Finite Group Databasebeta

← #17968903

Subgroups — #17968903

Order 4352.

1210 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×#11763779C211yesmaximal, characteristic
2C22×C22⋊#3624C211yesmaximal, characteristic
3#14544526C214yesmaximal, complemented
4#14544521C212yesmaximal, complemented
5C22⋊C4×#3624C212yesmaximal, complemented
6#14544514C214yesmaximal, complemented
7#14531162C211yesmaximal, characteristic
8C22×#63247C2212yes—
9C23×#3624C2211yescharacteristic
10C23×#3624C2211yescharacteristic
11#11763779C418yescomplemented
12C2×C22⋊#3624C418yescomplemented
13C2×C22⋊#3624C2212yes—
14C2×C22⋊#3624C2212yes—
15C2×C22⋊#3624C2212yes—
16C2×C22⋊#3624C2212yes—
17C2×#10587542C2214yescomplemented
18C2×C4×#3624C2214yescomplemented
19#11763675C2214yescomplemented
20#11763675C2214yescomplemented
21C24×Dic17C2211yescharacteristic
22C2×#10587563C2212yes—
23#11762670C412yes—
24#11762667C412yes—
25#11762667C412yes—
26#11762665C411yescharacteristic
27C22⋊C4×Dic17C2212yes—
28C22⋊C4×Dic17C2212yes—
29C17×#379C411yescharacteristic
30C25×C17C2×C411yescharacteristic
31C23×Dic17—42no—
32C23×Dic17—42no—
33C23×Dic17C2×C411yescharacteristic
34C23×Dic17C2×C411yescharacteristic
35C23×Dic17C2×C411yescharacteristic
36C23×Dic17C2×C414yes—
37C23×Dic17C2×C414yes—
38C23×Dic17C2311yescharacteristic
39C23×Dic17C2311yescharacteristic
40C23×Dic17C2311yescharacteristic
41C2×#63247—42no—
42C2×#63247—42no—
43C2×#63247C2×C418yes—
44C22×#3624—42no—
45C22×#3624—42no—
46C22×#3624C2×C414yes—
47C22×#3624C2×C414yes—
48C22×#3624C2312yes—
49C22×#3624C2314yes—
50C22×#3624C2312yes—
51C2×C22⋊C4×C17C2×C412yes—
52C2×C22⋊C4×C17C2×C412yes—
53C2×C22⋊C4×C17C2×C412yes—
54C2×C22⋊Dic17C2×C412yes—
55C2×C22⋊Dic17C2×C412yes—
56C2×C22⋊Dic17C2×C412yes—
57C2×#63225C2×C414yes—
58C2×C4×Dic17C2314yes—
59#10587563—42no—
60#10587563—42no—
61C22⋊#3624C2×C418yes—
62C22⋊#3624C2×C418yes—
63C22⋊#3624C2×C418yes—
64C22⋊#3624C2×C418yes—
65#10587542—44no—
66C4×#3624—44no—
67#10587502C2×C414yes—
68#10587502C2×C414yes—
69C24×C17—24no—
70C24×C17—42no—
71C24×C17—42no—
72C24×C17C4214yes—
73C24×C17C22×C411yescharacteristic
74C24×C17C22×C411yescharacteristic
75C24×C17C22×C411yescharacteristic
76C22×C4×C17C22×C414yes—
77C22×Dic17—22no—
78C22×Dic17—22no—
79C22×Dic17—22no—
80C22×Dic17—22no—
81C22×Dic17—22no—
82C22×Dic17—22no—
83C22×Dic17—44no—
84C22×Dic17—44no—
85C22×Dic17—48no—
86C22×Dic17—42no—
87C22×Dic17—42no—
88C22×Dic17—42no—
89C22×Dic17C4214yes—
90C22×Dic17C4214yes—
91C22×Dic17C4214yes—
92C22×Dic17C22×C414yes—
93C22×Dic17C22×C411yescharacteristic
94C22×Dic17C22×C411yescharacteristic
95C22×Dic17C22×C411yescharacteristic
96C22×Dic17C22×C412yes—
97C22×Dic17C22×C412yes—
98C22×Dic17C22×C412yes—
99C22×Dic17C22×C412yes—
100C22×Dic17C22×C412yes—
101C22×Dic17C22×C412yes—
102C22×Dic17C2411yescharacteristic
103#63247—48no—
104C2×#3624—24no—
105C2×#3624—24no—
106C2×#3624—24no—
107C2×#3624—24no—
108C2×#3624—44no—
109C2×#3624—44no—
110C2×#3624C22×C414yes—
111C2×#3624C22×C414yes—
112C2×#3624C22×C418yes—
113C22⋊C4×C17—24no—
114C22⋊C4×C17—42no—
115C22⋊C4×C17—42no—
116C22⋊Dic17—24no—
117C22⋊Dic17—42no—
118C22⋊Dic17—42no—
119#63225—44no—
120C4×Dic17—24no—
121C4×Dic17—24no—
122#20527—171nomaximal, Sylow
123C23×C17—22no—
124C23×C17—22no—
125C23×C17—22no—
126C23×C17—22no—
127C23×C17—22no—
128C23×C17—22no—
129C23×C17—22no—
130C23×C17—22no—
131C23×C17—28no—
132C23×C17—44no—
133C23×C17—44no—
134C23×C17—416no—
135C23×C17C2×C4212yes—
136C23×C17C2×C4212yes—
137C23×C17C2×C4212yes—
138C23×C17#13114yes—
139C23×C17C23×C411yescharacteristic
140C2×C4×C17—24no—
141C2×C4×C17—24no—
142C2×C4×C17—24no—
143C2×Dic17—22no—
144C2×Dic17—22no—
145C2×Dic17—22no—
146C2×Dic17—22no—
147C2×Dic17—22no—
148C2×Dic17—24no—
149C2×Dic17—24no—
150C2×Dic17—22no—
151C2×Dic17—22no—
152C2×Dic17—22no—
153C2×Dic17—22no—
154C2×Dic17—24no—
155C2×Dic17—24no—
156C2×Dic17—24no—
157C2×Dic17—24no—
158C2×Dic17—44no—
159C2×Dic17—44no—
160C2×Dic17—44no—
161C2×Dic17C2×C4212yes—
162C2×Dic17C2×C4212yes—
163C2×Dic17C2×C4212yes—
164C2×Dic17C23×C411yescharacteristic
165C2×Dic17C23×C411yescharacteristic
166C2×Dic17H(5,𝔽2)12yes—
167C2×Dic17H(5,𝔽2)12yes—
168#3624—24no—
169#3624—24no—
170#3624—28no—
171C2×#407—171no—
172C22×C22⋊C8—171no—
173#2280—171no—
174#1753—174no—
175#1752—172no—
176C8×C22⋊C4—172no—
177#1750—174no—
178C22×C17—22no—
179C22×C17—28no—
180C22×C17—22no—
181C22×C17—22no—
182C22×C17—22no—
183C22×C17—22no—
184C22×C17—22no—
185C22×C17—22no—
186C22×C17—22no—
187C22×C17—416no—
188C22×C17—44no—
189C22×C17—44no—
190C22×C17#40514yes—
191C22×C17C22×C4211yescharacteristic
192C22×C17#51812yes—
193C22×C17C2×#13112yes—
194C22×C17#56812yes—
195C4×C17—44no—
196Dic17—22no—
197Dic17—22no—
198Dic17—22no—
199Dic17—44no—
200Dic17#51811yescharacteristic
201Dic17#51811yescharacteristic
202Dic17#51812yes—
203C24×C4—171no—
204C22×M16—172no—
205C23×C8—171no—
206C23×C8—171no—
207C2×#117—172no—
208#407—178no—
209C2×C22⋊C8—178no—
210C2×C22⋊C8—172no—
211C2×C22⋊C8—172no—
212C2×C22⋊C8—172no—
213C2×C22⋊C8—172no—
214C2×#97—174no—
215C2×C4×C8—174no—
216#381—172no—
217#380—172no—
218#380—172no—
219#379—171no—
220#379#59511yescharacteristic
221C4×C22⋊C4—172no—
222C4×C22⋊C4—172no—
223#336—174no—
224#336—174no—
225C2×C17—24no—
226C2×C17—42no—
227C2×C17—42no—
228C2×C17#227311yescharacteristic
229C2×C17C2×#40511yescharacteristic
230C2×C17#287311yescharacteristic
231C2×C17#297614yes—
232C25C2×#59511yescharacteristic
233C23×C4—171no—
234C23×C4—171no—
235C23×C4—171no—
236C23×C4—174no—
237C23×C4—174no—
238C23×C4—171no—
239C23×C4—171no—
240C23×C4—171no—
241C23×C4—682no—
242C23×C4—682no—
243C2×M16—178no—
244C2×M16—682no—
245C2×M16—682no—
246C22×C8—172no—
247C22×C8—172no—
248C22×C8—174no—
249C22×C8—174no—
250C22×C8—174no—
251C22×C8—682no—
252C22×C8—682no—
253#117—682no—
254#117—682no—
255C2×C4⋊C4—174no—
256C2×C22⋊C4—172no—
257C2×C22⋊C4—172no—
258C2×C22⋊C4—172no—
259C2×C22⋊C4C2×#59512yes—
260C2×C22⋊C4C2×#59512yes—
261C2×C22⋊C4C2×#59512yes—
262C2×C42—174no—
263C22⋊C8—178no—
264C22⋊C8—178no—
265C22⋊C8—178no—
266C22⋊C8—178no—
267#97—684no—
268C4×C8—684no—
269#95—174no—
270#95—174no—
271C17#2052711yescharacteristic, complemented, Sylow
272C24—24no—
273C24—42no—
274C24—42no—
275C24C4×#59514yes—
276C24C22×#59511yescharacteristic
277C24C22×#59511yescharacteristic
278C24C22×#59511yescharacteristic
279C22×C4—171no—
280C22×C4—172no—
281C22×C4—172no—
282C22×C4—172no—
283C22×C4—172no—
284C22×C4—172no—
285C22×C4—172no—
286C22×C4—171no—
287C22×C4—171no—
288C22×C4—171no—
289C22×C4—174no—
290C22×C4—174no—
291C22×C4—174no—
292C22×C4—174no—
293C22×C4—342no—
294C22×C4—342no—
295C22×C4—342no—
296C22×C4—342no—
297C22×C4—342no—
298C22×C4—342no—
299C22×C4—684no—
300C22×C4—684no—
301C22×C4—688no—
302C22×C4—682no—
303C22×C4—682no—
304C22×C4—682no—
305C22×C4C22×#59514yes—
306M16—688no—
307C2×C8—174no—
308C2×C8—174no—
309C2×C8—178no—
310C2×C8—344no—
311C2×C8—344no—
312C2×C8—344no—
313C2×C8—344no—
314C2×C8—684no—
315C2×C8—684no—
316C4⋊C4—684no—
317C22⋊C4—24no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—344no—
321C22⋊C4—682no—
322C22⋊C4—682no—
323C42—344no—
324C42—344no—
325C23—22no—
326C23—22no—
327C23—28no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—44no—
335C23—44no—
336C23—416no—
337C23C2×C4×#59512yes—
338C23C2×C4×#59512yes—
339C23C2×C4×#59512yes—
340C23#1058767814yes—
341C23C23×#59511yescharacteristic
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—171no—
346C2×C4—171no—
347C2×C4—172no—
348C2×C4—172no—
349C2×C4—172no—
350C2×C4—172no—
351C2×C4—172no—
352C2×C4—342no—
353C2×C4—342no—
354C2×C4—344no—
355C2×C4—344no—
356C2×C4—342no—
357C2×C4—342no—
358C2×C4—344no—
359C2×C4—342no—
360C2×C4—342no—
361C2×C4—342no—
362C2×C4—342no—
363C2×C4—342no—
364C2×C4—344no—
365C2×C4—344no—
366C2×C4—344no—
367C2×C4—684no—
368C2×C4—684no—
369C2×C4—684no—
370C8—344no—
371C8—344no—
372C8—348no—
373C22—22no—
374C22—22no—
375C22—28no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—416no—
383C22—44no—
384C22—44no—
385C22#1176377314yes—
386C22C22×C4×#59511yescharacteristic
387C22#1176387612yes—
388C22C2×#1058767812yes—
389C22#1176388712yes—
390C4—44no—
391C4—171no—
392C4—171no—
393C4—172no—
394C4—342no—
395C4—342no—
396C4—342no—
397C4—684no—
398C2—24no—
399C2—42no—
400C2—42no—
401C2#1454522011yescharacteristic
402C2C2×#1176377311yescharacteristic
403C2#1454534811yescharacteristic
404C2#1454565014yes—
subgroupG/H
center, FrattiniC23C23×#595
commutatorC22×C17C22×C42
FittingC17×#379C4
socleC23×C17C23×C4
prime radicalC25×C17C2×C4
cosocleC22×Dic17C24
hypercenter#379#595
hypocenterC17#20527
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
3. #14544526C2semidirect
4. #14544521C2semidirect
5. C22⋊C4×#3624C2semidirect
6. #14544514C2semidirect
11. #11763779C4semidirect
12. C2×C22⋊#3624C4semidirect
17. C2×#10587542C22semidirect
18. C2×C4×#3624C22semidirect
19. #11763675C22semidirect
20. #11763675C22semidirect
271. C17#20527semidirect

Maximal subgroups

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

subgroupindex
C2×#117637792
C22×C22⋊#36242
#145445262
#145445212
C22⋊C4×#36242
#145445142
#145311622
#2052717