Finite Group Databasebeta

← #16925317

Subgroups — #16925317

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537478C211yesmaximal, characteristic
2#14528560C211yesmaximal, characteristic, complemented
3#14527276C211yesmaximal, characteristic
4#14527273C211yesmaximal, characteristic
5#14526672C211yesmaximal, characteristic, complemented
6#14526670C211yesmaximal, characteristic, complemented
7C17×#1752C211yesmaximal, characteristic, complemented
8C2×#10587587C411yescharacteristic, complemented
9#11763320C2211yescharacteristic
10#11762880—21no—
11#11762880—21no—
12C8×C22⋊D17—21no—
13C8×C22⋊D17—21no—
14#11762835C411yescharacteristic, complemented
15#11762775C411yescharacteristic, complemented
16#11762773C2211yescharacteristic
17C2×C22⋊#3622C2211yescharacteristic
18#11762764C411yescharacteristic, complemented
19C2×#10587486—21no—
20C4×C22⋊Dic17C2211yescharacteristic
21#11762563—21no—
22#11762547C412yescomplemented
23#11762545C412yescomplemented
24#11762544C412yescomplemented
25#11762542C412yescomplemented
26C23×C8×C17C2211yescharacteristic
27C2×C17×C22⋊C8C2211yescharacteristic
28C2×C17×#97—21no—
29C4×C22⋊C4×C17C2211yescharacteristic
30C17×#336—21no—
31C23×C4×C17C2311yescharacteristic
32C2×#63252C2×C411yescharacteristic
33C22×C8×C17—21no—
34C22×C8×C17—42no—
35C22×C8×C17C2×C411yescharacteristic
36C22×C8×C17C2×C411yescharacteristic
37C22×C8×C17D411yescharacteristic
38C22×C8×C17D411yescharacteristic
39C22×C8×C17D411yescharacteristic
40C22×C8×C17D411yescharacteristic
41C2×C22⋊C4×C17—21no—
42C2×C42×C17—21no—
43#10587639—21no—
44C2×C22⋊Dic17—21no—
45#10587616C2×C411yescharacteristic
46#10587616C2×C411yescharacteristic
47#10587615—21no—
48C4×C22⋊D17—21no—
49C4×C22⋊D17—21no—
50#10587612C2×C411yescharacteristic
51#10587610C2×C411yescharacteristic
52#10587610C2×C411yescharacteristic
53C2×C4×Dic17—21no—
54C2×#63223C2×C411yescharacteristic
55C2×#63223C2×C411yescharacteristic
56C22×#3622—21no—
57#10587587—42no—
58C2×#63218—21no—
59C2×C8×D17—21no—
60C17×C22⋊C8—21no—
61C17×C22⋊C8C2×C412yes—
62C17×#97—42no—
63C17×#95C2×C412yes—
64#10587502C2×C412yes—
65C22⋊#3622—21no—
66C22⋊#3622—21no—
67C22⋊#3622—21no—
68C22⋊#3622C2×C412yes—
69#10587491—21no—
70#10587491—21no—
71#10587486—21no—
72#10587486—42no—
73#10587485—21no—
74#10587485—21no—
75C8×Dic17—21no—
76C24×C17#4111yescharacteristic
77C22×C4×C17—22no—
78C22×C4×C17—41no—
79C22×C4×C17—41no—
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C22×C411yescharacteristic
82C22×C4×C17C22×C411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17#4111yescharacteristic
86C2×C22⋊D17—21no—
87C2×C22⋊D17—21no—
88C2×C22⋊D17—21no—
89C22×Dic17—22no—
90#63252—41no—
91#63252—41no—
92C2×D68C4211yescharacteristic
93C2×C4×D17—21no—
94C2×Dic34C4211yescharacteristic
95C2×C8×C17—22no—
96C2×C8×C17—22no—
97C2×C8×C17—22no—
98C2×C8×C17—22no—
99C2×C8×C17—41no—
100C2×C8×C17—41no—
101C2×C8×C17—41no—
102C2×C8×C17—41no—
103C2×C8×C17—41no—
104C2×C8×C17C22⋊C411yescharacteristic
105C2×C8×C17C22⋊C411yescharacteristic
106C2×C8×C17C22⋊C411yescharacteristic
107C2×C8×C17C22⋊C411yescharacteristic
108C22⋊C4×C17—42no—
109C42×C17—41no—
110C42×C17—41no—
111C22⋊Dic17—21no—
112C22⋊Dic17—21no—
113C22⋊Dic17—21no—
114C22⋊Dic17—42no—
115#63227—21no—
116#63227—21no—
117C4⋊Dic17C4211yescharacteristic
118C4⋊Dic17C4211yescharacteristic
119#63225—21no—
120#63225—21no—
121C4×Dic17—21no—
122C4×Dic17—41no—
123C4×Dic17—41no—
124#63223—41no—
125#63223—41no—
126C2×#3622—21no—
127C2×#3622—21no—
128C2×#3622—22no—
129C2×#3622—41no—
130C2×#3622—41no—
131#63218—42no—
132C8×D17—42no—
133#8239—171nomaximal, Sylow
134C23×C17—22no—
135C23×C17—22no—
136C23×C17—41no—
137C23×C17#11711yescharacteristic
138C23×C17C4×D411yescharacteristic
139C23×C17C4×D411yescharacteristic
140C22×D17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17C2×C4211yescharacteristic
154C2×C4×C17C2×C22⋊C411yescharacteristic
155C2×C4×C17C4×D411yescharacteristic
156C2×C4×C17C4×D411yescharacteristic
157C22⋊D17—42no—
158C22⋊D17—42no—
159C22⋊D17—42no—
160C2×Dic17—21no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166D68—41no—
167C4×D17—41no—
168C4×D17—41no—
169Dic34—41no—
170C8×C17—21no—
171C8×C17—21no—
172C8×C17—21no—
173C8×C17—42no—
174C8×C17#9912yes—
175#3622—42no—
176#3622—41no—
177#3622—41no—
178#2929—171no—
179#1752—171no—
180#1752D1711yescharacteristic, complemented
181#1501—171no—
182#1501—171no—
183#1298—171no—
184#1297—171no—
185C22×C17—21no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—44no—
193C22×C17C4×C22⋊C411yescharacteristic
194C22×C17#44412yes—
195C22×C17#44412yes—
196C2×D17—41no—
197C2×D17—41no—
198C2×D17—41no—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—21no—
202C4×C17—82no—
203C4×C17C2×#9911yescharacteristic
204C4×C17#41011yescharacteristic
205Dic17—41no—
206Dic17—41no—
207Dic17—41no—
208Dic17—82no—
209C2×#131—171no—
210C23×C8C2×D1711yescharacteristic
211#525—171no—
212#435—341no—
213#435—341no—
214C8×D4—341no—
215C8×D4—341no—
216#424—171no—
217#408—171no—
218#407—171no—
219C2×C22⋊C8—171no—
220C2×C22⋊C8C2×D1711yescharacteristic
221#405—171no—
222C2×#97—21no—
223C2×#97—341no—
224C4×C22⋊C4—171no—
225C4×C22⋊C4C2×D1711yescharacteristic
226#336—21no—
227#336—341no—
228#328—172no—
229#328—172no—
230#327—172no—
231#327—172no—
232C2×C17—22no—
233C2×C17—22no—
234C2×C17—41no—
235C2×C17C4×#9911yescharacteristic
236C2×C17#176511yescharacteristic
237C2×C17#176511yescharacteristic
238D17—81no—
239C2×#41—171no—
240C23×C4C22×D1711yescharacteristic
241#131—682no—
242C2×M16—171no—
243C2×M16—171no—
244C2×M16—341no—
245C22×C8—21no—
246C22×C8—42no—
247C22×C8—341no—
248C22×C8—341no—
249C22×C8C4×D1711yescharacteristic
250C22×C8C4×D1711yescharacteristic
251C22×C8D6811yescharacteristic
252C22×C8D6811yescharacteristic
253C22×C8C22⋊D1711yescharacteristic
254C22×C8C22⋊D1711yescharacteristic
255#123—341no—
256C22⋊Q8—171no—
257C22⋊Q8—171no—
258#121—171no—
259#121—171no—
260#120—341no—
261C4×D4—341no—
262C4×D4—341no—
263#117—171no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—341no—
266C2×C42—21no—
267C2×C42—341no—
268C4⋊C8—341no—
269C4⋊C8—341no—
270C22⋊C8—21no—
271C22⋊C8—172no—
272C22⋊C8—341no—
273C22⋊C8—341no—
274C22⋊C8—341no—
275C22⋊C8—341no—
276C22⋊C8—341no—
277C22⋊C8C4×D1712yes—
278#97—42no—
279#97—341no—
280#97—682no—
281C4×C8—341no—
282#95—172no—
283#95C4×D1712yes—
284C17#823911yescharacteristic, complemented, Sylow
285C24#6325211yescharacteristic
286#41—681no—
287#41—681no—
288C2×Q8—171no—
289C2×D4—171no—
290C2×D4—341no—
291C2×D4—341no—
292C2×D4—341no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—341no—
297C22×C4—342no—
298C22×C4C2×C4×D1711yescharacteristic
299C22×C4C2×C4×D1711yescharacteristic
300C22×C4C2×C4×D1711yescharacteristic
301C22×C4C2×D6811yescharacteristic
302C22×C4#6325211yescharacteristic
303C22×C4C2×C22⋊D1711yescharacteristic
304M16—682no—
305M16—681no—
306M16—681no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—342no—
317C2×C8—341no—
318C2×C8—341no—
319C2×C8—681no—
320C2×C8—681no—
321C2×C8—682no—
322C2×C8#6322711yescharacteristic
323C2×C8#6322711yescharacteristic
324C2×C8#6322711yescharacteristic
325C2×C8#6322711yescharacteristic
326C4⋊C4—171no—
327C4⋊C4—171no—
328C4⋊C4—341no—
329C4⋊C4—341no—
330C22⋊C4—42no—
331C22⋊C4—341no—
332C22⋊C4—341no—
333C22⋊C4—341no—
334C22⋊C4—341no—
335C22⋊C4—341no—
336C22⋊C4—682no—
337C42—41no—
338C42—41no—
339C42—341no—
340C42—681no—
341C42—681no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—341no—
346C23#1058755811yescharacteristic
347C23C4×D6811yescharacteristic
348C23C4×C22⋊D1711yescharacteristic
349Q8—681no—
350D4—681no—
351D4—682no—
352D4—682no—
353D4—682no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—341no—
367C2×C4—341no—
368C2×C4—341no—
369C2×C4—681no—
370C2×C4—681no—
371C2×C4—682no—
372C2×C4—682no—
373C2×C4—682no—
374C2×C4C42×D1711yescharacteristic
375C2×C4C4×D6811yescharacteristic
376C2×C4C2×#6322711yescharacteristic
377C2×C4C4×C22⋊D1711yescharacteristic
378C8—21no—
379C8—21no—
380C8—21no—
381C8—42no—
382C8—682no—
383C8—681no—
384C8—681no—
385C8#1058747712yes—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22—681no—
395C22—681no—
396C22—681no—
397C22C4×#6322711yescharacteristic
398C22#1176292912yes—
399C22#1176293212yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—82no—
404C4—681no—
405C4—681no—
406C4—681no—
407C4—1362no—
408C4C2×#1058747711yescharacteristic
409C4#1176278211yescharacteristic
410C2—22no—
411C2—22no—
412C2—41no—
413C2—1361no—
414C2C4×#1058747711yescharacteristic
415C2#1452862811yescharacteristic
416C2#1452863011yescharacteristic
subgroupG/H
centerC2×C4C2×#63227
commutatorC2×C4×C17C2×C42
FrattiniC23×C4C22×D17
FittingC17×#1752C2
socleC22×C17C4×C22⋊C4
prime radical#11763320C22
cosocleC23×C4×C17C23
hypercenter#1752D17
hypocenterC17#8239
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #1752

Complements

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

NH ≅ G/Ntype
2. #14528560C2semidirect
5. #14526672C2semidirect
6. #14526670C2semidirect
7. C17×#1752C2semidirect
8. C2×#10587587C4semidirect
14. #11762835C4semidirect
15. #11762775C4semidirect
18. #11762764C4semidirect
22. #11762547C4semidirect
23. #11762545C4semidirect
24. #11762544C4semidirect
25. #11762542C4semidirect
180. #1752D17semidirect
284. C17#8239semidirect

Maximal subgroups

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

subgroupindex
#145374782
#145285602
#145272762
#145272732
#145266722
#145266702
C17×#17522
#823917