Finite Group Databasebeta

← #17194518

Subgroups — #17194518

Order 4352.

2250 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531749C216yesmaximal, complemented
2#14531705C212yesmaximal, complemented
3#14531661C216yesmaximal
4Q8×C22⋊Dic17C212yesmaximal
5#14531435C216yesmaximal, complemented
6#14531182C213yesmaximal
7#14531182C213yesmaximal
8#14531149C212yesmaximal, complemented
9C22×C4×Q8×C17C211yesmaximal, characteristic
10C2×Q8×Dic17C2214yes—
11#11762716C22112yes—
12#11762708C22112yes—
13#11762695C22124yes—
14#11762677C2216yes—
15#11762677C2216yes—
16#11762675C2216yes—
17#11762675C2216yes—
18#11762672C2211yescharacteristic
19#11762672C2213yes—
20#11762669C2216yes—
21#11762669C2216yes—
22C4⋊C4×Dic17C22112yes—
23C4×C4⋊Dic17C2216yes—
24C4×C4⋊Dic17C2216yes—
25C4×C22⋊Dic17C2216yes—
26C4×C22⋊Dic17C2216yes—
27#11762655C2216yes—
28#11762655C2216yes—
29C23×Q8×C17C2211yescharacteristic
30C2×C4×Q8×C17C4116yescomplemented
31C2×C4×Q8×C17C2211yescharacteristic
32C2×C4×Q8×C17C2213yes—
33C2×C4×Q8×C17C2211yescharacteristic
34C2×C4×Q8×C17C2213yes—
35C22×C4⋊C4×C17C2213yes—
36C22×C42×C17C2213yes—
37C22×Q8×C17—28no—
38C22×Q8×C17C2×C418yes—
39C22×Q8×C17C2313yes—
40C22×Q8×C17C2311yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2313yes—
43C23×C4×C17C2313yes—
44C4×Q8×C17C2×C4124yes—
45C4×Q8×C17C2×C4124yes—
46C4×Q8×C17C2×C418yes—
47C4×Q8×C17C2×C418yes—
48C2×C4⋊C4×C17C2×C4124yes—
49C2×C4⋊C4×C17C2316yes—
50C2×C4⋊C4×C17C2313yes—
51C2×C4⋊C4×C17C2313yes—
52C2×C42×C17C2×C4124yes—
53C2×C42×C17C2313yes—
54C2×C42×C17C2313yes—
55C2×C42×C17C2313yes—
56C2×C42×C17C2313yes—
57C2×C22⋊Dic17C2312yes—
58C2×C22⋊Dic17C2316yes—
59Q8×Dic17—216no—
60C2×C4⋊Dic17C23112yes—
61C2×C4⋊Dic17C23112yes—
62C2×C4×Dic17C23112yes—
63C2×C4×Dic17C23112yes—
64#10587502C23112yes—
65#10587502C23112yes—
66#10587502C23112yes—
67#10587502C23112yes—
68#10587502C2312yes—
69#10587502C2316yes—
70#10587502C2316yes—
71#10587502C2312yes—
72C24×C17C2411yescharacteristic
73C2×Q8×C17—232no—
74C2×Q8×C17—212no—
75C2×Q8×C17C22×C4112yes—
76C2×Q8×C17C22×C414yes—
77C2×Q8×C17#4118yes—
78C22×C4×C17—24no—
79C22×C4×C17—212no—
80C22×C4×C17—46no—
81C22×C4×C17C22×C4112yes—
82C22×C4×C17C22×C414yes—
83C22×C4×C17C22×C4112yes—
84C22×C4×C17C2413yes—
85C22×C4×C17C2411yescharacteristic
86C22×C4×C17C2411yescharacteristic
87C22×C4×C17C2413yes—
88C22×C4×C17C2413yes—
89C22×C4×C17C2413yes—
90C22×Dic17C2414yes—
91C22×Dic17C24112yes—
92C4⋊C4×C17C22×C4124yes—
93C4⋊C4×C17C22×C4112yes—
94C4⋊C4×C17C22×C4112yes—
95C42×C17C22×C4112yes—
96C42×C17C22×C4112yes—
97C42×C17C22×C4112yes—
98C42×C17C22×C4112yes—
99C22⋊Dic17—24no—
100C22⋊Dic17—212no—
101C4⋊Dic17—224no—
102C4⋊Dic17—412no—
103C4×Dic17—224no—
104C4×Dic17—412no—
105#34192—171nomaximal, Sylow
106C23×C17—22no—
107C23×C17—22no—
108C23×C17C23×C412yes—
109C23×C172−1+414yes—
110C23×C17C2511yescharacteristic
111Q8×C17—224no—
112Q8×C17#117116yes—
113C2×C4×C17—28no—
114C2×C4×C17—26no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—26no—
118C2×C4×C17—26no—
119C2×C4×C17—224no—
120C2×C4×C17—412no—
121C2×C4×C17C23×C412yes—
122C2×C4×C17C23×C412yes—
123C2×C4×C17C23×C416yes—
124C2×C4×C17C23×C416yes—
125C2×C4×C17C23×C416yes—
126C2×C4×C17C23×C416yes—
127C2×C4×C17C2×#41112yes—
128C2×Dic17—24no—
129C2×Dic17—212no—
130C2×Dic17—24no—
131C2×Dic17—212no—
132C2×Dic17—212no—
133C2×Dic17—24no—
134C22×C4×Q8D1711yescharacteristic
135#2356—176no—
136#2351—172no—
137#2345—176no—
138Q8×C22⋊C4—172no—
139#2320—176no—
140#2283—173no—
141#2283—173no—
142#2276—172no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—28no—
146C22×C17#51914yes—
147C22×C17C24×C411yescharacteristic
148C22×C17C22×#4112yes—
149C22×C17C2×2−1+412yes—
150C22×C17#58512yes—
151C4×C17—26no—
152C4×C17—22no—
153C4×C17—22no—
154C4×C17—26no—
155C4×C17—26no—
156C4×C17C2×#117112yes—
157Dic17—44no—
158Dic17—412no—
159C23×Q8C2×D1711yescharacteristic
160C2×C4×Q8—174no—
161C2×C4×Q8Dic17116yescomplemented
162C2×C4×Q8C2×D1713yes—
163C2×C4×Q8C2×D1711yescharacteristic
164C2×C4×Q8C2×D1713yes—
165C2×C4×Q8C2×D1711yescharacteristic
166C22×C4⋊C4C2×D1713yes—
167C22×C42C2×D1713yes—
168#391—1712no—
169#389—1712no—
170#387—1724no—
171#383—176no—
172#383—176no—
173#382—176no—
174#382—176no—
175#381—171no—
176#381—173no—
177#380—176no—
178#380—176no—
179C4×C4⋊C4—1712no—
180C4×C4⋊C4—176no—
181C4×C4⋊C4—176no—
182C4×C22⋊C4—176no—
183C4×C22⋊C4—176no—
184#376—176no—
185#376—176no—
186C2×C17—24no—
187C2×C17C22×#11711yescharacteristic
188C2×C17C2×#51911yescharacteristic
189C2×C17#342811yescharacteristic
190C2×C17#351614yes—
191C22×Q8—28no—
192C22×Q8C2×Dic1718yes—
193C22×Q8C22×D1713yes—
194C22×Q8C22×D1711yescharacteristic
195C23×C4C22×D1713yes—
196C23×C4C22×D1711yescharacteristic
197C23×C4C22×D1713yes—
198C4×Q8—3416no—
199C4×Q8C2×Dic17124yes—
200C4×Q8C2×Dic17124yes—
201C4×Q8C2×Dic1718yes—
202C4×Q8C2×Dic1718yes—
203C2×C4⋊C4—1712no—
204C2×C4⋊C4—1712no—
205C2×C4⋊C4C2×Dic17124yes—
206C2×C4⋊C4C22×D1713yes—
207C2×C4⋊C4C22×D1716yes—
208C2×C4⋊C4C22×D1713yes—
209C2×C22⋊C4—172no—
210C2×C22⋊C4—176no—
211C2×C42—1712no—
212C2×C42—1712no—
213C2×C42C2×Dic17124yes—
214C2×C42C22×D1713yes—
215C2×C42C22×D1713yes—
216C2×C42C22×D1713yes—
217C2×C42C22×D1713yes—
218#95—172no—
219#95—176no—
220#95—1712no—
221#95—176no—
222#95—1712no—
223#95—1712no—
224#95—1712no—
225#95—172no—
226C17#3419211yescharacteristic, complemented, Sylow
227C24C23×D1711yescharacteristic
228C2×Q8—232no—
229C2×Q8—212no—
230C2×Q8#6325218yes—
231C2×Q8C22×Dic17112yes—
232C2×Q8C22×Dic1714yes—
233C22×C4—24no—
234C22×C4—212no—
235C22×C4—46no—
236C22×C4—174no—
237C22×C4—1712no—
238C22×C4C22×Dic1714yes—
239C22×C4C22×Dic17112yes—
240C22×C4C22×Dic17112yes—
241C22×C4C23×D1711yescharacteristic
242C22×C4C23×D1711yescharacteristic
243C22×C4C23×D1713yes—
244C22×C4C23×D1713yes—
245C22×C4C23×D1713yes—
246C22×C4C23×D1713yes—
247C4⋊C4—3424no—
248C4⋊C4—6812no—
249C4⋊C4C22×Dic17124yes—
250C4⋊C4C22×Dic17112yes—
251C4⋊C4C22×Dic17112yes—
252C22⋊C4—344no—
253C22⋊C4—3412no—
254C42—3424no—
255C42—6812no—
256C42C22×Dic17112yes—
257C42C22×Dic17112yes—
258C42C22×Dic17112yes—
259C42C22×Dic17112yes—
260C23—22no—
261C23—22no—
262C23#1058769414yes—
263C23C23×Dic1712yes—
264C23C24×D1711yescharacteristic
265Q8—224no—
266Q8#10587612116yes—
267C2×C4—28no—
268C2×C4—26no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—26no—
272C2×C4—26no—
273C2×C4—224no—
274C2×C4—412no—
275C2×C4—344no—
276C2×C4—3412no—
277C2×C4—344no—
278C2×C4—3412no—
279C2×C4—3412no—
280C2×C4—344no—
281C2×C4C2×#63252112yes—
282C2×C4C23×Dic1712yes—
283C2×C4C23×Dic1712yes—
284C2×C4C23×Dic1716yes—
285C2×C4C23×Dic1716yes—
286C2×C4C23×Dic1716yes—
287C2×C4C23×Dic1716yes—
288C22—22no—
289C22—22no—
290C22—28no—
291C22#1176328714yes—
292C22C24×Dic1711yescharacteristic
293C22C22×#6325212yes—
294C22C2×#1058769412yes—
295C22#1176364512yes—
296C4—26no—
297C4—22no—
298C4—22no—
299C4—26no—
300C4—26no—
301C4—684no—
302C4—6812no—
303C4C2×#10587612112yes—
304C2—24no—
305C2C22×#1058761211yescharacteristic
306C2C2×#1176328711yescharacteristic
307C2#1454198711yescharacteristic
308C2#1454277714yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC22×C4×Q8×C17C2
socle, cosocleC23×C17C25
prime radicalC24×C17C24
hypercenterC22×C4×Q8D17
hypocenterC17#34192
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #14531749C2semidirect
2. #14531705C2semidirect
5. #14531435C2semidirect
8. #14531149C2semidirect
30. C2×C4×Q8×C17C4semidirect
161. C2×C4×Q8Dic17semidirect
226. C17#34192semidirect

Maximal subgroups

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

subgroupindex
#145317492
#145317052
#145316612
Q8×C22⋊Dic172
#145314352
#145311822
#145311492
C22×C4×Q8×C172
#3419217