Finite Group Databasebeta

← #17037286

Subgroups — #17037286

Order 4352.

858 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic68C211yesmaximal, characteristic
2#14531261C211yesmaximal, characteristic
3C42×Dic34C211yesmaximal, characteristic
4#14529419C212yesmaximal
5#14529356C212yesmaximal
6#14529124C212yesmaximal
7#14529055C212yesmaximal
8C4×#10587488C211yesmaximal, characteristic
9C4×#10587487C211yesmaximal, characteristic
10C4×#10587487C211yesmaximal, characteristic
11C17×#1744C211yesmaximal, characteristic
12C22×Dic68C2211yescharacteristic
13C2×C4×Dic34C2212yes—
14C2×C4×Dic34C2211yescharacteristic
15C2×C4×Dic34C2211yescharacteristic
16C4×Dic68C418yescomplemented
17C2×#10587489C2211yescharacteristic
18C2×#10587488C2212yes—
19C2×#10587487C2212yes—
20C2×#10587487C2212yes—
21C2×#10587487C2211yescharacteristic
22C2×#10587487C2211yescharacteristic
23#11762715C2212yes—
24#11762707C2212yes—
25#11762693—22no—
26C4×#63225—22no—
27C4×#63225—21no—
28C4×#63225—21no—
29C4×C4⋊Dic17C2212yes—
30C4×C4⋊Dic17C2211yescharacteristic
31C4×C4⋊Dic17C2211yescharacteristic
32#11762655—21no—
33C42×Dic17—21no—
34#11762572C2212yes—
35#11762572C2212yes—
36#11762572C2212yes—
37C2×C4×C8×C17C2211yescharacteristic
38C2×C4×C8×C17C2211yescharacteristic
39C43×C17C2211yescharacteristic
40C17×#336C2212yes—
41C17×#336C2212yes—
42C22×Dic34C2311yescharacteristic
43C22×Dic34C2311yescharacteristic
44C22×C8×C17C2312yes—
45C22×C8×C17C2311yescharacteristic
46C22×C8×C17C2311yescharacteristic
47C2×C42×C17D412yes—
48C2×C42×C17D411yescharacteristic
49C2×C42×C17D411yescharacteristic
50C2×C42×C17C2312yes—
51C2×C42×C17C2311yescharacteristic
52C2×C4⋊Dic17C2312yes—
53C2×C4⋊Dic17C2311yescharacteristic
54C2×C4⋊Dic17C2311yescharacteristic
55C2×C4⋊Dic17C2312yes—
56C2×#63225—21no—
57C2×#63225—22no—
58C2×#63225—21no—
59C2×#63225—22no—
60C2×C4×Dic17—22no—
61C2×C4×Dic17—21no—
62C2×C4×Dic17—21no—
63C2×C4×Dic17—22no—
64C2×C4×Dic17—21no—
65C2×C4×Dic17—21no—
66C2×Dic68—24no—
67C2×Dic68C2×C414yes—
68C4×Dic34—24no—
69C4×Dic34—24no—
70C4×Dic34—22no—
71C4×Dic34—22no—
72C4×Dic34C2×C414yes—
73C4×Dic34C2×C414yes—
74C4×C8×C17—22no—
75C4×C8×C17C2×C414yes—
76#10587502—21no—
77#10587502—22no—
78#10587502—22no—
79#10587502—22no—
80#10587502—22no—
81#10587502—21no—
82#10587489C2×C414yes—
83#10587488—24no—
84#10587487—24no—
85#10587487—24no—
86#10587487C2×C414yes—
87#10587487C2×C414yes—
88C22×C4×C17C2×D412yes—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D412yes—
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2411yescharacteristic
93C22×Dic17—21no—
94C22×Dic17—21no—
95C22×Dic17—21no—
96C22×Dic17—22no—
97C22×Dic17—21no—
98C22×Dic17—22no—
99C2×Dic34—21no—
100C2×Dic34—22no—
101C2×Dic34—22no—
102C2×Dic34—22no—
103C2×Dic34—21no—
104C2×Dic34C22×C412yes—
105C2×Dic34C22×C412yes—
106C2×Dic34#4114yes—
107C2×C8×C17—22no—
108C2×C8×C17—22no—
109C2×C8×C17—22no—
110C2×C8×C17—22no—
111C2×C8×C17—22no—
112C2×C8×C17C22×C412yes—
113C2×C8×C17C22×C412yes—
114C42×C17—24no—
115C42×C17—22no—
116C42×C17—22no—
117C42×C17—22no—
118C42×C17—22no—
119C42×C17C22×C412yes—
120C42×C17#4112yes—
121C4⋊Dic17—21no—
122C4⋊Dic17—21no—
123C4⋊Dic17—22no—
124C4⋊Dic17—22no—
125C4⋊Dic17—22no—
126C4⋊Dic17C22×C412yes—
127C4⋊Dic17C22×C412yes—
128C4⋊Dic17#4114yes—
129#63225—28no—
130#63225—24no—
131#63225—24no—
132#63225—44no—
133C4×Dic17—28no—
134C4×Dic17—24no—
135C4×Dic17—24no—
136C4×Dic17—44no—
137C4×Dic17—42no—
138C4×Dic17—42no—
139Dic68—28no—
140#21116—171nomaximal, Sylow
141C23×C17C22×D411yescharacteristic
142C2×C4×C17—22no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17C4×D412yes—
149C2×C4×C17C4×D412yes—
150C2×C4×C17#13514yes—
151C2×C4×C17#13712yes—
152C2×C4×C17#13712yes—
153C2×C4×C17C23×C411yescharacteristic
154C2×C4×C17C2×#4111yescharacteristic
155C2×C4×C17C2×#4112yes—
156C2×C4×C17C2×#4112yes—
157C2×Dic17—22no—
158C2×Dic17—24no—
159C2×Dic17—22no—
160C2×Dic17—22no—
161C2×Dic17—22no—
162C2×Dic17—24no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—42no—
168C2×Dic17—42no—
169C2×Dic17—42no—
170C2×Dic17—42no—
171Dic34—22no—
172Dic34—24no—
173Dic34—22no—
174Dic34—22no—
175Dic34—42no—
176Dic34#11714yes—
177C8×C17—22no—
178C8×C17—22no—
179C8×C17—42no—
180C2×C4×Q16—171no—
181#2295—171no—
182C42×Q8—171no—
183#1927—172no—
184#1918—172no—
185#1878—172no—
186#1865—172no—
187C4×#106—171no—
188C4×#103—171no—
189C4×#103—171no—
190#1744D1711yescharacteristic
191C22×C17C2×C4×D411yescharacteristic
192C22×C17#52512yes—
193C22×C17C2×#13512yes—
194C22×C17C2×#13711yescharacteristic
195C22×C17C2×#13711yescharacteristic
196C4×C17—24no—
197C4×C17—22no—
198C4×C17—42no—
199C4×C17#44114yes—
200C4×C17C2×#11711yescharacteristic
201C4×C17C4×#4111yescharacteristic
202C4×C17#52912yes—
203Dic17—24no—
204Dic17—44no—
205Dic17—42no—
206Dic17—42no—
207Dic17—42no—
208Dic17—82no—
209C22×Q16—171no—
210C2×C4×Q8—171no—
211C2×C4×Q8—171no—
212C2×C4×Q8—172no—
213C4×Q16—178no—
214C2×#107—171no—
215C2×#106—172no—
216C2×#103—171no—
217C2×#103—171no—
218C2×#103—172no—
219C2×#103—172no—
220C2×C4×C8C2×D1711yescharacteristic
221C2×C4×C8C2×D1711yescharacteristic
222#391—172no—
223#389—172no—
224#387—342no—
225C4×C4⋊C4—172no—
226C4×C4⋊C4—171no—
227C4×C4⋊C4—171no—
228C4×C4⋊C4—342no—
229C4×C4⋊C4—341no—
230C4×C4⋊C4—341no—
231#376—341no—
232C43—341no—
233C43C2×D1711yescharacteristic
234#340—172no—
235#340—172no—
236#340—172no—
237#336C2×D1712yes—
238#336C2×D1712yes—
239C2×C17#229011yescharacteristic
240C2×C17C4×#13511yescharacteristic
241C2×C17C2×#44111yescharacteristic
242C2×C17#310712yes—
243C2×C17#310812yes—
244C22×Q8—171no—
245C22×Q8—171no—
246C2×Q16—174no—
247C2×Q16—344no—
248C22×C8C22×D1712yes—
249C22×C8C22×D1711yescharacteristic
250C22×C8C22×D1711yescharacteristic
251C4×Q8—174no—
252C4×Q8—174no—
253C4×Q8—344no—
254C4×Q8—344no—
255C4×Q8—342no—
256C4×Q8—342no—
257C2×C4⋊C4—172no—
258C2×C4⋊C4—171no—
259C2×C4⋊C4—171no—
260C2×C4⋊C4—172no—
261C2×C4⋊C4—342no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4—342no—
265C2×C42—342no—
266C2×C42—341no—
267C2×C42—341no—
268C2×C42—342no—
269C2×C42—341no—
270C2×C42—341no—
271C2×C42D6812yes—
272C2×C42D6811yescharacteristic
273C2×C42D6811yescharacteristic
274C2×C42C22×D1712yes—
275C2×C42C22×D1711yescharacteristic
276#107—174no—
277#106—344no—
278#103—174no—
279#103—174no—
280#103—344no—
281#103—344no—
282C4×C8—22no—
283C4×C8C4×D1714yes—
284#95—342no—
285#95—342no—
286#95—342no—
287#95—341no—
288#95—341no—
289#95—342no—
290C17#2111611yescharacteristic, complemented, Sylow
291C2×Q8—172no—
292C2×Q8—172no—
293C2×Q8—174no—
294C2×Q8—341no—
295C2×Q8—341no—
296C2×Q8—342no—
297C2×Q8—342no—
298C2×Q8—342no—
299C22×C4—341no—
300C22×C4—341no—
301C22×C4—341no—
302C22×C4—342no—
303C22×C4—341no—
304C22×C4—342no—
305C22×C4C2×D6812yes—
306C22×C4C2×D6811yescharacteristic
307C22×C4C2×D6812yes—
308C22×C4C2×D6811yescharacteristic
309C22×C4C23×D1711yescharacteristic
310Q16—348no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8C2×C4×D1712yes—
317C2×C8C2×C4×D1712yes—
318C4⋊C4—172no—
319C4⋊C4—172no—
320C4⋊C4—174no—
321C4⋊C4—348no—
322C4⋊C4—341no—
323C4⋊C4—341no—
324C4⋊C4—342no—
325C4⋊C4—344no—
326C4⋊C4—344no—
327C4⋊C4—342no—
328C4⋊C4—342no—
329C4⋊C4—684no—
330C42—22no—
331C42—22no—
332C42—24no—
333C42—22no—
334C42—22no—
335C42—348no—
336C42—344no—
337C42—344no—
338C42—684no—
339C42—682no—
340C42—682no—
341C42C2×C4×D1712yes—
342C42#6325212yes—
343C23C22×D6811yescharacteristic
344Q8—174no—
345Q8—342no—
346Q8—344no—
347Q8—342no—
348Q8—342no—
349Q8—682no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—342no—
357C2×C4—344no—
358C2×C4—342no—
359C2×C4—342no—
360C2×C4—342no—
361C2×C4—344no—
362C2×C4—682no—
363C2×C4—682no—
364C2×C4—682no—
365C2×C4—682no—
366C2×C4—682no—
367C2×C4—682no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4C4×D6812yes—
371C2×C4C4×D6812yes—
372C2×C4#1058759014yes—
373C2×C4#1058759512yes—
374C2×C4#1058759512yes—
375C2×C4C22×C4×D1711yescharacteristic
376C2×C4C2×#6325211yescharacteristic
377C2×C4#41×D1712yes—
378C2×C4#41×D1712yes—
379C8—22no—
380C8—22no—
381C8—42no—
382C22C2×C4×D6811yescharacteristic
383C22#1176332112yes—
384C22C2×#1058759012yes—
385C22C2×#1058759511yescharacteristic
386C22C2×#1058759511yescharacteristic
387C4—22no—
388C4—24no—
389C4—42no—
390C4—344no—
391C4—682no—
392C4—682no—
393C4—684no—
394C4—682no—
395C4—1362no—
396C4#1176291714yes—
397C4#117×D1711yescharacteristic
398C4#1176327711yescharacteristic
399C4#1176333712yes—
400C2#1453122811yescharacteristic
401C2C4×#1058759011yescharacteristic
402C2C2×#1176291711yescharacteristic
403C2#1453885612yes—
404C2#1453886612yes—
subgroupG/H
centerC22×C4C2×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1744C2
socle, prime radicalC23×C17C22×D4
cosocleC22×C4×C17C24
hypercenter#1744D17
hypocenterC17#21116
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1744

Complements

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

NH ≅ G/Ntype
16. C4×Dic68C4semidirect
290. C17#21116semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic682
#145312612
C42×Dic342
#145294192
#145293562
#145291242
#145290552
C4×#105874882
C4×#105874872
C17×#17442
#2111617