228#343C2×D1711yescharacteristic
229C2×C17—21no—
230C2×C17—21no—
231C2×C17#233911yescharacteristic
232C2×C17C2×#41311yescharacteristic
233C2×C17#288911yescharacteristic
234D17—44no—
235C22×Q8—172no—
236C22×Q8C22×D1711yescharacteristic
237C22×D4—171no—
238C23×C4—172no—
239C2×M16—174no—
240C2×M16—344no—
241C2×M16—348no—
242C2×M16—684no—
243C2×M16C22×D1712yes—
244C2×M16C22×D1712yes—
245C22×C8—344no—
246C4⋊Q8—342no—
247C4⋊Q8—342no—
248#126—348no—
249#126—348no—
250#124—344no—
251#124—344no—
252C22⋊Q8—344no—
253C22⋊Q8—344no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—344no—
257C4×Q8—342no—
258C4×Q8—342no—
259C4×D4—172no—
260C4×D4—174no—
261C4×D4—172no—
262#117—344no—
263#117—344no—
264C2×C4⋊C4—171no—
265C2×C4⋊C4—174no—
266C2×C4⋊C4—172no—
267C2×C4⋊C4—172no—
268C2×C4⋊C4C4×D1712yes—
269C2×C4⋊C4C22×D1711yescharacteristic
270C2×C22⋊C4—172no—
271C2×C22⋊C4—174no—
272C2×C22⋊C4—174no—
273C2×C42—174no—
274C2×C42C4×D1712yes—
275C2×C42C22×D1711yescharacteristic
276C4⋊C8—342no—
277C4⋊C8—342no—
278#101—24no—
279#101—682no—
280C22⋊C8—344no—
281C22⋊C8—344no—
282#97—22no—
283C4×C8—22no—
284C17#2248411yescharacteristic, complemented, Sylow
285C24—172no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—342no—
291C2×Q8—342no—
292C2×Q8—684no—
293C2×D4—172no—
294C2×D4—172no—
295C2×D4—682no—
296C22×C4—172no—
297C22×C4—174no—
298C22×C4—174no—
299C22×C4—174no—
300C22×C4—344no—
301C22×C4—342no—
302C22×C4—342no—
303C22×C4C2×C4×D1712yes—
304C22×C4C2×C4×D1711yescharacteristic
305C22×C4C2×C4×D1712yes—
306C22×C4C2×C4×D1711yescharacteristic
307C22×C4C23×D1711yescharacteristic
308M16—22no—
309M16—22no—
310M16—22no—
311M16—344no—
312M16—344no—
313M16—688no—
314M16#6325214yes—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—344no—
318C2×C8—688no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—172no—
322C4⋊C4—171no—
323C4⋊C4—171no—
324C4⋊C4—344no—
325C4⋊C4—344no—
326C4⋊C4—344no—
327C4⋊C4—344no—
328C22⋊C4—174no—
329C22⋊C4—174no—
330C22⋊C4—348no—
331C22⋊C4—348no—
332C42—22no—
333C42—22no—
334C42—344no—
335C42—344no—
336C42D4×D1712yes—
337C42D4×D1712yes—
338C23—174no—
339C23—342no—
340C23—342no—
341C23—682no—
342C23—682no—
343C23C22×C4×D1711yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—684no—
347Q8—684no—
348D4—344no—
349D4—682no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—172no—
358C2×C4—172no—
359C2×C4—342no—
360C2×C4—342no—
361C2×C4—344no—
362C2×C4—344no—
363C2×C4—344no—
364C2×C4—344no—
365C2×C4—344no—
366C2×C4—684no—
367C2×C4C22⋊C4×D1711yescharacteristic
368C2×C4C22⋊C4×D1712yes—
369C2×C4C22⋊C4×D1711yescharacteristic
370C2×C4C2×#6325212yes—
371C2×C4C2×D4×D1712yes—
372C2×C4#1058769611yescharacteristic
373C2×C4#1058769711yescharacteristic
374C8—24no—
375C8—42no—
376C8—684no—
377C22—21no—
378C22—21no—
379C22—344no—
380C22—344no—
381C22—344no—
382C22—1362no—
383C22—1362no—
384C22C2×C22⋊C4×D1711yescharacteristic
385C22C4×#6325211yescharacteristic
386C22#1176328911yescharacteristic
387C4—21no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—342no—
392C4—342no—
393C4—684no—
394C4#1176279212yes—
395C4#1176345512yes—
396C4#1176346412yes—
397C2—21no—
398C2—21no—
399C2—684no—
400C2#1453161511yescharacteristic
401C2C2×#1176279211yescharacteristic
402C2#1453726511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D17
commutatorC23×C17C23×C4
FrattiniC22×C4C23×D17
FittingC4×C17×#101C2
socleC22×C17C22×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC22×C4×C17C24
hypercenterC4×#101D17
hypocenterC17#22484
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
3. #14529773C2semidirect
4. #14529249C2semidirect
5. #14528866C2semidirect
6. #14528780C2semidirect
7. #14528521C2semidirect
8. C4×C17×#101C2semidirect
182. C4×#101D17semidirect
284. C17#22484semidirect

Maximal subgroups

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

subgroupindex
C2×#117635322
M16×D682
#145297732
#145292492
#145288662
#145287802
#145285212
C4×C17×#1012
#2248417