Finite Group Databasebeta

← #17968613

Subgroups — #17968613

Order 4352.

1126 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
1C22×#10587534C211yesmaximal, characteristic
2C2×#11763772C211yesmaximal, characteristic
3#14545221C212yesmaximal
4#14544600C214yesmaximal, complemented
5#107×#595C214yesmaximal, complemented
6#14544473C212yesmaximal
7D17×#427C211yesmaximal, characteristic
8C2×D17⋊M16C2211yescharacteristic
9C2×#10587677C2211yescharacteristic
10C22×C4⋊#595C2212yes—
11C2×#10587534C418yescomplemented
12C2×#10587534C2212yes—
13C2×#10587534C2212yes—
14#11763772C418yescomplemented
15C2×#10587532C2212yes—
16C2×C8×#595C2212yes—
17C4⋊C4×#595C2218yes—
18C4×C4⋊#595—24no—
19C22⋊C4×#595—24no—
20#11763682C2218yes—
21C22×C8×D17C2211yescharacteristic
22#117×D17C2212yes—
23#11762847C411yescharacteristic, complemented
24#11762846C412yescomplemented
25#11762845C412yescomplemented
26#11762844C412yescomplemented
27#107×D17C2212yes—
28#106×D17C2212yes—
29C17×#427C411yescharacteristic, complemented
30C23×#595—22no—
31C22×C4×D17C2311yescharacteristic
32C2×C22⋊#595D412yes—
33C2×#63247—21no—
34#10587677C2×C414yes—
35#10587677Q814yes—
36C2×C4⋊#595—22no—
37C2×C4⋊#595—24no—
38C2×C4⋊#595C2×C418yes—
39C2×C4⋊#595D412yes—
40C2×C4⋊#595C2314yes—
41C2×C4×#595—28no—
42C2×C4×#595C2312yes—
43D17⋊M16—22no—
44D17⋊M16C2×C414yes—
45C2×#63243—21no—
46C2×#63242C2312yes—
47C22×C8×C17C2×C411yescharacteristic
48C17×#117C2×C412yes—
49#10587612C2×C412yes—
50C22×#3622C2×C411yescharacteristic
51C2×C8×D17C2×C414yes—
52C2×C8×D17C2×C414yes—
53C2×C8×D17C2×C411yescharacteristic
54C2×C8×D17C2×C411yescharacteristic
55C2×C8×D17C2311yescharacteristic
56C2×C8×D17C2311yescharacteristic
57#10587578C2×C414yes—
58C4⋊C4×D17C2314yes—
59C22⋊C4×D17—22no—
60#10587563—22no—
61#10587558—22no—
62C42×D17—22no—
63(C2×#595)⋊C4—28no—
64#10587534C2×C418yes—
65#10587534C2×C418yes—
66#10587532C2×C418yescomplemented
67C8×#595C2×C418yescomplemented
68C17×#107C2×C412yescomplemented
69C17×#106C2×C412yescomplemented
70#10587489C2×C412yescomplemented
71#10587488C2×C412yescomplemented
72#10587479C2×C414yescomplemented
73#10587478C2×C414yescomplemented
74C23×D17C2×D411yescharacteristic
75C22×#595—24no—
76C22×#595—28no—
77C22×#595—84no—
78C22×#595C2×D414yes—
79C22×C4×C17C22×C411yescharacteristic
80C22×Dic17#4111yescharacteristic
81C2×C4×D17—24no—
82C2×C4×D17C22×C411yescharacteristic
83C2×C4×D17C22×C412yes—
84C2×C4×D17C22×C412yes—
85C2×C4×D17C2×D411yescharacteristic
86C2×C4×D17C2×Q812yes—
87C2×C4×D17#4111yescharacteristic
88C2×C4×D17#4112yes—
89C2×C4×D17C2411yescharacteristic
90C22⋊#595C4⋊C418yes—
91#63247—24no—
92C2×#3624—22no—
93C4⋊#595—28no—
94C4⋊#595—24no—
95C4⋊#595C4⋊C418yes—
96C4⋊#595C22×C418yes—
97C4×#595—88no—
98C4×#595C22×C414yes—
99C4×#595C2×Q814yes—
100#63243—24no—
101#63242—22no—
102#63242C22×C414yes—
103C2×C8×C17C4214yescomplemented
104C2×C8×C17C22×C411yescharacteristic
105C2×C8×C17C22×C411yescharacteristic
106C4⋊C4×C17C22×C414yes—
107C22⋊C4×C17—22no—
108C42×C17—22no—
109C22⋊Dic17—22no—
110#63227—24no—
111C4⋊Dic17C22×C414yes—
112#63225—24no—
113C4×Dic17—22no—
114C4×Dic17—24no—
115C2×#3622C4214yescomplemented
116C2×#3622C22×C411yescharacteristic
117C2×#3622C22×C411yescharacteristic
118C8×D17C4214yescomplemented
119C8×D17C4214yescomplemented
120C8×D17C22×C412yes—
121C8×D17C22×C412yes—
122C8×D17C22×C412yes—
123C8×D17C22×C412yes—
124#20455—171nomaximal, Sylow
125C23×C17C4×D411yescharacteristic
126C22×D17—22no—
127C22×D17—22no—
128C22×D17C2×C4⋊C412yes—
129C22×D17C4×D412yes—
130C22×D17C4×D411yescharacteristic
131C22×D17C22×D411yescharacteristic
132C2×#595—24no—
133C2×#595—24no—
134C2×#595—28no—
135C2×#595—48no—
136C2×#595—88no—
137C2×#595C2×C4⋊C418yes—
138C2×C4×C17—24no—
139C2×C4×C17C2×C4212yes—
140C2×C4×C17C4×D411yescharacteristic
141C2×C4×C17C4×Q812yes—
142C2×C4×C17C23×C411yescharacteristic
143C2×Dic17—24no—
144C2×Dic17#11712yes—
145C2×Dic17C4×D411yescharacteristic
146C2×Dic17C4×Q812yes—
147C2×Dic17C2×#4111yescharacteristic
148C4×D17—44no—
149C4×D17—44no—
150C4×D17C2×C4212yes—
151C4×D17C2×C4⋊C412yes—
152C4×D17#11712yes—
153C4×D17C4×D412yes—
154C4×D17C4×Q812yes—
155C4×D17C4×Q812yes—
156C4×D17C23×C411yescharacteristic
157C4×D17C23×C411yescharacteristic
158C4×D17C22×Q811yescharacteristic
159C4×D17C2×#4111yescharacteristic
160#3624—24no—
161C8×C17C2×C4212yes—
162C8×C17C2×C4212yes—
163#3622C2×C4212yes—
164#3622C2×C4212yes—
165C2×#427—171no—
166C22×#107—171no—
167C2×#405—171no—
168#2274—172no—
169#1776—174no—
170C4×#107—174no—
171#1736—172no—
172C22×C17—22no—
173C22×C17C4×C4⋊C412yes—
174C22×C17C2×C4×D411yescharacteristic
175C2×D17—22no—
176C2×D17—24no—
177C2×D17—22no—
178C2×D17—22no—
179C2×D17C4×C4⋊C412yes—
180C2×D17C22×C4⋊C411yescharacteristic
181C2×D17C2×C4×D411yescharacteristic
182C2×D17#57612yes—
183C2×D17#57812yes—
184#595—28no—
185#595—48no—
186C4×C17—44no—
187C4×C17C4×C4⋊C412yes—
188C4×C17C22×C4211yescharacteristic
189C4×C17C2×C4×Q811yescharacteristic
190Dic17—44no—
191Dic17C4×C4⋊C412yes—
192Dic17C2×#11711yescharacteristic
193Dic17C2×C4×Q811yescharacteristic
194C22×M16—171no—
195C23×C8—171no—
196C2×#117—171no—
197C2×#117—172no—
198C22×C4⋊C4—172no—
199#427—171no—
200#427—172no—
201#427—172no—
202#427—172no—
203#427#59511yescharacteristic, complemented
204C2×#107—178no—
205C2×#107—172no—
206C2×#107—172no—
207C2×#107—172no—
208C2×#106—172no—
209#405—178no—
210C2×#97—172no—
211C2×C4×C8—172no—
212C4×C4⋊C4—178no—
213C4×C4⋊C4—344no—
214C4×C22⋊C4—344no—
215#340—178no—
216C2×C17—22no—
217C2×C17C2×C4×C4⋊C411yescharacteristic
218C2×C17#295112yes—
219D17—22no—
220D17#291312yes—
221D17#294811yescharacteristic
222D17#295011yescharacteristic
223C23×C4—171no—
224C23×C4—342no—
225C2×M16—174no—
226C2×M16—341no—
227C2×M16—342no—
228C2×M16—341no—
229C22×C8—171no—
230C22×C8—174no—
231C22×C8—174no—
232C22×C8—171no—
233C22×C8—171no—
234C22×C8—171no—
235C22×C8—171no—
236C22×C8—172no—
237C22×C8C2×#59511yescharacteristic
238#117—174no—
239#117—174no—
240#117—172no—
241#117—174no—
242#117—342no—
243#117—342no—
244#117C2×#59512yes—
245C2×C4⋊C4—172no—
246C2×C4⋊C4—178no—
247C2×C4⋊C4—174no—
248C2×C4⋊C4—174no—
249C2×C4⋊C4—344no—
250C2×C4⋊C4—342no—
251C2×C22⋊C4—172no—
252C2×C22⋊C4—342no—
253C2×C42—172no—
254C2×C42—342no—
255C2×C42—348no—
256#107—178no—
257#107—178no—
258#107—172no—
259#107—174no—
260#107C2×#59512yescomplemented
261#106—172no—
262#106—174no—
263#106C2×#59512yescomplemented
264#97—178no—
265C4×C8—178no—
266#95—348no—
267C17#2045511yescharacteristic, complemented, Sylow
268C24—171no—
269C22×C4—171no—
270C22×C4—172no—
271C22×C4—172no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—172no—
275C22×C4—172no—
276C22×C4—171no—
277C22×C4—171no—
278C22×C4—174no—
279C22×C4—344no—
280C22×C4—344no—
281C22×C4—348no—
282C22×C4—1364no—
283C22×C4C22×#59511yescharacteristic
284M16—344no—
285M16—344no—
286C2×C8—174no—
287C2×C8—171no—
288C2×C8—171no—
289C2×C8—172no—
290C2×C8—172no—
291C2×C8—172no—
292C2×C8—172no—
293C2×C8—174no—
294C2×C8—174no—
295C2×C8—174no—
296C2×C8—342no—
297C2×C8—342no—
298C2×C8C4×#59514yescomplemented
299C2×C8C22×#59511yescharacteristic
300C2×C8C22×#59511yescharacteristic
301C4⋊C4—178no—
302C4⋊C4—178no—
303C4⋊C4—174no—
304C4⋊C4—344no—
305C4⋊C4—348no—
306C4⋊C4—344no—
307C4⋊C4C22×#59514yes—
308C22⋊C4—22no—
309C22⋊C4—178no—
310C22⋊C4—344no—
311C22⋊C4—342no—
312C42—22no—
313C42—174no—
314C42—174no—
315C42—344no—
316C42—342no—
317C42—1368no—
318C23—171no—
319C23—172no—
320C23—172no—
321C23—171no—
322C23—342no—
323C23—342no—
324C23D4×#59511yescharacteristic
325C2×C4—24no—
326C2×C4—172no—
327C2×C4—172no—
328C2×C4—171no—
329C2×C4—171no—
330C2×C4—172no—
331C2×C4—172no—
332C2×C4—172no—
333C2×C4—172no—
334C2×C4—171no—
335C2×C4—171no—
336C2×C4—172no—
337C2×C4—171no—
338C2×C4—171no—
339C2×C4—172no—
340C2×C4—178no—
341C2×C4—344no—
342C2×C4—344no—
343C2×C4—348no—
344C2×C4—344no—
345C2×C4—684no—
346C2×C4—684no—
347C2×C4—688no—
348C2×C4—1368no—
349C2×C4C2×C4×#59512yes—
350C2×C4D4×#59511yescharacteristic
351C2×C4Q8×#59512yes—
352C2×C4C23×#59511yescharacteristic
353C8—172no—
354C8—172no—
355C8—344no—
356C8C2×C4×#59512yes—
357C8C2×C4×#59512yes—
358C22—22no—
359C22—171no—
360C22—171no—
361C22—172no—
362C22—172no—
363C22—172no—
364C22—342no—
365C22—344no—
366C22—342no—
367C22—342no—
368C22C4⋊C4×#59512yes—
369C22C2×D4×#59511yescharacteristic
370C4—44no—
371C4—171no—
372C4—171no—
373C4—172no—
374C4—348no—
375C4—688no—
376C4—684no—
377C4C4⋊C4×#59512yes—
378C4C22×C4×#59511yescharacteristic
379C4C2×Q8×#59511yescharacteristic
380C2—22no—
381C2—172no—
382C2—171no—
383C2—171no—
384C2—342no—
385C2C2×C4⋊C4×#59511yescharacteristic
386C2#1454554612yes—
subgroupG/H
centerC22C2×D4×#595
commutatorC4×C17C22×C42
FrattiniC2×C4C23×#595
FittingC17×#427C4
socleC22×C17C2×C4×D4
prime radicalC23×D17C2×D4
cosocleC2×C4×D17C24
hypercenter#427#595
hypocenterC17#20455
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
4. #14544600C2semidirect
5. #107×#595C2semidirect
11. C2×#10587534C4semidirect
14. #11763772C4semidirect
23. #11762847C4semidirect
24. #11762846C4semidirect
25. #11762845C4semidirect
26. #11762844C4semidirect
29. C17×#427C4semidirect
66. #10587532C2×C4semidirect
67. C8×#595C2×C4semidirect
68. C17×#107C2×C4semidirect
69. C17×#106C2×C4semidirect
70. #10587489C2×C4semidirect
71. #10587488C2×C4semidirect
72. #10587479C2×C4semidirect
73. #10587478C2×C4semidirect
103. C2×C8×C17C42semidirect
115. C2×#3622C42semidirect
118. C8×D17C42semidirect
119. C8×D17C42semidirect
203. #427#595semidirect
260. #107C2×#595semidirect
263. #106C2×#595semidirect
267. C17#20455semidirect
298. C2×C8C4×#595semidirect

Maximal subgroups

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

subgroupindex
C22×#105875342
C2×#117637722
#145452212
#145446002
#107×#5952
#145444732
D17×#4272
#2045517