Finite Group Databasebeta

← #16934187

Subgroups — #16934187

Order 4352.

626 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763181C211yesmaximal, characteristic
2#14529745C211yesmaximal, characteristic, complemented
3#14529744C211yesmaximal, characteristic, complemented
4#14529591C211yesmaximal, characteristic
5#14528943C211yesmaximal, characteristic, complemented
6C2×#11762531C211yesmaximal, characteristic, complemented
7C17×#1957C211yesmaximal, characteristic
8C22×#63229—22no—
9C2×#10587555C2211yescharacteristic
10#11763181—22no—
11#11763181C414yescomplemented
12C2×#10587478—22no—
13C2×#10587503—22no—
14C2×C4×#3622C2211yescharacteristic
15C2×C4×#3622C2211yescharacteristic
16#11762679C2211yescharacteristic
17#11762572—22no—
18#11762571—22no—
19#11762531—22no—
20C2×C17×#127C2211yescharacteristic
21C2×C17×#102—22no—
22C2×C4×C8×C17C2211yescharacteristic
23C17×#384C2211yescharacteristic
24C22×D4×C17—21no—
25C22×D4×C17—21no—
26C22×D4×C17—21no—
27C22×Dic34—21no—
28C22×C8×C17D412yes—
29C17×#127—21no—
30C17×#127—21no—
31C17×#127—21no—
32C17×#127C2×C412yes—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—21no—
36C2×C42×C17C2311yescharacteristic
37C2×#63229—28no—
38C2×#63229—44no—
39C2×#63229—44no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C22×#3622D412yes—
46C22×#3622D412yes—
47#10587555—21no—
48#10587555—21no—
49#10587555—21no—
50#10587555C2×C412yes—
51C17×#102—44no—
52C4×C8×C17—21no—
53C4×C8×C17—21no—
54#10587503—44no—
55#10587478—44no—
56C4×#3622—21no—
57C4×#3622—21no—
58C4×#3622C2×C412yes—
59C4×#3622D412yes—
60C24×C17—41no—
61C2×D4×C17—22no—
62C2×D4×C17—22no—
63C2×D4×C17—22no—
64C2×D4×C17—41no—
65C2×D4×C17—41no—
66C2×D4×C17—41no—
67C2×D4×C17—42no—
68C2×D4×C17—42no—
69C2×D4×C17—42no—
70C2×D4×C17—42no—
71C2×D4×C17—42no—
72C2×D4×C17—42no—
73C22×C4×C17—41no—
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×Dic17—41no—
78C22×Dic17—41no—
79C2×Dic34—22no—
80C2×Dic34—42no—
81C2×Dic34—42no—
82C2×Dic34—41no—
83C2×C8×C17—22no—
84C2×C8×C17—22no—
85C2×C8×C17—22no—
86C4⋊C4×C17—42no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17—42no—
89C42×C17C22×C411yescharacteristic
90C42×C17C2×D411yescharacteristic
91C42×C17#4111yescharacteristic
92C42×C17#4111yescharacteristic
93#63229—416no—
94C4⋊Dic17—22no—
95C4⋊Dic17—22no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98C4⋊Dic17—42no—
99C4⋊Dic17—42no—
100C4⋊Dic17—42no—
101C2×#3622—22no—
102C2×#3622—22no—
103C2×#3622—22no—
104C2×#3622C22⋊C414yes—
105C2×#3622D814yes—
106C2×#3622SD1614yes—
107#9842—171nomaximal, Sylow
108C23×C17—42no—
109C23×C17—81no—
110C23×C17—81no—
111C23×C17—81no—
112C23×C17—81no—
113C23×C17—81no—
114C23×C17—81no—
115C23×C17#12711yescharacteristic
116D4×C17—44no—
117D4×C17—44no—
118D4×C17—44no—
119D4×C17—82no—
120D4×C17—82no—
121D4×C17—82no—
122C2×C4×C17—81no—
123C2×C4×C17—81no—
124C2×C4×C17—81no—
125C2×C4×C17C2×C22⋊C411yescharacteristic
126C2×C4×C17C4×D411yescharacteristic
127C2×C4×C17C4×D411yescharacteristic
128C2×C4×C17#12111yescharacteristic
129C2×C4×C17#12111yescharacteristic
130C2×C4×C17#12411yescharacteristic
131C2×C4×C17C2×D812yes—
132C2×C4×C17C2×SD1612yes—
133C2×C4×C17AGL(1,ℤ/8)12yes—
134C2×C4×C17AGL(1,ℤ/8)12yes—
135C2×C4×C17#13712yes—
136C2×C4×C17#13712yes—
137C2×Dic17—42no—
138C2×Dic17—81no—
139C2×Dic17—81no—
140C2×Dic17—81no—
141C2×Dic17—81no—
142C2×Dic17—81no—
143Dic34—44no—
144Dic34—82no—
145C8×C17—42no—
146#3622—42no—
147#3622#10218yescomplemented
148C2×#492—171no—
149#1990—171no—
150#1990—171no—
151#1957—171no—
152#1957D1711yescharacteristic
153#1845—171no—
154C2×#322—171no—
155C22×C17—82no—
156C22×C17—82no—
157C22×C17—82no—
158C22×C17—161no—
159C22×C17—161no—
160C22×C17—161no—
161C22×C17—161no—
162C22×C17#39011yescharacteristic
163C22×C17#49211yescharacteristic
164C22×C17#49311yescharacteristic
165C22×C17#49611yescharacteristic
166C22×C17#49611yescharacteristic
167C22×C17#49711yescharacteristic
168C22×C17#49711yescharacteristic
169C4×C17—161no—
170C4×C17C2×#10212yes—
171C4×C17#44012yes—
172C4×C17#44012yes—
173C4×C17#46812yes—
174C4×C17#46812yes—
175C4×C17#48612yes—
176Dic17—82no—
177Dic17—161no—
178C22×SD16—342no—
179C2×C4⋊Q8—171no—
180C2×#127C2×D1711yescharacteristic
181#492—174no—
182#492—342no—
183C2×#107—342no—
184C2×#102—22no—
185C2×#102—342no—
186C2×C4×C8—171no—
187C2×C4×C8—171no—
188C2×C4×C8C2×D1711yescharacteristic
189#384—171no—
190#384C2×D1711yescharacteristic
191#340—342no—
192#340—342no—
193#322—342no—
194C2×C17—162no—
195C2×C17#171411yescharacteristic
196C2×C17#171411yescharacteristic
197C2×C17#172111yescharacteristic
198C2×C17#172211yescharacteristic
199C2×C17#195711yescharacteristic
200C2×C17#197311yescharacteristic
201C2×C17#197311yescharacteristic
202C22×Q8—341no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—21no—
206C2×SD16—348no—
207C2×SD16—684no—
208C2×SD16—684no—
209C22×C8—172no—
210C22×C8—172no—
211C22×C8C22⋊D1712yes—
212C4⋊Q8—172no—
213C4⋊Q8—341no—
214C4⋊Q8—341no—
215C4⋊Q8—341no—
216#127—21no—
217#127—21no—
218#127—21no—
219#127C4×D1712yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—341no—
228C2×C42C22×D1711yescharacteristic
229#107—684no—
230#102—44no—
231#102—684no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—172no—
235C4×C8—172no—
236C4×C8—341no—
237C4×C8—341no—
238C17#984211yescharacteristic, complemented, Sylow
239C24—41no—
240C2×Q8—342no—
241C2×Q8—681no—
242C2×Q8—682no—
243C2×Q8—682no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—42no—
256C22×C4—41no—
257C22×C4—681no—
258C22×C4—681no—
259C22×C4D4×D1711yescharacteristic
260C22×C4D4×D1711yescharacteristic
261C22×C4C2×C22⋊D1711yescharacteristic
262SD16—6816no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—174no—
267C2×C8—174no—
268C2×C8—174no—
269C2×C8—342no—
270C2×C8—342no—
271C2×C8—342no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—342no—
276C4⋊C4—342no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—682no—
280C4⋊C4—682no—
281C4⋊C4—682no—
282C42C2×C4×D1711yescharacteristic
283C42#6325211yescharacteristic
284C42D4×D1711yescharacteristic
285C42#6325411yescharacteristic
286C23—42no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23#1058762611yescharacteristic
294Q8—684no—
295Q8—1362no—
296D4—44no—
297D4—44no—
298D4—44no—
299D4—82no—
300D4—82no—
301D4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—682no—
306C2×C4—1361no—
307C2×C4—1361no—
308C2×C4—1361no—
309C2×C4—1361no—
310C2×C4—1361no—
311C2×C4C22⋊C4×D1711yescharacteristic
312C2×C4#1058756711yescharacteristic
313C2×C4#1058757011yescharacteristic
314C2×C4#1058757111yescharacteristic
315C2×C4D8×D1712yes—
316C2×C4#1058759712yes—
317C2×C4SD16×D1712yes—
318C2×C4#1058760112yes—
319C2×C4C4×C22⋊D1711yescharacteristic
320C2×C4#1058761812yes—
321C2×C4#1058762511yescharacteristic
322C2×C4#1058763712yes—
323C8—42no—
324C8—178no—
325C8—682no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—161no—
330C22—161no—
331C22—161no—
332C22—161no—
333C22#1176270911yescharacteristic
334C22#1176318011yescharacteristic
335C22#1176318411yescharacteristic
336C22#1176319611yescharacteristic
337C22#1176319711yescharacteristic
338C22#1176320011yescharacteristic
339C22#1176320411yescharacteristic
340C4—161no—
341C4—2721no—
342C4—1362no—
343C4#102×D1712yes—
344C4#1176290612yes—
345C4#1176290912yes—
346C4#1176305312yes—
347C4#1176305412yes—
348C4#1176315412yes—
349C2—162no—
350C2#1452840411yescharacteristic
351C2#1452840511yescharacteristic
352C2#1452844911yescharacteristic
353C2#1452845711yescharacteristic
354C2#1452958811yescharacteristic
355C2#1452966411yescharacteristic
356C2#1452966611yescharacteristic
subgroupG/H
centerC23#10587626
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1957C2
socleC23×C17#127
prime radicalC2×C17×#127C22
cosocleC2×C42×C17C23
hypercenter#1957D17
hypocenterC17#9842
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. #14529745C2semidirect
3. #14529744C2semidirect
5. #14528943C2semidirect
6. C2×#11762531C2semidirect
11. #11763181C4semidirect
147. #3622#102semidirect
238. C17#9842semidirect

Maximal subgroups

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

subgroupindex
C2×#117631812
#145297452
#145297442
#145295912
#145289432
C2×#117625312
C17×#19572
#984217