Finite Group Databasebeta

← #17062462

Subgroups — #17062462

Order 4352.

1334 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763435C211yesmaximal, characteristic
2#14531396C212yesmaximal
3#14530137C218yesmaximal, complemented
4C2×#11762576C211yesmaximal, characteristic
5C2×#11762575C212yesmaximal
6C17×#2315C211yesmaximal, characteristic
7#11763435—22no—
8#11763435—24no—
9#11763435C2214yes—
10C2×#10587610—22no—
11C2×#10587564—24no—
12C22×C22⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14#11762682C2218yes—
15#11762681—24no—
16#11762667—24no—
17#11762576C2214yes—
18#11762575C2218yes—
19C22×C22⋊C4×C17C2211yescharacteristic
20C22×C22⋊C4×C17C2211yescharacteristic
21C22×C22⋊C4×C17C2211yescharacteristic
22C17×#385—22no—
23C17×#385C2214yes—
24C17×#380—22no—
25C25×C17C2311yescharacteristic
26C23×C4×C17—21no—
27C23×C4×C17D412yes—
28C23×C4×C17C2312yes—
29C23×Dic17—22no—
30C23×Dic17D414yes—
31C22×Dic34—22no—
32C2×C4⋊C4×C17—24no—
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—22no—
40C2×C22⋊C4×C17—22no—
41C2×C22⋊C4×C17D412yes—
42C2×C22⋊C4×C17Q814yescomplemented
43C2×C22⋊C4×C17C2312yes—
44C2×C22⋊C4×C17C2312yes—
45C2×C22⋊Dic17—22no—
46C2×C22⋊Dic17—22no—
47C2×C22⋊Dic17—22no—
48C2×C22⋊Dic17—22no—
49C2×C22⋊Dic17—22no—
50C2×C22⋊Dic17—24no—
51C2×C22⋊Dic17D414yes—
52C2×C22⋊Dic17Q818yescomplemented
53C2×C22⋊Dic17C2314yes—
54C2×C22⋊Dic17C2314yes—
55C2×C4⋊Dic17—22no—
56#10587610—28no—
57#10587610—28no—
58C2×#63225—24no—
59C2×#63225—28no—
60#10587564—216no—
61#10587564—216no—
62C17×#95—24no—
63#10587502—28no—
64C24×C17—82no—
65C24×C17C2×D412yes—
66C24×C17C2×D412yes—
67C24×C17C2×D411yescharacteristic
68C24×C17C2×D411yescharacteristic
69C24×C17C2×Q812yes—
70C24×C17C2×Q814yes—
71C24×C17#4112yes—
72C24×C17C2411yescharacteristic
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—22no—
76C22×C4×C17—24no—
77C22×C4×C17—24no—
78C22×C4×C17—22no—
79C22×C4×C17—82no—
80C22×C4×C17—82no—
81C22×C4×C17—82no—
82C22×C4×C17C2×D414yes—
83C22×Dic17—24no—
84C22×Dic17—24no—
85C22×Dic17—24no—
86C22×Dic17—28no—
87C22×Dic17—84no—
88C22×Dic17—84no—
89C22×Dic17C2×D418yes—
90C2×Dic34—28no—
91C2×Dic34—84no—
92C4⋊C4×C17—84no—
93C22⋊C4×C17—24no—
94C22⋊C4×C17—24no—
95C22⋊C4×C17—24no—
96C22⋊C4×C17—22no—
97C22⋊C4×C17—41no—
98C22⋊C4×C17—42no—
99C22⋊C4×C17—41no—
100C22⋊C4×C17C2×Q814yescomplemented
101C22⋊Dic17—24no—
102C22⋊Dic17—24no—
103C22⋊Dic17—28no—
104C22⋊Dic17—24no—
105C22⋊Dic17—24no—
106C22⋊Dic17—24no—
107C22⋊Dic17C2×Q818yescomplemented
108C4⋊Dic17—28no—
109#63225—216no—
110#63225—88no—
111#23565—171nomaximal, Sylow
112C23×C17—44no—
113C23×C17—44no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17—42no—
117C23×C17—82no—
118C23×C17—84no—
119C23×C17—82no—
120C23×C17#12011yescharacteristic
121C23×C17#12012yes—
122C23×C17#12011yescharacteristic
123C23×C17C22⋊Q814yes—
124C23×C17C22⋊Q814yes—
125C23×C17C22⋊Q812yes—
126C23×C17C22⋊Q812yes—
127C23×C17C4⋊Q814yes—
128C23×C17C4⋊Q814yes—
129C23×C17C4⋊Q812yes—
130C23×C17C4⋊Q812yes—
131C23×C17C22×D412yes—
132C23×C17C22×D411yescharacteristic
133C23×C17C22×Q811yescharacteristic
134C23×C17C22×Q812yes—
135C23×C17C2×#4111yescharacteristic
136C2×C4×C17—24no—
137C2×C4×C17—24no—
138C2×C4×C17—44no—
139C2×C4×C17—44no—
140C2×C4×C17—44no—
141C2×C4×C17—44no—
142C2×C4×C17—48no—
143C2×C4×C17—42no—
144C2×C4×C17—82no—
145C2×C4×C17—84no—
146C2×Dic17—28no—
147C2×Dic17—28no—
148C2×Dic17—48no—
149C2×Dic17—48no—
150C2×Dic17—48no—
151C2×Dic17—84no—
152C2×Dic17—88no—
153Dic34—88no—
154C2×#543—171no—
155#2315—172no—
156#2315D1711yescharacteristic
157#2062—178no—
158C2×#342—172no—
159C2×#342—171no—
160C22×C17—22no—
161C22×C17—24no—
162C22×C17—24no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—44no—
166C22×C17—42no—
167C22×C17—44no—
168C22×C17—42no—
169C22×C17—44no—
170C22×C17—44no—
171C22×C17—42no—
172C22×C17—42no—
173C22×C17—41no—
174C22×C17—42no—
175C22×C17—41no—
176C22×C17#39512yes—
177C22×C17#39514yes—
178C22×C17#39512yes—
179C22×C17C2×#12011yescharacteristic
180C22×C17C2×C22⋊Q812yes—
181C22×C17C2×C22⋊Q811yescharacteristic
182C22×C17C2×C4⋊Q811yescharacteristic
183C22×C17C2×C4⋊Q812yes—
184C4×C17—44no—
185C4×C17—44no—
186C4×C17—82no—
187Dic17—48no—
188Dic17—48no—
189#543—174no—
190#543—344no—
191#543—342no—
192C2×C22⋊Q8—344no—
193C2×C22⋊Q8—342no—
194C22×C22⋊C4—172no—
195C22×C22⋊C4—172no—
196C22×C22⋊C4C2×D1711yescharacteristic
197C22×C22⋊C4C2×D1711yescharacteristic
198C22×C22⋊C4C2×D1711yescharacteristic
199#385—22no—
200#385—178no—
201#385—344no—
202#385C2×D1714yes—
203#380—22no—
204#380—344no—
205#342—178no—
206#342—174no—
207C2×C17—22no—
208C2×C17—21no—
209C2×C17—21no—
210C2×C17—22no—
211C2×C17—22no—
212C2×C17—24no—
213C2×C17—22no—
214C2×C17C2×#39511yescharacteristic
215C2×C17#302612yes—
216C25C22×D1711yescharacteristic
217C22×Q8—342no—
218C23×C4—21no—
219C23×C4—174no—
220C23×C4—342no—
221C23×C4C22⋊D1712yes—
222C23×C4C22×D1712yes—
223C22⋊Q8—3416no—
224C22⋊Q8—3416no—
225C22⋊Q8—348no—
226C22⋊Q8—348no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—344no—
229C2×C4⋊C4—342no—
230C2×C4⋊C4—348no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—178no—
240C2×C22⋊C4—174no—
241C2×C22⋊C4—174no—
242C2×C22⋊C4—174no—
243C2×C22⋊C4—344no—
244C2×C22⋊C4—342no—
245C2×C22⋊C4—342no—
246C2×C22⋊C4—342no—
247C2×C22⋊C4—342no—
248C2×C22⋊C4—342no—
249C2×C22⋊C4Dic3414yescomplemented
250C2×C22⋊C4C22⋊D1712yes—
251C2×C22⋊C4C22×D1712yes—
252C2×C22⋊C4C22×D1712yes—
253#95—24no—
254#95—348no—
255C17#2356511yescharacteristic, complemented, Sylow
256C24—82no—
257C24C2×Dic3412yes—
258C24#6325212yes—
259C24D4×D1712yes—
260C24D4×D1712yes—
261C24Q8×D1714yes—
262C24C2×C22⋊D1711yescharacteristic
263C24C2×C22⋊D1711yescharacteristic
264C24C23×D1711yescharacteristic
265C2×Q8—348no—
266C2×Q8—1364no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—178no—
277C22×C4—344no—
278C22×C4—344no—
279C22×C4—348no—
280C22×C4—344no—
281C22×C4—1364no—
282C22×C4—1364no—
283C22×C4C2×C22⋊D1714yes—
284C4⋊C4—84no—
285C4⋊C4—3416no—
286C4⋊C4—348no—
287C4⋊C4—1368no—
288C22⋊C4—22no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—178no—
296C22⋊C4—344no—
297C22⋊C4—344no—
298C22⋊C4—344no—
299C22⋊C4—348no—
300C22⋊C4—344no—
301C22⋊C4—344no—
302C22⋊C4C2×Dic3414yescomplemented
303C23—42no—
304C23—42no—
305C23—42no—
306C23—44no—
307C23—44no—
308C23—82no—
309C23—82no—
310C23—84no—
311C23#1058757414yes—
312C23#1058757414yes—
313C23#1058758214yes—
314C23#1058758214yes—
315C23#1058761012yes—
316C23#1058761012yes—
317C23#1058762312yes—
318C23#1058762311yescharacteristic
319C23#1058762311yescharacteristic
320C23#1058763012yes—
321C23#1058763012yes—
322C23C22×Dic3411yescharacteristic
323C23C2×#6325211yescharacteristic
324C23C2×D4×D1712yes—
325C23C2×Q8×D1712yes—
326C23C22×C22⋊D1711yescharacteristic
327Q8—1368no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—84no—
338C2×C4—348no—
339C2×C4—348no—
340C2×C4—688no—
341C2×C4—688no—
342C2×C4—688no—
343C2×C4—1364no—
344C2×C4—1368no—
345C22—22no—
346C22—24no—
347C22—22no—
348C22—22no—
349C22—24no—
350C22—42no—
351C22—44no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22#1176272912yes—
362C22#1176272914yes—
363C22#1176272912yes—
364C22C2×#1058762311yescharacteristic
365C22C2×#1058758212yes—
366C22C2×#1058761011yescharacteristic
367C22C2×#1058763011yescharacteristic
368C22C2×#1058757412yes—
369C4—44no—
370C4—44no—
371C4—82no—
372C4—688no—
373C4—688no—
374C2—21no—
375C2—22no—
376C2—22no—
377C2—22no—
378C2—24no—
379C2—21no—
380C2—22no—
381C2C2×#1176272911yescharacteristic
382C2#1453818012yes—
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC24×C17C24
FrattiniC24C23×D17
FittingC17×#2315C2
socleC22×C17C2×#120
prime radicalC25×C17C23
hypercenter#2315D17
hypocenterC17#23565
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C25 ◁ #2315

Complements

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

NH ≅ G/Ntype
3. #14530137C2semidirect
42. C2×C22⋊C4×C17Q8semidirect
52. C2×C22⋊Dic17Q8semidirect
100. C22⋊C4×C17C2×Q8semidirect
107. C22⋊Dic17C2×Q8semidirect
249. C2×C22⋊C4Dic34semidirect
255. C17#23565semidirect
302. C22⋊C4C2×Dic34semidirect

Maximal subgroups

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

subgroupindex
C2×#117634352
#145313962
#145301372
C2×#117625762
C2×#117625752
C17×#23152
#2356517