Finite Group Databasebeta

← #16937865

Subgroups — #16937865

Order 4352.

530 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533702C211yesmaximal, characteristic
2#14528912C211yesmaximal, characteristic, complemented
3#14528789C211yesmaximal, characteristic, complemented
4#14528785C211yesmaximal, characteristic
5C2×#11762538C211yesmaximal, characteristic
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#10587632—21no—
9C2×C4×D68C2211yescharacteristic
10C2×C4⋊#3622C2211yescharacteristic
11C2×#10587506—21no—
12C2×#10587492—21no—
13#11762731—22no—
14#11762711—21no—
15#11762707—21no—
16#11762679C2211yescharacteristic
17#11762679C2211yescharacteristic
18#11762666—21no—
19#11762572—21no—
20#11762572—22no—
21#11762571—21no—
22#11762538—22no—
23C2×C4×Q8×C17C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C4×C8×C17C2211yescharacteristic
26C17×Q8⋊C8—22no—
27C22×Q8×C17D411yescharacteristic
28C22×D68D411yescharacteristic
29C22×C4×D17—21no—
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C4×Q8×C17—21no—
33C4×Q8×C17—21no—
34C4×Q8×C17—21no—
35C4×Q8×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C42×C17—21no—
39C2×C42×C17C2311yescharacteristic
40#10587632—42no—
41#10587632—42no—
42C2×#63227—22no—
43C2×#63227—22no—
44C2×#63227—21no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—21no—
47C2×C4⋊Dic17—21no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17D411yescharacteristic
51C2×C4⋊Dic17D412yes—
52C2×#63225—22no—
53C22×#3622—21no—
54C4×D68—21no—
55C4×D68—21no—
56C4×D68—21no—
57C4×D68—21no—
58C17×C4⋊C8—21no—
59C17×C4⋊C8—21no—
60C4×C8×C17—21no—
61C4×C8×C17D412yes—
62#10587506—42no—
63#10587502—22no—
64#10587492—42no—
65C4⋊#3622—21no—
66C4⋊#3622—21no—
67C23×D17—21no—
68C2×Q8×C17—21no—
69C2×Q8×C17—21no—
70C2×Q8×C17—21no—
71C2×Q8×C17—21no—
72C2×Q8×C17—21no—
73C2×Q8×C17SD1612yes—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—21no—
80C22×Dic17—22no—
81C22×Dic17—41no—
82C2×D68—21no—
83C2×D68—21no—
84C2×D68—21no—
85C2×D68—21no—
86C2×D68—21no—
87C2×D68Q1612yes—
88C2×C4×D17—41no—
89C2×C4×D17—41no—
90C2×C4×D17—41no—
91C2×C4×D17—41no—
92C2×C4×D17—41no—
93C2×C4×D17—41no—
94C2×C8×C17—22no—
95C2×C8×C17—41no—
96C2×C8×C17—41no—
97C2×C8×C17—41no—
98C2×C8×C17—41no—
99C2×C8×C17—41no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C42×C17—41no—
105C42×C17—41no—
106C42×C17C2×D411yescharacteristic
107C42×C17#4111yescharacteristic
108C42×C17#4111yescharacteristic
109C42×C17#4111yescharacteristic
110#63227—44no—
111#63227—41no—
112#63227—41no—
113#63227—82no—
114C4⋊Dic17—21no—
115C4⋊Dic17—21no—
116C4⋊Dic17—41no—
117C4⋊Dic17—41no—
118C4⋊Dic17—41no—
119C4⋊Dic17—41no—
120C4⋊Dic17—42no—
121C4⋊Dic17—42no—
122C4⋊Dic17—41no—
123C4⋊Dic17—41no—
124C4⋊Dic17—41no—
125C4⋊Dic17—41no—
126#63225—44no—
127C2×#3622—41no—
128C2×#3622—41no—
129C2×#3622—41no—
130#10436—171nomaximal, Sylow
131C23×C17#12011yescharacteristic
132C22×D17—22no—
133C22×D17—41no—
134C22×D17—41no—
135C22×D17—41no—
136C22×D17—41no—
137C22×D17—41no—
138C22×D17—41no—
139Q8×C17—22no—
140Q8×C17—22no—
141Q8×C17—41no—
142Q8×C17—41no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—22no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17#12111yescharacteristic
152C2×C4×C17#12111yescharacteristic
153C2×C4×C17#12111yescharacteristic
154C2×C4×C17#12311yescharacteristic
155C2×C4×C17#12311yescharacteristic
156C2×C4×C17#12411yescharacteristic
157C2×C4×C17C2×SD1611yescharacteristic
158C2×C4×C17C2×Q1611yescharacteristic
159C2×C4×C17#13511yescharacteristic
160C2×C4×C17#13511yescharacteristic
161C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C17#13711yescharacteristic
165C2×Dic17—41no—
166C2×Dic17—41no—
167C2×Dic17—42no—
168C2×Dic17—41no—
169C2×Dic17—42no—
170C2×Dic17—42no—
171C2×Dic17—81no—
172C2×Dic17—81no—
173C2×Dic17—81no—
174D68—22no—
175D68—22no—
176D68—41no—
177D68—41no—
178C4×D17—81no—
179C4×D17—81no—
180C4×D17—81no—
181C4×D17—81no—
182C8×C17—42no—
183C8×C17—81no—
184#3622—81no—
185#2538—171no—
186#1834—171no—
187#1803—171no—
188#1802—171no—
189C2×Q8⋊C8D1711yescharacteristic, complemented
190C2×D4⋊C8—171no—
191#1296—171no—
192C22×C17#39411yescharacteristic
193C22×C17#44911yescharacteristic
194C22×C17#44911yescharacteristic
195C22×C17#45011yescharacteristic
196C22×C17#45111yescharacteristic
197C22×C17#45411yescharacteristic
198C22×C17#45511yescharacteristic
199C2×D17—42no—
200C2×D17—42no—
201C2×D17—42no—
202C2×D17—81no—
203C2×D17—81no—
204C2×D17—81no—
205C2×D17—81no—
206C4×C17—21no—
207C4×C17—21no—
208C4×C17—42no—
209C4×C17—81no—
210C4×C17#46511yescharacteristic
211C4×C17#46711yescharacteristic
212C4×C17#46811yescharacteristic
213C4×C17#46911yescharacteristic
214C4×C17#48111yescharacteristic
215C4×C17#48211yescharacteristic
216C4×C17#48411yescharacteristic
217C4×C17#48511yescharacteristic
218Dic17—82no—
219Dic17—81no—
220Dic17—161no—
221C2×C22⋊Q8—341no—
222C2×C4×Q8C2×D1711yescharacteristic
223C2×C4×D4—171no—
224C2×C4⋊C8—171no—
225C2×C4⋊C8C2×D1711yescharacteristic
226C2×#103—341no—
227C2×#102—341no—
228C2×C4×C8C2×D1711yescharacteristic
229#396—342no—
230#390—341no—
231#389—341no—
232#384—171no—
233#384—171no—
234#380—341no—
235#340—341no—
236#340—342no—
237#340—341no—
238Q8⋊C8—22no—
239D4⋊C8—342no—
240C2×C17#162811yescharacteristic
241C2×C17#163711yescharacteristic
242C2×C17#164211yescharacteristic
243C2×C17#164411yescharacteristic
244C2×C17#203411yescharacteristic
245C2×C17#203611yescharacteristic
246C2×C17#204011yescharacteristic
247D17—82no—
248C22×Q8D6811yescharacteristic
249C22×D4—171no—
250C23×C4—341no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—341no—
254C22⋊Q8—682no—
255C22⋊Q8—682no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×D4—341no—
261C4×D4—341no—
262C4×D4—341no—
263C4×D4—341no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—171no—
266C2×C4⋊C4—172no—
267C2×C4⋊C4—342no—
268C2×C4⋊C4—341no—
269C2×C4⋊C4—341no—
270C2×C4⋊C4—341no—
271C2×C4⋊C4—341no—
272C2×C4⋊C4—341no—
273C2×C4⋊C4D6811yescharacteristic
274C2×C22⋊C4—342no—
275C2×C22⋊C4—341no—
276C2×C22⋊C4—342no—
277C2×C42—21no—
278C2×C42C22×D1711yescharacteristic
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—341no—
282C4⋊C8—341no—
283#103—682no—
284#102—682no—
285C4×C8—21no—
286C4×C8C22⋊D1712yes—
287#95—342no—
288C17#1043611yescharacteristic, complemented, Sylow
289C24—341no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8#6321912yes—
296C2×D4—172no—
297C2×D4—341no—
298C2×D4—341no—
299C2×D4—341no—
300C2×D4—341no—
301C2×D4—341no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—341no—
305C22×C4—342no—
306C22×C4—681no—
307C22×C4—681no—
308C22×C4—681no—
309C22×C4—681no—
310C22×C4—681no—
311C22×C4—681no—
312C22×C4—681no—
313C22×C4C2×D6811yescharacteristic
314C22×C4D4×D1711yescharacteristic
315C22×C4D4×D1711yescharacteristic
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—681no—
323C2×C8—681no—
324C2×C8—681no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—341no—
330C4⋊C4—341no—
331C4⋊C4—682no—
332C4⋊C4—684no—
333C4⋊C4—682no—
334C4⋊C4—681no—
335C4⋊C4—681no—
336C4⋊C4—681no—
337C4⋊C4—681no—
338C4⋊C4—681no—
339C4⋊C4—681no—
340C4⋊C4—681no—
341C4⋊C4—681no—
342C22⋊C4—684no—
343C22⋊C4—681no—
344C22⋊C4—681no—
345C22⋊C4—1362no—
346C42—41no—
347C42—41no—
348C42#6325211yescharacteristic
349C42#6325411yescharacteristic
350C42#6325411yescharacteristic
351C42C2×C22⋊D1711yescharacteristic
352C23—342no—
353C23—681no—
354C23—681no—
355C23—681no—
356C23—681no—
357C23—681no—
358C23—681no—
359C23#1058756811yescharacteristic
360Q8—22no—
361Q8—22no—
362Q8—41no—
363Q8—41no—
364D4—342no—
365D4—342no—
366D4—681no—
367D4—681no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—681no—
377C2×C4—682no—
378C2×C4—682no—
379C2×C4—681no—
380C2×C4—681no—
381C2×C4—682no—
382C2×C4—1361no—
383C2×C4—1361no—
384C2×C4—1361no—
385C2×C4—1361no—
386C2×C4—1361no—
387C2×C4—1361no—
388C2×C4—1361no—
389C2×C4#1058756911yescharacteristic
390C2×C4#1058757111yescharacteristic
391C2×C4#1058757211yescharacteristic
392C2×C4C2×#6321911yescharacteristic
393C2×C4#1058759011yescharacteristic
394C2×C4#1058759411yescharacteristic
395C2×C4#1058759411yescharacteristic
396C2×C4#1058760011yescharacteristic
397C2×C4#1058760111yescharacteristic
398C2×C4Q16×D1711yescharacteristic
399C2×C4#1058760511yescharacteristic
400C2×C4#1058761611yescharacteristic
401C2×C4#1058762411yescharacteristic
402C2×C4#1058762511yescharacteristic
403C8—42no—
404C8—81no—
405C8—1361no—
406C22—682no—
407C22—682no—
408C22—682no—
409C22—1361no—
410C22—1361no—
411C22—1361no—
412C22—1361no—
413C22#1176272511yescharacteristic
414C22#1176295611yescharacteristic
415C22#1176295711yescharacteristic
416C22#1176296211yescharacteristic
417C22#1176296611yescharacteristic
418C22#1176298311yescharacteristic
419C22#1176298811yescharacteristic
420C4—21no—
421C4—21no—
422C4—42no—
423C4—81no—
424C4—2721no—
425C4—1361no—
426C4—1362no—
427C4#1176303811yescharacteristic
428C4#1176304711yescharacteristic
429C4#1176305211yescharacteristic
430C4#1176306211yescharacteristic
431C4#1176312811yescharacteristic
432C4#1176313411yescharacteristic
433C4#1176314311yescharacteristic
434C4#1176314911yescharacteristic
435C2—1362no—
436C2#1452789711yescharacteristic
437C2#1452795111yescharacteristic
438C2#1452798711yescharacteristic
439C2#1452800011yescharacteristic
440C2#1452997411yescharacteristic
441C2#1452998611yescharacteristic
442C2#1453001211yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×Q8⋊C8C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenterC2×Q8⋊C8D17
hypocenterC17#10436
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #14528912C2semidirect
3. #14528789C2semidirect
6. #14526668C2semidirect
7. C2×C17×Q8⋊C8C2semidirect
189. C2×Q8⋊C8D17semidirect
288. C17#10436semidirect

Maximal subgroups

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

subgroupindex
#145337022
#145289122
#145287892
#145287852
C2×#117625382
#145266682
C2×C17×Q8⋊C82
#1043617