Finite Group Databasebeta

← #17957956

Subgroups — #17957956

Order 4352.

1426 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763687C212yesmaximal
2C2×#11763687C212yesmaximal
3#14543958C212yesmaximal
4C22×#10587582C211yesmaximal, characteristic
5C22×C22⋊#595C2214yes—
6C2×(C2×#595)⋊C4—24no—
7#11763687—24no—
8#11763687—24no—
9C23×Dic34C411yescharacteristic, complemented
10C23×C4×D17C2211yescharacteristic
11C2×#10587582—24no—
12C2×#10587582—24no—
13C2×#10587582—24no—
14C22×#63225C411yescharacteristic, complemented
15C22×#63225C411yescharacteristic, complemented
16C22×#63227C2211yescharacteristic
17C22×#63227C2211yescharacteristic
18C22×C4⋊C4×C17C411yescharacteristic, complemented
19C24×D17C2311yescharacteristic
20C23×#595—24no—
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17C2×C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17C2×C411yescharacteristic
25C23×Dic17C2×C411yescharacteristic
26C23×Dic17C2×C411yescharacteristic
27C22×C4×D17—22no—
28C22×C4×D17—22no—
29C22×C4×D17—44no—
30C22×C4×D17D414yes—
31C22×Dic34—22no—
32C22×Dic34—22no—
33C22×Dic34—22no—
34C22×Dic34—44no—
35C2×C22⋊#595—24no—
36C2×C22⋊#595—24no—
37C2×C22⋊#595—24no—
38C2×C22⋊#595—24no—
39C2×C22⋊#595—24no—
40C2×C22⋊#595Q818yescomplemented
41C2×C4⋊C4×C17—22no—
42C2×C4⋊C4×C17—22no—
43C2×C4⋊C4×C17—22no—
44C2×#63227—22no—
45C2×#63227—22no—
46C2×#63227—22no—
47C2×#63227—22no—
48C2×#63227D414yes—
49C2×#63227D414yes—
50C2×#63225—22no—
51C2×#63225—22no—
52C2×#63225—22no—
53C2×#63225—22no—
54C2×#63225—22no—
55C2×#63225—22no—
56#10587582—232no—
57(C2×#595)⋊C4—48no—
58C24×C17C22×C411yescharacteristic
59C23×D17—42no—
60C23×D17—42no—
61C23×D17C2×D412yes—
62C23×D17C2×D412yes—
63C23×D17C2×D412yes—
64C23×D17C2×Q814yes—
65C23×D17#4114yes—
66C22×#595—28no—
67C22×#595—48no—
68C22×#595—48no—
69C22×#595—44no—
70C22×#595—44no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—42no—
78C22×C4×C17—42no—
79C22×C4×C17—42no—
80C22×C4×C17C22⋊C412yes—
81C22×C4×C17C22⋊C412yes—
82C22×C4×C17C22⋊C412yes—
83C22×Dic17—21no—
84C22×Dic17—21no—
85C22×Dic17—24no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—42no—
91C22×Dic17—42no—
92C22×Dic17C22⋊C412yes—
93C22×Dic17C22⋊C412yes—
94C22×Dic17C22⋊C412yes—
95C2×C4×D17—28no—
96C2×C4×D17—28no—
97C2×C4×D17—44no—
98C2×C4×D17—48no—
99C2×C4×D17—48no—
100C2×Dic34—28no—
101C2×Dic34—48no—
102C2×Dic34—48no—
103C2×Dic34—44no—
104C2×Dic34—44no—
105C22⋊#595—28no—
106C22⋊#595—28no—
107C22⋊#595—44no—
108C22⋊#595—44no—
109C4⋊C4×C17—28no—
110#63227—28no—
111#63227—28no—
112#63225—28no—
113#63225—28no—
114#11381—171nomaximal, Sylow
115C23×C17—22no—
116C23×C17—22no—
117C23×C17C2×C22⋊C411yescharacteristic
118C23×C17C2×C22⋊C411yescharacteristic
119C23×C17C2×C22⋊C411yescharacteristic
120C23×C17C4×Q814yes—
121C22×D17—24no—
122C22×D17—24no—
123C22×D17—42no—
124C22×D17—42no—
125C22×D17—42no—
126C22×D17—42no—
127C22×D17—48no—
128C22×D17—48no—
129C22×D17#12014yes—
130C22×D17C22⋊Q814yes—
131C22×D17C22⋊Q814yes—
132C22×D17C22⋊Q814yes—
133C22×D17#12412yes—
134C22×D17#12412yes—
135C22×D17#12412yes—
136C22×D17C4⋊Q812yes—
137C22×D17C4⋊Q812yes—
138C22×D17C4⋊Q812yes—
139C2×#595—416no—
140C2×#595—48no—
141C2×#595—88no—
142C2×#595—84no—
143C2×#595—84no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—24no—
148C2×C4×C17—42no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×Dic17—24no—
157C2×Dic17—24no—
158C2×Dic17—22no—
159C2×Dic17—22no—
160C2×Dic17—22no—
161C2×Dic17—22no—
162C2×Dic17—42no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—42no—
168C2×Dic17#9914yes—
169C4×D17—416no—
170Dic34—416no—
171C22×C22⋊Q8—171no—
172C2×#342—172no—
173C2×#342—172no—
174C22⋊#95—172no—
175C22×C17—28no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17#37911yescharacteristic
181C22×C17#39112yes—
182C22×C17#39112yes—
183C22×C17#39112yes—
184C22×C17C2×#9912yes—
185C22×C17#41012yes—
186C2×D17—28no—
187C2×D17—24no—
188C2×D17—24no—
189C2×D17—24no—
190C2×D17—24no—
191C2×D17—44no—
192C2×D17—44no—
193C2×D17—42no—
194C2×D17—42no—
195C2×D17—42no—
196C2×D17—42no—
197C2×D17#39314yes—
198C2×D17#39514yes—
199#595—88no—
200C4×C17—42no—
201C4×C17—42no—
202C4×C17—42no—
203Dic17—24no—
204Dic17—42no—
205Dic17—42no—
206C23×Q8—171no—
207C24×C4—171no—
208C2×C22⋊Q8—344no—
209C2×C22⋊Q8—344no—
210C2×C22⋊Q8—344no—
211C22×C4⋊C4—171no—
212C22×C4⋊C4—171no—
213C22×C4⋊C4#59511yescharacteristic, complemented
214C22×C22⋊C4—171no—
215C22×C22⋊C4—171no—
216C22×C22⋊C4—174no—
217C2×#95—344no—
218#342—344no—
219#342—344no—
220C2×C17—22no—
221C2×C17—22no—
222C2×C17#144611yescharacteristic
223C2×C17#178111yescharacteristic
224C2×C17#178211yescharacteristic
225C2×C17#199614yes—
226D17—24no—
227D17—24no—
228C25—171no—
229C22×Q8—342no—
230C22×Q8—342no—
231C22×Q8—342no—
232C22×Q8—684no—
233C23×C4—171no—
234C23×C4—171no—
235C23×C4—171no—
236C23×C4—174no—
237C23×C4—342no—
238C23×C4—342no—
239C23×C4—344no—
240C23×C4—684no—
241C23×C4C2×#59511yescharacteristic
242C23×C4C2×#59511yescharacteristic
243C23×C4C2×#59511yescharacteristic
244C22⋊Q8—3432no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—342no—
249C2×C4⋊C4—342no—
250C2×C4⋊C4—342no—
251C2×C4⋊C4—342no—
252C2×C4⋊C4—342no—
253C2×C4⋊C4—342no—
254C2×C22⋊C4—174no—
255C2×C22⋊C4—174no—
256C2×C22⋊C4—178no—
257C2×C22⋊C4—342no—
258C2×C22⋊C4—342no—
259C2×C22⋊C4—342no—
260C2×C22⋊C4—342no—
261C2×C22⋊C4—344no—
262C2×C22⋊C4—344no—
263C2×C22⋊C4—344no—
264C2×C22⋊C4—344no—
265C2×C22⋊C4—344no—
266#95—688no—
267C17#1138111yescharacteristic, complemented, Sylow
268C24—172no—
269C24—174no—
270C24—174no—
271C24—172no—
272C24—172no—
273C24—682no—
274C24—682no—
275C24C22×#59511yescharacteristic
276C2×Q8—348no—
277C2×Q8—684no—
278C2×Q8—688no—
279C2×Q8—684no—
280C2×Q8—688no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—172no—
291C22×C4—172no—
292C22×C4—172no—
293C22×C4—348no—
294C22×C4—348no—
295C22×C4—344no—
296C22×C4—341no—
297C22×C4—341no—
298C22×C4—341no—
299C22×C4—341no—
300C22×C4—341no—
301C22×C4—341no—
302C22×C4—348no—
303C22×C4—684no—
304C22×C4—688no—
305C22×C4—688no—
306C22×C4—682no—
307C22×C4—682no—
308C22×C4—684no—
309C22×C4—684no—
310C22×C4—688no—
311C22×C4—688no—
312C22×C4C22⋊#59512yes—
313C22×C4C22⋊#59512yes—
314C22×C4C22⋊#59512yes—
315C4⋊C4—28no—
316C4⋊C4—348no—
317C4⋊C4—348no—
318C22⋊C4—348no—
319C22⋊C4—348no—
320C22⋊C4—348no—
321C22⋊C4—348no—
322C22⋊C4—684no—
323C22⋊C4—684no—
324C23—22no—
325C23—22no—
326C23—174no—
327C23—174no—
328C23—172no—
329C23—172no—
330C23—174no—
331C23—174no—
332C23—172no—
333C23—172no—
334C23—172no—
335C23—172no—
336C23—344no—
337C23—344no—
338C23—682no—
339C23—682no—
340C23—688no—
341C23—688no—
342C23—682no—
343C23—682no—
344C23Q8×#59514yes—
345C23C2×C22⋊#59511yescharacteristic
346C23C2×C22⋊#59511yescharacteristic
347C23C2×C22⋊#59511yescharacteristic
348Q8—6816no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—174no—
362C2×C4—344no—
363C2×C4—344no—
364C2×C4—342no—
365C2×C4—342no—
366C2×C4—342no—
367C2×C4—342no—
368C2×C4—6816no—
369C2×C4—682no—
370C2×C4—682no—
371C2×C4—682no—
372C2×C4—682no—
373C2×C4—682no—
374C2×C4—682no—
375C2×C4—688no—
376C2×C4—6816no—
377C2×C4—1368no—
378C2×C4—1364no—
379C2×C4—1364no—
380C22—28no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—174no—
386C22—174no—
387C22—348no—
388C22—344no—
389C22—344no—
390C22—344no—
391C22—344no—
392C22—682no—
393C22—682no—
394C22—684no—
395C22—684no—
396C22—682no—
397C22—682no—
398C22#1176375011yescharacteristic
399C22(C2×#595)⋊Q812yes—
400C22(C2×#595)⋊Q812yes—
401C22(C2×#595)⋊Q812yes—
402C22D17⋊#9912yes—
403C22#1176378812yes—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—344no—
408C4—682no—
409C4—682no—
410C4—1368no—
411C2—22no—
412C2—22no—
413C2—344no—
414C2—344no—
415C2#1454397411yescharacteristic
416C2#1454461111yescharacteristic
417C2#1454461311yescharacteristic
418C2#1454497514yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radical, cosocleC24×D17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#11381
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
9. C23×Dic34C4semidirect
14. C22×#63225C4semidirect
15. C22×#63225C4semidirect
18. C22×C4⋊C4×C17C4semidirect
40. C2×C22⋊#595Q8semidirect
213. C22×C4⋊C4#595semidirect
267. C17#11381semidirect

Maximal subgroups

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

subgroupindex
C2×#117636872
#145439582
C22×#105875822
#1138117