Finite Group Databasebeta

← #16938180

Subgroups — #16938180

Order 4352.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763100C211yesmaximal, characteristic
2#14529744C211yesmaximal, characteristic, complemented
3#14529577C211yesmaximal, characteristic
4#14528781C211yesmaximal, characteristic
5C2×#11762561C211yesmaximal, characteristic, complemented
6C2×#11762557C211yesmaximal, characteristic, complemented
7C17×#1296C211yesmaximal, characteristic, complemented
8C2×#10587555C2211yescharacteristic
9C2×C4×D68C2211yescharacteristic
10#11763100—22no—
11#11763100—22no—
12C2×#10587488—21no—
13C2×C4⋊#3622C2211yescharacteristic
14C2×#10587492—21no—
15C2×#10587480—21no—
16C2×#10587491—21no—
17C2×#10587491—21no—
18C2×C4×#3622C2211yescharacteristic
19#11762571—21no—
20#11762571—21no—
21#11762571—21no—
22#11762561—22no—
23#11762557—22no—
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C17×C4⋊C8C2211yescharacteristic
26C17×#384C2211yescharacteristic
27C22×D68D411yescharacteristic
28C22×C4×D17—21no—
29C22×Dic34—21no—
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C42×C17C2311yescharacteristic
36C2×#63227—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C22×#3622—21no—
41C22×#3622D411yescharacteristic
42C22×#3622D411yescharacteristic
43C4×D68—21no—
44C4×D68—21no—
45C4×D68—21no—
46C4×D68—21no—
47#10587555—21no—
48#10587555—21no—
49#10587555—21no—
50#10587555D412yes—
51C17×C4⋊C8—21no—
52C17×C4⋊C8—21no—
53C17×C4⋊C8—21no—
54C17×C4⋊C8D412yes—
55#10587492—24no—
56#10587491—42no—
57#10587491—42no—
58#10587488—24no—
59#10587480—42no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4×#3622—21no—
63C4×#3622—21no—
64C23×D17—21no—
65C22×C4×C17—41no—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17#4111yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—41no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68—21no—
76C2×D68Q1612yes—
77C2×C4×D17—41no—
78C2×C4×D17—41no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×Dic34—22no—
84C2×Dic34—41no—
85C2×Dic34—81no—
86C2×Dic34—81no—
87C2×C8×C17—22no—
88C2×C8×C17—41no—
89C2×C8×C17—41no—
90C2×C8×C17—41no—
91C2×C8×C17—41no—
92C2×C8×C17—41no—
93C4⋊C4×C17—41no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—81no—
98C42×C17C2×D411yescharacteristic
99C42×C17C2×D411yescharacteristic
100C42×C17#4111yescharacteristic
101C42×C17#4111yescharacteristic
102#63227—41no—
103#63227—41no—
104C4⋊Dic17—21no—
105C4⋊Dic17—22no—
106C4⋊Dic17—22no—
107C4⋊Dic17—21no—
108C4⋊Dic17—41no—
109C4⋊Dic17—41no—
110C2×#3622—21no—
111C2×#3622—21no—
112C2×#3622—21no—
113C2×#3622—21no—
114C2×#3622—21no—
115C2×#3622—41no—
116C2×#3622—41no—
117C2×#3622—41no—
118C2×#3622SD1612yes—
119#10481—171nomaximal, Sylow
120C23×C17#12111yescharacteristic
121C22×D17—22no—
122C22×D17—41no—
123C22×D17—41no—
124C22×D17—41no—
125C22×D17—41no—
126C22×D17—41no—
127C22×D17—41no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17—81no—
133C2×C4×C17#12011yescharacteristic
134C2×C4×C17#12111yescharacteristic
135C2×C4×C17#12111yescharacteristic
136C2×C4×C17#12311yescharacteristic
137C2×C4×C17#12311yescharacteristic
138C2×C4×C17#12411yescharacteristic
139C2×C4×C17C2×SD1611yescharacteristic
140C2×C4×C17C2×Q1611yescharacteristic
141C2×C4×C17#13511yescharacteristic
142C2×C4×C17#13511yescharacteristic
143C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C17#13711yescharacteristic
146C2×C4×C17#13711yescharacteristic
147C2×Dic17—42no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—81no—
152C2×Dic17—81no—
153D68—22no—
154D68—22no—
155D68—41no—
156D68—41no—
157C4×D17—81no—
158C4×D17—81no—
159C4×D17—81no—
160C4×D17—81no—
161Dic34—82no—
162Dic34—82no—
163C8×C17—42no—
164C8×C17—81no—
165#3622—22no—
166#3622—41no—
167#3622—81no—
168C2×#476—171no—
169#1990—171no—
170#1954—171no—
171#1802—171no—
172C2×#335—171no—
173C2×#333—171no—
174#1296D1711yescharacteristic, complemented
175C22×C17#39411yescharacteristic
176C22×C17#46111yescharacteristic
177C22×C17#46311yescharacteristic
178C22×C17#46711yescharacteristic
179C22×C17#46811yescharacteristic
180C22×C17#47111yescharacteristic
181C22×C17#47211yescharacteristic
182C2×D17—42no—
183C2×D17—42no—
184C2×D17—42no—
185C2×D17—81no—
186C2×D17—81no—
187C2×D17—81no—
188C2×D17—81no—
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—161no—
192C4×C17#44911yescharacteristic
193C4×C17#45111yescharacteristic
194C4×C17#46011yescharacteristic
195C4×C17#46411yescharacteristic
196C4×C17#46511yescharacteristic
197C4×C17#46811yescharacteristic
198C4×C17#48411yescharacteristic
199C4×C17#48511yescharacteristic
200Dic17—82no—
201Dic17—81no—
202C2×C4⋊Q8—171no—
203C2×C4×D4—171no—
204#476—342no—
205#476—342no—
206C2×#106—341no—
207C2×C4⋊C8—171no—
208C2×C4⋊C8C2×D1711yescharacteristic
209C2×C4⋊C8C2×D1711yescharacteristic
210C2×#102—341no—
211C2×#102—341no—
212C2×C22⋊C8—341no—
213C2×C22⋊C8—341no—
214C2×C4×C8—171no—
215#384C2×D1711yescharacteristic
216#340—341no—
217#340—341no—
218#340—341no—
219#335—342no—
220#333—342no—
221C2×C17#167511yescharacteristic
222C2×C17#168311yescharacteristic
223C2×C17#169111yescharacteristic
224C2×C17#169711yescharacteristic
225C2×C17#203611yescharacteristic
226C2×C17#205111yescharacteristic
227C2×C17#205511yescharacteristic
228D17—82no—
229C22×Q8—341no—
230C22×D4—171no—
231C23×C4—341no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—171no—
235C22×C8—171no—
236C22×C8—341no—
237C4⋊Q8—172no—
238C4⋊Q8—341no—
239C4⋊Q8—341no—
240C4⋊Q8—341no—
241C4×D4—341no—
242C4×D4—341no—
243C4×D4—341no—
244C4×D4—341no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—171no—
249C2×C4⋊C4—341no—
250C2×C4⋊C4—341no—
251C2×C22⋊C4—341no—
252C2×C42C22×D1711yescharacteristic
253#106—344no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—341no—
258C4⋊C8—341no—
259C4⋊C8D6812yes—
260#102—344no—
261#102—682no—
262C22⋊C8—682no—
263C22⋊C8—682no—
264C4×C8—341no—
265C4×C8—341no—
266C17#1048111yescharacteristic, complemented, Sylow
267C24—341no—
268C2×Q8—342no—
269C2×Q8—681no—
270C2×Q8—1361no—
271C2×Q8—1361no—
272C2×D4—172no—
273C2×D4—341no—
274C2×D4—341no—
275C2×D4—341no—
276C2×D4—341no—
277C2×D4—341no—
278C22×C4—41no—
279C22×C4—341no—
280C22×C4—681no—
281C22×C4—681no—
282C22×C4—681no—
283C22×C4—681no—
284C22×C4—681no—
285C22×C4—681no—
286C22×C4—681no—
287C22×C4#6325211yescharacteristic
288C22×C4D4×D1711yescharacteristic
289C22×C4D4×D1711yescharacteristic
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—172no—
297C2×C8—341no—
298C2×C8—341no—
299C2×C8—341no—
300C2×C8—341no—
301C2×C8—341no—
302C2×C8—681no—
303C2×C8—681no—
304C2×C8—681no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—81no—
310C4⋊C4—342no—
311C4⋊C4—341no—
312C4⋊C4—342no—
313C4⋊C4—341no—
314C4⋊C4—681no—
315C4⋊C4—681no—
316C22⋊C4—681no—
317C22⋊C4—681no—
318C42C2×D6811yescharacteristic
319C42#6325211yescharacteristic
320C42D4×D1711yescharacteristic
321C42#6325611yescharacteristic
322C23—342no—
323C23—681no—
324C23—681no—
325C23—681no—
326C23—681no—
327C23—681no—
328C23—681no—
329C23#1058757011yescharacteristic
330Q8—1362no—
331Q8—1362no—
332D4—342no—
333D4—342no—
334D4—681no—
335D4—681no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—681no—
342C2×C4—682no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—1361no—
346C2×C4—1361no—
347C2×C4—1361no—
348C2×C4—1361no—
349C2×C4—1361no—
350C2×C4—1361no—
351C2×C4#1058756111yescharacteristic
352C2×C4#1058756811yescharacteristic
353C2×C4#1058757011yescharacteristic
354C2×C4#1058758011yescharacteristic
355C2×C4#1058758011yescharacteristic
356C2×C4#1058758111yescharacteristic
357C2×C4#1058759711yescharacteristic
358C2×C4#1058759711yescharacteristic
359C2×C4SD16×D1711yescharacteristic
360C2×C4#1058760111yescharacteristic
361C2×C4#1058760211yescharacteristic
362C2×C4#1058760211yescharacteristic
363C2×C4Q16×D1711yescharacteristic
364C2×C4#1058760411yescharacteristic
365C8—42no—
366C8—81no—
367C8—342no—
368C8—681no—
369C8—1361no—
370C22—682no—
371C22—682no—
372C22—682no—
373C22—1361no—
374C22—1361no—
375C22—1361no—
376C22—1361no—
377C22#1176272711yescharacteristic
378C22#1176301311yescharacteristic
379C22#1176302311yescharacteristic
380C22#1176304911yescharacteristic
381C22#1176305411yescharacteristic
382C22#1176307511yescharacteristic
383C22#1176307811yescharacteristic
384C4—21no—
385C4—21no—
386C4—161no—
387C4—1362no—
388C4—1361no—
389C4#1176295511yescharacteristic
390C4#1176296611yescharacteristic
391C4#1176300811yescharacteristic
392C4#1176303411yescharacteristic
393C4#1176303911yescharacteristic
394C4#1176305611yescharacteristic
395C4#1176314611yescharacteristic
396C4#1176315311yescharacteristic
397C2—1362no—
398C2#1452818211yescharacteristic
399C2#1452823411yescharacteristic
400C2#1452828211yescharacteristic
401C2#1452831611yescharacteristic
402C2#1452999011yescharacteristic
403C2#1453007911yescharacteristic
404C2#1453010311yescharacteristic
subgroupG/H
centerC23#10587570
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1296C2
socleC23×C17#121
prime radicalC22×D68D4
hypercenter#1296D17
hypocenterC17#10481
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #14529744C2semidirect
5. C2×#11762561C2semidirect
6. C2×#11762557C2semidirect
7. C17×#1296C2semidirect
174. #1296D17semidirect
266. C17#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#117631002
#145297442
#145295772
#145287812
C2×#117625612
C2×#117625572
C17×#12962
#1048117