Finite Group Databasebeta

← #16931613

Subgroups — #16931613

Order 4352.

492 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531599C211yesmaximal, characteristic
2#14529729C211yesmaximal, characteristic, complemented
3#14527382C211yesmaximal, characteristic
4#14527335C211yesmaximal, characteristic, complemented
5#14526650C211yesmaximal, characteristic, complemented
6#14526639C211yesmaximal, characteristic, complemented
7C17×#3215C211yesmaximal, characteristic
8#11762866C2211yescharacteristic
9C4×#63223C2211yescharacteristic
10#11762716—21no—
11#11762702—21no—
12#11762702—21no—
13#11762689—21no—
14#11762684—21no—
15#11762677C2211yescharacteristic
16C4×C4⋊Dic17C2211yescharacteristic
17C22⋊C4×Dic17—21no—
18#11762568—21no—
19#11762568—21no—
20#11762549—22no—
21#11762546—21no—
22#11762546—21no—
23C17×#533C2211yescharacteristic
24C2×C17×#124C2211yescharacteristic
25C17×#497C412yescomplemented
26C17×#489C412yescomplemented
27C17×#464—22no—
28C17×#451—21no—
29C17×#450—21no—
30C17×#432C2211yescharacteristic
31C22×Q8×C17D411yescharacteristic
32C22×D4×C17D411yescharacteristic
33C23×Dic17—21no—
34C2×Q16×C17—22no—
35C2×SD16×C17—22no—
36C17×C4⋊Q8C2×C411yescharacteristic, complemented
37C17×C4⋊Q8C2×C411yescharacteristic, complemented
38C17×#125—21no—
39C17×#124—21no—
40C17×#124—21no—
41C17×#124C2×C412yes—
42C17×#124D412yes—
43C17×C22⋊Q8—21no—
44C17×C22⋊Q8—21no—
45C4×Q8×C17—21no—
46C4×Q8×C17—21no—
47C17×#117—21no—
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—21no—
50C2×C42×C17C2311yescharacteristic
51C2×C22⋊Dic17—21no—
52C2×C22⋊Dic17—21no—
53C2×C22⋊Dic17—21no—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17—21no—
56C2×C4×Dic17—21no—
57C2×C4×Dic17D411yescharacteristic
58C2×C4×Dic17D411yescharacteristic
59C2×#63223—21no—
60C17×C4⋊C8—21no—
61C17×#103—22no—
62C17×#102—22no—
63C17×C22⋊C8—21no—
64C17×#97C2×C412yescomplemented
65#10587502—21no—
66#10587502—21no—
67#10587502—21no—
68#10587502—21no—
69#10587502—21no—
70#10587502D411yescharacteristic
71#10587502D411yescharacteristic
72C22⋊#3622—21no—
73C4⋊#3622—21no—
74#10587474—21no—
75#10587474—21no—
76C4×#3622—21no—
77C24×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—21no—
81C2×Q8×C17—81no—
82C2×Q8×C17C22⋊C412yes—
83C2×D4×C17—21no—
84C2×D4×C17—81no—
85C2×D4×C17C22⋊C412yes—
86C22×C4×C17—21no—
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×Dic17—21no—
91C22×Dic17—21no—
92C22×Dic17—21no—
93C22×Dic17—21no—
94C22×Dic17—42no—
95C22×Dic17—81no—
96C22×Dic17—81no—
97Q16×C17—82no—
98SD16×C17—82no—
99C2×C8×C17—22no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—41no—
105C4⋊C4×C17—41no—
106C22⋊C4×C17—22no—
107C22⋊C4×C17—22no—
108C22⋊C4×C17—22no—
109C22⋊C4×C17—22no—
110C22⋊C4×C17—41no—
111C42×C17—41no—
112C42×C17C22×C411yescharacteristic
113C42×C17C2×D411yescharacteristic
114C42×C17#4111yescharacteristic
115C42×C17#4111yescharacteristic
116C22⋊Dic17—42no—
117C22⋊Dic17—81no—
118C22⋊Dic17—81no—
119C4⋊Dic17—42no—
120C4⋊Dic17—81no—
121C4×Dic17—41no—
122C4×Dic17—41no—
123C4×Dic17—81no—
124#63223—22no—
125#63223—41no—
126C2×#3622—41no—
127C2×#3622—41no—
128#9379—171nomaximal, Sylow
129C23×C17—22no—
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—41no—
134C23×C17—41no—
135C23×C17—41no—
136C23×C17#12011yescharacteristic
137Q8×C17—42no—
138Q8×C17—42no—
139Q8×C17—81no—
140Q8×C17—81no—
141D4×C17—42no—
142D4×C17—81no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—22no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17C2×C22⋊C411yescharacteristic
153C2×C4×C17C4×D411yescharacteristic
154C2×C4×C17C4×D411yescharacteristic
155C2×C4×C17#12111yescharacteristic
156C2×C4×C17#12111yescharacteristic
157C2×C4×C17#12311yescharacteristic
158C2×Dic17—22no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—41no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—41no—
170C2×Dic17—84no—
171C8×C17—42no—
172#3622—42no—
173#3622—81no—
174#3215D1711yescharacteristic
175#2337—171no—
176#1986—171no—
177#1527—171no—
178#1515—171no—
179#1288—171no—
180#1285—171no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—42no—
184C22×C17—42no—
185C22×C17—42no—
186C22×C17—81no—
187C22×C17—81no—
188C22×C17—81no—
189C22×C17—81no—
190C22×C17#38611yescharacteristic
191C22×C17#45511yescharacteristic
192C22×C17#45611yescharacteristic
193C4×C17—21no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—42no—
200C4×C17—81no—
201C4×C17—81no—
202Dic17—42no—
203Dic17—81no—
204Dic17—81no—
205Dic17—81no—
206#533C2×D1711yescharacteristic
207C2×#124C2×D1711yescharacteristic
208#497Dic1712yescomplemented
209#489Dic1712yescomplemented
210#464—22no—
211#451—21no—
212#450—21no—
213#432—171no—
214#432C2×D1711yescharacteristic
215C4×M16—171no—
216#391—341no—
217#388—341no—
218#388—341no—
219#386—341no—
220#385—341no—
221#383—171no—
222C4×C4⋊C4—171no—
223C4×C22⋊C4—341no—
224#339—341no—
225#339—341no—
226#329—342no—
227#328—341no—
228#328—341no—
229C2×C17—21no—
230C2×C17—21no—
231C2×C17—82no—
232C2×C17#188211yescharacteristic
233C2×C17#190611yescharacteristic
234C2×C17#191111yescharacteristic
235C22×Q8C22⋊D1711yescharacteristic
236C22×D4C22⋊D1711yescharacteristic
237C23×C4—341no—
238C2×Q16—22no—
239C2×SD16—22no—
240C2×M16—341no—
241C4⋊Q8C2×Dic1711yescharacteristic, complemented
242C4⋊Q8C2×Dic1711yescharacteristic, complemented
243#125—21no—
244#124—21no—
245#124—21no—
246#124C2×Dic1712yes—
247#124C22⋊D1712yes—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252#117—21no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4—341no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—341no—
258C2×C22⋊C4—341no—
259C2×C22⋊C4—341no—
260C2×C42—171no—
261C2×C42—171no—
262C2×C42—341no—
263C2×C42C22×D1711yescharacteristic
264C4⋊C8—21no—
265C4⋊C8—341no—
266#103—22no—
267#102—22no—
268C22⋊C8—21no—
269C22⋊C8—341no—
270#97—341no—
271#97—341no—
272#97C2×Dic1712yescomplemented
273C4×C8—341no—
274#95—171no—
275#95—171no—
276#95—341no—
277#95—341no—
278#95—341no—
279#95—341no—
280#95—341no—
281C17#937911yescharacteristic, complemented, Sylow
282C24—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—81no—
287C2×Q8C22⋊Dic1712yes—
288C2×D4—21no—
289C2×D4—81no—
290C2×D4C22⋊Dic1712yes—
291C22×C4—21no—
292C22×C4—341no—
293C22×C4—341no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4—682no—
297C22×C4—1361no—
298C22×C4—1361no—
299C22×C4D4×D1711yescharacteristic
300C22×C4D4×D1711yescharacteristic
301C22×C4C2×C22⋊D1711yescharacteristic
302Q16—82no—
303SD16—82no—
304M16—342no—
305M16—681no—
306C2×C8—22no—
307C2×C8—681no—
308C2×C8—681no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—682no—
316C4⋊C4—1361no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—682no—
323C22⋊C4—1361no—
324C22⋊C4—1361no—
325C42—41no—
326C42—681no—
327C42—681no—
328C42—1361no—
329C42#6325411yescharacteristic
330C42#6325411yescharacteristic
331C42C22×Dic1711yescharacteristic
332C42C2×C22⋊D1711yescharacteristic
333C23—22no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23#1058762311yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345D4—42no—
346D4—81no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—342no—
357C2×C4—681no—
358C2×C4—681no—
359C2×C4—681no—
360C2×C4—681no—
361C2×C4—681no—
362C2×C4—681no—
363C2×C4—681no—
364C2×C4—681no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—1364no—
369C2×C4D4×Dic1711yescharacteristic
370C2×C4D4×Dic1711yescharacteristic
371C2×C4#1058762111yescharacteristic
372C2×C4#1058762511yescharacteristic
373C2×C4#1058762511yescharacteristic
374C2×C4C2×C22⋊Dic1711yescharacteristic
375C8—42no—
376C8—682no—
377C8—1361no—
378C22—21no—
379C22—21no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22#1176268911yescharacteristic
388C22#1176298511yescharacteristic
389C22#1176299011yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—682no—
400C4—1361no—
401C4—1361no—
402C4—1361no—
403C2—21no—
404C2—21no—
405C2—82no—
406C2#1452915511yescharacteristic
407C2#1452928711yescharacteristic
408C2#1452931711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#3215C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#3215D17
hypocenterC17#9379
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
2. #14529729C2semidirect
4. #14527335C2semidirect
5. #14526650C2semidirect
6. #14526639C2semidirect
25. C17×#497C4semidirect
26. C17×#489C4semidirect
36. C17×C4⋊Q8C2×C4semidirect
37. C17×C4⋊Q8C2×C4semidirect
64. C17×#97C2×C4semidirect
208. #497Dic17semidirect
209. #489Dic17semidirect
241. C4⋊Q8C2×Dic17semidirect
242. C4⋊Q8C2×Dic17semidirect
272. #97C2×Dic17semidirect
281. C17#9379semidirect

Maximal subgroups

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

subgroupindex
#145315992
#145297292
#145273822
#145273352
#145266502
#145266392
C17×#32152
#937917