Finite Group Databasebeta

← #16938147

Subgroups — #16938147

Order 4352.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763011C211yesmaximal, characteristic
2#14529746C211yesmaximal, characteristic, complemented
3#14529577C211yesmaximal, characteristic
4#14528785C211yesmaximal, characteristic
5C2×#11762559C211yesmaximal, characteristic, complemented
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×#332C211yesmaximal, characteristic, complemented
8C22×#63230—21no—
9C2×C4×D68C2211yescharacteristic
10#11763011—22no—
11#11763011—22no—
12C2×C4⋊#3622C2211yescharacteristic
13C2×#10587506—21no—
14C2×#10587492—21no—
15C2×#10587491—21no—
16C2×#10587491—21no—
17C2×C4×#3622C2211yescharacteristic
18#11762679C2211yescharacteristic
19#11762572—21no—
20#11762571—21no—
21#11762571—21no—
22#11762559—22no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C17×C4⋊C8C2211yescharacteristic
26C17×#332—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×#63230—24no—
40C2×#63230—42no—
41C2×#63230—42no—
42C2×#63227—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17D411yescharacteristic
47C22×#3622—21no—
48C22×#3622D411yescharacteristic
49C22×#3622D411yescharacteristic
50C4×D68—21no—
51C4×D68—21no—
52C4×D68—21no—
53C4×D68—21no—
54C17×C4⋊C8—21no—
55C17×C4⋊C8—21no—
56C17×C4⋊C8—21no—
57C17×C4⋊C8—21no—
58#10587506—24no—
59#10587492—42no—
60#10587491—42no—
61#10587491—42no—
62C4⋊#3622—21no—
63C4⋊#3622D412yes—
64C4×#3622—21no—
65C4×#3622—21no—
66C23×D17—21no—
67C2×Q8×C17—22no—
68C2×Q8×C17—41no—
69C2×Q8×C17—41no—
70C2×Q8×C17—41no—
71C2×Q8×C17—41no—
72C2×Q8×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17#4111yescharacteristic
77C22×Dic17—21no—
78C22×Dic17—41no—
79C2×D68—21no—
80C2×D68—21no—
81C2×D68—21no—
82C2×D68—21no—
83C2×D68—21no—
84C2×D68SD1612yes—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×C4×D17—41no—
88C2×C4×D17—41no—
89C2×C4×D17—41no—
90C2×C4×D17—41no—
91C2×C8×C17—41no—
92C2×C8×C17—41no—
93C2×C8×C17—41no—
94C2×C8×C17—41no—
95C2×C8×C17—41no—
96C2×C8×C17—41no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—22no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—41no—
101C42×C17C2×D411yescharacteristic
102C42×C17C2×D411yescharacteristic
103C42×C17#4111yescharacteristic
104C42×C17#4111yescharacteristic
105#63230—48no—
106#63227—41no—
107#63227—41no—
108C4⋊Dic17—21no—
109C4⋊Dic17—21no—
110C4⋊Dic17—41no—
111C4⋊Dic17—41no—
112C4⋊Dic17—41no—
113C4⋊Dic17—41no—
114C4⋊Dic17—81no—
115C2×#3622—21no—
116C2×#3622—21no—
117C2×#3622—22no—
118C2×#3622—21no—
119C2×#3622—21no—
120C2×#3622—21no—
121C2×#3622—41no—
122C2×#3622—41no—
123C2×#3622SD1612yescomplemented
124#10477—171nomaximal, Sylow
125C23×C17#12111yescharacteristic
126C22×D17—22no—
127C22×D17—41no—
128C22×D17—41no—
129C22×D17—41no—
130C22×D17—41no—
131C22×D17—41no—
132C22×D17—41no—
133Q8×C17—42no—
134Q8×C17—42no—
135Q8×C17—81no—
136Q8×C17—81no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—42no—
140C2×C4×C17—81no—
141C2×C4×C17—81no—
142C2×C4×C17#12011yescharacteristic
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×C17C2×SD1611yescharacteristic
150C2×C4×C17#13511yescharacteristic
151C2×C4×C17#13511yescharacteristic
152C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C17#13711yescharacteristic
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—81no—
160C2×Dic17—81no—
161C2×Dic17—81no—
162D68—22no—
163D68—22no—
164D68—41no—
165D68—41no—
166C4×D17—81no—
167C4×D17—81no—
168C4×D17—81no—
169C4×D17—81no—
170C8×C17—81no—
171C8×C17—81no—
172#3622—22no—
173#3622—42no—
174#3622—41no—
175C2×#461—171no—
176#1990—171no—
177#1954—171no—
178#1802—171no—
179C2×#334—171no—
180C2×#332D1711yescharacteristic, complemented
181#1296—171no—
182C22×C17#39411yescharacteristic
183C22×C17#46111yescharacteristic
184C22×C17#46311yescharacteristic
185C22×C17#46511yescharacteristic
186C22×C17#46911yescharacteristic
187C22×C17#47111yescharacteristic
188C22×C17#47311yescharacteristic
189C2×D17—42no—
190C2×D17—42no—
191C2×D17—42no—
192C2×D17—81no—
193C2×D17—81no—
194C2×D17—81no—
195C2×D17—81no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—82no—
199C4×C17#44911yescharacteristic
200C4×C17#45011yescharacteristic
201C4×C17#46011yescharacteristic
202C4×C17#46411yescharacteristic
203C4×C17#46511yescharacteristic
204C4×C17#46811yescharacteristic
205C4×C17#48111yescharacteristic
206C4×C17#48211yescharacteristic
207Dic17—81no—
208Dic17—161no—
209C22×SD16—341no—
210C2×C4⋊Q8C2×D1711yescharacteristic
211C2×C4×D4—171no—
212#461—342no—
213#461—342no—
214C2×C4⋊C8—171no—
215C2×C4⋊C8C2×D1711yescharacteristic
216C2×C4⋊C8C2×D1711yescharacteristic
217C2×#103—341no—
218C2×#102—341no—
219C2×C22⋊C8—341no—
220C2×C22⋊C8—341no—
221C2×C4×C8—171no—
222#384—171no—
223#340—341no—
224#340—341no—
225#340—341no—
226#334—342no—
227#332—22no—
228C2×C17#167111yescharacteristic
229C2×C17#167911yescharacteristic
230C2×C17#169311yescharacteristic
231C2×C17#170111yescharacteristic
232C2×C17#203411yescharacteristic
233C2×C17#205211yescharacteristic
234C2×C17#205511yescharacteristic
235D17—82no—
236C22×Q8—21no—
237C22×D4—171no—
238C23×C4—341no—
239C2×SD16—344no—
240C2×SD16—682no—
241C2×SD16—682no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—171no—
245C22×C8—171no—
246C22×C8—341no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8C22⋊D1712yes—
251C4×D4—341no—
252C4×D4—341no—
253C4×D4—341no—
254C4×D4—341no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—171no—
258C2×C4⋊C4—341no—
259C2×C4⋊C4—341no—
260C2×C4⋊C4—341no—
261C2×C22⋊C4—341no—
262C2×C42C22×D1711yescharacteristic
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—172no—
268C4⋊C8—341no—
269#103—344no—
270#102—682no—
271C22⋊C8—682no—
272C22⋊C8—682no—
273C4×C8—341no—
274C4×C8—341no—
275C17#1047711yescharacteristic, complemented, Sylow
276C24—341no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—172no—
284C2×D4—341no—
285C2×D4—341no—
286C2×D4—341no—
287C2×D4—341no—
288C2×D4—341no—
289C22×C4—41no—
290C22×C4—341no—
291C22×C4—681no—
292C22×C4—681no—
293C22×C4—681no—
294C22×C4—681no—
295C22×C4—681no—
296C22×C4—681no—
297C22×C4—681no—
298C22×C4#6325211yescharacteristic
299C22×C4D4×D1711yescharacteristic
300C22×C4D4×D1711yescharacteristic
301SD16—688no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—172no—
309C2×C8—341no—
310C2×C8—341no—
311C2×C8—341no—
312C2×C8—341no—
313C2×C8—341no—
314C2×C8—342no—
315C2×C8—681no—
316C2×C8—681no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—341no—
322C4⋊C4—341no—
323C4⋊C4—681no—
324C4⋊C4—681no—
325C4⋊C4—681no—
326C4⋊C4—681no—
327C4⋊C4—1361no—
328C22⋊C4—681no—
329C22⋊C4—681no—
330C42#6325211yescharacteristic
331C42D4×D1711yescharacteristic
332C42#6325411yescharacteristic
333C42C2×C22⋊D1711yescharacteristic
334C23—342no—
335C23—681no—
336C23—681no—
337C23—681no—
338C23—681no—
339C23—681no—
340C23—681no—
341C23#1058757011yescharacteristic
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346D4—342no—
347D4—342no—
348D4—681no—
349D4—681no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—681no—
356C2×C4—681no—
357C2×C4—681no—
358C2×C4—1361no—
359C2×C4—1361no—
360C2×C4—1361no—
361C2×C4—1361no—
362C2×C4—1361no—
363C2×C4—1361no—
364C2×C4—1361no—
365C2×C4#1058756911yescharacteristic
366C2×C4#1058757011yescharacteristic
367C2×C4#1058757111yescharacteristic
368C2×C4#1058759711yescharacteristic
369C2×C4#1058759811yescharacteristic
370C2×C4SD16×D1711yescharacteristic
371C2×C4SD16×D1711yescharacteristic
372C2×C4#1058760011yescharacteristic
373C2×C4#1058760011yescharacteristic
374C2×C4#1058760411yescharacteristic
375C2×C4#1058760511yescharacteristic
376C2×C4#1058761511yescharacteristic
377C2×C4#1058762311yescharacteristic
378C2×C4#1058762511yescharacteristic
379C8—81no—
380C8—81no—
381C8—342no—
382C8—682no—
383C8—681no—
384C22—682no—
385C22—682no—
386C22—682no—
387C22—1361no—
388C22—1361no—
389C22—1361no—
390C22—1361no—
391C22#1176272411yescharacteristic
392C22#1176301311yescharacteristic
393C22#1176302711yescharacteristic
394C22#1176303711yescharacteristic
395C22#1176306111yescharacteristic
396C22#1176307511yescharacteristic
397C22#1176308311yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—2721no—
402C4—1361no—
403C4#1176295111yescharacteristic
404C4#1176295811yescharacteristic
405C4#1176300711yescharacteristic
406C4#1176303311yescharacteristic
407C4#1176304111yescharacteristic
408C4#1176305811yescharacteristic
409C4#1176312611yescharacteristic
410C4#1176313111yescharacteristic
411C2—1362no—
412C2#1452815811yescharacteristic
413C2#1452821011yescharacteristic
414C2#1452829611yescharacteristic
415C2#1452834811yescharacteristic
416C2#1452997111yescharacteristic
417C2#1453008311yescharacteristic
418C2#1453009811yescharacteristic
subgroupG/H
centerC23#10587570
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#332C2
socleC23×C17#121
prime radicalC22×D68D4
hypercenterC2×#332D17
hypocenterC17#10477
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #14529746C2semidirect
5. C2×#11762559C2semidirect
6. #14526668C2semidirect
7. C2×C17×#332C2semidirect
123. C2×#3622SD16semidirect
180. C2×#332D17semidirect
275. C17#10477semidirect

Maximal subgroups

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

subgroupindex
C2×#117630112
#145297462
#145295772
#145287852
C2×#117625592
#145266682
C2×C17×#3322
#1047717