Finite Group Databasebeta

← #17197186

Subgroups — #17197186

Order 4352.

2410 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531965C212yesmaximal, complemented
2#14531961C212yesmaximal, complemented
3#14531896C214yesmaximal, complemented
4#14531713C218yesmaximal, complemented
5#14531551C218yesmaximal, complemented
6#14531376C214yesmaximal, complemented
7#14531182C212yesmaximal, complemented
8C2×C17×#532C211yesmaximal, characteristic
9C2×#10587612C2214yes—
10C22×C4⋊Dic17C2214yes—
11#11762716C2218yes—
12#11762712C2218yes—
13#11762702C22116yes—
14#11762695C22116yes—
15#11762689C2218yes—
16#11762689C2218yes—
17#11762689C22116yes—
18#11762677C2214yes—
19#11762675C2214yes—
20#11762672C2214yes—
21#11762669C2214yes—
22#11762669C2218yes—
23C4×C4⋊Dic17C2214yes—
24C22⋊C4×Dic17C22116yes—
25C4×C22⋊Dic17C2214yes—
26#11762655C2214yes—
27C22×#41×C17C2212yes—
28C17×#532C4116yescomplemented
29C2×C17×C4⋊Q8C2211yescharacteristic
30C2×C17×#127C2211yescharacteristic
31C2×C17×#124C2212yes—
32C2×C17×#121C2214yes—
33C2×C4×D4×C17C2214yes—
34C22×C42×C17C2211yescharacteristic
35C2×#41×C17—44no—
36C2×#41×C17—44no—
37C2×#41×C17C2×C4116yescomplemented
38C22×Q8×C17C2312yes—
39C22×D4×C17C2314yes—
40C22×D4×C17C2312yes—
41C22×D4×C17C2314yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2314yes—
44C23×C4×C17C2312yes—
45C23×Dic17C2318yes—
46C17×C4⋊Q8C2×C418yescomplemented
47C17×#127C2×C418yes—
48C17×#124C2×C4116yescomplemented
49C17×#121C2×C4132yescomplemented
50C4×D4×C17C2×C4132yescomplemented
51C2×C4⋊C4×C17C2314yes—
52C2×C22⋊C4×C17C2318yes—
53C2×C42×C17C2×C418yescomplemented
54C2×C42×C17C2312yes—
55C2×C42×C17C2311yescharacteristic
56C2×C42×C17C2311yescharacteristic
57C2×C22⋊Dic17C2318yes—
58C2×C22⋊Dic17C23116yes—
59C2×C22⋊Dic17C2318yes—
60#10587612—48no—
61C2×C4⋊Dic17—48no—
62C2×C4⋊Dic17C2318yes—
63C2×C4⋊Dic17C2318yes—
64C2×C4×Dic17C2318yes—
65C2×C4×Dic17C2318yes—
66#10587502C2318yes—
67#10587502C2318yes—
68#10587502C2318yes—
69#10587502C23116yes—
70#10587502C2318yes—
71C24×C17C2414yes—
72C24×C17C2411yescharacteristic
73#41×C17—432no—
74C2×Q8×C17—42no—
75C2×Q8×C17—42no—
76C2×Q8×C17C22×C418yes—
77C2×D4×C17—44no—
78C2×D4×C17—44no—
79C2×D4×C17—42no—
80C2×D4×C17—42no—
81C2×D4×C17—44no—
82C2×D4×C17—44no—
83C2×D4×C17C22×C418yes—
84C2×D4×C17C22×C4116yes—
85C2×D4×C17C22×C4116yes—
86C22×C4×C17—24no—
87C22×C4×C17—28no—
88C22×C4×C17—28no—
89C22×C4×C17—44no—
90C22×C4×C17C22×C414yes—
91C22×C4×C17C22×C4116yes—
92C22×C4×C17C22×C418yes—
93C22×C4×C17C2411yescharacteristic
94C22×C4×C17C2412yes—
95C22×C4×C17C2414yes—
96C22×C4×C17C2412yes—
97C22×C4×C17C2411yescharacteristic
98C22×Dic17—416no—
99C22×Dic17—48no—
100C22×Dic17C24116yes—
101C4⋊C4×C17C22×C4116yes—
102C22⋊C4×C17C22×C4132yes—
103C42×C17C22×C414yes—
104C42×C17C22×C414yes—
105C42×C17C22×C418yes—
106C22⋊Dic17—48no—
107C22⋊Dic17—48no—
108C22⋊Dic17—416no—
109C4⋊Dic17—216no—
110C4⋊Dic17—48no—
111C4×Dic17—48no—
112C4×Dic17—48no—
113#34369—171nomaximal, Sylow
114C23×C17—24no—
115C23×C17—24no—
116C23×C17—22no—
117C23×C17—24no—
118C23×C17—28no—
119C23×C17—28no—
120C23×C17C23×C418yes—
121C23×C17C23×C412yes—
122C23×C17C2511yescharacteristic
123Q8×C17—48no—
124D4×C17—416no—
125D4×C17—48no—
126D4×C17—416no—
127C2×C4×C17—22no—
128C2×C4×C17—24no—
129C2×C4×C17—24no—
130C2×C4×C17—24no—
131C2×C4×C17—24no—
132C2×C4×C17—232no—
133C2×C4×C17—28no—
134C2×C4×C17—48no—
135C2×C4×C17C23×C414yes—
136C2×C4×C17C23×C414yes—
137C2×C4×C17C23×C412yes—
138C2×C4×C17C23×C418yes—
139C2×C4×C17C23×C412yes—
140C2×C4×C17H(5,𝔽2)14yes—
141C2×Dic17—216no—
142C2×Dic17—216no—
143C2×Dic17—216no—
144C2×Dic17—816no—
145C2×#532D1711yescharacteristic
146#2381—172no—
147#2380—172no—
148#2372—174no—
149#2352—178no—
150#2333—178no—
151#2311—174no—
152#2283—172no—
153C22×C17—22no—
154C22×C17—28no—
155C22×C17—24no—
156C22×C17—216no—
157C22×C17—44no—
158C22×C17—48no—
159C22×C17—44no—
160C22×C17—44no—
161C22×C17C24×C411yescharacteristic
162C22×C17C2×H(5,𝔽2)12yes—
163C22×C17#58514yes—
164C4×C17—22no—
165C4×C17—28no—
166C4×C17—24no—
167C4×C17—24no—
168C4×C17#51814yes—
169Dic17—416no—
170C22×#41C2×D1712yes—
171#532Dic17116yescomplemented
172C2×C4⋊Q8C2×D1711yescharacteristic
173C2×#127C2×D1711yescharacteristic
174C2×#124C2×D1712yes—
175C2×#121C2×D1714yes—
176C2×C4×D4C2×D1714yes—
177C2×#117—174no—
178C22×C4⋊C4—174no—
179C22×C42C2×D1711yescharacteristic
180#391—178no—
181#390—178no—
182#388—1716no—
183#387—1716no—
184#386—178no—
185#386—178no—
186#386—1716no—
187#383—174no—
188#382—174no—
189#381—174no—
190#380—174no—
191#380—178no—
192C4×C4⋊C4—174no—
193C4×C22⋊C4—174no—
194C4×C22⋊C4—1716no—
195#376—174no—
196C2×C17—42no—
197C2×C17—48no—
198C2×C17C2×#51811yescharacteristic
199C2×C17#342812yes—
200C2×C17#355114yes—
201C2×#41—44no—
202C2×#41—44no—
203C2×#41C2×Dic17116yescomplemented
204C22×Q8C22×D1712yes—
205C22×D4C22×D1714yes—
206C22×D4C22×D1714yes—
207C22×D4C22×D1712yes—
208C23×C4—178no—
209C23×C4C22×D1714yes—
210C23×C4C22×D1711yescharacteristic
211C23×C4C22×D1712yes—
212C4⋊Q8C2×Dic1718yescomplemented
213#127C2×Dic1718yes—
214#124C2×Dic17116yescomplemented
215#121C2×Dic17132yescomplemented
216C4×D4C2×Dic17132yescomplemented
217#117—688no—
218C2×C4⋊C4—178no—
219C2×C4⋊C4—178no—
220C2×C4⋊C4—688no—
221C2×C4⋊C4C22×D1714yes—
222C2×C22⋊C4—178no—
223C2×C22⋊C4—178no—
224C2×C22⋊C4—1716no—
225C2×C22⋊C4C22×D1718yes—
226C2×C42—178no—
227C2×C42—178no—
228C2×C42C2×Dic1718yescomplemented
229C2×C42C22×D1712yes—
230C2×C42C22×D1711yescharacteristic
231C2×C42C22×D1711yescharacteristic
232#95—178no—
233#95—178no—
234#95—1716no—
235#95—178no—
236#95—178no—
237C17#3436911yescharacteristic, complemented, Sylow
238C24C23×D1711yescharacteristic
239C24C23×D1714yes—
240#41—432no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8C22×Dic1718yes—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4C22×Dic17116yes—
251C2×D4C22×Dic17116yes—
252C2×D4C22×Dic1718yes—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—1716no—
258C22×C4—688no—
259C22×C4—6816no—
260C22×C4C22×Dic1714yes—
261C22×C4C22×Dic17116yes—
262C22×C4C22×Dic1718yes—
263C22×C4C23×D1712yes—
264C22×C4C23×D1711yescharacteristic
265C22×C4C23×D1712yes—
266C22×C4C23×D1714yes—
267C22×C4C23×D1711yescharacteristic
268C4⋊C4—3416no—
269C4⋊C4—688no—
270C4⋊C4C22×Dic17116yes—
271C22⋊C4—6816no—
272C22⋊C4—688no—
273C22⋊C4—688no—
274C22⋊C4C22×Dic17132yes—
275C42—688no—
276C42—688no—
277C42C22×Dic1718yes—
278C42C22×Dic1714yes—
279C42C22×Dic1714yes—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—24no—
284C23—28no—
285C23—28no—
286C23C23×Dic1718yes—
287C23C23×Dic1712yes—
288C23C24×D1711yescharacteristic
289Q8—48no—
290D4—48no—
291D4—416no—
292D4—416no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—28no—
299C2×C4—232no—
300C2×C4—48no—
301C2×C4—3416no—
302C2×C4—3416no—
303C2×C4—3416no—
304C2×C4—13616no—
305C2×C4#1058769114yes—
306C2×C4C23×Dic1714yes—
307C2×C4C23×Dic1714yes—
308C2×C4C23×Dic1712yes—
309C2×C4C23×Dic1718yes—
310C2×C4C23×Dic1712yes—
311C22—22no—
312C22—28no—
313C22—24no—
314C22—216no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—44no—
319C22C24×Dic1711yescharacteristic
320C22C2×#1058769112yes—
321C22#1176364514yes—
322C4—22no—
323C4—28no—
324C4—24no—
325C4—24no—
326C4—6816no—
327C4#1176328414yes—
328C2—42no—
329C2—48no—
330C2C2×#1176328411yescharacteristic
331C2#1454198712yes—
332C2#1454321114yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
Fitting, prime radicalC2×C17×#532C2
socle, cosocleC23×C17C25
hypercenterC2×#532D17
hypocenterC17#34369
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. #14531965C2semidirect
2. #14531961C2semidirect
3. #14531896C2semidirect
4. #14531713C2semidirect
5. #14531551C2semidirect
6. #14531376C2semidirect
7. #14531182C2semidirect
28. C17×#532C4semidirect
37. C2×#41×C17C2×C4semidirect
46. C17×C4⋊Q8C2×C4semidirect
48. C17×#124C2×C4semidirect
49. C17×#121C2×C4semidirect
50. C4×D4×C17C2×C4semidirect
53. C2×C42×C17C2×C4semidirect
171. #532Dic17semidirect
203. C2×#41C2×Dic17semidirect
212. C4⋊Q8C2×Dic17semidirect
214. #124C2×Dic17semidirect
215. #121C2×Dic17semidirect
216. C4×D4C2×Dic17semidirect
228. C2×C42C2×Dic17semidirect
237. C17#34369semidirect

Maximal subgroups

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

subgroupindex
#145319652
#145319612
#145318962
#145317132
#145315512
#145313762
#145311822
C2×C17×#5322
#3436917