Finite Group Databasebeta

← #16929165

Subgroups — #16929165

Order 4352.

608 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763011C211yesmaximal, characteristic
2#14531731C211yesmaximal, characteristic
3#14529746C211yesmaximal, characteristic, complemented
4C2×#11762536C211yesmaximal, characteristic
5#14526668C211yesmaximal, characteristic, complemented
6#14526625C211yesmaximal, characteristic, complemented
7C17×#1988C211yesmaximal, characteristic, complemented
8C22×#63230—21no—
9C2×C4×D68C2211yescharacteristic
10C2×#10587578—21no—
11#11763011—22no—
12#11763011C414yescomplemented
13C2×C4⋊#3622C2211yescharacteristic
14C2×#10587506—21no—
15C2×C4×#3622C2211yescharacteristic
16#11762711—21no—
17#11762701—22no—
18#11762679C2211yescharacteristic
19#11762666—21no—
20C4×C4⋊Dic17—21no—
21#11762571—22no—
22#11762536—22no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C4×C4⋊C4×C17C2211yescharacteristic
26C17×#340—21no—
27C17×#340—21no—
28C22×Q8×C17—21no—
29C22×D68D411yescharacteristic
30C22×C4×D17—21no—
31C22×C8×C17—21no—
32C17×C4⋊Q8—21no—
33C17×C4⋊Q8—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8C2×C412yes—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17D411yescharacteristic
39C2×C4⋊C4×C17D411yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17C2311yescharacteristic
42C2×#63230—24no—
43C2×#63230—42no—
44C2×#63230—42no—
45C2×#63227—22no—
46C2×#63227—22no—
47C2×#63227—21no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17D411yescharacteristic
51C2×C4⋊Dic17D412yes—
52C2×C4×Dic17—22no—
53C22×#3622—21no—
54C22×#3622—21no—
55#10587578—42no—
56#10587578—42no—
57C4×D68—21no—
58C4×D68—21no—
59C4×D68C2×C412yes—
60C4×D68D412yes—
61C17×C4⋊C8—21no—
62C17×C4⋊C8—21no—
63C17×#95—21no—
64#10587506—24no—
65#10587502—22no—
66C4⋊#3622—21no—
67C4⋊#3622C2×C412yes—
68C4×#3622—21no—
69C4×#3622—21no—
70C23×D17—21no—
71C2×Q8×C17—22no—
72C2×Q8×C17—41no—
73C2×Q8×C17—41no—
74C2×Q8×C17—41no—
75C2×Q8×C17—41no—
76C2×Q8×C17—41no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—41no—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×D411yescharacteristic
83C22×Dic17—21no—
84C22×Dic17—22no—
85C2×D68—21no—
86C2×D68—21no—
87C2×D68—21no—
88C2×D68—21no—
89C2×D68—21no—
90C2×D68C22⋊C412yes—
91C2×C4×D17—22no—
92C2×C4×D17—22no—
93C2×C4×D17—41no—
94C2×C4×D17—41no—
95C2×C4×D17—41no—
96C2×C4×D17—41no—
97C2×C8×C17—41no—
98C2×C8×C17—41no—
99C2×C8×C17—41no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—22no—
102C4⋊C4×C17—22no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—41no—
106C4⋊C4×C17—41no—
107C4⋊C4×C17SD1612yescomplemented
108C42×C17—41no—
109C42×C17—41no—
110C42×C17C22×C411yescharacteristic
111C42×C17C2×D411yescharacteristic
112C42×C17#4111yescharacteristic
113C42×C17#4111yescharacteristic
114#63230—48no—
115#63227—22no—
116#63227—22no—
117#63227—44no—
118#63227—82no—
119C4⋊Dic17—21no—
120C4⋊Dic17—41no—
121C4⋊Dic17—41no—
122C4⋊Dic17—41no—
123C4⋊Dic17—41no—
124C4⋊Dic17—42no—
125C4⋊Dic17C22⋊C412yes—
126C4×Dic17—44no—
127C2×#3622—22no—
128C2×#3622—22no—
129C2×#3622—41no—
130C2×#3622—41no—
131C2×#3622—41no—
132C2×#3622—41no—
133#8988—171nomaximal, Sylow
134C23×C17#12011yescharacteristic
135C22×D17—22no—
136C22×D17—41no—
137C22×D17—41no—
138C22×D17—41no—
139C22×D17—41no—
140C22×D17—41no—
141C22×D17—41no—
142Q8×C17—42no—
143Q8×C17—42no—
144Q8×C17—81no—
145Q8×C17—81no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—22no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—42no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—81no—
156C2×C4×C17—81no—
157C2×C4×C17C2×C22⋊C411yescharacteristic
158C2×C4×C17C4×D411yescharacteristic
159C2×C4×C17C4×D411yescharacteristic
160C2×C4×C17#12111yescharacteristic
161C2×C4×C17#12111yescharacteristic
162C2×C4×C17#12311yescharacteristic
163C2×C4×C17C2×SD1611yescharacteristic
164C2×C4×C17#13511yescharacteristic
165C2×C4×C17#13511yescharacteristic
166C2×C4×C17#13511yescharacteristic
167C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C17#13711yescharacteristic
171C2×Dic17—22no—
172C2×Dic17—41no—
173C2×Dic17—41no—
174C2×Dic17—42no—
175C2×Dic17—42no—
176C2×Dic17—42no—
177D68—22no—
178D68—22no—
179D68—41no—
180D68—41no—
181C4×D17—42no—
182C4×D17—42no—
183C4×D17—42no—
184C4×D17—42no—
185C8×C17—81no—
186#3622—42no—
187#3622—42no—
188C2×#461—171no—
189#2354—171no—
190#1990—171no—
191#1988D1711yescharacteristic, complemented
192C2×D4⋊C8—171no—
193#1296—171no—
194#1279—171no—
195C22×C17#38611yescharacteristic
196C22×C17#44911yescharacteristic
197C22×C17#44911yescharacteristic
198C22×C17#45011yescharacteristic
199C22×C17#45211yescharacteristic
200C22×C17#45511yescharacteristic
201C22×C17#45611yescharacteristic
202C2×D17—42no—
203C2×D17—42no—
204C2×D17—42no—
205C2×D17—81no—
206C2×D17—81no—
207C2×D17—81no—
208C2×D17—81no—
209C4×C17—21no—
210C4×C17—21no—
211C4×C17—42no—
212C4×C17—82no—
213C4×C17—81no—
214C4×C17#41611yescharacteristic
215C4×C17#41811yescharacteristic
216C4×C17C4×SD1611yescharacteristic
217C4×C17#44011yescharacteristic
218C4×C17#46111yescharacteristic
219C4×C17#46311yescharacteristic
220C4×C17#48211yescharacteristic
221C4×C17#48211yescharacteristic
222Dic17—42no—
223Dic17—82no—
224C22×SD16—341no—
225C2×C4⋊Q8C2×D1711yescharacteristic
226C2×C4×D4—171no—
227C2×#117—341no—
228#461—174no—
229#461—342no—
230C2×C4⋊C8—171no—
231C2×C4⋊C8C2×D1711yescharacteristic
232C2×#103—341no—
233C2×C4×C8—171no—
234#390—341no—
235#388—342no—
236#384—171no—
237#380—341no—
238C4×C4⋊C4—341no—
239C4×C4⋊C4C2×D1711yescharacteristic
240#340—21no—
241#340—21no—
242#340—342no—
243D4⋊C8—342no—
244C2×C17#164211yescharacteristic
245C2×C17#164611yescharacteristic
246C2×C17#165011yescharacteristic
247C2×C17#165411yescharacteristic
248C2×C17#187511yescharacteristic
249C2×C17#187711yescharacteristic
250C2×C17#188211yescharacteristic
251D17—82no—
252C22×Q8—21no—
253C22×D4—171no—
254C23×C4—341no—
255C2×SD16—344no—
256C2×SD16—682no—
257C2×SD16—682no—
258C22×C8—21no—
259C22×C8—341no—
260C22×C8—341no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8C4×D1712yes—
265C4×D4—172no—
266C4×D4—172no—
267C4×D4—341no—
268C4×D4—341no—
269#117—682no—
270#117—682no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—171no—
274C2×C4⋊C4—172no—
275C2×C4⋊C4—341no—
276C2×C4⋊C4—341no—
277C2×C4⋊C4C22⋊D1711yescharacteristic
278C2×C4⋊C4C22⋊D1711yescharacteristic
279C2×C22⋊C4—341no—
280C2×C22⋊C4—342no—
281C2×C22⋊C4—342no—
282C2×C42—21no—
283C2×C42—342no—
284C2×C42C22×D1711yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—172no—
288C4⋊C8—341no—
289#103—344no—
290C4×C8—341no—
291C4×C8—341no—
292#95—21no—
293#95—342no—
294C17#898811yescharacteristic, complemented, Sylow
295C24—341no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×D4—172no—
303C2×D4—341no—
304C2×D4—341no—
305C2×D4—341no—
306C2×D4—341no—
307C2×D4—341no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—341no—
312C22×C4—342no—
313C22×C4—342no—
314C22×C4—342no—
315C22×C4—681no—
316C22×C4—681no—
317C22×C4—681no—
318C22×C4—681no—
319C22×C4D4×D1711yescharacteristic
320C22×C4D4×D1711yescharacteristic
321C22×C4C2×C22⋊D1711yescharacteristic
322SD16—688no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—342no—
327C2×C8—342no—
328C2×C8—681no—
329C2×C8—681no—
330C2×C8—681no—
331C2×C8—681no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—172no—
340C4⋊C4—341no—
341C4⋊C4—682no—
342C4⋊C4—681no—
343C4⋊C4—681no—
344C4⋊C4—681no—
345C4⋊C4—681no—
346C4⋊C4#6323012yescomplemented
347C22⋊C4—342no—
348C22⋊C4—342no—
349C22⋊C4—684no—
350C22⋊C4—1362no—
351C42—41no—
352C42—41no—
353C42—684no—
354C42C2×C4×D1711yescharacteristic
355C42#6325211yescharacteristic
356C42D4×D1711yescharacteristic
357C42#6325411yescharacteristic
358C23—342no—
359C23—681no—
360C23—681no—
361C23—681no—
362C23—681no—
363C23—681no—
364C23—681no—
365C23#1058762311yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—81no—
369Q8—81no—
370D4—342no—
371D4—342no—
372D4—681no—
373D4—681no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—342no—
386C2×C4—681no—
387C2×C4—681no—
388C2×C4—682no—
389C2×C4—682no—
390C2×C4—682no—
391C2×C4—682no—
392C2×C4—682no—
393C2×C4—682no—
394C2×C4—682no—
395C2×C4C22⋊C4×D1711yescharacteristic
396C2×C4#1058756711yescharacteristic
397C2×C4#1058756911yescharacteristic
398C2×C4#1058757011yescharacteristic
399C2×C4#1058759711yescharacteristic
400C2×C4#1058759811yescharacteristic
401C2×C4#1058760011yescharacteristic
402C2×C4#1058760211yescharacteristic
403C2×C4C4×C22⋊D1711yescharacteristic
404C2×C4#1058762411yescharacteristic
405C2×C4C2×#6323011yescharacteristic
406C2×C4#1058762811yescharacteristic
407C2×C4#1058763511yescharacteristic
408C2×C4#1058763611yescharacteristic
409C8—81no—
410C8—682no—
411C8—682no—
412C22—682no—
413C22—682no—
414C22—682no—
415C22—1361no—
416C22—1361no—
417C22—1361no—
418C22—1361no—
419C22#1176268511yescharacteristic
420C22#1176295111yescharacteristic
421C22#1176295311yescharacteristic
422C22#1176295911yescharacteristic
423C22#1176296911yescharacteristic
424C22#1176298511yescharacteristic
425C22#1176299011yescharacteristic
426C4—21no—
427C4—21no—
428C4—42no—
429C4—81no—
430C4—82no—
431C4—682no—
432C4—1362no—
433C4#1176280211yescharacteristic
434C4#1176281211yescharacteristic
435C4C4×#6323011yescharacteristic
436C4#1176290711yescharacteristic
437C4#1176301111yescharacteristic
438C4#1176302111yescharacteristic
439C4#1176313111yescharacteristic
440C4#1176313211yescharacteristic
441C2—1362no—
442C2#1452798311yescharacteristic
443C2#1452801111yescharacteristic
444C2#1452803911yescharacteristic
445C2#1452806711yescharacteristic
446C2#1452910311yescharacteristic
447C2#1452912111yescharacteristic
448C2#1452915011yescharacteristic
subgroupG/H
centerC23#10587623
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1988C2
socleC23×C17#120
prime radicalC22×D68D4
cosocleC2×C42×C17C23
hypercenter#1988D17
hypocenterC17#8988
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
3. #14529746C2semidirect
5. #14526668C2semidirect
6. #14526625C2semidirect
7. C17×#1988C2semidirect
12. #11763011C4semidirect
107. C4⋊C4×C17SD16semidirect
191. #1988D17semidirect
294. C17#8988semidirect
346. C4⋊C4#63230semidirect

Maximal subgroups

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

subgroupindex
C2×#117630112
#145317312
#145297462
C2×#117625362
#145266682
#145266252
C17×#19882
#898817