Finite Group Databasebeta

← #16933709

Subgroups — #16933709

Order 4352.

408 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539109C211yesmaximal, characteristic
2#14528946C211yesmaximal, characteristic, complemented
3#14527348C211yesmaximal, characteristic, complemented
4#14527345C211yesmaximal, characteristic
5#14527343C211yesmaximal, characteristic
6#14527077C211yesmaximal, 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
16C2×#10587478—21no—
17C2×#10587479—21no—
18#11762803—21no—
19#11762803—21no—
20C2×C4×#3622C2211yescharacteristic
21#11762675C2211yescharacteristic
22#11762546—22no—
23#11762544—22no—
24#11762542—22no—
25#11762531—21no—
26#11762530—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—
68#10587479—42no—
69#10587478—42no—
70C4⋊#3622—21no—
71C4×#3622—21no—
72C4×#3622C2×C411yescharacteristic
73C4×#3622C2×C411yescharacteristic
74C4×#3622D411yescharacteristic
75C4×#3622D411yescharacteristic
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C22×C4×C17—41no—
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C2×C22⋊D17—41no—
83C22×Dic17—41no—
84#63252—41no—
85#63252—41no—
86#63252—41no—
87#63252—41no—
88C2×D68—21no—
89C2×D68—21no—
90C2×D68—21no—
91C2×C4×D17—41no—
92C2×Dic34—21no—
93C2×C8×C17—22no—
94C4⋊C4×C17—21no—
95C4⋊C4×C17—21no—
96C4⋊C4×C17—21no—
97C4⋊C4×C17—21no—
98C4⋊C4×C17—41no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C22⋊C4×C17—41no—
103C42×C17—41no—
104C42×C17C22×C411yescharacteristic
105C42×C17C2×D411yescharacteristic
106C42×C17#4111yescharacteristic
107C42×C17#4111yescharacteristic
108#63227—41no—
109#63227—41no—
110C4⋊Dic17—21no—
111C4⋊Dic17—21no—
112C4⋊Dic17—41no—
113C4⋊Dic17—41no—
114C2×#3622—21no—
115C2×#3622—21no—
116C2×#3622—21no—
117C2×#3622—21no—
118C2×#3622—22no—
119C2×#3622C22⋊C411yescharacteristic
120C2×#3622C22⋊C411yescharacteristic
121C2×#3622C22⋊C411yescharacteristic
122C2×#3622C22⋊C411yescharacteristic
123#9758—171nomaximal, Sylow
124C23×C17#12711yescharacteristic
125C22×D17—41no—
126Q8×C17—81no—
127Q8×C17—81no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—81no—
133C2×C4×C17—81no—
134C2×C4×C17—81no—
135C2×C4×C17C2×C22⋊C411yescharacteristic
136C2×C4×C17C4×D411yescharacteristic
137C2×C4×C17C4×D411yescharacteristic
138C2×C4×C17#12111yescharacteristic
139C2×C4×C17#12111yescharacteristic
140C2×C4×C17#12411yescharacteristic
141C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C17#13711yescharacteristic
144C2×C4×C17#13711yescharacteristic
145C22⋊D17—81no—
146C22⋊D17—81no—
147C2×Dic17—41no—
148C2×Dic17—81no—
149C2×Dic17—81no—
150C2×Dic17—81no—
151D68—41no—
152D68—41no—
153D68—81no—
154D68—81no—
155C4×D17—81no—
156C4×D17—81no—
157Dic34—41no—
158Dic34—41no—
159C8×C17—42no—
160#3622—21no—
161#3622—21no—
162#3622—21no—
163#3622—21no—
164#3622—42no—
165#3131—171no—
166#1846—171no—
167#1518—171no—
168#1518D1711yescharacteristic, complemented
169#1517—171no—
170#1517—171no—
171#1452—171no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17#39011yescharacteristic
175C22×C17#49611yescharacteristic
176C22×C17#49711yescharacteristic
177C2×D17—81no—
178C2×D17—81no—
179C2×D17—81no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—21no—
184C4×C17—81no—
185C4×C17—81no—
186C4×C17—161no—
187C4×C17#41711yescharacteristic
188C4×C17#41711yescharacteristic
189C4×C17#48911yescharacteristic
190C4×C17#48911yescharacteristic
191Dic17—81no—
192Dic17—161no—
193#533C2×D1711yescharacteristic
194#532—171no—
195#488—341no—
196#488—341no—
197#487—171no—
198#487—171no—
199#486—171no—
200#486—171no—
201#431—171no—
202#431C2×D1711yescharacteristic
203C2×#107—341no—
204C2×#106—341no—
205#416—341no—
206#416—341no—
207C2×C4×C8—171no—
208#382—171no—
209#382C2×D1711yescharacteristic
210#328—22no—
211#328—342no—
212#327—342no—
213#327—342no—
214#322—341no—
215#322—341no—
216C2×C17—21no—
217C2×C17—21no—
218C2×C17#195911yescharacteristic
219C2×C17#197611yescharacteristic
220C2×C17#197611yescharacteristic
221D17—161no—
222C2×#41—341no—
223C22×C8—171no—
224C22×C8—171no—
225C4⋊Q8—171no—
226C4⋊Q8C4×D1711yescharacteristic
227C4⋊Q8C4×D1711yescharacteristic
228#127—171no—
229#125—21no—
230#124—341no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233#121—341no—
234#121—341no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×D4—341no—
238C4×D4—341no—
239#117—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—341no—
242C2×C42C22×D1711yescharacteristic
243#107—682no—
244#106—682no—
245C4⋊C8—21no—
246C4⋊C8—341no—
247#103—341no—
248#103—341no—
249#103—341no—
250#103—341no—
251#102—341no—
252#102—341no—
253#102—341no—
254#102—341no—
255C22⋊C8—172no—
256C22⋊C8C22⋊D1712yes—
257C4×C8—21no—
258C4×C8—171no—
259C4×C8—171no—
260C4×C8—171no—
261C4×C8—171no—
262C4×C8—341no—
263#95—21no—
264#95—21no—
265#95—341no—
266#95—341no—
267C17#975811yescharacteristic, complemented, Sylow
268#41—681no—
269#41—681no—
270#41—681no—
271#41—681no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—341no—
275C2×D4—341no—
276C2×D4—341no—
277C2×D4—341no—
278C2×D4—681no—
279C22×C4—41no—
280C22×C4—681no—
281C22×C4—681no—
282C22×C4D4×D1711yescharacteristic
283C22×C4D4×D1711yescharacteristic
284C22×C4C2×C22⋊D1711yescharacteristic
285C2×C8—22no—
286C2×C8—171no—
287C2×C8—171no—
288C2×C8—171no—
289C2×C8—171no—
290C2×C8—341no—
291C2×C8—341no—
292C2×C8—341no—
293C2×C8—341no—
294C2×C8—342no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—341no—
304C4⋊C4—341no—
305C4⋊C4—681no—
306C4⋊C4—681no—
307C22⋊C4—41no—
308C22⋊C4—681no—
309C22⋊C4—681no—
310C42—41no—
311C42C2×C4×D1711yescharacteristic
312C42#6325211yescharacteristic
313C42D4×D1711yescharacteristic
314C42#6325411yescharacteristic
315C23—681no—
316C23#1058762611yescharacteristic
317Q8—81no—
318Q8—81no—
319Q8—681no—
320Q8—681no—
321D4—681no—
322D4—681no—
323D4—1361no—
324D4—1361no—
325D4—1361no—
326D4—1361no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—681no—
335C2×C4—1361no—
336C2×C4—1361no—
337C2×C4—1361no—
338C2×C4—1361no—
339C2×C4—1361no—
340C2×C4C22⋊C4×D1711yescharacteristic
341C2×C4#1058756711yescharacteristic
342C2×C4#1058757011yescharacteristic
343C2×C4#1058757111yescharacteristic
344C2×C4#1058760011yescharacteristic
345C2×C4#1058760011yescharacteristic
346C2×C4#1058760411yescharacteristic
347C2×C4#1058760411yescharacteristic
348C2×C4C4×C22⋊D1711yescharacteristic
349C2×C4#1058762511yescharacteristic
350C8—42no—
351C8—341no—
352C8—341no—
353C8—341no—
354C8—341no—
355C8—682no—
356C22—21no—
357C22—21no—
358C22—1361no—
359C22—1361no—
360C22—1361no—
361C22#1176270911yescharacteristic
362C22#1176319511yescharacteristic
363C22#1176320211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—161no—
371C4—2721no—
372C4—1361no—
373C4#1176280911yescharacteristic
374C4#1176280911yescharacteristic
375C4#1176317011yescharacteristic
376C4#1176317011yescharacteristic
377C2—21no—
378C2—21no—
379C2—2721no—
380C2#1452959711yescharacteristic
381C2#1452967911yescharacteristic
382C2#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#9758
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. #14528946C2semidirect
3. #14527348C2semidirect
6. #14527077C2semidirect
7. C17×#1518C2semidirect
11. #11763161C4semidirect
12. #11763161C4semidirect
13. #11763157C4semidirect
14. #11763157C4semidirect
168. #1518D17semidirect
267. C17#9758semidirect

Maximal subgroups

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

subgroupindex
#145391092
#145289462
#145273482
#145273452
#145273432
#145270772
C17×#15182
#975817