Finite Group Databasebeta

← #16938175

Subgroups — #16938175

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×#11763096C211yesmaximal, characteristic
2#14529746C211yesmaximal, characteristic, complemented
3#14529577C211yesmaximal, characteristic
4#14528785C211yesmaximal, characteristic
5C2×#11762561C211yesmaximal, characteristic, complemented
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×#333C211yesmaximal, characteristic, complemented
8C2×C4×D68C2211yescharacteristic
9#11763096—22no—
10#11763096—22no—
11C2×#10587479—21no—
12C2×C4⋊#3622C2211yescharacteristic
13C2×#10587492—21no—
14C2×#10587480—21no—
15C2×#10587491—21no—
16C2×#10587491—21no—
17C2×C4×#3622C2211yescharacteristic
18#11762679C2211yescharacteristic
19#11762572—21no—
20#11762571—21no—
21#11762571—21no—
22#11762561—22no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C17×C4⋊C8C2211yescharacteristic
26C17×#333—22no—
27C22×Q8×C17—21no—
28C22×D68D411yescharacteristic
29C22×C4×D17—21no—
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C17×C4⋊Q8—21no—
33C17×C4⋊Q8—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8D412yes—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C42×C17C2311yescharacteristic
39C2×#63227—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17D411yescharacteristic
44C22×#3622—21no—
45C22×#3622D411yescharacteristic
46C22×#3622D411yescharacteristic
47C4×D68—21no—
48C4×D68—21no—
49C4×D68—21no—
50C4×D68—21no—
51C17×C4⋊C8—21no—
52C17×C4⋊C8—21no—
53C17×C4⋊C8—21no—
54C17×C4⋊C8—21no—
55#10587492—42no—
56#10587491—42no—
57#10587491—42no—
58#10587480—24no—
59#10587479—24no—
60C4⋊#3622—21no—
61C4⋊#3622D412yes—
62C4×#3622—21no—
63C4×#3622—21no—
64C23×D17—21no—
65C2×Q8×C17—22no—
66C2×Q8×C17—41no—
67C2×Q8×C17—81no—
68C2×Q8×C17—81no—
69C22×C4×C17—41no—
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17#4111yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—41no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68—21no—
78C2×D68—21no—
79C2×D68—21no—
80C2×D68Q1612yes—
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×C8×C17—41no—
88C2×C8×C17—41no—
89C2×C8×C17—41no—
90C2×C8×C17—41no—
91C2×C8×C17—41no—
92C2×C8×C17—41no—
93C4⋊C4×C17—22no—
94C4⋊C4×C17—22no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C42×C17C2×D411yescharacteristic
98C42×C17C2×D411yescharacteristic
99C42×C17#4111yescharacteristic
100C42×C17#4111yescharacteristic
101#63227—41no—
102#63227—41no—
103C4⋊Dic17—21no—
104C4⋊Dic17—21no—
105C4⋊Dic17—41no—
106C4⋊Dic17—41no—
107C4⋊Dic17—41no—
108C4⋊Dic17—41no—
109C4⋊Dic17—81no—
110C2×#3622—21no—
111C2×#3622—21no—
112C2×#3622—22no—
113C2×#3622—21no—
114C2×#3622—21no—
115C2×#3622—21no—
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—
128Q8×C17—82no—
129Q8×C17—82no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17—81no—
134C2×C4×C17—81no—
135C2×C4×C17#12011yescharacteristic
136C2×C4×C17#12111yescharacteristic
137C2×C4×C17#12111yescharacteristic
138C2×C4×C17#12311yescharacteristic
139C2×C4×C17#12311yescharacteristic
140C2×C4×C17#12411yescharacteristic
141C2×C4×C17C2×SD1611yescharacteristic
142C2×C4×C17C2×Q1611yescharacteristic
143C2×C4×C17#13511yescharacteristic
144C2×C4×C17#13511yescharacteristic
145C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C17#13711yescharacteristic
148C2×C4×C17#13711yescharacteristic
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—81no—
153C2×Dic17—81no—
154C2×Dic17—81no—
155D68—22no—
156D68—22no—
157D68—41no—
158D68—41no—
159C4×D17—81no—
160C4×D17—81no—
161C4×D17—81no—
162C4×D17—81no—
163C8×C17—81no—
164C8×C17—81no—
165#3622—22no—
166#3622—42no—
167#3622—41no—
168C2×#476—171no—
169#1990—171no—
170#1954—171no—
171#1802—171no—
172C2×#335—171no—
173C2×#333D1711yescharacteristic, complemented
174#1296—171no—
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—82no—
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—81no—
201Dic17—161no—
202C2×C4⋊Q8C2×D1711yescharacteristic
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#384—171no—
216#340—341no—
217#340—341no—
218#340—341no—
219#335—342no—
220#333—22no—
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—21no—
230C22×D4—171no—
231C23×C4—341no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—171no—
235C22×C8—171no—
236C22×C8—341no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8C22⋊D1712yes—
241C4×D4—341no—
242C4×D4—341no—
243C4×D4—341no—
244C4×D4—341no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—171no—
248C2×C4⋊C4—341no—
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—21no—
258C4⋊C8—172no—
259C4⋊C8—341no—
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—22no—
269C2×Q8—41no—
270C2×Q8—81no—
271C2×Q8—81no—
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—41no—
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—342no—
303C2×C8—681no—
304C2×C8—681no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—341no—
310C4⋊C4—341no—
311C4⋊C4—681no—
312C4⋊C4—681no—
313C4⋊C4—681no—
314C4⋊C4—681no—
315C4⋊C4—1361no—
316C22⋊C4—681no—
317C22⋊C4—681no—
318C42#6325211yescharacteristic
319C42D4×D1711yescharacteristic
320C42#6325411yescharacteristic
321C42C2×C22⋊D1711yescharacteristic
322C23—342no—
323C23—681no—
324C23—681no—
325C23—681no—
326C23—681no—
327C23—681no—
328C23—681no—
329C23#1058757011yescharacteristic
330Q8—82no—
331Q8—82no—
332D4—342no—
333D4—342no—
334D4—681no—
335D4—681no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—681no—
344C2×C4—1361no—
345C2×C4—1361no—
346C2×C4—1361no—
347C2×C4—1361no—
348C2×C4—1361no—
349C2×C4—1361no—
350C2×C4—1361no—
351C2×C4#1058756911yescharacteristic
352C2×C4#1058757011yescharacteristic
353C2×C4#1058757111yescharacteristic
354C2×C4SD16×D1711yescharacteristic
355C2×C4#1058760011yescharacteristic
356C2×C4#1058760011yescharacteristic
357C2×C4#1058760211yescharacteristic
358C2×C4Q16×D1711yescharacteristic
359C2×C4#1058760411yescharacteristic
360C2×C4#1058760411yescharacteristic
361C2×C4#1058760511yescharacteristic
362C2×C4#1058761511yescharacteristic
363C2×C4#1058762311yescharacteristic
364C2×C4#1058762511yescharacteristic
365C8—81no—
366C8—81no—
367C8—342no—
368C8—682no—
369C8—681no—
370C22—682no—
371C22—682no—
372C22—682no—
373C22—1361no—
374C22—1361no—
375C22—1361no—
376C22—1361no—
377C22#1176272411yescharacteristic
378C22#1176301311yescharacteristic
379C22#1176302711yescharacteristic
380C22#1176304911yescharacteristic
381C22#1176305811yescharacteristic
382C22#1176307111yescharacteristic
383C22#1176307811yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—2721no—
388C4—1361no—
389C4#1176295311yescharacteristic
390C4#1176296311yescharacteristic
391C4#1176300711yescharacteristic
392C4#1176303311yescharacteristic
393C4#1176304111yescharacteristic
394C4#1176305811yescharacteristic
395C4#1176314311yescharacteristic
396C4#1176314811yescharacteristic
397C2—1362no—
398C2#1452818211yescharacteristic
399C2#1452823811yescharacteristic
400C2#1452828611yescharacteristic
401C2#1452832011yescharacteristic
402C2#1452998511yescharacteristic
403C2#1453007611yescharacteristic
404C2#1453009811yescharacteristic
subgroupG/H
centerC23#10587570
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#333C2
socleC23×C17#121
prime radicalC22×D68D4
hypercenterC2×#333D17
hypocenterC17#10481
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
2. #14529746C2semidirect
5. C2×#11762561C2semidirect
6. #14526668C2semidirect
7. C2×C17×#333C2semidirect
173. C2×#333D17semidirect
266. C17#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
#145297462
#145295772
#145287852
C2×#117625612
#145266682
C2×C17×#3332
#1048117