Finite Group Databasebeta

← #16933704

Subgroups — #16933704

Order 4352.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539109C211yesmaximal, characteristic
2#14528914C211yesmaximal, characteristic, complemented
3#14527348C211yesmaximal, characteristic, complemented
4#14527345C211yesmaximal, characteristic
5#14527343C211yesmaximal, characteristic
6#14527068C211yesmaximal, characteristic, complemented
7C17×#1518C211yesmaximal, characteristic, complemented
8#11763359C2211yescharacteristic
9#11763166—21no—
10#11763166—21no—
11#11763161C411yescharacteristic, complemented
12#11763161C411yescharacteristic, complemented
13#11763157C411yescharacteristic, complemented
14#11763157C411yescharacteristic, complemented
15#11762862C2211yescharacteristic
16#11762803—21no—
17#11762803—21no—
18C2×C4×#3622C2211yescharacteristic
19#11762675C2211yescharacteristic
20#11762571—21no—
21#11762571—21no—
22#11762546—22no—
23#11762544—22no—
24#11762542—22no—
25#11762531—21no—
26C8×#3622—21no—
27C17×#533C2211yescharacteristic
28C17×#431C2211yescharacteristic
29C17×#382C2211yescharacteristic
30C17×#328—22no—
31C2×#63252—21no—
32C17×C4⋊Q8C2×C411yescharacteristic
33C17×C4⋊Q8C2×C411yescharacteristic
34C17×#125—21no—
35C17×C22⋊Q8—21no—
36C17×C22⋊Q8—21no—
37C4×Q8×C17—21no—
38C4×Q8×C17—21no—
39C17×#117—21no—
40C2×C4⋊C4×C17—21no—
41C2×C42×C17C2311yescharacteristic
42#10587616—21no—
43#10587616—21no—
44C2×C4⋊Dic17—21no—
45C22×#3622D411yescharacteristic
46C22×#3622D411yescharacteristic
47#10587561—21no—
48#10587560C2×C411yescharacteristic
49C4×D68—21no—
50C4×D68—21no—
51#10587555C2×C411yescharacteristic
52C17×C4⋊C8—21no—
53C17×C22⋊C8D412yes—
54C4×C8×C17—21no—
55C17×#95—21no—
56C17×#95—21no—
57#10587502—21no—
58#10587502—21no—
59C22⋊#3622D412yes—
60#10587481—21no—
61#10587481—21no—
62#10587481—21no—
63#10587481—21no—
64#10587480—21no—
65#10587480—21no—
66#10587480—21no—
67#10587480—21no—
68C4⋊#3622—21no—
69C4×#3622—21no—
70C4×#3622C2×C411yescharacteristic
71C4×#3622C2×C411yescharacteristic
72C4×#3622D411yescharacteristic
73C4×#3622D411yescharacteristic
74C2×Q8×C17—21no—
75C2×Q8×C17—21no—
76C22×C4×C17—41no—
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C2×C22⋊D17—41no—
81C22×Dic17—41no—
82#63252—41no—
83#63252—41no—
84#63252—41no—
85#63252—41no—
86C2×D68—21no—
87C2×D68—21no—
88C2×D68—21no—
89C2×C4×D17—41no—
90C2×Dic34—21no—
91C2×C8×C17—22no—
92C4⋊C4×C17—21no—
93C4⋊C4×C17—21no—
94C4⋊C4×C17—21no—
95C4⋊C4×C17—21no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—41no—
98C4⋊C4×C17—41no—
99C4⋊C4×C17—41no—
100C22⋊C4×C17—41no—
101C42×C17—41no—
102C42×C17C22×C411yescharacteristic
103C42×C17C2×D411yescharacteristic
104C42×C17#4111yescharacteristic
105C42×C17#4111yescharacteristic
106#63227—41no—
107#63227—41no—
108C4⋊Dic17—21no—
109C4⋊Dic17—21no—
110C4⋊Dic17—41no—
111C4⋊Dic17—41no—
112C2×#3622—21no—
113C2×#3622—21no—
114C2×#3622—21no—
115C2×#3622—21no—
116C2×#3622—22no—
117C2×#3622C22⋊C411yescharacteristic
118C2×#3622C22⋊C411yescharacteristic
119C2×#3622C22⋊C411yescharacteristic
120C2×#3622C22⋊C411yescharacteristic
121#9757—171nomaximal, Sylow
122C23×C17#12711yescharacteristic
123C22×D17—41no—
124Q8×C17—81no—
125Q8×C17—81no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17—81no—
133C2×C4×C17C2×C22⋊C411yescharacteristic
134C2×C4×C17C4×D411yescharacteristic
135C2×C4×C17C4×D411yescharacteristic
136C2×C4×C17#12111yescharacteristic
137C2×C4×C17#12111yescharacteristic
138C2×C4×C17#12411yescharacteristic
139C2×C4×C17#13511yescharacteristic
140C2×C4×C17#13511yescharacteristic
141C2×C4×C17#13511yescharacteristic
142C2×C4×C17#13511yescharacteristic
143C22⋊D17—81no—
144C22⋊D17—81no—
145C2×Dic17—41no—
146C2×Dic17—81no—
147C2×Dic17—81no—
148C2×Dic17—81no—
149D68—41no—
150D68—41no—
151D68—81no—
152D68—81no—
153C4×D17—81no—
154C4×D17—81no—
155Dic34—41no—
156Dic34—41no—
157C8×C17—42no—
158#3622—21no—
159#3622—21no—
160#3622—21no—
161#3622—21no—
162#3622—42no—
163#3131—171no—
164#1835—171no—
165#1518—171no—
166#1518D1711yescharacteristic, complemented
167#1517—171no—
168#1517—171no—
169#1450—171no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17#39011yescharacteristic
173C22×C17#49511yescharacteristic
174C22×C17#49511yescharacteristic
175C2×D17—81no—
176C2×D17—81no—
177C2×D17—81no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—81no—
183C4×C17—81no—
184C4×C17—161no—
185C4×C17#41611yescharacteristic
186C4×C17#41611yescharacteristic
187C4×C17#48811yescharacteristic
188C4×C17#48811yescharacteristic
189Dic17—81no—
190Dic17—161no—
191#533C2×D1711yescharacteristic
192#532—171no—
193#488—341no—
194#488—341no—
195#487—171no—
196#487—171no—
197#486—171no—
198#486—171no—
199#431—171no—
200#431C2×D1711yescharacteristic
201#416—341no—
202#416—341no—
203C2×C4×C8—171no—
204#382—171no—
205#382C2×D1711yescharacteristic
206#340—341no—
207#340—341no—
208#328—22no—
209#328—342no—
210#327—342no—
211#327—342no—
212#322—341no—
213C82—341no—
214C2×C17—21no—
215C2×C17—21no—
216C2×C17#195811yescharacteristic
217C2×C17#197611yescharacteristic
218C2×C17#197611yescharacteristic
219D17—161no—
220C2×#41—341no—
221C22×C8—171no—
222C22×C8—171no—
223C4⋊Q8—171no—
224C4⋊Q8C4×D1711yescharacteristic
225C4⋊Q8C4×D1711yescharacteristic
226#127—171no—
227#125—21no—
228#124—341no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—341no—
232#121—341no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—341no—
236C4×D4—341no—
237#117—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—341no—
240C2×C42C22×D1711yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—341no—
243#103—341no—
244#103—341no—
245#103—341no—
246#103—341no—
247#102—341no—
248#102—341no—
249#102—341no—
250#102—341no—
251C22⋊C8—172no—
252C22⋊C8C22⋊D1712yes—
253C4×C8—21no—
254C4×C8—171no—
255C4×C8—171no—
256C4×C8—171no—
257C4×C8—171no—
258C4×C8—341no—
259#95—21no—
260#95—21no—
261#95—341no—
262#95—341no—
263C17#975711yescharacteristic, complemented, Sylow
264#41—681no—
265#41—681no—
266#41—681no—
267#41—681no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—341no—
271C2×D4—341no—
272C2×D4—341no—
273C2×D4—341no—
274C2×D4—681no—
275C22×C4—41no—
276C22×C4—681no—
277C22×C4—681no—
278C22×C4D4×D1711yescharacteristic
279C22×C4D4×D1711yescharacteristic
280C22×C4C2×C22⋊D1711yescharacteristic
281C2×C8—22no—
282C2×C8—171no—
283C2×C8—171no—
284C2×C8—171no—
285C2×C8—171no—
286C2×C8—341no—
287C2×C8—341no—
288C2×C8—341no—
289C2×C8—341no—
290C2×C8—342no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—341no—
300C4⋊C4—341no—
301C4⋊C4—681no—
302C4⋊C4—681no—
303C22⋊C4—41no—
304C22⋊C4—681no—
305C22⋊C4—681no—
306C42—41no—
307C42C2×C4×D1711yescharacteristic
308C42#6325211yescharacteristic
309C42D4×D1711yescharacteristic
310C42#6325411yescharacteristic
311C23—681no—
312C23#1058762611yescharacteristic
313Q8—81no—
314Q8—81no—
315Q8—681no—
316Q8—681no—
317D4—681no—
318D4—681no—
319D4—1361no—
320D4—1361no—
321D4—1361no—
322D4—1361no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—681no—
331C2×C4—1361no—
332C2×C4—1361no—
333C2×C4—1361no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4C22⋊C4×D1711yescharacteristic
337C2×C4#1058756711yescharacteristic
338C2×C4#1058757011yescharacteristic
339C2×C4#1058757111yescharacteristic
340C2×C4#1058760211yescharacteristic
341C2×C4#1058760211yescharacteristic
342C2×C4#1058760511yescharacteristic
343C2×C4#1058760511yescharacteristic
344C2×C4C4×C22⋊D1711yescharacteristic
345C2×C4#1058762511yescharacteristic
346C8—42no—
347C8—341no—
348C8—341no—
349C8—341no—
350C8—341no—
351C8—682no—
352C22—21no—
353C22—21no—
354C22—1361no—
355C22—1361no—
356C22—1361no—
357C22#1176270911yescharacteristic
358C22#1176319011yescharacteristic
359C22#1176319411yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—81no—
366C4—161no—
367C4—2721no—
368C4—1361no—
369C4#1176280411yescharacteristic
370C4#1176280411yescharacteristic
371C4#1176316511yescharacteristic
372C4#1176316511yescharacteristic
373C2—21no—
374C2—21no—
375C2—2721no—
376C2#1452959211yescharacteristic
377C2#1452967911yescharacteristic
378C2#1452968111yescharacteristic
subgroupG/H
centerC22#11762709
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1518C2
socleC22×C17#390
prime radical#11763359C22
cosocleC2×C42×C17C23
hypercenter#1518D17
hypocenterC17#9757
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. #14528914C2semidirect
3. #14527348C2semidirect
6. #14527068C2semidirect
7. C17×#1518C2semidirect
11. #11763161C4semidirect
12. #11763161C4semidirect
13. #11763157C4semidirect
14. #11763157C4semidirect
166. #1518D17semidirect
263. C17#9757semidirect

Maximal subgroups

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

subgroupindex
#145391092
#145289142
#145273482
#145273452
#145273432
#145270682
C17×#15182
#975717