Finite Group Databasebeta

← #16934169

Subgroups — #16934169

Order 4352.

562 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14529746C211yesmaximal, characteristic, complemented
2#14529591C211yesmaximal, characteristic
3#14529591C211yesmaximal, characteristic
4#14529587C211yesmaximal, characteristic, complemented
5#14528912C211yesmaximal, characteristic, complemented
6C2×#11762530C211yesmaximal, characteristic, complemented
7C2×C17×#486C211yesmaximal, characteristic
8C2×#10587506—22no—
9C2×#10587503—22no—
10C2×#10587503—22no—
11C2×C4×#3622C2211yescharacteristic
12C2×C4×#3622C2211yescharacteristic
13#11762679C2211yescharacteristic
14#11762679C2211yescharacteristic
15#11762572—22no—
16#11762571—22no—
17#11762530—22no—
18C2×C17×C4⋊Q8C2211yescharacteristic
19C2×C17×#127C2211yescharacteristic
20C17×#486—22no—
21C17×#486C414yescomplemented
22C2×C17×#102—22no—
23C2×C4×C8×C17C2211yescharacteristic
24C22×Q8×C17—21no—
25C22×D4×C17—21no—
26C22×D4×C17—21no—
27C22×D4×C17—21no—
28C22×C8×C17D412yes—
29C17×C4⋊Q8—21no—
30C17×C4⋊Q8—21no—
31C17×C4⋊Q8—21no—
32C17×C4⋊Q8C2×C412yes—
33C17×#127—21no—
34C17×#127—21no—
35C17×#127—21no—
36C17×#127C2×C412yes—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C42×C17C2311yescharacteristic
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17—21no—
46C22×#3622D412yes—
47C22×#3622D412yes—
48C17×#102—28no—
49C4×C8×C17C2×C412yes—
50C4×C8×C17D412yes—
51#10587506—44no—
52#10587503—44no—
53#10587503—44no—
54C4×#3622—21no—
55C4×#3622—21no—
56C4×#3622—21no—
57C4×#3622—21no—
58C24×C17—41no—
59C2×Q8×C17—22no—
60C2×Q8×C17—41no—
61C2×Q8×C17—42no—
62C2×Q8×C17—42no—
63C2×D4×C17—22no—
64C2×D4×C17—22no—
65C2×D4×C17—22no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C2×D4×C17—41no—
69C2×D4×C17—42no—
70C2×D4×C17—42no—
71C2×D4×C17—42no—
72C2×D4×C17—42no—
73C2×D4×C17—42no—
74C2×D4×C17—42no—
75C22×C4×C17—41no—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—41no—
80C22×Dic17—41no—
81C2×C8×C17—22no—
82C2×C8×C17—22no—
83C2×C8×C17C22⋊C414yes—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C42×C17C22×C411yescharacteristic
89C42×C17C2×D411yescharacteristic
90C42×C17#4111yescharacteristic
91C42×C17#4111yescharacteristic
92C4⋊Dic17—42no—
93C4⋊Dic17—42no—
94C4⋊Dic17—42no—
95C4⋊Dic17—42no—
96C4⋊Dic17—42no—
97C4⋊Dic17—42no—
98C2×#3622—22no—
99C2×#3622—22no—
100C2×#3622—22no—
101C2×#3622—22no—
102C2×#3622—22no—
103C2×#3622—22no—
104#9838—171nomaximal, Sylow
105C23×C17—42no—
106C23×C17—81no—
107C23×C17—81no—
108C23×C17—81no—
109C23×C17—81no—
110C23×C17—81no—
111C23×C17—81no—
112C23×C17#12711yescharacteristic
113Q8×C17—44no—
114Q8×C17—82no—
115D4×C17—44no—
116D4×C17—44no—
117D4×C17—44no—
118D4×C17—82no—
119D4×C17—82no—
120D4×C17—82no—
121C2×C4×C17—42no—
122C2×C4×C17—81no—
123C2×C4×C17—81no—
124C2×C4×C17C2×C22⋊C411yescharacteristic
125C2×C4×C17C4×D411yescharacteristic
126C2×C4×C17C4×D411yescharacteristic
127C2×C4×C17#12111yescharacteristic
128C2×C4×C17#12111yescharacteristic
129C2×C4×C17#12411yescharacteristic
130C2×C4×C17#13512yes—
131C2×C4×C17#13512yes—
132C2×C4×C17AGL(1,ℤ/8)12yes—
133C2×C4×C17AGL(1,ℤ/8)12yes—
134C2×C4×C17AGL(1,ℤ/8)12yes—
135C2×C4×C17#13712yes—
136C2×Dic17—81no—
137C2×Dic17—81no—
138C2×Dic17—81no—
139C2×Dic17—81no—
140C2×Dic17—81no—
141C2×Dic17—81no—
142C8×C17—24no—
143#3622—42no—
144#3622—42no—
145C2×#486D1711yescharacteristic
146#1990—171no—
147#1957—171no—
148#1957—171no—
149#1956—171no—
150#1834—171no—
151C2×#322—171no—
152C22×C17—82no—
153C22×C17—82no—
154C22×C17—82no—
155C22×C17—161no—
156C22×C17—161no—
157C22×C17—161no—
158C22×C17—161no—
159C22×C17#39011yescharacteristic
160C22×C17#49511yescharacteristic
161C22×C17#49511yescharacteristic
162C22×C17#49611yescharacteristic
163C22×C17#49611yescharacteristic
164C22×C17#49611yescharacteristic
165C22×C17#49711yescharacteristic
166C4×C17—82no—
167C4×C17#41612yes—
168C4×C17#44012yes—
169C4×C17#44212yes—
170C4×C17#46812yes—
171C4×C17#46912yes—
172C4×C17#48812yes—
173Dic17—161no—
174Dic17—161no—
175C2×C4⋊Q8C2×D1711yescharacteristic
176C2×#127C2×D1711yescharacteristic
177#486—22no—
178#486Dic1714yescomplemented
179C2×#103—342no—
180C2×#102—22no—
181C2×#102—342no—
182C2×#102—342no—
183C2×C4×C8—171no—
184C2×C4×C8—171no—
185C2×C4×C8C2×D1711yescharacteristic
186#384—171no—
187#384—171no—
188#340—342no—
189#340—342no—
190#322—342no—
191C2×C17—162no—
192C2×C17#171711yescharacteristic
193C2×C17#171711yescharacteristic
194C2×C17#171911yescharacteristic
195C2×C17#172011yescharacteristic
196C2×C17#195811yescharacteristic
197C2×C17#197211yescharacteristic
198C2×C17#197311yescharacteristic
199C22×Q8—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×C8—172no—
204C22×C8—172no—
205C22×C8C22⋊D1712yes—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C2×Dic1712yes—
210#127—21no—
211#127—21no—
212#127—21no—
213#127C2×Dic1712yes—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C42C22×D1711yescharacteristic
223#103—684no—
224#102—28no—
225#102—684no—
226#102—684no—
227C4×C8—341no—
228C4×C8—341no—
229C4×C8—341no—
230C4×C8—341no—
231C4×C8C2×Dic1712yes—
232C4×C8C22⋊D1712yes—
233C17#983811yescharacteristic, complemented, Sylow
234C24—41no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—42no—
251C22×C4—41no—
252C22×C4—681no—
253C22×C4—681no—
254C22×C4D4×D1711yescharacteristic
255C22×C4D4×D1711yescharacteristic
256C22×C4C2×C22⋊D1711yescharacteristic
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—342no—
260C2×C8—342no—
261C2×C8—342no—
262C2×C8—342no—
263C2×C8—342no—
264C2×C8—342no—
265C2×C8C22⋊Dic1714yes—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—682no—
271C4⋊C4—682no—
272C4⋊C4—682no—
273C4⋊C4—682no—
274C4⋊C4—682no—
275C4⋊C4—682no—
276C42#6325411yescharacteristic
277C42#6325411yescharacteristic
278C42C22×Dic1711yescharacteristic
279C42C2×C22⋊D1711yescharacteristic
280C23—42no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23#1058762611yescharacteristic
288Q8—44no—
289Q8—82no—
290D4—44no—
291D4—44no—
292D4—44no—
293D4—82no—
294D4—82no—
295D4—82no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—1361no—
300C2×C4—1361no—
301C2×C4—1361no—
302C2×C4—1361no—
303C2×C4—1361no—
304C2×C4—1361no—
305C2×C4#1058759712yes—
306C2×C4#1058759712yes—
307C2×C4#1058760012yes—
308C2×C4#1058760112yes—
309C2×C4D4×Dic1711yescharacteristic
310C2×C4D4×Dic1711yescharacteristic
311C2×C4#1058762211yescharacteristic
312C2×C4#1058762411yescharacteristic
313C2×C4#1058762411yescharacteristic
314C2×C4#1058763612yes—
315C2×C4#1058763612yes—
316C2×C4C2×C22⋊Dic1711yescharacteristic
317C8—24no—
318C8—682no—
319C8—682no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—161no—
324C22—161no—
325C22—161no—
326C22—161no—
327C22#1176271211yescharacteristic
328C22#1176319111yescharacteristic
329C22#1176319111yescharacteristic
330C22#1176319511yescharacteristic
331C22#1176319711yescharacteristic
332C22#1176319711yescharacteristic
333C22#1176320011yescharacteristic
334C4—82no—
335C4—2721no—
336C4—2721no—
337C4#1176280512yes—
338C4#1176291012yes—
339C4#1176292112yes—
340C4#1176305212yes—
341C4#1176305912yes—
342C4#1176316312yes—
343C2—162no—
344C2#1452842011yescharacteristic
345C2#1452842411yescharacteristic
346C2#1452843411yescharacteristic
347C2#1452844511yescharacteristic
348C2#1452959611yescharacteristic
349C2#1452966111yescharacteristic
350C2#1452966711yescharacteristic
subgroupG/H
centerC23#10587626
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC2×C17×#486C2
socleC23×C17#127
prime radicalC2×C17×#127C22
cosocleC2×C42×C17C23
hypercenterC2×#486D17
hypocenterC17#9838
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. #14529746C2semidirect
4. #14529587C2semidirect
5. #14528912C2semidirect
6. C2×#11762530C2semidirect
21. C17×#486C4semidirect
178. #486Dic17semidirect
233. C17#9838semidirect

Maximal subgroups

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

subgroupindex
#145297462
#145295912
#145295872
#145289122
C2×#117625302
C2×C17×#4862
#983817