Finite Group Databasebeta

← #16931074

Subgroups — #16931074

Order 4352.

480 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531533C211yesmaximal, characteristic, complemented
2#14529617C211yesmaximal, characteristic
3#14527469C211yesmaximal, characteristic, complemented
4#14527344C211yesmaximal, characteristic
5#14526643C211yesmaximal, characteristic, complemented
6#14526612C211yesmaximal, characteristic, complemented
7C17×#3227C211yesmaximal, characteristic
8#11762862C2211yescharacteristic
9#11762833C2211yescharacteristic
10C2×#10587508—21no—
11C2×#10587508—21no—
12#11762716—22no—
13#11762708—21no—
14#11762695—22no—
15#11762675C2211yescharacteristic
16C4⋊C4×Dic17—21no—
17#11762655C2211yescharacteristic
18#11762555—22no—
19#11762543—22no—
20C17×#532C2211yescharacteristic
21C2×C17×C4⋊Q8C2211yescharacteristic
22C17×#498C412yescomplemented
23C17×#486C412yescomplemented
24C17×#481—22no—
25C17×#476—22no—
26C17×#431C2211yescharacteristic
27C2×#41×C17—21no—
28C22×Q8×C17—21no—
29C17×C4⋊Q8—21no—
30C17×C4⋊Q8—21no—
31C17×C4⋊Q8C2×C411yescharacteristic
32C17×C4⋊Q8C2×C412yes—
33C17×C4⋊Q8D412yes—
34C17×#127C2×C411yescharacteristic
35C17×#124—21no—
36C17×#121—21no—
37C17×#121—21no—
38C4×D4×C17—21no—
39C4×D4×C17—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17D412yes—
42C2×C42×C17C2311yescharacteristic
43C2×C4⋊Dic17—22no—
44C2×C4⋊Dic17—21no—
45C2×C4×Dic17—22no—
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C2×#63223—21no—
49C17×#106—24no—
50C17×C4⋊C8—21no—
51C17×#102—24no—
52C17×C22⋊C8—21no—
53C4×C8×C17C2×C412yes—
54#10587508—42no—
55#10587508—42no—
56#10587502—22no—
57#10587502—22no—
58#10587502—21no—
59#10587502—21no—
60#10587502—21no—
61#10587502D412yes—
62C22⋊#3622—21no—
63C4⋊#3622—21no—
64C4⋊#3622—21no—
65C4⋊#3622—21no—
66C4×#3622—21no—
67#41×C17—41no—
68#41×C17—41no—
69#41×C17—41no—
70#41×C17—41no—
71C2×Q8×C17—22no—
72C2×Q8×C17—22no—
73C2×Q8×C17—21no—
74C2×Q8×C17—82no—
75C2×D4×C17—21no—
76C2×D4×C17—21no—
77C2×D4×C17—21no—
78C2×D4×C17—41no—
79C22×C4×C17—22no—
80C22×C4×C17—41no—
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×D411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×Dic17—21no—
85C22×Dic17—21no—
86C22×Dic17—22no—
87C2×C8×C17—22no—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—22no—
90C4⋊C4×C17—22no—
91C4⋊C4×C17—21no—
92C4⋊C4×C17—21no—
93C4⋊C4×C17C22⋊C414yes—
94C22⋊C4×C17—41no—
95C22⋊C4×C17—41no—
96C42×C17C22×C411yescharacteristic
97C42×C17C2×D411yescharacteristic
98C42×C17#4111yescharacteristic
99C42×C17#4111yescharacteristic
100C4⋊Dic17—41no—
101C4⋊Dic17—41no—
102C4⋊Dic17—82no—
103C4×Dic17—21no—
104C4×Dic17—21no—
105C4×Dic17—21no—
106C4×Dic17—82no—
107C4×Dic17SD1612yes—
108#63223—41no—
109#63223—41no—
110C2×#3622—41no—
111C2×#3622—41no—
112#9302—171nomaximal, Sylow
113C23×C17—41no—
114C23×C17#12011yescharacteristic
115Q8×C17—41no—
116Q8×C17—41no—
117Q8×C17—82no—
118Q8×C17—82no—
119D4×C17—42no—
120D4×C17—41no—
121D4×C17—41no—
122D4×C17—81no—
123D4×C17—81no—
124D4×C17—81no—
125C2×C4×C17—24no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—41no—
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#12311yescharacteristic
139C2×C4×C17C2×SD1611yescharacteristic
140C2×C4×C17#13511yescharacteristic
141C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C17#13711yescharacteristic
143C2×Dic17—22no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—42no—
148C2×Dic17—42no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—42no—
152C8×C17—42no—
153#3622—81no—
154#3622—81no—
155#3227D1711yescharacteristic
156#2331—171no—
157#1963—171no—
158#1549—171no—
159#1517—171no—
160#1286—171no—
161#1274—171no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—81no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17#38611yescharacteristic
168C22×C17#44811yescharacteristic
169C22×C17#44911yescharacteristic
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—44no—
175C4×C17—81no—
176C4×C17C4×SD1611yescharacteristic
177C4×C17#44011yescharacteristic
178C4×C17#46011yescharacteristic
179C4×C17#46411yescharacteristic
180Dic17—42no—
181Dic17—82no—
182Dic17—81no—
183#532C2×D1711yescharacteristic
184C2×C4⋊Q8C2×D1711yescharacteristic
185#498Dic1712yescomplemented
186#486Dic1712yescomplemented
187#481—22no—
188#476—22no—
189#431—171no—
190#431C2×D1711yescharacteristic
191#423—171no—
192C2×#104—341no—
193C2×#104—341no—
194#391—342no—
195#389—341no—
196#387—342no—
197#382—171no—
198C4×C4⋊C4—341no—
199#376—171no—
200#332—342no—
201#327—342no—
202C2×C17—21no—
203C2×C17—21no—
204C2×C17—161no—
205C2×C17#188011yescharacteristic
206C2×C17#190411yescharacteristic
207C2×C17#191211yescharacteristic
208C2×#41—21no—
209C22×Q8—21no—
210C2×M16—341no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8C2×Dic1711yescharacteristic
214C4⋊Q8C2×Dic1712yes—
215C4⋊Q8C22⋊D1712yes—
216#127C2×Dic1711yescharacteristic
217#124—21no—
218#121—21no—
219#121—21no—
220C4×D4—21no—
221C4×D4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—342no—
225C2×C4⋊C4C22⋊D1712yes—
226C2×C42—171no—
227C2×C42—171no—
228C2×C42—342no—
229C2×C42C22×D1711yescharacteristic
230#106—24no—
231C4⋊C8—21no—
232C4⋊C8—341no—
233C4⋊C8—341no—
234C4⋊C8—341no—
235#104—682no—
236#104—682no—
237#102—24no—
238C22⋊C8—21no—
239C22⋊C8—341no—
240C4×C8—341no—
241C4×C8C2×Dic1712yes—
242#95—172no—
243#95—342no—
244#95—342no—
245#95—341no—
246#95—341no—
247#95—341no—
248C17#930211yescharacteristic, complemented, Sylow
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—21no—
256C2×Q8—82no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—341no—
264C22×C4—342no—
265C22×C4—341no—
266C22×C4D4×D1711yescharacteristic
267C22×C4D4×D1711yescharacteristic
268C22×C4C2×C22⋊D1711yescharacteristic
269M16—681no—
270M16—681no—
271C2×C8—22no—
272C2×C8—681no—
273C2×C8—681no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C4⋊C4—1362no—
282C4⋊C4C22⋊Dic1714yes—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C42—172no—
286C42—341no—
287C42—341no—
288C42—341no—
289C42—1362no—
290C42#6325411yescharacteristic
291C42#6325411yescharacteristic
292C42C22×Dic1711yescharacteristic
293C42C2×C22⋊D1711yescharacteristic
294C23—41no—
295C23#1058762311yescharacteristic
296Q8—41no—
297Q8—41no—
298Q8—82no—
299Q8—82no—
300D4—42no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—81no—
306C2×C4—24no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—342no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—682no—
318C2×C4—681no—
319C2×C4—682no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—682no—
323C2×C4SD16×D1711yescharacteristic
324C2×C4#1058760011yescharacteristic
325C2×C4#1058760111yescharacteristic
326C2×C4#1058760211yescharacteristic
327C2×C4D4×Dic1711yescharacteristic
328C2×C4D4×Dic1711yescharacteristic
329C2×C4#1058762111yescharacteristic
330C2×C4#1058762511yescharacteristic
331C2×C4#1058762511yescharacteristic
332C2×C4C2×C22⋊Dic1711yescharacteristic
333C8—42no—
334C8—1361no—
335C8—1361no—
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1176268911yescharacteristic
342C22#1176294611yescharacteristic
343C22#1176295311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—44no—
349C4—81no—
350C4—682no—
351C4—1362no—
352C4—1361no—
353C4SD16×Dic1711yescharacteristic
354C4#1176291011yescharacteristic
355C4#1176300711yescharacteristic
356C4#1176302911yescharacteristic
357C2—21no—
358C2—21no—
359C2—161no—
360C2#1452914111yescharacteristic
361C2#1452927311yescharacteristic
362C2#1452932211yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#3227C2
socleC22×C17#386
prime radicalC17×#532C22
cosocleC2×C42×C17C23
hypercenter#3227D17
hypocenterC17#9302
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
1. #14531533C2semidirect
3. #14527469C2semidirect
5. #14526643C2semidirect
6. #14526612C2semidirect
22. C17×#498C4semidirect
23. C17×#486C4semidirect
185. #498Dic17semidirect
186. #486Dic17semidirect
248. C17#9302semidirect

Maximal subgroups

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

subgroupindex
#145315332
#145296172
#145274692
#145273442
#145266432
#145266122
C17×#32272
#930217