Finite Group Databasebeta

← #16937713

Subgroups — #16937713

Order 4352.

516 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533953C211yesmaximal, characteristic
2#14529586C211yesmaximal, characteristic, complemented
3#14529547C211yesmaximal, characteristic, complemented
4#14528785C211yesmaximal, characteristic
5C2×#11762557C211yesmaximal, characteristic, complemented
6C2×#11762538C211yesmaximal, characteristic
7C17×#1279C211yesmaximal, characteristic, complemented
8C2×#10587555C2211yescharacteristic
9C2×C4×D68C2211yescharacteristic
10C2×#10587479—21no—
11C2×C4⋊#3622C2211yescharacteristic
12C2×#10587487—22no—
13C2×#10587492—21no—
14#11762722—22no—
15#11762711—21no—
16#11762679C2211yescharacteristic
17#11762675—21no—
18#11762666—21no—
19#11762666—21no—
20#11762572—21no—
21#11762571—21no—
22#11762557—22no—
23#11762538—22no—
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C4×C8×C17C2211yescharacteristic
26C4×C4⋊C4×C17C2211yescharacteristic
27C22×D68D411yescharacteristic
28C22×C4×D17—21no—
29C22×Dic34D412yes—
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C42×C17—21no—
35C2×C42×C17C2311yescharacteristic
36C2×#63227—22no—
37C2×#63227—22no—
38C2×#63227—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17D411yescharacteristic
45C22×#3622—21no—
46C4×D68—21no—
47C4×D68—21no—
48C4×D68—21no—
49C4×D68—21no—
50#10587555—21no—
51#10587555—21no—
52#10587555—21no—
53#10587555—21no—
54C17×C4⋊C8—21no—
55C17×C4⋊C8—21no—
56C4×C8×C17—21no—
57C4×C8×C17—21no—
58C17×#95—21no—
59#10587502—22no—
60#10587502—22no—
61#10587492—42no—
62#10587487—44no—
63#10587479—42no—
64C4⋊#3622—21no—
65C4⋊#3622D412yes—
66C23×D17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×Dic17—22no—
73C22×Dic17—21no—
74C22×Dic17—41no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68—21no—
78C2×D68—21no—
79C2×D68—21no—
80C2×D68SD1612yes—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17—41no—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×Dic34—22no—
88C2×Dic34—22no—
89C2×Dic34—42no—
90C2×Dic34—42no—
91C2×C8×C17—22no—
92C2×C8×C17—41no—
93C2×C8×C17—41no—
94C2×C8×C17—41no—
95C2×C8×C17—41no—
96C2×C8×C17—41no—
97C4⋊C4×C17—21no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C42×C17—41no—
102C42×C17—41no—
103C42×C17C2×D411yescharacteristic
104C42×C17#4111yescharacteristic
105C42×C17#4111yescharacteristic
106C42×C17#4111yescharacteristic
107#63227—41no—
108#63227—41no—
109#63227—82no—
110#63227—82no—
111C4⋊Dic17—21no—
112C4⋊Dic17—21no—
113C4⋊Dic17—41no—
114C4⋊Dic17—41no—
115C4⋊Dic17—41no—
116C4⋊Dic17—41no—
117C4⋊Dic17—42no—
118C4⋊Dic17—41no—
119C4⋊Dic17—41no—
120C4⋊Dic17—41no—
121C4⋊Dic17—41no—
122C2×#3622—22no—
123C2×#3622—41no—
124C2×#3622—41no—
125#10412—171nomaximal, Sylow
126C23×C17#12011yescharacteristic
127C22×D17—22no—
128C22×D17—41no—
129C22×D17—41no—
130C22×D17—41no—
131C22×D17—41no—
132C22×D17—41no—
133C22×D17—41no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17#12111yescharacteristic
143C2×C4×C17#12111yescharacteristic
144C2×C4×C17#12111yescharacteristic
145C2×C4×C17#12311yescharacteristic
146C2×C4×C17#12311yescharacteristic
147C2×C4×C17#12411yescharacteristic
148C2×C4×C17C2×SD1611yescharacteristic
149C2×C4×C17#13511yescharacteristic
150C2×C4×C17#13511yescharacteristic
151C2×C4×C17#13511yescharacteristic
152C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C17#13711yescharacteristic
154C2×C4×C17#13711yescharacteristic
155C2×C4×C17#13711yescharacteristic
156C2×Dic17—42no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—42no—
161C2×Dic17—42no—
162C2×Dic17—81no—
163C2×Dic17—81no—
164C2×Dic17—81no—
165D68—22no—
166D68—22no—
167D68—41no—
168D68—41no—
169C4×D17—81no—
170C4×D17—81no—
171C4×D17—81no—
172C4×D17—81no—
173Dic34—44no—
174Dic34—82no—
175C8×C17—42no—
176C8×C17—81no—
177#3622—42no—
178#2563—171no—
179#1956—171no—
180#1948—171no—
181#1802—171no—
182C2×#333—171no—
183C2×D4⋊C8—171no—
184#1279D1711yescharacteristic, complemented
185C22×C17#39411yescharacteristic
186C22×C17#44811yescharacteristic
187C22×C17#44911yescharacteristic
188C22×C17#45211yescharacteristic
189C22×C17#45211yescharacteristic
190C22×C17#45311yescharacteristic
191C22×C17#45511yescharacteristic
192C2×D17—42no—
193C2×D17—42no—
194C2×D17—42no—
195C2×D17—81no—
196C2×D17—81no—
197C2×D17—81no—
198C2×D17—81no—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—81no—
202C4×C17—81no—
203C4×C17#46311yescharacteristic
204C4×C17#46411yescharacteristic
205C4×C17#46511yescharacteristic
206C4×C17#47011yescharacteristic
207C4×C17#48311yescharacteristic
208C4×C17#48511yescharacteristic
209C4×C17#48811yescharacteristic
210C4×C17#48911yescharacteristic
211Dic17—82no—
212Dic17—81no—
213Dic17—161no—
214C2×C4⋊Q8—171no—
215C2×C4×D4—171no—
216C2×#106—341no—
217C2×C4⋊C8—171no—
218C2×C4⋊C8C2×D1711yescharacteristic
219C2×#103—342no—
220C2×#102—341no—
221C2×C4×C8C2×D1711yescharacteristic
222#393—342no—
223#390—341no—
224#384—171no—
225#382—341no—
226#380—341no—
227#380—341no—
228C4×C4⋊C4C2×D1711yescharacteristic
229#340—341no—
230#340—341no—
231#333—342no—
232D4⋊C8—342no—
233C2×C17#162211yescharacteristic
234C2×C17#163411yescharacteristic
235C2×C17#164311yescharacteristic
236C2×C17#164611yescharacteristic
237C2×C17#203511yescharacteristic
238C2×C17#203711yescharacteristic
239C2×C17#204211yescharacteristic
240D17—82no—
241C22×Q8—172no—
242C22×D4—171no—
243C23×C4—341no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—341no—
247C4⋊Q8—341no—
248C4⋊Q8—341no—
249C4⋊Q8—341no—
250C4⋊Q8—341no—
251C4×D4—341no—
252C4×D4—341no—
253C4×D4—341no—
254C4×D4—341no—
255C2×C4⋊C4—171no—
256C2×C4⋊C4—341no—
257C2×C4⋊C4—341no—
258C2×C4⋊C4—341no—
259C2×C4⋊C4—341no—
260C2×C4⋊C4—341no—
261C2×C4⋊C4D6811yescharacteristic
262C2×C4⋊C4D6811yescharacteristic
263C2×C22⋊C4—341no—
264C2×C22⋊C4—342no—
265C2×C22⋊C4—342no—
266C2×C42—21no—
267C2×C42C22×D1711yescharacteristic
268#106—682no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—172no—
272C4⋊C8—341no—
273#103—684no—
274#102—682no—
275C4×C8—21no—
276C4×C8—21no—
277#95—21no—
278#95—342no—
279#95—342no—
280C17#1041211yescharacteristic, complemented, Sylow
281C24—341no—
282C2×Q8—342no—
283C2×Q8—342no—
284C2×Q8—682no—
285C2×Q8—682no—
286C2×D4—172no—
287C2×D4—341no—
288C2×D4—341no—
289C2×D4—341no—
290C2×D4—341no—
291C2×D4—341no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—341no—
295C22×C4—342no—
296C22×C4—681no—
297C22×C4—681no—
298C22×C4—681no—
299C22×C4—681no—
300C22×C4—681no—
301C22×C4—681no—
302C22×C4—681no—
303C22×C4C2×D6811yescharacteristic
304C22×C4D4×D1711yescharacteristic
305C22×C4D4×D1711yescharacteristic
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—342no—
313C2×C8—681no—
314C2×C8—681no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—341no—
320C4⋊C4—341no—
321C4⋊C4—682no—
322C4⋊C4—681no—
323C4⋊C4—681no—
324C4⋊C4—681no—
325C4⋊C4—681no—
326C4⋊C4—681no—
327C4⋊C4—681no—
328C4⋊C4—681no—
329C4⋊C4—681no—
330C22⋊C4—681no—
331C22⋊C4—681no—
332C22⋊C4—1362no—
333C22⋊C4—1362no—
334C42—41no—
335C42—41no—
336C42#6325211yescharacteristic
337C42#6325211yescharacteristic
338C42D4×D1711yescharacteristic
339C42#6325611yescharacteristic
340C23—342no—
341C23—681no—
342C23—681no—
343C23—681no—
344C23—681no—
345C23—681no—
346C23—681no—
347C23#1058756811yescharacteristic
348Q8—684no—
349Q8—1362no—
350D4—342no—
351D4—342no—
352D4—681no—
353D4—681no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—681no—
363C2×C4—681no—
364C2×C4—682no—
365C2×C4—681no—
366C2×C4—682no—
367C2×C4—682no—
368C2×C4—1361no—
369C2×C4—1361no—
370C2×C4—1361no—
371C2×C4—1361no—
372C2×C4—1361no—
373C2×C4—1361no—
374C2×C4—1361no—
375C2×C4#1058756111yescharacteristic
376C2×C4#1058757011yescharacteristic
377C2×C4#1058757011yescharacteristic
378C2×C4#1058758011yescharacteristic
379C2×C4#1058758011yescharacteristic
380C2×C4#1058758111yescharacteristic
381C2×C4#1058759011yescharacteristic
382C2×C4#1058759011yescharacteristic
383C2×C4#1058759411yescharacteristic
384C2×C4#1058759511yescharacteristic
385C2×C4#1058759811yescharacteristic
386C2×C4SD16×D1711yescharacteristic
387C2×C4#1058760111yescharacteristic
388C2×C4#1058760411yescharacteristic
389C8—42no—
390C8—81no—
391C8—682no—
392C22—682no—
393C22—682no—
394C22—682no—
395C22—1361no—
396C22—1361no—
397C22—1361no—
398C22—1361no—
399C22#1176272711yescharacteristic
400C22#1176294911yescharacteristic
401C22#1176295611yescharacteristic
402C22#1176297211yescharacteristic
403C22#1176297311yescharacteristic
404C22#1176297811yescharacteristic
405C22#1176298811yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—2721no—
411C4—1362no—
412C4—1361no—
413C4#1176302511yescharacteristic
414C4#1176303211yescharacteristic
415C4#1176303711yescharacteristic
416C4#1176306611yescharacteristic
417C4#1176314111yescharacteristic
418C4#1176315211yescharacteristic
419C4#1176316411yescharacteristic
420C4#1176316911yescharacteristic
421C2—1362no—
422C2#1452785911yescharacteristic
423C2#1452793111yescharacteristic
424C2#1452799411yescharacteristic
425C2#1452801411yescharacteristic
426C2#1452998311yescharacteristic
427C2#1452999711yescharacteristic
428C2#1453002611yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1279C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenter#1279D17
hypocenterC17#10412
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #14529586C2semidirect
3. #14529547C2semidirect
5. C2×#11762557C2semidirect
7. C17×#1279C2semidirect
184. #1279D17semidirect
280. C17#10412semidirect

Maximal subgroups

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

subgroupindex
#145339532
#145295862
#145295472
#145287852
C2×#117625572
C2×#117625382
C17×#12792
#1041217