Finite Group Databasebeta

← #16955802

Subgroups — #16955802

Order 4352.

2154 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63227C213yesmaximal
2C2×#11762654C211yesmaximal, characteristic, complemented
3C2×#95×D17C211yesmaximal, characteristic
4#14527011C213yesmaximal, complemented
5#14527008C213yesmaximal, complemented
6#14527005C213yesmaximal
7C4×C17×#95C211yesmaximal, characteristic, complemented
8C23×C4×D17C2213yes—
9C22×#63227C2213yes—
10C22×#63227C2211yescharacteristic
11C22×C4×Dic17C2213yes—
12C4×#63227C4124yescomplemented
13#11762654—24no—
14C2×#10587502C2213yes—
15C2×#10587502C2213yes—
16C2×#10587502C2216yes—
17C2×#10587502C2213yes—
18C2×#10587502C2213yes—
19#95×D17—24no—
20C22×C42×C17C2213yes—
21C2×C17×#95C2213yes—
22C2×C17×#95C2211yescharacteristic
23C24×D17C2311yescharacteristic
24C23×C4×C17C2313yes—
25C23×C4×C17C2311yescharacteristic
26C23×C4×C17C2313yes—
27C23×Dic17C2313yes—
28C23×Dic17C2313yes—
29C23×Dic17C2311yescharacteristic
30C22×C4×D17—212no—
31C22×C4×D17—26no—
32C22×C4×D17—26no—
33C22×C4×D17—46no—
34C22×C4×D17C2×C4112yes—
35C2×C42×C17—46no—
36C2×C42×C17C2×C4112yescomplemented
37C2×#63227—26no—
38C2×#63227—26no—
39C2×#63227C2×C4112yes—
40C2×#63227C2×C4112yes—
41C2×C4×Dic17—26no—
42C2×C4×Dic17—26no—
43C2×C4×Dic17C2×C4112yescomplemented
44C17×#95—26no—
45C17×#95D414yes—
46#10587502—26no—
47#10587502—26no—
48#10587502—46no—
49#10587502C2×C4112yescomplemented
50#10587502C2×C4112yescomplemented
51C24×C17C2411yescharacteristic
52C23×D17—21no—
53C23×D17—23no—
54C23×D17—23no—
55C23×D17—21no—
56C23×D17C22×C416yes—
57C23×D17#4116yes—
58C23×D17#4112yes—
59C22×C4×C17—26no—
60C22×C4×C17—26no—
61C22×C4×C17—23no—
62C22×C4×C17—21no—
63C22×C4×C17—26no—
64C22×C4×C17—23no—
65C22×C4×C17—23no—
66C22×C4×C17—26no—
67C22×C4×C17C22×C416yes—
68C22×C4×C17C22×C416yes—
69C22×C4×C17C22×C416yes—
70C22×C4×C17C2×D416yes—
71C22×C4×C17#4116yes—
72C22×Dic17—21no—
73C22×Dic17—23no—
74C22×Dic17—23no—
75C22×Dic17—23no—
76C22×Dic17—23no—
77C22×Dic17—26no—
78C22×Dic17—23no—
79C22×Dic17—23no—
80C22×Dic17—23no—
81C22×Dic17—26no—
82C22×Dic17—23no—
83C22×Dic17—23no—
84C22×Dic17C22×C416yes—
85C22×Dic17C22×C416yes—
86C22×Dic17C22×C416yes—
87C2×C4×D17—224no—
88C2×C4×D17—224no—
89C2×C4×D17—412no—
90C2×C4×D17—424no—
91C2×C4×D17—424no—
92C42×C17—412no—
93#63227—224no—
94#63227C42116yes—
95C4×Dic17—224no—
96#13809—171nomaximal, Sylow
97C23×C17C23×C413yes—
98C23×C17C22×D411yescharacteristic
99C23×C17C2×#4113yes—
100C23×C17C2×#4113yes—
101C23×C17C2×#4111yescharacteristic
102C23×C17H(5,𝔽2)13yes—
103C23×C172−1+411yescharacteristic
104C22×D17—26no—
105C22×D17—26no—
106C22×D17—26no—
107C22×D17—26no—
108C22×D17—26no—
109C22×D17—22no—
110C22×D17—212no—
111C22×D17—26no—
112C22×D17—26no—
113C22×D17C2×C4214yes—
114C22×D17#117112yes—
115C22×D17#117112yes—
116C2×C4×C17—26no—
117C2×C4×C17—26no—
118C2×C4×C17—26no—
119C2×C4×C17—26no—
120C2×C4×C17—212no—
121C2×C4×C17—212no—
122C2×C4×C17—26no—
123C2×C4×C17—26no—
124C2×C4×C17—26no—
125C2×C4×C17—46no—
126C2×C4×C17—46no—
127C2×C4×C17C2×C4214yes—
128C2×C4×C17C4×D4112yes—
129C2×Dic17—26no—
130C2×Dic17—26no—
131C2×Dic17—26no—
132C2×Dic17—26no—
133C2×Dic17—26no—
134C2×Dic17—26no—
135C2×Dic17—26no—
136C2×Dic17—26no—
137C2×Dic17—46no—
138C2×Dic17—43no—
139C2×Dic17—43no—
140C2×Dic17—46no—
141C2×Dic17—43no—
142C2×Dic17—43no—
143C2×Dic17C2×C4214yes—
144C4×D17—448no—
145C2×C4×C22⋊C4—173no—
146C2×#376—171no—
147C22×#95—171no—
148#1434—173no—
149#1434—173no—
150C22⋊#95—173no—
151C4×#95D1711yescharacteristic, complemented
152C22×C17C22×C4211yescharacteristic
153C22×C17C2×#11713yes—
154C22×C17C2×#11713yes—
155C22×C17C2×C4×D413yes—
156C22×C17C4×#4113yes—
157C22×C17#51813yes—
158C22×C17#52013yes—
159C22×C17#54613yes—
160C22×C17#54811yescharacteristic
161C22×C17#55116yes—
162C22×C17#55213yes—
163C22×C17#55313yes—
164C2×D17—212no—
165C2×D17—212no—
166C2×D17—212no—
167C2×D17—212no—
168C2×D17—24no—
169C2×D17—24no—
170C2×D17#37618yescomplemented
171C4×C17—212no—
172C4×C17—24no—
173C4×C17—46no—
174Dic17—24no—
175Dic17—46no—
176Dic17—46no—
177C24×C4—173no—
178C22×C22⋊C4—171no—
179C22×C22⋊C4—173no—
180C22×C42—173no—
181C22×C42C2×D1713yes—
182C4×C22⋊C4—1724no—
183#376—344no—
184C2×#95—173no—
185C2×#95—173no—
186C2×#95—173no—
187C2×#95—173no—
188C2×#95—176no—
189C2×#95—344no—
190C2×#95C2×D1713yes—
191C2×#95C2×D1711yescharacteristic
192C2×C17C2×#37611yescharacteristic
193C2×C17C42×D411yescharacteristic
194C2×C17#227511yescharacteristic
195C2×C17#234213yes—
196C2×C17#234313yes—
197C2×C17#235213yes—
198C2×C17#235313yes—
199D17—28no—
200C25—171no—
201C23×C4—171no—
202C23×C4—173no—
203C23×C4—173no—
204C23×C4—1712no—
205C23×C4—3412no—
206C23×C4—346no—
207C23×C4—346no—
208C23×C4—686no—
209C23×C4C22×D1713yes—
210C23×C4C22×D1711yescharacteristic
211C23×C4C22×D1713yes—
212C2×C22⋊C4—1712no—
213C2×C22⋊C4—1712no—
214C2×C22⋊C4—346no—
215C2×C22⋊C4—346no—
216C2×C42—46no—
217C2×C42—1712no—
218C2×C42—346no—
219C2×C42—346no—
220C2×C42C4×D17112yescomplemented
221#95—26no—
222#95—1712no—
223#95—1712no—
224#95—346no—
225#95—346no—
226#95—686no—
227#95D6814yes—
228C17#1380911yescharacteristic, complemented, Sylow
229C24—176no—
230C24—172no—
231C24—176no—
232C24—341no—
233C24—343no—
234C24—343no—
235C24—341no—
236C24C23×D1711yescharacteristic
237C22×C4—23no—
238C22×C4—26no—
239C22×C4—21no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—23no—
244C22×C4—23no—
245C22×C4—176no—
246C22×C4—176no—
247C22×C4—176no—
248C22×C4—3424no—
249C22×C4—341no—
250C22×C4—343no—
251C22×C4—343no—
252C22×C4—343no—
253C22×C4—343no—
254C22×C4—343no—
255C22×C4—343no—
256C22×C4—346no—
257C22×C4—343no—
258C22×C4—343no—
259C22×C4—343no—
260C22×C4—346no—
261C22×C4—3424no—
262C22×C4—6812no—
263C22×C4—6824no—
264C22×C4—6824no—
265C22×C4C2×C4×D1716yes—
266C22×C4C2×C4×D1716yes—
267C22×C4C2×C4×D1716yes—
268C22×C4C2×D6816yes—
269C22×C4#6325216yes—
270C22⋊C4—1716no—
271C22⋊C4—3424no—
272C42—412no—
273C42—3424no—
274C23—174no—
275C23—1712no—
276C23—1712no—
277C23—342no—
278C23—3412no—
279C23—346no—
280C23—346no—
281C23—346no—
282C23—346no—
283C23—346no—
284C23—346no—
285C23—346no—
286C23C22×C4×D1713yes—
287C23C22×D6811yescharacteristic
288C23C2×#6325213yes—
289C23#1058769113yes—
290C23#1058769411yescharacteristic
291C23#41×D1711yescharacteristic
292C23#41×D1713yes—
293C2×C4—212no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—26no—
297C2×C4—26no—
298C2×C4—212no—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—26no—
302C2×C4—46no—
303C2×C4—46no—
304C2×C4—174no—
305C2×C4—346no—
306C2×C4—346no—
307C2×C4—346no—
308C2×C4—346no—
309C2×C4—346no—
310C2×C4—346no—
311C2×C4—346no—
312C2×C4—346no—
313C2×C4—6848no—
314C2×C4—683no—
315C2×C4—686no—
316C2×C4—683no—
317C2×C4—683no—
318C2×C4—686no—
319C2×C4—683no—
320C2×C4C42×D1714yes—
321C2×C4C4×D68112yes—
322C22—178no—
323C22—3412no—
324C22—344no—
325C22—3412no—
326C22—344no—
327C22—3412no—
328C22—3412no—
329C22C2×C42×D1711yescharacteristic
330C22#117×D1713yes—
331C22#117×D1713yes—
332C22C2×C4×D6813yes—
333C22C4×#6325213yes—
334C22#1176328213yes—
335C22#1176329013yes—
336C22#1176345413yes—
337C22#1176347111yescharacteristic
338C22#1176348816yes—
339C22#1176350013yes—
340C22#1176350913yes—
341C4—212no—
342C4—24no—
343C4—46no—
344C4—344no—
345C4—686no—
346C4—686no—
347C2—348no—
348C2D17×#37611yescharacteristic
349C2C42×D6811yescharacteristic
350C2#1453114211yescharacteristic
351C2#1453163613yes—
352C2#1453164713yes—
353C2#1453171613yes—
354C2#1453172713yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC4×C17×#95C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC4×#95D17
hypocenterC17#13809
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. C2×#11762654C2semidirect
4. #14527011C2semidirect
5. #14527008C2semidirect
7. C4×C17×#95C2semidirect
12. C4×#63227C4semidirect
36. C2×C42×C17C2×C4semidirect
43. C2×C4×Dic17C2×C4semidirect
49. #10587502C2×C4semidirect
50. #10587502C2×C4semidirect
151. C4×#95D17semidirect
170. C2×D17#376semidirect
220. C2×C42C4×D17semidirect
228. C17#13809semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#632272
C2×#117626542
C2×#95×D172
#145270112
#145270082
#145270052
C4×C17×#952
#1380917