Finite Group Databasebeta

← #16956156

Subgroups — #16956156

Order 4352.

1966 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×#11762746C211yesmaximal, characteristic
2C2×C4⋊C4×Dic17C213yesmaximal, complemented
3C2×#11762656C213yesmaximal
4#14527053C213yesmaximal, complemented
5#14527026C213yesmaximal
6#95×Dic17C211yesmaximal, characteristic, complemented
7C17×#1444C211yesmaximal, characteristic, complemented
8C23×C4×D17C2211yescharacteristic
9C22×C4⋊Dic17C2213yes—
10C22×#63227C2213yes—
11C22×#63227C2213yes—
12C22×C4×Dic17C2213yes—
13C22×C4×Dic17C2213yes—
14#11762746C418yescomplemented
15C4⋊C4×Dic17—212no—
16#11762656—212no—
17C2×#10587502C2213yes—
18C2×#10587502C2213yes—
19C2×#10587502C2216yes—
20C22×C4⋊C4×C17C2213yes—
21C2×C17×#95C2213yes—
22C2×C17×#95C2211yescharacteristic
23C24×D17C2311yescharacteristic
24C23×C4×C17C2311yescharacteristic
25C23×C4×C17C2313yes—
26C23×C4×C17C2313yes—
27C23×Dic17C2313yes—
28C23×Dic17C2313yes—
29C23×Dic17C2311yescharacteristic
30C22×C4×D17—26no—
31C22×C4×D17—26no—
32C22×C4×D17Q814yes—
33C2×C4⋊C4×C17—26no—
34C2×C4⋊C4×C17—212no—
35C2×#63227—26no—
36C2×#63227—26no—
37C2×#63227—26no—
38C2×#63227—26no—
39C2×#63227—26no—
40C2×#63227C2×C4112yes—
41C2×C4⋊Dic17—26no—
42C2×C4⋊Dic17—26no—
43C2×C4⋊Dic17C2×C4112yescomplemented
44C2×C4×Dic17—26no—
45C2×C4×Dic17—212no—
46C2×C4×Dic17—212no—
47C2×C4×Dic17D4112yes—
48C17×#95—26no—
49C17×#95C2×C414yescomplemented
50#10587502—26no—
51#10587502—26no—
52#10587502—212no—
53C24×C17C2411yescharacteristic
54C23×D17—23no—
55C23×D17—23no—
56C23×D17—23no—
57C23×D17—23no—
58C23×D17C22×C412yes—
59C23×D17C2×Q812yes—
60C23×D17#4112yes—
61C22×C4×C17—26no—
62C22×C4×C17—26no—
63C22×C4×C17—23no—
64C22×C4×C17—26no—
65C22×C4×C17—23no—
66C22×C4×C17—23no—
67C22×C4×C17—23no—
68C22×C4×C17—26no—
69C22×C4×C17C22×C416yes—
70C22×C4×C17C2×D416yes—
71C22×C4×C17C2×Q812yes—
72C22×C4×C17#4116yes—
73C22×C4×C17#4116yes—
74C22×Dic17—26no—
75C22×Dic17—23no—
76C22×Dic17—26no—
77C22×Dic17—26no—
78C22×Dic17—26no—
79C22×Dic17—23no—
80C22×Dic17—23no—
81C22×Dic17—23no—
82C22×Dic17C22×C416yes—
83C22×Dic17C2×D416yes—
84C22×Dic17C2×D416yes—
85C22×Dic17C2×Q812yes—
86C22×Dic17#4116yes—
87C2×C4×D17—224no—
88C2×C4×D17—412no—
89C2×C4×D17—44no—
90C4⋊C4×C17—224no—
91#63227—224no—
92#63227—224no—
93C4⋊Dic17—224no—
94C4×Dic17—224no—
95C4×Dic17—412no—
96C4×#399—171nomaximal, Sylow
97C23×C17C23×C411yescharacteristic
98C23×C17C22×D413yes—
99C23×C17C22×Q811yescharacteristic
100C23×C17C2×#4113yes—
101C23×C17C2×#4113yes—
102C23×C17C2×#4111yescharacteristic
103C23×C17C2×#4113yes—
104C22×D17—26no—
105C22×D17—26no—
106C22×D17—26no—
107C22×D17—26no—
108C22×D17—26no—
109C22×D17—26no—
110C22×D17—43no—
111C22×D17—43no—
112C22×D17—46no—
113C22×D17—41no—
114C22×D17—43no—
115C22×D17C4×Q814yes—
116C2×C4×C17—212no—
117C2×C4×C17—26no—
118C2×C4×C17—26no—
119C2×C4×C17—212no—
120C2×C4×C17—26no—
121C2×C4×C17—212no—
122C2×C4×C17—26no—
123C2×C4×C17—43no—
124C2×C4×C17—43no—
125C2×C4×C17—46no—
126C2×C4×C17—41no—
127C2×C4×C17—43no—
128C2×C4×C17C4×D4112yes—
129C2×Dic17—212no—
130C2×Dic17—26no—
131C2×Dic17—26no—
132C2×Dic17—26no—
133C2×Dic17—26no—
134C2×Dic17—46no—
135C2×Dic17—46no—
136C2×Dic17—43no—
137C2×Dic17—46no—
138C2×Dic17—43no—
139C2×Dic17C22⋊Q8112yes—
140C2×Dic17#123112yes—
141C2×Dic17#12714yes—
142C4×D17—416no—
143C2×#399—171no—
144C2×C4×C4⋊C4—173no—
145C2×C4×C22⋊C4—173no—
146#1445—173no—
147#1444D1711yescharacteristic, complemented
148#1439—173no—
149C4×#95—171no—
150C22×C17C2×C4×D413yes—
151C22×C17C2×C4×Q811yescharacteristic
152C22×C17C4×#4113yes—
153C22×C17C2×C22⋊Q813yes—
154C22×C17C2×#12313yes—
155C22×C17#52513yes—
156C22×C17#52513yes—
157C22×C17#52913yes—
158C22×C17#52916yes—
159C22×C17C2×#12711yescharacteristic
160C22×C17#53213yes—
161C22×C17#53313yes—
162C2×D17—212no—
163C2×D17—42no—
164C2×D17—42no—
165C2×D17—46no—
166C2×D17—46no—
167C2×D17—42no—
168C2×D17—46no—
169C4×C17—212no—
170C4×C17—42no—
171C4×C17—46no—
172Dic17—46no—
173Dic17—46no—
174Dic17#39918yescomplemented
175C24×C4—171no—
176C22×C4⋊C4—173no—
177C22×C4⋊C4C2×D1713yes—
178C22×C22⋊C4—173no—
179C22×C22⋊C4—173no—
180C22×C42—173no—
181C22×C42—173no—
182#399—178no—
183C4×C4⋊C4—3412no—
184C4×C22⋊C4—3412no—
185C2×#95—173no—
186C2×#95—176no—
187C2×#95—173no—
188C2×#95C2×D1713yes—
189C2×#95C2×D1711yescharacteristic
190C2×C17C4×#12313yes—
191C2×C17C4×C22⋊Q813yes—
192C2×C17C4×#12711yescharacteristic
193C2×C17C2×#39911yescharacteristic
194C2×C17#239513yes—
195C2×C17#239613yes—
196C2×C17#239711yescharacteristic
197D17—44no—
198C25—171no—
199C23×C4—171no—
200C23×C4—173no—
201C23×C4—173no—
202C23×C4—174no—
203C23×C4—346no—
204C23×C4—346no—
205C23×C4C22×D1713yes—
206C23×C4C22×D1711yescharacteristic
207C23×C4C22×D1713yes—
208C2×C4⋊C4—26no—
209C2×C4⋊C4—212no—
210C2×C4⋊C4—1712no—
211C2×C4⋊C4—346no—
212C2×C4⋊C4—346no—
213C2×C22⋊C4—1712no—
214C2×C22⋊C4—346no—
215C2×C22⋊C4—346no—
216C2×C22⋊C4—346no—
217C2×C22⋊C4—346no—
218C2×C22⋊C4—346no—
219C2×C42—1712no—
220C2×C42—3412no—
221C2×C42—346no—
222C2×C42—3412no—
223#95—26no—
224#95—346no—
225#95—3412no—
226#95—346no—
227#95C4×D1714yescomplemented
228C17C4×#39911yescharacteristic, complemented, Sylow
229C24—172no—
230C24—172no—
231C24—172no—
232C24—343no—
233C24—343no—
234C24—343no—
235C24—343no—
236C24C23×D1711yescharacteristic
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—23no—
240C22×C4—23no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—23no—
244C22×C4—23no—
245C22×C4—176no—
246C22×C4—172no—
247C22×C4—176no—
248C22×C4—176no—
249C22×C4—176no—
250C22×C4—3424no—
251C22×C4—343no—
252C22×C4—346no—
253C22×C4—343no—
254C22×C4—346no—
255C22×C4—343no—
256C22×C4—346no—
257C22×C4—346no—
258C22×C4—343no—
259C22×C4—6812no—
260C22×C4—684no—
261C22×C4C2×C4×D1716yes—
262C22×C4D4×D1716yes—
263C22×C4#6325416yes—
264C22×C4Q8×D1712yes—
265C22×C4#6325616yes—
266C4⋊C4—224no—
267C4⋊C4—3424no—
268C22⋊C4—3424no—
269C22⋊C4—3424no—
270C42—3424no—
271C42—6812no—
272C23—174no—
273C23—346no—
274C23—346no—
275C23—346no—
276C23—346no—
277C23—346no—
278C23—346no—
279C23—683no—
280C23—683no—
281C23—686no—
282C23—681no—
283C23—683no—
284C23C22×C4×D1711yescharacteristic
285C23C2×D4×D1713yes—
286C23C2×#6325413yes—
287C23C2×Q8×D1711yescharacteristic
288C23C2×#6325613yes—
289C23#41×D1711yescharacteristic
290C23#41×D1713yes—
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—212no—
295C2×C4—26no—
296C2×C4—212no—
297C2×C4—26no—
298C2×C4—43no—
299C2×C4—43no—
300C2×C4—46no—
301C2×C4—43no—
302C2×C4—41no—
303C2×C4—1712no—
304C2×C4—1712no—
305C2×C4—174no—
306C2×C4—346no—
307C2×C4—3412no—
308C2×C4—346no—
309C2×C4—346no—
310C2×C4—346no—
311C2×C4—6816no—
312C2×C4—686no—
313C2×C4—686no—
314C2×C4—683no—
315C2×C4—683no—
316C2×C4—686no—
317C2×C4#10587579112yes—
318C22—3412no—
319C22—682no—
320C22—682no—
321C22—686no—
322C22—686no—
323C22—682no—
324C22—686no—
325C22C2×#1058757913yes—
326C22C4×Q8×D1711yescharacteristic
327C22C4×#6325413yes—
328C22C22⋊Q8×D1713yes—
329C22#123×D1713yes—
330C22#1176331513yes—
331C22#1176331913yes—
332C22#1176334413yes—
333C22#1176334516yes—
334C22#127×D1711yescharacteristic
335C22#1176336013yes—
336C22#1176336413yes—
337C4—212no—
338C4—42no—
339C4—46no—
340C4—178no—
341C4—686no—
342C4—686no—
343C2—684no—
344C2#1453130313yes—
345C2#1453131413yes—
346C2#1453134611yescharacteristic
347C2D17×#39911yescharacteristic
348C2#1453204813yes—
349C2#1453205513yes—
350C2#1453206211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1444C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1444D17
hypocenterC17C4×#399
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic17C2semidirect
4. #14527053C2semidirect
6. #95×Dic17C2semidirect
7. C17×#1444C2semidirect
14. #11762746C4semidirect
43. C2×C4⋊Dic17C2×C4semidirect
49. C17×#95C2×C4semidirect
147. #1444D17semidirect
174. Dic17#399semidirect
227. #95C4×D17semidirect
228. C17C4×#399semidirect

Maximal subgroups

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

subgroupindex
C2×#117627462
C2×C4⋊C4×Dic172
C2×#117626562
#145270532
#145270262
#95×Dic172
C17×#14442
C4×#39917