Finite Group Databasebeta

← #17183100

Subgroups — #17183100

Order 4352.

3866 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 3866 conjugacy classes of subgroups — too many to draw. Only its 315 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
1C2×#11762996C214yesmaximal, complemented
2#14532099C213yesmaximal, complemented
3#14528732C214yesmaximal, complemented
4#14528702C213yesmaximal, complemented
5C2×C17×#585C211yesmaximal, characteristic, complemented
6C2×#10587623C2216yescomplemented
7C22×#63227C2211yescharacteristic, complemented
8#11762996—216no—
9C2×#10587505C22112yescomplemented
10#11762720—23no—
11#11762717—23no—
12#11762689C2216yescomplemented
13#11762688—26no—
14#11762672C2213yescomplemented
15C17×#585—216no—
16C2×C17×2−1+4—23no—
17C2×C17×H(5,𝔽2)—23no—
18C2×C17×H(5,𝔽2)C2214yes—
19C22×#41×C17—26no—
20C22×#41×C17C2213yes—
21C24×D17C2311yescharacteristic
22C17×2−1+4—248no—
23C17×H(5,𝔽2)—248no—
24C17×H(5,𝔽2)—224no—
25C17×H(5,𝔽2)D4116yescomplemented
26C2×#41×C17—24no—
27C2×#41×C17—212no—
28C2×#41×C17—248no—
29C2×#41×C17—248no—
30C2×#41×C17—224no—
31C2×#41×C17—26no—
32C2×#41×C17—212no—
33C2×#41×C17—26no—
34C22×Q8×C17—26no—
35C22×Q8×C17D413yescomplemented
36C22×D4×C17—26no—
37C22×D4×C17—212no—
38C22×D4×C17D413yescomplemented
39C22×D4×C17C2316yes—
40C23×C4×C17—24no—
41C23×C4×C17D416yescomplemented
42C23×C4×C17C2311yescharacteristic
43C22×C22⋊D17—26no—
44C23×Dic17C2311yescharacteristic
45C2×C22⋊Dic17—26no—
46C2×C22⋊Dic17C2316yes—
47#10587623—224no—
48C2×#63227—23no—
49C2×#63227—26no—
50C2×#63227—23no—
51#10587505—224no—
52#10587502—23no—
53#10587502—26no—
54#10587502—23no—
55C24×C17—24no—
56C24×C17C2×D416yes—
57C24×C17C2411yescharacteristic
58C23×D17—23no—
59C23×D17—24no—
60C23×D17—42no—
61C23×D17—81no—
62#41×C17—232no—
63#41×C17—248no—
64#41×C17—296no—
65#41×C17—248no—
66#41×C17—248no—
67#41×C17—248no—
68C2×Q8×C17—24no—
69C2×Q8×C17—248no—
70C2×Q8×C17—26no—
71C2×Q8×C17—212no—
72C2×D4×C17—212no—
73C2×D4×C17—248no—
74C2×D4×C17—248no—
75C2×D4×C17—248no—
76C2×D4×C17—212no—
77C2×D4×C17—212no—
78C2×D4×C17—212no—
79C2×D4×C17—26no—
80C2×D4×C17C2×D4124yes—
81C22×C4×C17—212no—
82C22×C4×C17—232no—
83C22×C4×C17—212no—
84C22×C4×C17—24no—
85C22×C4×C17—26no—
86C22×C4×C17—26no—
87C22×C4×C17—212no—
88C22×C4×C17C2×D413yes—
89C22×C4×C17C2×D416yes—
90C22×C4×C17C2×D413yes—
91C2×C22⋊D17—412no—
92C2×C22⋊D17—412no—
93C2×C22⋊D17—86no—
94C22×Dic17—23no—
95C22×Dic17—26no—
96C22×Dic17—81no—
97C22⋊Dic17—212no—
98C22⋊Dic17—86no—
99#63227—41no—
100#63227—43no—
101#63227—412no—
102#32899—171nomaximal, Sylow
103C23×C17—216no—
104C23×C17—26no—
105C23×C17—26no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—46no—
109C23×C17—46no—
110C23×C17—84no—
111C23×C17—84no—
112C23×C17#120112yes—
113C23×C17C22×D413yes—
114C23×C17C22×D414yes—
115C22×D17—26no—
116C22×D17—46no—
117C22×D17—43no—
118C22×D17—46no—
119C22×D17—41no—
120C22×D17—88no—
121Q8×C17—216no—
122Q8×C17—224no—
123D4×C17—248no—
124D4×C17—224no—
125D4×C17—248no—
126C2×C4×C17—224no—
127C2×C4×C17—248no—
128C2×C4×C17—224no—
129C2×C4×C17—26no—
130C2×C4×C17—26no—
131C2×C4×C17—26no—
132C2×C4×C17—26no—
133C2×C4×C17#12011yescharacteristic
134C2×C4×C17#12013yes—
135C2×C4×C17#120112yes—
136C22⋊D17—824no—
137C2×Dic17—46no—
138C2×Dic17—46no—
139C2×Dic17—46no—
140C2×Dic17—46no—
141C2×Dic17—41no—
142C2×#585D1711yescharacteristic, complemented
143C2×B(4,𝔽2)—174no—
144#2404—173no—
145#1790—174no—
146#1782—173no—
147C22×C17—23no—
148C22×C17—23no—
149C22×C17—212no—
150C22×C17—44no—
151C22×C17—412no—
152C22×C17—46no—
153C22×C17—46no—
154C22×C17—86no—
155C22×C17—816no—
156C22×C17C2×#12016yes—
157C22×C17C2×#12011yescharacteristic
158C2×D17—412no—
159C2×D17—42no—
160C2×D17—86no—
161C2×D17—81no—
162C2×D17—81no—
163C4×C17—21no—
164C4×C17—212no—
165C4×C17—23no—
166Dic17—86no—
167Dic17—81no—
168#585—216no—
169C2×2−1+4—23no—
170C2×H(5,𝔽2)—23no—
171C2×H(5,𝔽2)C2×D1714yes—
172C22×#41—26no—
173C22×#41C2×D1713yes—
174C2×#120—176no—
175C22×C22⋊C4—171no—
176B(4,𝔽2)—3416no—
177C2×#99—1712no—
178#393—343no—
179#392—343no—
180#386—176no—
181#386—346no—
182#381—173no—
183C2×C17—26no—
184C2×C17—412no—
185C2×C17#284611yescharacteristic
186C2×C17#302511yescharacteristic
187C2×C17#302711yescharacteristic
188D17—82no—
189C25—171no—
1902−1+4—248no—
191H(5,𝔽2)—248no—
192H(5,𝔽2)—224no—
193H(5,𝔽2)C22⋊D17116yescomplemented
194C2×#41—24no—
195C2×#41—212no—
196C2×#41—248no—
197C2×#41—248no—
198C2×#41—224no—
199C2×#41—26no—
200C2×#41—212no—
201C2×#41—26no—
202C22×Q8—26no—
203C22×Q8C22⋊D1713yescomplemented
204C22×D4—26no—
205C22×D4—212no—
206C22×D4—346no—
207C22×D4C22⋊D1713yescomplemented
208C22×D4C22×D1716yes—
209C23×C4—24no—
210C23×C4—171no—
211C23×C4C22⋊D1716yescomplemented
212C23×C4C22×D1711yescharacteristic
213#120—3424no—
214C2×C22⋊C4—176no—
215C2×C22⋊C4—343no—
216C2×C22⋊C4—343no—
217C2×C22⋊C4—346no—
218C2×C22⋊C4—346no—
219#99—3424no—
220#95—343no—
221#95—343no—
222#95—346no—
223C17#3289911yescharacteristic, complemented, Sylow
224C24—24no—
225C24—343no—
226C24—344no—
227C24—682no—
228C24—1361no—
229C24C2×C22⋊D1716yes—
230C24C23×D1711yescharacteristic
231#41—232no—
232#41—248no—
233#41—296no—
234#41—248no—
235#41—248no—
236#41—248no—
237C2×Q8—24no—
238C2×Q8—248no—
239C2×Q8—26no—
240C2×Q8—212no—
241C2×D4—212no—
242C2×D4—248no—
243C2×D4—248no—
244C2×D4—212no—
245C2×D4—212no—
246C2×D4—248no—
247C2×D4—212no—
248C2×D4—26no—
249C2×D4—6812no—
250C2×D4—6812no—
251C2×D4—1366no—
252C2×D4C2×C22⋊D17124yes—
253C22×C4—212no—
254C22×C4—232no—
255C22×C4—24no—
256C22×C4—212no—
257C22×C4—26no—
258C22×C4—26no—
259C22×C4—212no—
260C22×C4—346no—
261C22×C4—343no—
262C22×C4—1361no—
263C22×C4C2×C22⋊D1713yes—
264C22×C4C2×C22⋊D1716yes—
265C22×C4C2×C22⋊D1713yes—
266C22⋊C4—3412no—
267C22⋊C4—681no—
268C22⋊C4—683no—
269C22⋊C4—6812no—
270C22⋊C4—1366no—
271C23—216no—
272C23—26no—
273C23—26no—
274C23—41no—
275C23—41no—
276C23—46no—
277C23—46no—
278C23—84no—
279C23—84no—
280C23—346no—
281C23—686no—
282C23—683no—
283C23—686no—
284C23—681no—
285C23—1368no—
286C23#10587639112yes—
287C23C22×C22⋊D1713yes—
288C23C22×C22⋊D1714yes—
289Q8—216no—
290Q8—224no—
291D4—248no—
292D4—224no—
293D4—248no—
294D4—13624no—
295C2×C4—224no—
296C2×C4—248no—
297C2×C4—224no—
298C2×C4—26no—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—26no—
302C2×C4—686no—
303C2×C4—686no—
304C2×C4—686no—
305C2×C4—686no—
306C2×C4—681no—
307C2×C4#10587639112yes—
308C2×C4#1058763911yescharacteristic
309C2×C4#1058763913yes—
310C22—23no—
311C22—23no—
312C22—212no—
313C22—44no—
314C22—412no—
315C22—46no—
316C22—46no—
317C22—86no—
318C22—816no—
319C22—6812no—
320C22—682no—
321C22—1366no—
322C22—1361no—
323C22—1361no—
324C22C2×#1058763916yes—
325C22C2×#1058763911yescharacteristic
326C4—21no—
327C4—212no—
328C4—23no—
329C4—1366no—
330C4—1361no—
331C2—26no—
332C2—412no—
333C2—1362no—
334C2#1453706111yescharacteristic
335C2#1453817611yescharacteristic
336C2#1453818611yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC24×C17C24
FrattiniC24C23×D17
FittingC2×C17×#585C2
socleC22×C17C2×#120
hypercenterC2×#585D17
hypocenterC17#32899
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#585

Complements

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

NH ≅ G/Ntype
1. C2×#11762996C2semidirect
2. #14532099C2semidirect
3. #14528732C2semidirect
4. #14528702C2semidirect
5. C2×C17×#585C2semidirect
6. C2×#10587623C22semidirect
7. C22×#63227C22semidirect
9. C2×#10587505C22semidirect
12. #11762689C22semidirect
14. #11762672C22semidirect
25. C17×H(5,𝔽2)D4semidirect
35. C22×Q8×C17D4semidirect
38. C22×D4×C17D4semidirect
41. C23×C4×C17D4semidirect
142. C2×#585D17semidirect
193. H(5,𝔽2)C22⋊D17semidirect
203. C22×Q8C22⋊D17semidirect
207. C22×D4C22⋊D17semidirect
211. C23×C4C22⋊D17semidirect
223. C17#32899semidirect

Maximal subgroups

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

subgroupindex
C2×#117629962
#145320992
#145287322
#145287022
C2×C17×#5852
#3289917