Finite Group Databasebeta

← #16942588

Subgroups — #16942588

Order 4352.

798 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762675C211yesmaximal, characteristic
2C2×#11762572C212yesmaximal
3C2×#11762572C212yesmaximal
4#14527014C211yesmaximal, characteristic
5C2×C17×#336C211yesmaximal, characteristic
6C22×C4⋊Dic17C2211yescharacteristic
7C22×C4⋊Dic17C2212yes—
8C22×C4⋊Dic17C2211yescharacteristic
9#11762675—24no—
10C2×#10587502—22no—
11C2×#10587502—22no—
12#11762572—24no—
13#11762572—24no—
14C23×C8×C17C2212yes—
15C22×C42×C17C2211yescharacteristic
16C17×#336C414yescomplemented
17C23×C4×C17D412yes—
18C23×C4×C17C2311yescharacteristic
19C23×Dic17—22no—
20C23×Dic17—21no—
21C23×Dic17—21no—
22C22×C8×C17—22no—
23C22×C8×C17—22no—
24C22×C8×C17—22no—
25C22×C8×C17—22no—
26C22×C8×C17—22no—
27C22×C8×C17—22no—
28C22×C8×C17C2×C414yes—
29C2×C42×C17—22no—
30C2×C42×C17—22no—
31C2×C42×C17C2×C412yes—
32C2×C42×C17Q812yes—
33C2×C4⋊Dic17—22no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17—22no—
38C2×C4⋊Dic17—22no—
39C2×C4⋊Dic17—22no—
40C2×C4⋊Dic17—22no—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17—22no—
43C2×C4⋊Dic17—22no—
44C2×C4⋊Dic17D412yes—
45C2×C4⋊Dic17D414yes—
46C2×C4⋊Dic17Q812yes—
47#10587502—28no—
48#10587502—44no—
49C24×C17C2×D411yescharacteristic
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17C4⋊C414yes—
57C22×C4×C17C22×C411yescharacteristic
58C22×C4×C17C2×D411yescharacteristic
59C22×C4×C17C2×D412yes—
60C22×C4×C17C2×Q811yescharacteristic
61C22×C4×C17C2×Q811yescharacteristic
62C22×C4×C17#4112yes—
63C22×C4×C17#4112yes—
64C22×C4×C17#4112yes—
65C22×C4×C17#4111yescharacteristic
66C22×C4×C17#4111yescharacteristic
67C22×Dic17—24no—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—41no—
71C22×Dic17—42no—
72C22×Dic17—41no—
73C22×Dic17—41no—
74C22×Dic17—42no—
75C22×Dic17—41no—
76C22×Dic17—42no—
77C22×Dic17—42no—
78C22×Dic17—42no—
79C22×Dic17—42no—
80C22×Dic17—42no—
81C22×Dic17—42no—
82C22×Dic17—42no—
83C22×Dic17—42no—
84C2×C8×C17—24no—
85C2×C8×C17—24no—
86C2×C8×C17—24no—
87C2×C8×C17—42no—
88C2×C8×C17—42no—
89C2×C8×C17—42no—
90C2×C8×C17—42no—
91C42×C17—24no—
92C42×C17—24no—
93C4⋊Dic17—24no—
94C4⋊Dic17—24no—
95C4⋊Dic17—24no—
96C4⋊Dic17—24no—
97C4⋊Dic17—42no—
98C4⋊Dic17—42no—
99C4⋊Dic17—42no—
100C4⋊Dic17—42no—
101#11268—171nomaximal, Sylow
102C23×C17C2×C4⋊C411yescharacteristic
103C23×C17#12011yescharacteristic
104C23×C17#12111yescharacteristic
105C23×C17C22⋊Q811yescharacteristic
106C23×C17#12311yescharacteristic
107C23×C17#12411yescharacteristic
108C23×C17#12411yescharacteristic
109C23×C17#13512yes—
110C23×C17#13512yes—
111C23×C17AGL(1,ℤ/8)12yes—
112C23×C17#13712yes—
113C2×C4×C17—24no—
114C2×C4×C17—24no—
115C2×C4×C17—24no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17#11712yes—
121C2×C4×C17C4×D412yes—
122C2×C4×C17C4×D411yescharacteristic
123C2×C4×C17C4×Q811yescharacteristic
124C2×C4×C17#12112yes—
125C2×C4×C17C22⋊Q812yes—
126C2×C4×C17C22⋊Q812yes—
127C2×C4×C17#12312yes—
128C2×C4×C17#12312yes—
129C2×C4×C17#12412yes—
130C2×C4×C17#12512yes—
131C2×C4×C17#12512yes—
132C2×C4×C17#12511yescharacteristic
133C2×C4×C17#12612yes—
134C2×C4×C17#12612yes—
135C2×C4×C17C4⋊Q811yescharacteristic
136C2×Dic17—42no—
137C2×Dic17—44no—
138C2×Dic17—42no—
139C2×Dic17—44no—
140C2×Dic17—44no—
141C2×Dic17—44no—
142C2×Dic17—44no—
143C2×Dic17—82no—
144C2×Dic17—82no—
145C2×Dic17—82no—
146C2×Dic17—82no—
147C2×Dic17—81no—
148C2×Dic17—81no—
149C2×Dic17—82no—
150C2×Dic17—82no—
151C2×Dic17—81no—
152C2×Dic17—81no—
153C8×C17—44no—
154C2×#382—171no—
155C2×#340—172no—
156C2×#340—172no—
157C2×#336D1711yescharacteristic
158C4⋊#95—171no—
159C22×C17#38211yescharacteristic
160C22×C17#38511yescharacteristic
161C22×C17#38911yescharacteristic
162C22×C17#39311yescharacteristic
163C22×C17#39411yescharacteristic
164C22×C17#39711yescharacteristic
165C22×C17#40011yescharacteristic
166C22×C17#42712yes—
167C22×C17#42812yes—
168C22×C17#44912yes—
169C22×C17#45212yes—
170C22×C17#46312yes—
171C22×C17#46412yes—
172C22×C17#47812yes—
173C22×C17#47912yes—
174C22×C17#48212yes—
175C22×C17#48512yes—
176C22×C17#48811yescharacteristic
177C22×C17#48811yescharacteristic
178C22×C17#48812yes—
179C22×C17#48912yes—
180C22×C17#49011yescharacteristic
181C22×C17#49111yescharacteristic
182C4×C17—44no—
183C4×C17#38712yes—
184C4×C17#38812yes—
185C4×C17#39812yes—
186C4×C17#40012yes—
187Dic17—82no—
188Dic17—82no—
189Dic17—84no—
190C23×C8C2×D1712yes—
191C22×C4⋊C4—171no—
192C22×C4⋊C4—171no—
193C22×C4⋊C4—172no—
194C22×C42C2×D1711yescharacteristic
195#382—344no—
196C2×#95—342no—
197C2×#95—342no—
198#340—344no—
199#340—344no—
200#336Dic1714yescomplemented
201C2×C17#144211yescharacteristic
202C2×C17#183511yescharacteristic
203C2×C17#183611yescharacteristic
204C2×C17#185412yes—
205C2×C17#194912yes—
206C2×C17#202511yescharacteristic
207C2×C17#202611yescharacteristic
208C2×C17#204212yes—
209C2×C17#207311yescharacteristic
210C2×C17#207411yescharacteristic
211C2×C17#210112yes—
212C23×C4—342no—
213C23×C4—341no—
214C23×C4—341no—
215C23×C4D6812yes—
216C23×C4C22×D1711yescharacteristic
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8C2×Dic1714yes—
224C2×C4⋊C4—172no—
225C2×C4⋊C4—174no—
226C2×C4⋊C4—172no—
227C2×C4⋊C4—342no—
228C2×C4⋊C4—342no—
229C2×C4⋊C4—342no—
230C2×C4⋊C4—342no—
231C2×C4⋊C4—342no—
232C2×C4⋊C4—342no—
233C2×C4⋊C4—342no—
234C2×C4⋊C4—342no—
235C2×C4⋊C4—342no—
236C2×C4⋊C4—341no—
237C2×C4⋊C4—341no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42Dic3412yes—
241C2×C42C2×Dic1712yes—
242#95—348no—
243#95—684no—
244C17#1126811yescharacteristic, complemented, Sylow
245C24C2×D6811yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—344no—
253C22×C4—342no—
254C22×C4—342no—
255C22×C4—681no—
256C22×C4—682no—
257C22×C4—681no—
258C22×C4—681no—
259C22×C4—681no—
260C22×C4—682no—
261C22×C4—682no—
262C22×C4—682no—
263C22×C4—682no—
264C22×C4—682no—
265C22×C4—682no—
266C22×C4—682no—
267C22×C4—682no—
268C22×C4—682no—
269C22×C4C4⋊Dic1714yes—
270C22×C4C2×Dic3411yescharacteristic
271C22×C4#6325212yes—
272C22×C4#6325212yes—
273C22×C4D4×D1712yes—
274C22×C4D4×D1711yescharacteristic
275C22×C4#6325412yes—
276C22×C4#6325411yescharacteristic
277C22×C4Q8×D1711yescharacteristic
278C22×C4#6325611yescharacteristic
279C22×C4C22×Dic1711yescharacteristic
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—344no—
288C4⋊C4—344no—
289C4⋊C4—344no—
290C4⋊C4—344no—
291C4⋊C4—682no—
292C4⋊C4—682no—
293C4⋊C4—682no—
294C4⋊C4—682no—
295C42—24no—
296C42—24no—
297C23#1058756111yescharacteristic
298C23#1058756111yescharacteristic
299C23#1058756811yescharacteristic
300C23#1058757211yescharacteristic
301C23#1058758111yescharacteristic
302C23#1058758311yescharacteristic
303C23#1058759012yes—
304C23#1058759012yes—
305C23#1058759412yes—
306C23#1058759512yes—
307C23C2×C4⋊Dic1711yescharacteristic
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—684no—
316C2×C4—684no—
317C2×C4—682no—
318C2×C4—684no—
319C2×C4—682no—
320C2×C4—684no—
321C2×C4—684no—
322C2×C4—1362no—
323C2×C4—1362no—
324C2×C4—1362no—
325C2×C4—1362no—
326C2×C4—1361no—
327C2×C4—1361no—
328C2×C4—1362no—
329C2×C4—1362no—
330C2×C4—1361no—
331C2×C4—1361no—
332C2×C4#1058755612yes—
333C2×C4#1058756412yes—
334C2×C4#1058756512yes—
335C2×C4#1058756912yes—
336C2×C4#1058757012yes—
337C2×C4#1058757112yes—
338C2×C4#1058757411yescharacteristic
339C2×C4#1058757512yes—
340C2×C4#1058757611yescharacteristic
341C2×C4#1058758012yes—
342C2×C4#1058758212yes—
343C2×C4#1058758412yes—
344C2×C4#1058761212yes—
345C2×C4D4×Dic1712yes—
346C2×C4D4×Dic1711yescharacteristic
347C2×C4Q8×Dic1711yescharacteristic
348C8—44no—
349C22#1176267511yescharacteristic
350C22#1176268411yescharacteristic
351C22#1176270811yescharacteristic
352C22#1176272211yescharacteristic
353C22#1176272711yescharacteristic
354C22#1176273611yescharacteristic
355C22#1176274911yescharacteristic
356C22#1176284712yes—
357C22#1176285112yes—
358C22#1176295612yes—
359C22#1176297312yes—
360C22#1176302512yes—
361C22#1176303212yes—
362C22#1176311012yes—
363C22#1176311712yes—
364C22#1176313412yes—
365C22#1176315112yes—
366C22#1176316412yes—
367C22#1176316411yescharacteristic
368C22#1176316411yescharacteristic
369C22#1176316912yes—
370C22#1176317411yescharacteristic
371C22#1176317811yescharacteristic
372C4—44no—
373C4—1362no—
374C4—1362no—
375C4—1364no—
376C4#1176269512yes—
377C4#1176270212yes—
378C4#1176274312yes—
379C4#1176275112yes—
380C2#1452704111yescharacteristic
381C2#1452891511yescharacteristic
382C2#1452891811yescharacteristic
383C2#1452898512yes—
384C2#1452955412yes—
385C2#1452993111yescharacteristic
386C2#1452993611yescharacteristic
387C2#1453002612yes—
388C2#1453019911yescharacteristic
389C2#1453020411yescharacteristic
390C2#1453035312yes—
subgroupG/H
centerC24C2×D68
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#336C2
socle, prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenterC2×#336D17
hypocenterC17#11268
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
16. C17×#336C4semidirect
200. #336Dic17semidirect
244. C17#11268semidirect

Maximal subgroups

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

subgroupindex
C2×#117626752
C2×#117625722
#145270142
C2×C17×#3362
#1126817