Finite Group Databasebeta

← #16931072

Subgroups — #16931072

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#14539912C211yesmaximal, characteristic
2#14529616C211yesmaximal, characteristic
3#14527469C211yesmaximal, characteristic, complemented
4#14527343C211yesmaximal, characteristic
5#14526642C211yesmaximal, characteristic, complemented
6#14526613C211yesmaximal, characteristic, complemented
7C17×#2331C211yesmaximal, characteristic, complemented
8#11763359C2211yescharacteristic
9#11763205C412yescomplemented
10#11763157C412yescomplemented
11#11763127—22no—
12#11763096—22no—
13#11762862C2211yescharacteristic
14#11762862C2211yescharacteristic
15#11762833C2211yescharacteristic
16C2×#10587476—21no—
17C2×#10587476—21no—
18#11762555—22no—
19#11762542—22no—
20C2×C17×C4⋊Q8C2211yescharacteristic
21C17×#391—22no—
22C17×#389—21no—
23C17×#387—22no—
24C17×#382C2211yescharacteristic
25C4×C4⋊C4×C17—21no—
26C17×#376C2211yescharacteristic
27C22×Q8×C17—21no—
28C2×#63252—21no—
29C17×C4⋊Q8—21no—
30C17×C4⋊Q8—21no—
31C17×C4⋊Q8C2×C412yes—
32C17×C4⋊Q8D412yes—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—22no—
36C2×C4⋊C4×C17D412yes—
37C2×C42×C17—22no—
38C2×C42×C17D411yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×C42×C17C2311yescharacteristic
41#10587616—21no—
42#10587616—21no—
43C2×#63223—21no—
44#10587561—21no—
45#10587560C2×C411yescharacteristic
46C4×D68—21no—
47C4×D68—21no—
48#10587555C2×C411yescharacteristic
49C17×#95—22no—
50C17×#95—22no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54C17×#95D412yes—
55C22⋊#3622—21no—
56C22⋊#3622—21no—
57#10587480—24no—
58#10587479—24no—
59#10587476—42no—
60#10587476—42no—
61C4⋊#3622—21no—
62C4⋊#3622—21no—
63C4⋊#3622—21no—
64C4⋊#3622—21no—
65C4×#3622—21no—
66C4×#3622C2×C412yes—
67C2×Q8×C17—22no—
68C2×Q8×C17—22no—
69C2×Q8×C17—82no—
70C22×C4×C17—22no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—22no—
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C2×C22⋊D17—41no—
78#63252—41no—
79#63252—41no—
80#63252—41no—
81#63252—41no—
82C2×D68—21no—
83C2×D68—21no—
84C2×D68—21no—
85C2×C4×D17—41no—
86C2×Dic34—21no—
87C4⋊C4×C17—22no—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—22no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—82no—
93C4⋊C4×C17C22⋊C414yes—
94C42×C17—21no—
95C42×C17—21no—
96C42×C17—21no—
97C42×C17—82no—
98C42×C17SD1612yes—
99C42×C17C22×C411yescharacteristic
100C42×C17C2×D411yescharacteristic
101C42×C17#4111yescharacteristic
102C42×C17#4111yescharacteristic
103#63227—41no—
104#63227—41no—
105C4⋊Dic17—21no—
106C4⋊Dic17—21no—
107#63223—41no—
108#63223—41no—
109C2×#3622—22no—
110C2×#3622—41no—
111C2×#3622—41no—
112#9302—171nomaximal, Sylow
113C23×C17#12011yescharacteristic
114C22×D17—41no—
115Q8×C17—82no—
116Q8×C17—82no—
117C2×C4×C17—24no—
118C2×C4×C17—22no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—41no—
130C2×C4×C17—42no—
131C2×C4×C17C2×C22⋊C411yescharacteristic
132C2×C4×C17C4×D411yescharacteristic
133C2×C4×C17C4×D411yescharacteristic
134C2×C4×C17#12111yescharacteristic
135C2×C4×C17#12111yescharacteristic
136C2×C4×C17#12311yescharacteristic
137C2×C4×C17C2×SD1611yescharacteristic
138C2×C4×C17#13511yescharacteristic
139C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C17#13711yescharacteristic
141C22⋊D17—81no—
142C22⋊D17—81no—
143C2×Dic17—41no—
144D68—42no—
145D68—41no—
146D68—41no—
147D68—81no—
148C4×D17—81no—
149C4×D17—81no—
150Dic34—41no—
151Dic34—41no—
152#3622—42no—
153#3622—81no—
154#3622—81no—
155#3227—171no—
156#2331D1711yescharacteristic, complemented
157#1963—171no—
158#1549—171no—
159#1517—171no—
160#1286—171no—
161#1274—171no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17#38611yescharacteristic
165C22×C17#44811yescharacteristic
166C22×C17#44911yescharacteristic
167C2×D17—81no—
168C2×D17—81no—
169C2×D17—81no—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—42no—
175C4×C17—44no—
176C4×C17—82no—
177C4×C17—81no—
178C4×C17C4×SD1611yescharacteristic
179C4×C17#44011yescharacteristic
180C4×C17#46011yescharacteristic
181C4×C17#46411yescharacteristic
182Dic17—81no—
183#532—171no—
184C2×C4⋊Q8C2×D1711yescharacteristic
185#498—172no—
186#486—172no—
187#481—342no—
188#476—342no—
189#431—171no—
190#431—171no—
191#423—171no—
192C2×#104—341no—
193C2×#104—341no—
194#391—22no—
195#389—21no—
196#387—22no—
197#382C2×D1711yescharacteristic
198C4×C4⋊C4—21no—
199#376C2×D1711yescharacteristic
200#332—342no—
201#327—342no—
202C2×C17—21no—
203C2×C17—21no—
204C2×C17#188011yescharacteristic
205C2×C17#190411yescharacteristic
206C2×C17#191211yescharacteristic
207D17—161no—
208C2×#41—341no—
209C22×Q8—21no—
210C2×M16—341no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—171no—
214C4⋊Q8C4×D1712yes—
215C4⋊Q8D6812yes—
216#127—171no—
217#124—341no—
218#121—341no—
219#121—341no—
220C4×D4—341no—
221C4×D4—341no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4C22⋊D1712yes—
226C2×C42—22no—
227C2×C42C22⋊D1711yescharacteristic
228C2×C42C22⋊D1711yescharacteristic
229C2×C42C22×D1711yescharacteristic
230#106—344no—
231C4⋊C8—341no—
232C4⋊C8—341no—
233C4⋊C8—341no—
234C4⋊C8—341no—
235#104—682no—
236#104—682no—
237#102—344no—
238C22⋊C8—341no—
239C22⋊C8—341no—
240C4×C8—172no—
241C4×C8—341no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95C22⋊D1712yes—
248C17#930211yescharacteristic, complemented, Sylow
249#41—681no—
250#41—681no—
251#41—681no—
252#41—681no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—82no—
256C2×Q8—341no—
257C2×D4—341no—
258C2×D4—341no—
259C2×D4—341no—
260C2×D4—681no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—681no—
266C22×C4C2×C22⋊D1711yescharacteristic
267C22×C4C2×C22⋊D1711yescharacteristic
268C22×C4C2×C22⋊D1711yescharacteristic
269M16—681no—
270M16—681no—
271C2×C8—342no—
272C2×C8—681no—
273C2×C8—681no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4#6322714yes—
283C22⋊C4—681no—
284C22⋊C4—681no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—82no—
289C42#6323012yes—
290C42C2×C4×D1711yescharacteristic
291C42C2×D6811yescharacteristic
292C42#6325211yescharacteristic
293C42#6325211yescharacteristic
294C23—681no—
295C23#1058763911yescharacteristic
296Q8—82no—
297Q8—82no—
298Q8—681no—
299Q8—681no—
300D4—682no—
301D4—681no—
302D4—681no—
303D4—1361no—
304D4—1361no—
305D4—1361no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—681no—
321C2×C4—1361no—
322C2×C4—1361no—
323C2×C4C2×#6322711yescharacteristic
324C2×C4C4×C22⋊D1711yescharacteristic
325C2×C4C4×C22⋊D1711yescharacteristic
326C2×C4#1058761511yescharacteristic
327C2×C4#1058761611yescharacteristic
328C2×C4#1058761611yescharacteristic
329C2×C4C2×#6323011yescharacteristic
330C2×C4#1058762811yescharacteristic
331C2×C4#1058763511yescharacteristic
332C2×C4#1058763611yescharacteristic
333C8—682no—
334C8—1361no—
335C8—1361no—
336C22—21no—
337C22—21no—
338C22—1361no—
339C22—1361no—
340C22—1361no—
341C22#1176268811yescharacteristic
342C22#1176294811yescharacteristic
343C22#1176295411yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—44no—
350C4—82no—
351C4—81no—
352C4—1361no—
353C4C4×#6323011yescharacteristic
354C4#1176290711yescharacteristic
355C4#1176300911yescharacteristic
356C4#1176303111yescharacteristic
357C2—21no—
358C2—21no—
359C2—2721no—
360C2#1452913911yescharacteristic
361C2#1452927111yescharacteristic
362C2#1452932111yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2331C2
socleC22×C17#386
prime radical#11763359C22
cosocleC2×C42×C17C23
hypercenter#2331D17
hypocenterC17#9302
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #14527469C2semidirect
5. #14526642C2semidirect
6. #14526613C2semidirect
7. C17×#2331C2semidirect
9. #11763205C4semidirect
10. #11763157C4semidirect
156. #2331D17semidirect
248. C17#9302semidirect

Maximal subgroups

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

subgroupindex
#145399122
#145296162
#145274692
#145273432
#145266422
#145266132
C17×#23312
#930217