Finite Group Databasebeta

← #16930280

Subgroups — #16930280

Order 4352.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538256C211yesmaximal, characteristic
2#14527343C211yesmaximal, characteristic
3#14527278C211yesmaximal, characteristic
4#14527179C211yesmaximal, characteristic, complemented
5#14526687C211yesmaximal, characteristic, complemented
6#14526642C211yesmaximal, characteristic, complemented
7C17×#2356C211yesmaximal, characteristic, complemented
8C2×#10587628—21no—
9#11763359C2211yescharacteristic
10#11763031—22no—
11#11763019C412yescomplemented
12#11763009C412yescomplemented
13#11762862C2211yescharacteristic
14#11762862C2211yescharacteristic
15#11762833C2211yescharacteristic
16#11762808—21no—
17#11762542—21no—
18#11762542—21no—
19#11762542—22no—
20Q8⋊#3622—22no—
21C2×C4×Q8×C17C2211yescharacteristic
22C17×#391—21no—
23C17×#387—22no—
24C17×#383—21no—
25C17×#382C2211yescharacteristic
26C4×C4⋊C4×C17—21no—
27C4×C4⋊C4×C17C2211yescharacteristic
28C17×#376—21no—
29C22×Q8×C17D411yescharacteristic
30C2×#63252—21no—
31C4×Q8×C17—21no—
32C4×Q8×C17—21no—
33C4×Q8×C17C2×C412yes—
34C4×Q8×C17D412yes—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17D411yescharacteristic
39C2×C42×C17—21no—
40C2×C42×C17—22no—
41C2×C42×C17—21no—
42C2×C42×C17C2311yescharacteristic
43C2×#63231—22no—
44#10587628—42no—
45#10587628—42no—
46C2×#63230—22no—
47#10587616—21no—
48#10587616—21no—
49C2×#63223—21no—
50#10587561—21no—
51#10587560C2×C411yescharacteristic
52C4×D68—21no—
53C4×D68—21no—
54#10587555C2×C411yescharacteristic
55C17×#95—22no—
56C17×#95—22no—
57C17×#95—22no—
58C17×#95—21no—
59C17×#95D411yescharacteristic
60C17×#95D411yescharacteristic
61C17×#95D412yes—
62C22⋊#3622—21no—
63C22⋊#3622—21no—
64#10587481—22no—
65#10587480—22no—
66C4⋊#3622—21no—
67C4⋊#3622—21no—
68C4⋊#3622—21no—
69C4⋊#3622C2×C412yes—
70C4×#3622—21no—
71C4×#3622—21no—
72C2×Q8×C17—21no—
73C2×Q8×C17—21no—
74C2×Q8×C17—21no—
75C2×Q8×C17—21no—
76C2×Q8×C17—21no—
77C2×Q8×C17C22⋊C412yes—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—21no—
81C22×C4×C17—22no—
82C22×C4×C17—21no—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C2×C22⋊D17—41no—
87#63252—41no—
88#63252—41no—
89#63252—41no—
90#63252—41no—
91C2×D68—21no—
92C2×D68—21no—
93C2×D68—21no—
94C2×C4×D17—41no—
95C2×Dic34—21no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—42no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17C22⋊C412yes—
103C42×C17—22no—
104C42×C17—22no—
105C42×C17—44no—
106C42×C17—41no—
107C42×C17—41no—
108C42×C17C22×C411yescharacteristic
109C42×C17C2×D411yescharacteristic
110C42×C17#4111yescharacteristic
111C42×C17#4111yescharacteristic
112#63231—44no—
113#63230—44no—
114#63227—41no—
115#63227—41no—
116C4⋊Dic17—21no—
117C4⋊Dic17—21no—
118#63223—41no—
119#63223—41no—
120C2×#3622—22no—
121C2×#3622—41no—
122C2×#3622—41no—
123#9182—171nomaximal, Sylow
124C23×C17#12011yescharacteristic
125C22×D17—41no—
126Q8×C17—22no—
127Q8×C17—22no—
128Q8×C17—41no—
129Q8×C17—41no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—22no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—42no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—42no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—42no—
144C2×C4×C17—41no—
145C2×C4×C17—42no—
146C2×C4×C17C2×C22⋊C411yescharacteristic
147C2×C4×C17C4×D411yescharacteristic
148C2×C4×C17C4×D411yescharacteristic
149C2×C4×C17#12111yescharacteristic
150C2×C4×C17#12111yescharacteristic
151C2×C4×C17#12311yescharacteristic
152C2×C4×C17#13511yescharacteristic
153C2×C4×C17#13511yescharacteristic
154C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
156C22⋊D17—81no—
157C22⋊D17—81no—
158C2×Dic17—41no—
159D68—41no—
160D68—41no—
161D68—81no—
162D68—81no—
163C4×D17—81no—
164C4×D17—81no—
165Dic34—41no—
166Dic34—41no—
167#3622—42no—
168#3622—81no—
169#3622—81no—
170#3037—171no—
171#2356D1711yescharacteristic, complemented
172#1517—171no—
173#1502—171no—
174#1479—171no—
175#1303—171no—
176#1286—171no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17#38611yescharacteristic
180C22×C17#44911yescharacteristic
181C22×C17#44911yescharacteristic
182C2×D17—81no—
183C2×D17—81no—
184C2×D17—81no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—42no—
190C4×C17—42no—
191C4×C17—42no—
192C4×C17—81no—
193C4×C17—82no—
194C4×C17#41611yescharacteristic
195C4×C17#41811yescharacteristic
196C4×C17#48211yescharacteristic
197C4×C17#48211yescharacteristic
198Dic17—81no—
199C2×#137—341no—
200#532—171no—
201C2×C4×Q8C2×D1711yescharacteristic
202#464—342no—
203#462—172no—
204#460—172no—
205#431—171no—
206#431—171no—
207#423—171no—
208#417—341no—
209#391—21no—
210#387—22no—
211#383—21no—
212#382C2×D1711yescharacteristic
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2×D1711yescharacteristic
215#376—21no—
216#327—341no—
217#327—341no—
218#327—342no—
219Q8⋊C8—342no—
220C2×C17—21no—
221C2×C17—21no—
222C2×C17#187511yescharacteristic
223C2×C17#190511yescharacteristic
224C2×C17#190611yescharacteristic
225D17—161no—
226C2×#41—341no—
227C22×Q8C22⋊D1711yescharacteristic
228#137—682no—
229#137—682no—
230C2×Q16—342no—
231C2×SD16—342no—
232C2×M16—341no—
233C4⋊Q8—171no—
234#127—171no—
235#124—341no—
236#121—341no—
237#121—341no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C4×D1712yes—
241C4×Q8D6812yes—
242C4×D4—341no—
243C4×D4—341no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D1711yescharacteristic
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42C22×D1711yescharacteristic
252C4⋊C8—172no—
253C4⋊C8—341no—
254C4⋊C8—341no—
255C4⋊C8—341no—
256#103—342no—
257#102—342no—
258C22⋊C8—341no—
259C22⋊C8—341no—
260C4×C8—341no—
261C4×C8—341no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—21no—
266#95C22⋊D1711yescharacteristic
267#95C22⋊D1711yescharacteristic
268#95C22⋊D1712yes—
269C17#918211yescharacteristic, complemented, Sylow
270#41—681no—
271#41—681no—
272#41—681no—
273#41—681no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—341no—
280C2×Q8#6322712yes—
281C2×D4—341no—
282C2×D4—341no—
283C2×D4—341no—
284C2×D4—681no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—681no—
291C22×C4C2×C22⋊D1711yescharacteristic
292C22×C4C2×C22⋊D1711yescharacteristic
293C22×C4C2×C22⋊D1711yescharacteristic
294Q16—684no—
295SD16—684no—
296M16—681no—
297M16—681no—
298C2×C8—342no—
299C2×C8—681no—
300C2×C8—681no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—341no—
308C4⋊C4—341no—
309C4⋊C4#6322712yes—
310C22⋊C4—681no—
311C22⋊C4—681no—
312C42—22no—
313C42—22no—
314C42—44no—
315C42—41no—
316C42—41no—
317C42C2×C4×D1711yescharacteristic
318C42C2×D6811yescharacteristic
319C42#6325211yescharacteristic
320C42#6325211yescharacteristic
321C23—681no—
322C23#1058763911yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—41no—
326Q8—41no—
327Q8—681no—
328Q8—681no—
329D4—681no—
330D4—681no—
331D4—1361no—
332D4—1361no—
333D4—1361no—
334D4—1361no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—681no—
352C2×C4—1361no—
353C2×C4—1361no—
354C2×C4C2×#6322711yescharacteristic
355C2×C4C4×C22⋊D1711yescharacteristic
356C2×C4C4×C22⋊D1711yescharacteristic
357C2×C4#1058761511yescharacteristic
358C2×C4#1058761611yescharacteristic
359C2×C4#1058761611yescharacteristic
360C2×C4#1058763511yescharacteristic
361C2×C4#1058763511yescharacteristic
362C2×C4#1058763611yescharacteristic
363C2×C4#1058763611yescharacteristic
364C8—682no—
365C8—1361no—
366C8—1361no—
367C22—21no—
368C22—21no—
369C22—1361no—
370C22—1361no—
371C22—1361no—
372C22#1176268811yescharacteristic
373C22#1176295411yescharacteristic
374C22#1176295411yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—81no—
383C4—82no—
384C4—1361no—
385C4#1176280311yescharacteristic
386C4#1176281311yescharacteristic
387C4#1176313311yescharacteristic
388C4#1176313311yescharacteristic
389C2—21no—
390C2—21no—
391C2—2721no—
392C2#1452910611yescharacteristic
393C2#1452927811yescharacteristic
394C2#1452928511yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2356C2
socleC22×C17#386
prime radical#11763359C22
cosocleC2×C42×C17C23
hypercenter#2356D17
hypocenterC17#9182
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
4. #14527179C2semidirect
5. #14526687C2semidirect
6. #14526642C2semidirect
7. C17×#2356C2semidirect
11. #11763019C4semidirect
12. #11763009C4semidirect
171. #2356D17semidirect
269. C17#9182semidirect

Maximal subgroups

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

subgroupindex
#145382562
#145273432
#145272782
#145271792
#145266872
#145266422
C17×#23562
#918217