Finite Group Databasebeta

← #17951153

Subgroups — #17951153

Order 4352.

786 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763832C211yesmaximal, characteristic
2C2×#11763745C211yesmaximal, characteristic
3#14543657C214yesmaximal
4#14531188C211yesmaximal, characteristic
5C22×C4×#595—21no—
6#11763832—22no—
7#11763832C414yescomplemented
8C2×#10587544—21no—
9C2×#10587543C2212yes—
10C2×#10587542—21no—
11#11763745—22no—
12#11763745C414yescomplemented
13C2×(C2×#595)⋊C4—21no—
14C2×(C2×#595)⋊C4C2212yes—
15C2×#10587558—21no—
16C2×C42×D17C2211yescharacteristic
17#11762678C412yescomplemented
18#11762673—21no—
19#11762668—21no—
20#11762666C2212yes—
21C4×#63225—21no—
22C4×C4⋊Dic17C411yescharacteristic, complemented
23C4×C4⋊C4×C17C411yescharacteristic, complemented
24C23×#595—22no—
25C22×C4×D17D411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×C4×D17C2311yescharacteristic
28C22×#3624—22no—
29C2×C4×#595—28no—
30C2×C4×#595—24no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C42×C17—21no—
33C2×C42×C17C2×C411yescharacteristic
34C2×#63227—22no—
35C2×C4⋊Dic17C2×C412yes—
36C2×#63225—22no—
37C2×#63225—22no—
38C2×C4×Dic17—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42#10587558—44no—
43C42×D17C2×C412yes—
44C42×D17C2×C412yes—
45C42×D17D412yes—
46C42×D17Q812yes—
47(C2×#595)⋊C4—24no—
48(C2×#595)⋊C4—22no—
49(C2×#595)⋊C4C2×C414yes—
50#10587544—24no—
51#10587543—22no—
52#10587543C2×C414yes—
53#10587542—24no—
54C17×#95—21no—
55#10587502—21no—
56#10587502—22no—
57C23×D17C2×D411yescharacteristic
58C22×#595—28no—
59C22×#595—24no—
60C22×#595—44no—
61C22×#595—42no—
62C22×#595—42no—
63C22×C4×C17—22no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×Dic17—22no—
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C2×D411yescharacteristic
71C2×C4×D17—22no—
72C2×C4×D17—22no—
73C2×C4×D17—24no—
74C2×C4×D17—42no—
75C2×C4×D17C22⋊C412yes—
76C2×C4×D17C22⋊C412yes—
77C2×C4×D17C4⋊C412yes—
78C2×C4×D17C4⋊C412yes—
79C2×C4×D17C22×C411yescharacteristic
80C2×C4×D17C22×C411yescharacteristic
81C2×C4×D17C2×D411yescharacteristic
82C2×C4×D17C2×Q811yescharacteristic
83C2×#3624—24no—
84C2×#3624—44no—
85C4×#595—216no—
86C4⋊C4×C17—42no—
87C42×C17—42no—
88C42×C17C4212yes—
89C42×C17C4⋊C412yes—
90#63227—44no—
91C4⋊Dic17—42no—
92#63225—44no—
93#63225—44no—
94C4×Dic17—42no—
95C4×Dic17C4212yes—
96C4×Dic17C22⋊C412yes—
97C4×Dic17C22⋊C412yes—
98C4×Dic17C4⋊C412yes—
99C4×Dic17C4⋊C412yes—
100C4×Dic17C4⋊C412yes—
101#8022—171nomaximal, Sylow
102C23×C17C2×C22⋊C411yescharacteristic
103C22×D17—22no—
104C22×D17—22no—
105C22×D17C2×C22⋊C411yescharacteristic
106C22×D17C2×C4⋊C411yescharacteristic
107C22×D17AGL(1,ℤ/8)12yes—
108C22×D17#13712yes—
109C2×#595—28no—
110C2×#595—44no—
111C2×#595—48no—
112C2×#595—48no—
113C2×#595—44no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—22no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17#9512yes—
122C2×C4×C17#9512yes—
123C2×C4×C17C2×C4211yescharacteristic
124C2×C4×C17C2×C4⋊C411yescharacteristic
125C2×Dic17—22no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17#9512yes—
132C2×Dic17#9512yes—
133C2×Dic17#10414yes—
134C2×Dic17C2×C22⋊C411yescharacteristic
135C2×Dic17C2×C4⋊C411yescharacteristic
136C4×D17—24no—
137C4×D17—24no—
138C4×D17—24no—
139C4×D17—24no—
140C4×D17—41no—
141C4×D17—41no—
142C4×D17—41no—
143C4×D17—41no—
144#3624—44no—
145C2×#432—171no—
146#2285—171no—
147C2×#376—171no—
148#1294—174no—
149C22×C17C2×#9511yescharacteristic
150C22×C17#41012yes—
151C22×C17#41811yescharacteristic
152C22×C17#41911yescharacteristic
153C22×C17C2×#10412yes—
154C2×D17—21no—
155C2×D17—21no—
156C2×D17—21no—
157C2×D17—21no—
158C2×D17—24no—
159C2×D17—24no—
160C2×D17#41712yes—
161C2×D17#42812yes—
162#595—48no—
163C4×C17—22no—
164C4×C17—22no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—82no—
168Dic17—22no—
169Dic17—21no—
170Dic17—21no—
171Dic17—82no—
172Dic17#33914yes—
173C2×#117—341no—
174C22×C42—171no—
175C22×C42—341no—
176#432—174no—
177#432—342no—
178C2×C4⋊C8—341no—
179C2×C22⋊C8—172no—
180C2×#97—341no—
181#384—172no—
182#382—341no—
183#380—172no—
184#380—341no—
185C4×C4⋊C4—171no—
186C4×C4⋊C4—341no—
187C4×C4⋊C4#59511yescharacteristic, complemented
188#376—174no—
189#376—342no—
190C2×#95—172no—
191C2×#95—341no—
192C2×C17#150512yes—
193C2×C17#150612yes—
194C2×C17C2×#33911yescharacteristic
195C2×C17#173611yescharacteristic
196C2×C17#174011yescharacteristic
197D17—22no—
198D17—22no—
199C23×C4—171no—
200C23×C4—171no—
201C23×C4—171no—
202C23×C4—342no—
203C22×C8—342no—
204#117—684no—
205C2×C4⋊C4—172no—
206C2×C4⋊C4—342no—
207C2×C4⋊C4—342no—
208C2×C4⋊C4C2×#59512yes—
209C2×C22⋊C4—342no—
210C2×C42—21no—
211C2×C42—171no—
212C2×C42—171no—
213C2×C42—171no—
214C2×C42—172no—
215C2×C42—172no—
216C2×C42—172no—
217C2×C42—172no—
218C2×C42—344no—
219C2×C42—348no—
220C2×C42—341no—
221C2×C42C2×#59511yescharacteristic
222C4⋊C8—344no—
223C22⋊C8—174no—
224C22⋊C8—342no—
225#97—344no—
226#95—21no—
227#95—174no—
228#95—342no—
229#95—342no—
230#95—344no—
231#95—341no—
232C17#802211yescharacteristic, complemented, Sylow
233C24—171no—
234C22×C4—22no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—172no—
239C22×C4—172no—
240C22×C4—172no—
241C22×C4—172no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—171no—
245C22×C4—171no—
246C22×C4—344no—
247C22×C4—342no—
248C22×C4—342no—
249C22×C4—344no—
250C22×C4—348no—
251C22×C4—342no—
252C22×C4—682no—
253C22×C4—682no—
254C22×C4—682no—
255C22×C4—684no—
256C22×C4C22⋊#59511yescharacteristic
257C22×C4C22⋊#59511yescharacteristic
258C22×C4C22×#59511yescharacteristic
259C2×C8—344no—
260C2×C8—684no—
261C4⋊C4—42no—
262C4⋊C4—682no—
263C4⋊C4—684no—
264C4⋊C4—684no—
265C22⋊C4—684no—
266C42—42no—
267C42—172no—
268C42—172no—
269C42—172no—
270C42—172no—
271C42—172no—
272C42—172no—
273C42—3416no—
274C42—682no—
275C42C4×#59512yes—
276C42C4⋊#59512yes—
277C23—171no—
278C23—171no—
279C23—172no—
280C23—172no—
281C23—342no—
282C23—342no—
283C23C2×C22⋊#59511yescharacteristic
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—172no—
292C2×C4—174no—
293C2×C4—171no—
294C2×C4—172no—
295C2×C4—171no—
296C2×C4—344no—
297C2×C4—344no—
298C2×C4—344no—
299C2×C4—344no—
300C2×C4—342no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—348no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—684no—
309C2×C4—688no—
310C2×C4—688no—
311C2×C4—684no—
312C2×C4—682no—
313C2×C4—682no—
314C2×C4—682no—
315C2×C4(C2×#595)⋊C412yes—
316C2×C4(C2×#595)⋊C412yes—
317C2×C4C2×C4×#59511yescharacteristic
318C2×C4C2×C4⋊#59511yescharacteristic
319C8—684no—
320C22—172no—
321C22—172no—
322C22—344no—
323C22—344no—
324C22—341no—
325C22—341no—
326C22—341no—
327C22—341no—
328C22C2×(C2×#595)⋊C411yescharacteristic
329C22#1176378812yes—
330C22#1176380111yescharacteristic
331C22#1176380211yescharacteristic
332C22D17⋊#10412yes—
333C4—22no—
334C4—22no—
335C4—21no—
336C4—21no—
337C4—82no—
338C4—174no—
339C4—342no—
340C4—341no—
341C4—341no—
342C4—688no—
343C4—1362no—
344C2—342no—
345C2—342no—
346C2#1454417412yes—
347C2#1454417712yes—
348C2#1454446011yescharacteristic
349C2#1454447111yescharacteristic
350C2#1454448311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC4×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenterC4×C4⋊C4#595
hypocenterC17#8022
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #11763832C4semidirect
12. #11763745C4semidirect
17. #11762678C4semidirect
22. C4×C4⋊Dic17C4semidirect
23. C4×C4⋊C4×C17C4semidirect
187. C4×C4⋊C4#595semidirect
232. C17#8022semidirect

Maximal subgroups

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

subgroupindex
C2×#117638322
C2×#117637452
#145436572
#145311882
#802217