Finite Group Databasebeta

← #17050890

Subgroups — #17050890

Order 4352.

1014 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763516C211yesmaximal, characteristic
2M16×Dic34C212yesmaximal, complemented
3#14529775C214yesmaximal, complemented
4#14529773C212yesmaximal, complemented
5#14528776C211yesmaximal, characteristic
6#14528774C212yesmaximal
7#100×Dic17C212yesmaximal
8C4×C17×#100C211yesmaximal, characteristic
9#11763516—22no—
10#11763516—24no—
11#11763516—22no—
12C2×#10587555C2211yescharacteristic
13C2×#10587576C2212yes—
14C2×#10587610C412yes—
15C2×#10587564C414yes—
16C2×C4×Dic34C2211yescharacteristic
17C2×D4×Dic17C2212yes—
18C22×C4⋊Dic17C412yes—
19#11762939—22no—
20#11762937—24no—
21C8×Dic34—22no—
22#11762833C2212yes—
23#11762832C2214yes—
24C2×#10587504C2212yes—
25M16×Dic17C2214yes—
26#11762578C2212yes—
27#11762577C2214yes—
28#11762577C2214yes—
29C2×C4×D4×C17C2211yescharacteristic
30C2×C17×#100C2211yescharacteristic
31C2×C17×#100C2211yescharacteristic
32C4×M16×C17C2212yes—
33C17×#343C2212yes—
34C22×D4×C17C2311yescharacteristic
35C23×C4×C17C2×C412yes—
36C23×Dic17C2×C414yes—
37C22×Dic34C2×C412yes—
38C22×Dic34C2×C411yescharacteristic
39C2×M16×C17C2312yes—
40C2×M16×C17C2312yes—
41C4×D4×C17—22no—
42C4×D4×C17—22no—
43C2×C4⋊C4×C17C2311yescharacteristic
44C2×C22⋊C4×C17C2×C412yes—
45C2×C42×C17C2311yescharacteristic
46C2×C22⋊Dic17C2×C414yes—
47D4×Dic17—24no—
48D4×Dic17—24no—
49C2×C4⋊Dic17—24no—
50C2×C4⋊Dic17—22no—
51C2×C4⋊Dic17—22no—
52C2×C4⋊Dic17C2×C414yes—
53C2×C4⋊Dic17C2×C412yes—
54C2×C4⋊Dic17C2×C411yescharacteristic
55C2×C4⋊Dic17C2312yes—
56#10587610—24no—
57#10587610—24no—
58C2×#63225C2×C414yes—
59C2×#63225C2×C412yes—
60C2×C4×Dic17C2312yes—
61C2×#63223C2314yes—
62#10587576—24no—
63#10587576—24no—
64#10587564—28no—
65#10587564—28no—
66#10587555—22no—
67#10587555—22no—
68C4×Dic34D412yes—
69C4×Dic34D414yes—
70C4×Dic34D412yes—
71C17×#100—22no—
72C17×#100Q814yescomplemented
73C17×#97—22no—
74C4×C8×C17—22no—
75#10587504—24no—
76#10587486—24no—
77#10587485—24no—
78#10587485—24no—
79C8×Dic17—24no—
80C4⋊#3622—22no—
81C4⋊#3622—22no—
82C24×C17C22×C412yes—
83C2×D4×C17—22no—
84C2×D4×C17—22no—
85C2×D4×C17C2×Q812yes—
86C2×D4×C17#4112yes—
87C22×C4×C17—22no—
88C22×C4×C17—22no—
89C22×C4×C17—44no—
90C22×C4×C17C22×C411yescharacteristic
91C22×C4×C17C22×C412yes—
92C22×C4×C17C22×C411yescharacteristic
93C22×C4×C17C2411yescharacteristic
94C22×Dic17—24no—
95C22×Dic17—24no—
96C22×Dic17—84no—
97C22×Dic17C22×C414yes—
98C22×Dic17C22×C412yes—
99C22×Dic17C22×C412yes—
100C2×Dic34—24no—
101C2×Dic34—42no—
102C2×Dic34—82no—
103C2×Dic34C22⋊C412yes—
104C2×Dic34C22⋊C412yes—
105M16×C17—22no—
106M16×C17—22no—
107M16×C17—22no—
108M16×C17C2×Q814yes—
109C2×C8×C17—22no—
110C2×C8×C17—22no—
111C4⋊C4×C17—22no—
112C22⋊C4×C17—24no—
113C42×C17C2×D412yes—
114C42×C17C2×D412yes—
115C22⋊Dic17—28no—
116C4⋊Dic17—22no—
117C4⋊Dic17—22no—
118C4⋊Dic17—24no—
119C4⋊Dic17—24no—
120C4⋊Dic17—24no—
121C4⋊Dic17C22⋊C412yes—
122C4⋊Dic17C22⋊C411yescharacteristic
123C4⋊Dic17C22⋊C411yescharacteristic
124#63225—28no—
125#63225C22⋊C414yes—
126#63225C22⋊C414yes—
127C4×Dic17C2×D414yes—
128C4×Dic17C2×D414yes—
129#63223—24no—
130#63223—24no—
131C2×#3622—24no—
132#22515—171nomaximal, Sylow
133C23×C17—22no—
134C23×C17—22no—
135C23×C17—42no—
136C23×C17—42no—
137C23×C17C4×Q814yes—
138C23×C17C23×C411yescharacteristic
139D4×C17—24no—
140D4×C17—24no—
141C2×C4×C17—22no—
142C2×C4×C17—22no—
143C2×C4×C17—41no—
144C2×C4×C17—44no—
145C2×C4×C17—44no—
146C2×C4×C17—42no—
147C2×C4×C17C2×C22⋊C412yes—
148C2×C4×C17C2×C22⋊C411yescharacteristic
149C2×C4×C17C2×C22⋊C411yescharacteristic
150C2×C4×C17C22×D412yes—
151C2×C4×C17C22×Q811yescharacteristic
152C2×C4×C17C2×#4111yescharacteristic
153C2×C4×C172−1+412yes—
154C2×Dic17—24no—
155C2×Dic17—42no—
156C2×Dic17—42no—
157C2×Dic17—44no—
158C2×Dic17—84no—
159C2×Dic17—84no—
160C2×Dic17C2×C22⋊C414yes—
161C2×Dic17C2×C22⋊C412yes—
162C2×Dic17C2×C22⋊C412yes—
163Dic34—24no—
164Dic34—42no—
165Dic34—84no—
166C8×C17—24no—
167C8×C17—42no—
168#3622—44no—
169C2×#554—171no—
170Q8×M16—172no—
171#1998—174no—
172#1997—172no—
173#1800—172no—
174#1800—171no—
175C4×#100—172no—
176C4×#100D1711yescharacteristic
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—24no—
180C22×C17—44no—
181C22×C17—44no—
182C22×C17—42no—
183C22×C17—42no—
184C22×C17C22×C22⋊C411yescharacteristic
185C22×C17C2×C4×Q811yescharacteristic
186C22×C17#51911yescharacteristic
187C4×C17—21no—
188C4×C17—22no—
189C4×C17—42no—
190C4×C17#41312yes—
191C4×C17#54812yes—
192C4×C17D4×Q812yes—
193Dic17—24no—
194Dic17—22no—
195Dic17—22no—
196Dic17—44no—
197#554—342no—
198#554—342no—
199#554—344no—
200C2×C4⋊Q8—171no—
201C2×#125—172no—
202C2×C22⋊Q8—172no—
203C2×C22⋊Q8—174no—
204C2×C4×Q8—171no—
205C2×C4×D4—172no—
206C2×C4×D4C2×D1711yescharacteristic
207C22×C4⋊C4—172no—
208#446—342no—
209#446—344no—
210C8×Q8—342no—
211#423—174no—
212#423—172no—
213C2×#100—172no—
214C2×#100C2×D1711yescharacteristic
215C2×#100C2×D1711yescharacteristic
216C4×M16—174no—
217C4×M16C2×D1712yes—
218#343—174no—
219#343—174no—
220#343—172no—
221#343C2×D1712yes—
222C2×C17—21no—
223C2×C17—21no—
224C2×C17—44no—
225C2×C17Q8×C22⋊C411yescharacteristic
226C2×C17C2×#41311yescharacteristic
227C2×C17#288811yescharacteristic
228C22×Q8—171no—
229C22×Q8—172no—
230C22×D4C22×D1711yescharacteristic
231C23×C4—174no—
232C23×C4C4×D1712yes—
233C2×M16—174no—
234C2×M16C22×D1712yes—
235C2×M16C22×D1712yes—
236C4⋊Q8—342no—
237C4⋊Q8—342no—
238#125—344no—
239#125—344no—
240C22⋊Q8—348no—
241C22⋊Q8—348no—
242C22⋊Q8—344no—
243C22⋊Q8—344no—
244C4×Q8—172no—
245C4×Q8—174no—
246C4×Q8—172no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—344no—
250C4×D4—344no—
251C2×C4⋊C4—171no—
252C2×C4⋊C4—172no—
253C2×C4⋊C4—174no—
254C2×C4⋊C4—172no—
255C2×C4⋊C4—174no—
256C2×C4⋊C4—172no—
257C2×C4⋊C4—344no—
258C2×C4⋊C4—342no—
259C2×C4⋊C4—342no—
260C2×C4⋊C4C22×D1711yescharacteristic
261C2×C22⋊C4—174no—
262C2×C22⋊C4C4×D1712yes—
263C2×C42—172no—
264C2×C42C22×D1711yescharacteristic
265C4⋊C8—344no—
266C4⋊C8—344no—
267C4⋊C8—342no—
268C4⋊C8—342no—
269#100—22no—
270#100—344no—
271#100Dic3414yescomplemented
272#97—22no—
273#97—344no—
274C4×C8—22no—
275C4×C8—344no—
276C17#2251511yescharacteristic, complemented, Sylow
277C24C2×C4×D1712yes—
278C2×Q8—172no—
279C2×Q8—172no—
280C2×Q8—344no—
281C2×Q8—682no—
282C2×Q8—1362no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4C2×Dic3412yes—
286C2×D4#6325212yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—44no—
290C22×C4—172no—
291C22×C4—172no—
292C22×C4—174no—
293C22×C4—344no—
294C22×C4—344no—
295C22×C4—1364no—
296C22×C4C2×C4×D1711yescharacteristic
297C22×C4C2×C4×D1712yes—
298C22×C4C2×C4×D1711yescharacteristic
299C22×C4C23×D1711yescharacteristic
300M16—22no—
301M16—22no—
302M16—22no—
303M16—344no—
304M16—344no—
305M16C2×Dic3414yes—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—344no—
309C4⋊C4—22no—
310C4⋊C4—172no—
311C4⋊C4—171no—
312C4⋊C4—171no—
313C4⋊C4—174no—
314C4⋊C4—174no—
315C4⋊C4—344no—
316C4⋊C4—344no—
317C4⋊C4—348no—
318C4⋊C4—344no—
319C4⋊C4—342no—
320C4⋊C4—342no—
321C22⋊C4—24no—
322C22⋊C4—348no—
323C42—174no—
324C42—174no—
325C42D4×D1712yes—
326C42D4×D1712yes—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23C4×Dic3414yes—
332C23C22×C4×D1711yescharacteristic
333Q8—344no—
334Q8—682no—
335Q8—1364no—
336D4—24no—
337D4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—174no—
345C2×C4—172no—
346C2×C4—172no—
347C2×C4—344no—
348C2×C4—682no—
349C2×C4—682no—
350C2×C4—684no—
351C2×C4—1364no—
352C2×C4—1364no—
353C2×C4C22⋊C4×D1712yes—
354C2×C4C22⋊C4×D1711yescharacteristic
355C2×C4C22⋊C4×D1711yescharacteristic
356C2×C4C22×Dic3411yescharacteristic
357C2×C4C2×#6325211yescharacteristic
358C2×C4C2×D4×D1712yes—
359C2×C4#1058769712yes—
360C8—24no—
361C8—42no—
362C8—684no—
363C22—21no—
364C22—21no—
365C22—24no—
366C22—44no—
367C22—44no—
368C22—42no—
369C22—42no—
370C22C2×C22⋊C4×D1711yescharacteristic
371C22C2×C4×Dic3411yescharacteristic
372C22#1176328611yescharacteristic
373C4—21no—
374C4—22no—
375C4—42no—
376C4—344no—
377C4—342no—
378C4—342no—
379C4—684no—
380C4#1176279012yes—
381C4#1176346812yes—
382C4D4×Dic3412yes—
383C2—21no—
384C2—21no—
385C2—44no—
386C2C22⋊C4×Dic3411yescharacteristic
387C2C2×#1176279011yescharacteristic
388C2#1453726111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D17
commutatorC23×C17C23×C4
FrattiniC22×C4C23×D17
FittingC4×C17×#100C2
socleC22×C17C22×C22⋊C4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC4×#100D17
hypocenterC17#22515
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. M16×Dic34C2semidirect
3. #14529775C2semidirect
4. #14529773C2semidirect
72. C17×#100Q8semidirect
271. #100Dic34semidirect
276. C17#22515semidirect

Maximal subgroups

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

subgroupindex
C2×#117635162
M16×Dic342
#145297752
#145297732
#145287762
#145287742
#100×Dic172
C4×C17×#1002
#2251517