Finite Group Databasebeta

← #17000254

Subgroups — #17000254

Order 4352.

1098 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×#11763285C211yesmaximal, characteristic
2#14537869C211yesmaximal, characteristic, complemented
3#14527127C214yesmaximal, complemented
4#14527116C214yesmaximal
5#14527107C212yesmaximal, complemented
6#14527096C212yesmaximal
7C17×#2983C211yesmaximal, characteristic, complemented
8C22×Q8×D17C411yescharacteristic
9#11763285—24no—
10#11763285—24no—
11C2×Q8×Dic17C411yescharacteristic
12C2×C4×Dic34C2211yescharacteristic
13C2×C4×Dic34C2212yes—
14C2×#10587573C412yes—
15C2×#10587573C411yescharacteristic
16C2×#10587578C412yes—
17C2×#10587578C411yescharacteristic
18C2×#10587558C2211yescharacteristic
19C2×#10587558C2212yes—
20#11762940—24no—
21#11762866C2214yes—
22#11762862C2212yes—
23#11762534C2214yes—
24#11762534C2218yes—
25#11762532C2214yes—
26C2×C4×Q8×C17C2211yescharacteristic
27C17×#446—24no—
28C17×#432C2214yes—
29C17×#431C2212yes—
30C22×Q8×C17C2×C411yescharacteristic
31C2×Q8×D17—24no—
32C2×Q8×D17—24no—
33C2×Q8×D17—24no—
34C22×C4×D17C2×C412yes—
35C22×C4×D17C2×C411yescharacteristic
36C22×Dic34C2×C412yes—
37C22×Dic34C2×C411yescharacteristic
38C4×Q8×C17D414yes—
39C4×Q8×C17D414yes—
40C2×C4⋊C4×C17C2×C412yes—
41C2×C4⋊C4×C17C2×C411yescharacteristic
42C2×C42×C17C2311yescharacteristic
43C2×C42×C17C2312yes—
44Q8×Dic17—24no—
45C2×#63227C2×C411yescharacteristic
46C2×#63227C2×C412yes—
47C2×#63227C2311yescharacteristic
48C2×C4⋊Dic17C2×C412yes—
49C2×C4⋊Dic17C2×C411yescharacteristic
50C2×#63225C2×C411yescharacteristic
51C2×#63225C2×C412yes—
52C2×#63225C2311yescharacteristic
53C2×#63225C2312yes—
54C2×C4×Dic17C2×C412yes—
55C2×C4×Dic17C2×C411yescharacteristic
56C2×C4×Dic17C2×C412yes—
57C2×C4×Dic17C2×C412yes—
58C2×C4×Dic17C2×C412yes—
59#10587578—28no—
60#10587578—24no—
61#10587573—28no—
62#10587573—24no—
63#10587558—22no—
64#10587558—24no—
65#10587558—22no—
66#10587558—24no—
67C4×Dic34—22no—
68C4×Dic34—24no—
69C4×Dic34—22no—
70C4×Dic34—24no—
71C17×C4⋊C8—22no—
72C17×C4⋊C8—24no—
73C17×C22⋊C8C2314yes—
74C17×#97—24no—
75C4×C8×C17—22no—
76C22⋊#3622C2314yes—
77C4⋊#3622—22no—
78C4⋊#3622—24no—
79#10587474—24no—
80C4×#3622—22no—
81C23×D17C22×C411yescharacteristic
82C2×Q8×C17—22no—
83C2×Q8×C17—22no—
84C2×Q8×C17C22⋊C414yes—
85C22×C4×C17C22×C412yes—
86C22×C4×C17C22×C411yescharacteristic
87C22×C4×C17C22×C412yes—
88C22×C4×C17C22×C411yescharacteristic
89C22×C4×C17C2411yescharacteristic
90C22×Dic17C22×C411yescharacteristic
91C22×Dic17C22×C412yes—
92C22×Dic17C22×C411yescharacteristic
93C22×Dic17C22×C412yes—
94C22×Dic17C22×C411yescharacteristic
95Q8×D17—232no—
96C2×C4×D17—24no—
97C2×C4×D17—24no—
98C2×C4×D17—22no—
99C2×C4×D17—22no—
100C2×C4×D17—24no—
101C2×C4×D17M1614yes—
102C2×Dic34—24no—
103C2×Dic34—22no—
104C2×Dic34—22no—
105C2×Dic34—24no—
106C2×Dic34—24no—
107C2×Dic34M1614yes—
108C2×C8×C17—24no—
109C4⋊C4×C17C22⋊C418yes—
110C4⋊C4×C17C22⋊C414yes—
111C42×C17C2×D412yes—
112C42×C17C2×D414yes—
113C42×C17C2×D412yes—
114C42×C17C2×D414yes—
115#63227—22no—
116#63227—24no—
117#63227—21no—
118#63227—21no—
119C4⋊Dic17—24no—
120C4⋊Dic17—22no—
121#63225—22no—
122#63225—24no—
123#63225—22no—
124#63225—21no—
125#63225—22no—
126#63225—21no—
127C4×Dic17—24no—
128C4×Dic17—24no—
129C4×Dic17—24no—
130C4×Dic17—24no—
131C4×Dic17—22no—
132C2×#3622—24no—
133#17371—171nomaximal, Sylow
134C23×C17C23×C411yescharacteristic
135C22×D17—21no—
136C22×D17—21no—
137C22×D17—41no—
138C22×D17—41no—
139C22×D17C2×M1612yes—
140Q8×C17—28no—
141C2×C4×C17—22no—
142C2×C4×C17—22no—
143C2×C4×C17—41no—
144C2×C4×C17—42no—
145C2×C4×C17—41no—
146C2×C4×C17C2×C22⋊C412yes—
147C2×C4×C17C2×C22⋊C414yes—
148C2×C4×C17C2×C22⋊C414yes—
149C2×C4×C17C2×C22⋊C412yes—
150C2×C4×C17C2×M1612yes—
151C2×C4×C17C2×M1612yes—
152C2×C4×C17C22×D411yescharacteristic
153C2×C4×C17C22×D411yescharacteristic
154C2×Dic17—21no—
155C2×Dic17—22no—
156C2×Dic17—22no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17—22no—
160C2×Dic17—21no—
161C2×Dic17—42no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17C2×M1612yes—
165C2×Dic17C2×M1614yes—
166C4×D17—216no—
167C4×D17—28no—
168Dic34—216no—
169Dic34—28no—
170C8×C17—44no—
171#3622—44no—
172C2×#519—171no—
173#2983—171no—
174#2983D1711yescharacteristic, complemented
175#1467—174no—
176#1464—174no—
177#1461—172no—
178#1458—172no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17C22×C22⋊C411yescharacteristic
182C22×C17C22×M1611yescharacteristic
183C22×C17C22×M1611yescharacteristic
184C2×D17—22no—
185C2×D17—42no—
186C2×D17—41no—
187C2×D17—41no—
188C2×D17—42no—
189C4×C17—24no—
190C4×C17—21no—
191C4×C17—21no—
192C4×C17—22no—
193C4×C17—24no—
194C4×C17#40714yes—
195Dic17—22no—
196Dic17—24no—
197Dic17—42no—
198Dic17—41no—
199Dic17—41no—
200C23×Q8—171no—
201#519—344no—
202#519—344no—
203C2×C4×Q8—171no—
204C2×C4×Q8—171no—
205C2×C4×Q8—172no—
206C2×C4×Q8—171no—
207C2×C4×Q8—172no—
208C2×C4×Q8C2×D1711yescharacteristic
209C2×#117—171no—
210C2×#117—172no—
211C2×#117—171no—
212C2×#117—172no—
213#446—24no—
214#446—344no—
215#432—174no—
216#432C2×D1714yes—
217#431—172no—
218#431C2×D1712yes—
219#324—174no—
220#324—178no—
221#323—174no—
222C2×C17—21no—
223C2×C17—21no—
224C2×C17C2×#40711yescharacteristic
225C2×C17#288411yescharacteristic
226C2×C17#288911yescharacteristic
227D17—42no—
228C22×Q8—171no—
229C22×Q8—172no—
230C22×Q8—344no—
231C22×Q8—344no—
232C22×Q8—344no—
233C22×Q8C4×D1711yescharacteristic
234C23×C4—171no—
235C23×C4—172no—
236C4×Q8—344no—
237C4×Q8—348no—
238C4×Q8—344no—
239C4×Q8—342no—
240C4×Q8—344no—
241C4×Q8—342no—
242C4×Q8—344no—
243C4×Q8D6814yes—
244C4×Q8C22⋊D1714yes—
245#117—348no—
246#117—344no—
247#117—342no—
248#117—344no—
249#117—342no—
250#117—344no—
251C2×C4⋊C4—171no—
252C2×C4⋊C4—172no—
253C2×C4⋊C4—172no—
254C2×C4⋊C4—171no—
255C2×C4⋊C4—171no—
256C2×C4⋊C4—172no—
257C2×C4⋊C4C4×D1712yes—
258C2×C4⋊C4C4×D1711yescharacteristic
259C2×C22⋊C4—171no—
260C2×C22⋊C4—172no—
261C2×C22⋊C4—171no—
262C2×C42—172no—
263C2×C42—172no—
264C2×C42—172no—
265C2×C42—172no—
266C2×C42—171no—
267C2×C42C22×D1711yescharacteristic
268C2×C42C22×D1712yes—
269C4⋊C8—22no—
270C4⋊C8—24no—
271C4⋊C8—342no—
272C4⋊C8—344no—
273C22⋊C8—174no—
274C22⋊C8C22×D1714yes—
275#97—24no—
276#97—344no—
277C4×C8—22no—
278C4×C8—342no—
279C17#1737111yescharacteristic, complemented, Sylow
280C24—171no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—174no—
284C2×Q8—344no—
285C2×Q8—344no—
286C2×Q8—3432no—
287C2×Q8—342no—
288C2×Q8—344no—
289C2×Q8—342no—
290C2×Q8#6322714yes—
291C22×C4—171no—
292C22×C4—172no—
293C22×C4—174no—
294C22×C4—171no—
295C22×C4—172no—
296C22×C4—171no—
297C22×C4—344no—
298C22×C4—344no—
299C22×C4—342no—
300C22×C4—344no—
301C22×C4—342no—
302C22×C4C2×C4×D1712yes—
303C22×C4C2×C4×D1711yescharacteristic
304C22×C4C2×C4×D1712yes—
305C22×C4C2×C4×D1711yescharacteristic
306C22×C4C23×D1711yescharacteristic
307C2×C8—24no—
308C2×C8—344no—
309C4⋊C4—341no—
310C4⋊C4—342no—
311C4⋊C4—341no—
312C4⋊C4—342no—
313C4⋊C4—342no—
314C4⋊C4—344no—
315C4⋊C4—344no—
316C4⋊C4—342no—
317C4⋊C4#6322718yes—
318C4⋊C4#6322714yes—
319C22⋊C4—341no—
320C22⋊C4—341no—
321C22⋊C4—342no—
322C22⋊C4—344no—
323C42—344no—
324C42—344no—
325C42—344no—
326C42—344no—
327C42—342no—
328C42C2×D6812yes—
329C42C2×D6814yes—
330C42C2×C22⋊D1712yes—
331C42C2×C22⋊D1714yes—
332C23—172no—
333C23—341no—
334C23—341no—
335C23—681no—
336C23—681no—
337C23C22×C4×D1711yescharacteristic
338Q8—28no—
339Q8—3416no—
340Q8—348no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—172no—
347C2×C4—174no—
348C2×C4—3416no—
349C2×C4—341no—
350C2×C4—342no—
351C2×C4—341no—
352C2×C4—342no—
353C2×C4—348no—
354C2×C4—342no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—682no—
358C2×C4—681no—
359C2×C4—681no—
360C2×C4M16×D1712yes—
361C2×C4M16×D1712yes—
362C2×C4C2×#6322712yes—
363C2×C4C2×#6322714yes—
364C2×C4C2×#6322714yes—
365C2×C4C2×#6322712yes—
366C2×C4C22×D6811yescharacteristic
367C2×C4C22×C22⋊D1711yescharacteristic
368C8—44no—
369C8—684no—
370C22—21no—
371C22—21no—
372C22—342no—
373C22—682no—
374C22—682no—
375C22—681no—
376C22—681no—
377C22C22×#6322711yescharacteristic
378C22C2×M16×D1711yescharacteristic
379C22C2×M16×D1711yescharacteristic
380C4—24no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—24no—
385C4—342no—
386C4—344no—
387C4—682no—
388C4—681no—
389C4—681no—
390C4#1176277214yes—
391C2—21no—
392C2—21no—
393C2—682no—
394C2C2×#1176277211yescharacteristic
395C2#1453724711yescharacteristic
396C2#1453726611yescharacteristic
subgroupG/H
centerC22C22×#63227
commutatorC23×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2983C2
socleC22×C17C22×C22⋊C4
prime radicalC23×D17C22×C4
cosocleC22×C4×C17C24
hypercenter#2983D17
hypocenterC17#17371
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. #14537869C2semidirect
3. #14527127C2semidirect
5. #14527107C2semidirect
7. C17×#2983C2semidirect
174. #2983D17semidirect
279. C17#17371semidirect

Maximal subgroups

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

subgroupindex
C2×#117632852
#145378692
#145271272
#145271162
#145271072
#145270962
C17×#29832
#1737117