Finite Group Databasebeta

← #16956987

Subgroups — #16956987

Order 4352.

1342 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762677C211yesmaximal, characteristic
2C2×C4×C4⋊Dic17C211yesmaximal, characteristic
3#14527049C212yesmaximal
4#14527041C212yesmaximal
5#14527018C214yesmaximal
6#14527014C212yesmaximal
7#14527011C212yesmaximal
8C2×C17×#376C211yesmaximal, characteristic
9C22×C4⋊Dic17C2212yes—
10C22×C4⋊Dic17C2214yes—
11C22×C4×Dic17C2212yes—
12#11762677—24no—
13C4×C4⋊Dic17—24no—
14C2×#10587502C2211yescharacteristic
15C2×#10587502C2212yes—
16C2×#10587502C2214yes—
17C2×#10587502C2214yes—
18C2×#10587502C2211yescharacteristic
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2214yes—
22C22×C42×C17C2212yes—
23C22×C42×C17C2211yescharacteristic
24C17×#376C418yescomplemented
25C2×C17×#95C2212yes—
26C2×C17×#95C2211yescharacteristic
27C2×C17×#95C2211yescharacteristic
28C23×C4×C17C2312yes—
29C23×C4×C17C2311yescharacteristic
30C23×C4×C17C2311yescharacteristic
31C23×C4×C17C2311yescharacteristic
32C23×C4×C17C2312yes—
33C23×Dic17C2312yes—
34C23×Dic17C2312yes—
35C23×Dic17C2314yes—
36C2×C42×C17—28no—
37C2×C42×C17C2×C418yes—
38C2×C42×C17C2×C414yes—
39C2×C42×C17D414yes—
40C2×C42×C17Q814yes—
41C2×C4⋊Dic17—24no—
42C2×C4⋊Dic17—28no—
43C2×C4⋊Dic17—28no—
44C2×C4⋊Dic17—28no—
45C2×C4⋊Dic17—44no—
46C2×C4×Dic17—24no—
47C2×C4×Dic17—42no—
48C2×C4×Dic17—42no—
49C17×#95C2×C414yes—
50C17×#95C2×C418yes—
51C17×#95C2×C414yes—
52#10587502—24no—
53#10587502—22no—
54#10587502—22no—
55#10587502—24no—
56#10587502—24no—
57#10587502—28no—
58#10587502—28no—
59#10587502—44no—
60C24×C17C2411yescharacteristic
61C22×C4×C17—24no—
62C22×C4×C17—24no—
63C22×C4×C17—24no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—22no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—22no—
70C22×C4×C17—24no—
71C22×C4×C17—24no—
72C22×C4×C17—22no—
73C22×C4×C17—22no—
74C22×C4×C17C22×C414yes—
75C22×C4×C17C22×C412yes—
76C22×C4×C17C22×C412yes—
77C22×C4×C17C22×C412yes—
78C22×C4×C17C22×C414yes—
79C22×C4×C17C2×D412yes—
80C22×C4×C17C2×D412yes—
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×Q812yes—
83C22×C4×C17C2×Q812yes—
84C22×C4×C17C2×Q812yes—
85C22×C4×C17#4118yes—
86C22×Dic17—24no—
87C22×Dic17—24no—
88C22×Dic17—24no—
89C22×Dic17—24no—
90C22×Dic17—24no—
91C22×Dic17—24no—
92C22×Dic17—22no—
93C22×Dic17—22no—
94C22×Dic17—24no—
95C22×Dic17—24no—
96C22×Dic17—24no—
97C22×Dic17—24no—
98C22×Dic17—22no—
99C22×Dic17—22no—
100C22×Dic17—22no—
101C22×Dic17—22no—
102C22×Dic17#4118yes—
103C42×C17—216no—
104C42×C17C4⋊C4116yescomplemented
105C4⋊Dic17—216no—
106C4⋊Dic17—416no—
107C4×Dic17—48no—
108#13952—171nomaximal, Sylow
109C23×C17C23×C411yescharacteristic
110C23×C17C22×D411yescharacteristic
111C23×C17C22×Q811yescharacteristic
112C23×C17C2×#4114yes—
113C23×C17C2×#4114yes—
114C23×C17H(5,𝔽2)11yescharacteristic
115C23×C17H(5,𝔽2)11yescharacteristic
116C23×C172−1+411yescharacteristic
117C23×C172−1+411yescharacteristic
118C2×C4×C17—24no—
119C2×C4×C17—24no—
120C2×C4×C17—24no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—24no—
124C2×C4×C17—24no—
125C2×C4×C17—24no—
126C2×C4×C17—24no—
127C2×C4×C17—24no—
128C2×C4×C17—22no—
129C2×C4×C17—22no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—28no—
133C2×C4×C17—28no—
134C2×C4×C17—24no—
135C2×C4×C17—24no—
136C2×C4×C17—24no—
137C2×C4×C17—44no—
138C2×C4×C17—44no—
139C2×C4×C17C2×C4⋊C414yes—
140C2×C4×C17C2×C4⋊C414yes—
141C2×C4×C17C2×C4⋊C414yes—
142C2×C4×C17#11718yes—
143C2×Dic17—28no—
144C2×Dic17—28no—
145C2×Dic17—28no—
146C2×Dic17—44no—
147C2×Dic17—44no—
148C2×Dic17—44no—
149C2×Dic17—44no—
150C2×Dic17—42no—
151C2×Dic17—42no—
152C2×Dic17—42no—
153C2×Dic17—42no—
154C2×Dic17—42no—
155C2×Dic17—42no—
156C2×Dic17—44no—
157C2×Dic17—44no—
158C2×Dic17—44no—
159C2×Dic17—44no—
160C2×#383—171no—
161C2×C4×C4⋊C4—171no—
162C2×#376D1711yescharacteristic
163#1444—172no—
164#1442—172no—
165#1436—174no—
166C4⋊#95—172no—
167#1434—172no—
168C22×C17C22×C4⋊C411yescharacteristic
169C22×C17C2×#11712yes—
170C22×C17C4×#4112yes—
171C22×C17#52011yescharacteristic
172C22×C17#52011yescharacteristic
173C22×C17#52512yes—
174C22×C17#53212yes—
175C22×C17#53312yes—
176C22×C17#53312yes—
177C22×C17#53511yescharacteristic
178C22×C17#53611yescharacteristic
179C22×C17#54411yescharacteristic
180C22×C17#54411yescharacteristic
181C22×C17#55114yes—
182C22×C17#55214yes—
183C22×C17#55314yes—
184C22×C17#55614yes—
185C4×C17—24no—
186C4×C17—28no—
187C4×C17—24no—
188C4×C17—24no—
189C4×C17—44no—
190Dic17—48no—
191Dic17—82no—
192Dic17—82no—
193C22×C4⋊C4—172no—
194C22×C4⋊C4—174no—
195C22×C42—172no—
196C22×C42C2×D1712yes—
197C22×C42C2×D1711yescharacteristic
198#383—344no—
199C4×C4⋊C4—344no—
200#376Dic1718yescomplemented
201C2×#95—171no—
202C2×#95—172no—
203C2×#95—174no—
204C2×#95—174no—
205C2×#95—171no—
206C2×#95—172no—
207C2×#95—172no—
208C2×#95—174no—
209C2×#95C2×D1712yes—
210C2×#95C2×D1711yescharacteristic
211C2×#95C2×D1711yescharacteristic
212C2×C17#228511yescharacteristic
213C2×C17#229611yescharacteristic
214C2×C17#231711yescharacteristic
215C2×C17#234312yes—
216C2×C17#234712yes—
217C2×C17#260312yes—
218C2×C17#260412yes—
219C2×C17#260512yes—
220C2×C17#260612yes—
221C23×C4—174no—
222C23×C4—172no—
223C23×C4—172no—
224C23×C4C22×D1712yes—
225C23×C4C22×D1711yescharacteristic
226C23×C4C22×D1711yescharacteristic
227C23×C4C22×D1711yescharacteristic
228C23×C4C22×D1712yes—
229C2×C4⋊C4—348no—
230C2×C4⋊C4—344no—
231C2×C4⋊C4—348no—
232C2×C4⋊C4—348no—
233C2×C4⋊C4—684no—
234C2×C42—28no—
235C2×C42—344no—
236C2×C42—682no—
237C2×C42—682no—
238C2×C42Dic3414yes—
239C2×C42D6814yes—
240C2×C42C2×Dic1718yes—
241C2×C42C2×Dic1714yes—
242#95—348no—
243#95—344no—
244#95—342no—
245#95—342no—
246#95—344no—
247#95—344no—
248#95—348no—
249#95—684no—
250#95C2×Dic1714yes—
251#95C2×Dic1718yes—
252#95C2×Dic1714yes—
253C17#1395211yescharacteristic, complemented, Sylow
254C24C23×D1711yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—178no—
269C22×C4—344no—
270C22×C4—344no—
271C22×C4—344no—
272C22×C4—344no—
273C22×C4—344no—
274C22×C4—344no—
275C22×C4—342no—
276C22×C4—342no—
277C22×C4—344no—
278C22×C4—344no—
279C22×C4—344no—
280C22×C4—344no—
281C22×C4—342no—
282C22×C4—342no—
283C22×C4—342no—
284C22×C4—342no—
285C22×C4C2×Dic3412yes—
286C22×C4C2×Dic3412yes—
287C22×C4C2×Dic3412yes—
288C22×C4C2×D6812yes—
289C22×C4C2×D6812yes—
290C22×C4C2×D6812yes—
291C22×C4#6325218yes—
292C22×C4C22×Dic1714yes—
293C22×C4C22×Dic1712yes—
294C22×C4C22×Dic1712yes—
295C22×C4C22×Dic1712yes—
296C22×C4C22×Dic1714yes—
297C4⋊C4—3416no—
298C4⋊C4—6816no—
299C42—216no—
300C42—688no—
301C42C4⋊Dic17116yescomplemented
302C23C22×Dic3411yescharacteristic
303C23C22×D6811yescharacteristic
304C23C2×#6325214yes—
305C23#41×D1714yes—
306C23#1058769611yescharacteristic
307C23#1058769611yescharacteristic
308C23#1058769711yescharacteristic
309C23#1058769711yescharacteristic
310C23C23×Dic1711yescharacteristic
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—348no—
333C2×C4—348no—
334C2×C4—348no—
335C2×C4—684no—
336C2×C4—684no—
337C2×C4—684no—
338C2×C4—684no—
339C2×C4—684no—
340C2×C4—684no—
341C2×C4—684no—
342C2×C4—684no—
343C2×C4—682no—
344C2×C4—682no—
345C2×C4—682no—
346C2×C4—682no—
347C2×C4—682no—
348C2×C4—682no—
349C2×C4C2×C4⋊Dic1714yes—
350C2×C4C2×C4⋊Dic1714yes—
351C2×C4C2×C4⋊Dic1714yes—
352C2×C4#1058761218yes—
353C22C22×C4⋊Dic1711yescharacteristic
354C22C2×#1058761212yes—
355C22#41×Dic1712yes—
356C22#1176329211yescharacteristic
357C22#1176329211yescharacteristic
358C22#1176332112yes—
359C22#1176335912yes—
360C22#1176336212yes—
361C22#1176336612yes—
362C22#1176337811yescharacteristic
363C22#1176338511yescharacteristic
364C22#1176343711yescharacteristic
365C22#1176343711yescharacteristic
366C22#1176348314yes—
367C22#1176349214yes—
368C22#1176350314yes—
369C22#1176352714yes—
370C4—28no—
371C4—24no—
372C4—24no—
373C4—24no—
374C4—44no—
375C4—688no—
376C4—1362no—
377C4—1362no—
378C2#1453119111yescharacteristic
379C2#1453126811yescharacteristic
380C2#1453140711yescharacteristic
381C2#1453164412yes—
382C2#1453167712yes—
383C2#1453441112yes—
384C2#1453442212yes—
385C2#1453443212yes—
386C2#1453444112yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#376C2
socle, prime radical, cosocleC24×C17C24
hypercenterC2×#376D17
hypocenterC17#13952
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
24. C17×#376C4semidirect
104. C42×C17C4⋊C4semidirect
200. #376Dic17semidirect
253. C17#13952semidirect
301. C42C4⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117626772
C2×C4×C4⋊Dic172
#145270492
#145270412
#145270182
#145270142
#145270112
C2×C17×#3762
#1395217