Finite Group Databasebeta

← #17959381

Subgroups — #17959381

Order 4352.

590 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 74 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×#10587551C211yesmaximal, characteristic
2#14544043C211yesmaximal, characteristic
3#14543779C214yesmaximal, complemented
4#14532143C211yesmaximal, characteristic
5#11763830—21no—
6C2×#10587551—21no—
7C4×C22⋊#595—21no—
8#11763687C2212yes—
9#11763653C2212yes—
10#11763652—21no—
11C2×#10587632—21no—
12C2×C4×C22⋊D17C2211yescharacteristic
13#11762728C412yescomplemented
14#11762720C2212yes—
15#11762716—21no—
16#11762688—21no—
17#11762682—21no—
18#11762669C411yescharacteristic, complemented
19C23×Q8×C17C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×Q8×C17—41no—
22C22×Q8×C17—41no—
23C22×Q8×C17—44no—
24C22×Q8×C17C2×C412yes—
25C23×C4×C17—21no—
26C23×C4×C17C2×C411yescharacteristic
27C22×C22⋊D17D411yescharacteristic
28C22×C4×D17D411yescharacteristic
29C2×C22⋊#595—22no—
30C2×C4×#595—41no—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17D411yescharacteristic
33#10587632—44no—
34C4×C22⋊D17—41no—
35C4×C22⋊D17—41no—
36C2×#63227—22no—
37C2×#63227C2311yescharacteristic
38C2×C4⋊Dic17—21no—
39C2×#63225—22no—
40C2×#63225C2×C411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C22⋊#3624—22no—
43#10587551—24no—
44(C2×#595)⋊C4—41no—
45#10587542—41no—
46C4⋊#3624—41no—
47#10587502—22no—
48#10587502—22no—
49#10587502C2×C412yes—
50C24×C17C22⋊C411yescharacteristic
51C23×D17C2×D411yescharacteristic
52C22×#595—42no—
53C2×Q8×C17—24no—
54C2×Q8×C17—44no—
55C2×Q8×C17—42no—
56C2×Q8×C17—42no—
57C2×Q8×C17—44no—
58C2×Q8×C17—48no—
59C2×Q8×C17—44no—
60C2×Q8×C17—41no—
61C2×Q8×C17—41no—
62C22×C4×C17—22no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—44no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C2×C22⋊D17—22no—
71C2×C22⋊D17—22no—
72C2×C22⋊D17—41no—
73C22×Dic17—22no—
74C22×Dic17C22⋊C411yescharacteristic
75C22×Dic17C22⋊C411yescharacteristic
76C22×Dic17C2×D411yescharacteristic
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×C4×D17—42no—
80C22⋊#595—24no—
81C22⋊#595—42no—
82C2×#3624—42no—
83C4×#595—82no—
84C22⋊Dic17—41no—
85C22⋊Dic17—42no—
86#63227—41no—
87#63227—44no—
88C4⋊Dic17—42no—
89#63225—41no—
90#63225—44no—
91C4×Dic17—21no—
92C4×Dic17—21no—
93#11785—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17#10412yes—
99C23×C17C2×C22⋊C411yescharacteristic
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—41no—
103C22×D17—41no—
104C22×D17#13712yes—
105C2×#595—44no—
106C2×#595—82no—
107C2×#595—82no—
108Q8×C17—44no—
109Q8×C17—44no—
110Q8×C17—44no—
111Q8×C17—44no—
112C2×C4×C17—22no—
113C2×C4×C17—24no—
114C2×C4×C17—21no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—48no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—41no—
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#9911yescharacteristic
123C22⋊D17—44no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C4×D17—42no—
135C4×D17—81no—
136#3624—82no—
137#2409—171no—
138C4×#99—171no—
139#1466—171no—
140#1351—174no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—22no—
144C22×C17—42no—
145C22×C17—42no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17C2×#9911yescharacteristic
149C22×C17#41911yescharacteristic
150C22×C17C2×#10411yescharacteristic
151C2×D17—22no—
152C2×D17—41no—
153C2×D17—41no—
154C2×D17—42no—
155C2×D17—81no—
156#595—84no—
157C4×C17—21no—
158C4×C17—44no—
159C4×C17—41no—
160C4×C17AΣL(1,𝔽16)12yes—
161Dic17—41no—
162Dic17—41no—
163Dic17—41no—
164Dic17—82no—
165C23×Q8#59511yescharacteristic, complemented
166C2×C22⋊Q8—341no—
167C2×C4×D4—171no—
168#432—341no—
169C2×#99—341no—
170#395—172no—
171#393—172no—
172#391—341no—
173#386—341no—
174#385—341no—
175#380—171no—
176C4×C22⋊C4—341no—
177#342—172no—
178#323—172no—
179#323—341no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—42no—
183C2×C17#150811yescharacteristic
184C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
185C2×C17#212911yescharacteristic
186D17—42no—
187C22×Q8—21no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—44no—
191C22×Q8C2×#59512yes—
192C22×D4—171no—
193C23×C4—21no—
194C23×C4—171no—
195C23×C4C2×#59511yescharacteristic
196C22⋊Q8—684no—
197C4×D4—681no—
198C4×D4—681no—
199C2×C4⋊C4—171no—
200C2×C4⋊C4—342no—
201C2×C4⋊C4—341no—
202C2×C22⋊C4—171no—
203C2×C22⋊C4—171no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4—341no—
207C2×C42—171no—
208C2×C42—681no—
209C4⋊C8—681no—
210#99—344no—
211C22⋊C8—342no—
212#97—681no—
213#95—172no—
214#95—342no—
215#95—342no—
216#95—681no—
217C17#1178511yescharacteristic, complemented, Sylow
218C24—171no—
219C24C22⋊#59511yescharacteristic
220C2×Q8—24no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—48no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×D4—342no—
230C2×D4—342no—
231C2×D4—681no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—342no—
244C22×C4—682no—
245C22×C4—682no—
246C22×C4C22⋊#59511yescharacteristic
247C22×C4C22×#59511yescharacteristic
248C2×C8—682no—
249C4⋊C4—681no—
250C4⋊C4—684no—
251C4⋊C4—682no—
252C22⋊C4—344no—
253C22⋊C4—681no—
254C22⋊C4—684no—
255C22⋊C4—681no—
256C22⋊C4—682no—
257C22⋊C4—682no—
258C42—341no—
259C42—341no—
260C42—1362no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—172no—
266C23—341no—
267C23—341no—
268C23—681no—
269C23—681no—
270C23#1058755012yes—
271C23C2×C22⋊#59511yescharacteristic
272Q8—44no—
273Q8—44no—
274Q8—44no—
275Q8—44no—
276D4—684no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—48no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—682no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—684no—
297C2×C4—682no—
298C2×C4—1361no—
299C2×C4—1362no—
300C2×C4—1362no—
301C2×C4#1058755111yescharacteristic
302C2×C4#1058755111yescharacteristic
303C8—1362no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—42no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—342no—
312C22—682no—
313C22—681no—
314C22—681no—
315C22—1361no—
316C22C2×#1058755111yescharacteristic
317C22#1176380211yescharacteristic
318C22C2×#1058755011yescharacteristic
319C4—21no—
320C4—44no—
321C4—41no—
322C4—681no—
323C4—681no—
324C4—681no—
325C4—1362no—
326C4—1364no—
327C4#1176370912yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—682no—
332C2#1454418411yescharacteristic
333C2C2×#1176370911yescharacteristic
334C2#1454506611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC23×Q8×C17C4
socleC22×C17C2×#99
prime radicalC22×C22⋊D17D4
cosocleC2×#63227C23
hypercenterC23×Q8#595
hypocenterC17#11785
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #14543779C2semidirect
13. #11762728C4semidirect
18. #11762669C4semidirect
19. C23×Q8×C17C4semidirect
165. C23×Q8#595semidirect
217. C17#11785semidirect

Maximal subgroups

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

subgroupindex
C4×#105875512
#145440432
#145437792
#145321432
#1178517