Finite Group Databasebeta

← #17956505

Subgroups — #17956505

Order 4352.

566 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C4×#10587543C211yesmaximal, characteristic
2#14543960C211yesmaximal, characteristic
3#14543657C212yesmaximal
4#14543657C212yesmaximal
5#14537089C211yesmaximal, characteristic
6C2×#10587543C2212yes—
7C2×C4×#3624—21no—
8C2×(C2×#595)⋊C4—21no—
9C2×(C2×#595)⋊C4—21no—
10C2×(C2×#595)⋊C4C2212yes—
11#11763675—21no—
12C2×C42×D17C2211yescharacteristic
13#11762705C411yescharacteristic, complemented
14#11762705C411yescharacteristic, complemented
15#11762690—21no—
16#11762690—21no—
17#11762680—21no—
18#11762680—21no—
19#11762675C411yescharacteristic, complemented
20#11762666C2211yescharacteristic
21#11762666C2211yescharacteristic
22C2×C17×#125C411yescharacteristic, complemented
23C23×#595—22no—
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17C2311yescharacteristic
27C22×#3624—22no—
28C17×#125—42no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×#63227—21no—
35C2×#63227—21no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×#63225—21no—
39C2×#63225—21no—
40C2×C4×Dic17C2×C411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C42×D17—42no—
44(C2×#595)⋊C4—24no—
45(C2×#595)⋊C4—24no—
46(C2×#595)⋊C4—42no—
47#10587543—22no—
48#10587543Q814yes—
49C4×#3624—42no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C23×D17C2×D411yescharacteristic
57C22×#595—24no—
58C22×#595—42no—
59C22×#595—42no—
60C22×#595—42no—
61C22×#595—42no—
62C22×#595—42no—
63C22×#595—42no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C2×C4×D17—22no—
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—41no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×C4×D17C2×Q812yes—
83C2×C4×D17#4112yes—
84C2×#3624—24no—
85C2×#3624—42no—
86C2×#3624—42no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—42no—
90C4⋊C4×C17—42no—
91C42×C17—22no—
92#63227—42no—
93#63227—42no—
94C4⋊Dic17—41no—
95C4⋊Dic17—41no—
96#63225—42no—
97#63225—42no—
98C4×Dic17—22no—
99C4×Dic17—41no—
100C4×Dic17—41no—
101#9886—171nomaximal, Sylow
102C23×C17C2×C22⋊C411yescharacteristic
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17#12411yescharacteristic
108C22×D17C4⋊Q811yescharacteristic
109C22×D17#13512yes—
110C22×D17#13512yes—
111C2×#595—44no—
112C2×#595—44no—
113C2×#595—44no—
114C2×#595—82no—
115C2×#595—82no—
116C2×#595—82no—
117C2×#595—82no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17#9912yes—
132C2×C4×C17C4×Q812yes—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17#10412yes—
147C2×Dic17C22⋊Q812yes—
148C4×D17—42no—
149C4×D17—42no—
150C4×D17—44no—
151C4×D17—44no—
152#3624—44no—
153#2851—171no—
154C4×C22⋊C8—171no—
155#1434—171no—
156#1294—172no—
157#1294—172no—
158C22×C17#39111yescharacteristic
159C22×C17C2×#9911yescharacteristic
160C22×C17#41011yescharacteristic
161C22×C17#41611yescharacteristic
162C22×C17#41611yescharacteristic
163C22×C17C2×#10411yescharacteristic
164C22×C17#42111yescharacteristic
165C2×D17—21no—
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—22no—
170C2×D17—22no—
171C2×D17—22no—
172C2×D17—22no—
173C2×D17#48812yes—
174C2×D17#49912yes—
175#595—84no—
176C4×C17—22no—
177C4×C17—22no—
178C4×C17—41no—
179C4×C17—81no—
180C4×C17—81no—
181Dic17—22no—
182Dic17—22no—
183Dic17—41no—
184Dic17—81no—
185Dic17—81no—
186C2×#125#59511yescharacteristic, complemented
187C22×C42—171no—
188C2×C22⋊C8—172no—
189C2×C4×C8—341no—
190#389—171no—
191#389—171no—
192#387—341no—
193#387—341no—
194#385—341no—
195#385—341no—
196#382—171no—
197#380—171no—
198#380—171no—
199C2×#95—172no—
200C2×#95—341no—
201C2×#95—341no—
202#336—341no—
203C2×C17#150011yescharacteristic
204C2×C17#150111yescharacteristic
205C2×C17#151611yescharacteristic
206C2×C17#152111yescharacteristic
207C2×C17#198611yescharacteristic
208C2×C17#199111yescharacteristic
209C2×C17#199611yescharacteristic
210D17—22no—
211D17—22no—
212C23×C4—171no—
213C23×C4—171no—
214C23×C4—171no—
215C23×C4—342no—
216C22×C8—342no—
217#125—42no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4C2×#59511yescharacteristic
225C2×C4⋊C4C2×#59511yescharacteristic
226C2×C22⋊C4—341no—
227C2×C22⋊C4—341no—
228C2×C42—171no—
229C2×C42—171no—
230C2×C42—171no—
231C2×C42—682no—
232C2×C42C2×#59511yescharacteristic
233C22⋊C8—174no—
234C22⋊C8—342no—
235C4×C8—682no—
236#95—341no—
237#95—341no—
238#95—341no—
239#95—341no—
240#95—344no—
241#95—344no—
242#95—341no—
243#95—341no—
244#95—682no—
245C17#988611yescharacteristic, complemented, Sylow
246C24—171no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—171no—
250C22×C4—171no—
251C22×C4—172no—
252C22×C4—172no—
253C22×C4—171no—
254C22×C4—342no—
255C22×C4—342no—
256C22×C4—342no—
257C22×C4—342no—
258C22×C4—344no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—681no—
262C22×C4—681no—
263C22×C4—681no—
264C22×C4—681no—
265C22×C4—682no—
266C22×C4—682no—
267C22×C4—682no—
268C22×C4—682no—
269C22×C4—682no—
270C22×C4—682no—
271C22×C4C22⋊#59511yescharacteristic
272C22×C4C22⋊#59511yescharacteristic
273C22×C4C22×#59511yescharacteristic
274C2×C8—344no—
275C2×C8—682no—
276C2×C8—682no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—681no—
282C4⋊C4—681no—
283C4⋊C4—682no—
284C4⋊C4—682no—
285C22⋊C4—682no—
286C22⋊C4—682no—
287C42—22no—
288C42—342no—
289C42—681no—
290C42—681no—
291C23—171no—
292C23—171no—
293C23—172no—
294C23—172no—
295C23—341no—
296C23—341no—
297C23—341no—
298C23—341no—
299C23C2×C22⋊#59511yescharacteristic
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—172no—
314C2×C4—172no—
315C2×C4—341no—
316C2×C4—341no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—682no—
323C2×C4—682no—
324C2×C4—684no—
325C2×C4—684no—
326C2×C4—684no—
327C2×C4—684no—
328C2×C4—684no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4—1362no—
336C2×C4—1362no—
337C2×C4—1362no—
338C2×C4—1362no—
339C2×C4#1058755112yes—
340C2×C4Q8×#59512yes—
341C8—684no—
342C22—172no—
343C22—172no—
344C22—342no—
345C22—342no—
346C22—341no—
347C22—341no—
348C22—341no—
349C22—341no—
350C22—342no—
351C22—342no—
352C22(C2×#595)⋊Q811yescharacteristic
353C22C2×#1058755111yescharacteristic
354C22#1176379011yescharacteristic
355C22#1176379911yescharacteristic
356C22#1176379911yescharacteristic
357C22D17⋊#10411yescharacteristic
358C22#1176380611yescharacteristic
359C4—22no—
360C4—22no—
361C4—41no—
362C4—81no—
363C4—81no—
364C4—342no—
365C4—342no—
366C4—681no—
367C4—1364no—
368C4—1361no—
369C4—1361no—
370C2—342no—
371C2—342no—
372C2#1454415811yescharacteristic
373C2#1454415911yescharacteristic
374C2#1454421011yescharacteristic
375C2#1454422711yescharacteristic
376C2#1454496111yescharacteristic
377C2#1454496611yescharacteristic
378C2#1454497311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#125C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenterC2×#125#595
hypocenterC17#9886
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
13. #11762705C4semidirect
14. #11762705C4semidirect
19. #11762675C4semidirect
22. C2×C17×#125C4semidirect
186. C2×#125#595semidirect
245. C17#9886semidirect

Maximal subgroups

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

subgroupindex
C4×#105875432
#145439602
#145436572
#145370892
#988617