Finite Group Databasebeta

← #17022997

Subgroups — #17022997

Order 4352.

896 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 179 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
1#14539722C212yesmaximal, complemented
2#14539699C212yesmaximal, complemented
3C2×#11763186C211yesmaximal, characteristic, complemented
4#14528452C214yesmaximal, complemented
5#14528426C214yesmaximal, complemented
6#14527076C211yesmaximal, characteristic, complemented
7C17×#3131C211yesmaximal, characteristic, complemented
8C22×D136—22no—
9C2×#10587560C2212yes—
10#11763186—22no—
11#11763186C2214yes—
12#11763185C2214yescomplemented
13#11763183C2214yescomplemented
14#11763157C2214yescomplemented
15#11763157C2214yescomplemented
16#11763009—24no—
17#11763004—24no—
18#11762959—24no—
19#11762942—24no—
20#11762862C2212yes—
21#11762531C2214yescomplemented
22C17×#533C2211yescharacteristic
23C17×#532C2211yescharacteristic
24C17×#488—22no—
25C17×#487C2212yescomplemented
26C17×#486C2212yescomplemented
27C17×#416—22no—
28C2×C4×C8×C17C2211yescharacteristic
29C2×#41×C17—21no—
30C22×D68—22no—
31C22×D68—22no—
32C22×D68—22no—
33C22×C8×C17D412yes—
34C17×C4⋊Q8C2311yescharacteristic
35C17×C4⋊Q8C2312yes—
36C17×#127C2311yescharacteristic
37C17×#125—21no—
38C17×#124—21no—
39C17×C22⋊Q8—22no—
40C17×#121—22no—
41C4×Q8×C17—22no—
42C4×D4×C17—22no—
43C17×#117—21no—
44C2×C42×C17C2311yescharacteristic
45C2×#63230—28no—
46C2×#63228—28no—
47C2×D136—28no—
48C2×D136—48no—
49#10587560—22no—
50#10587560—22no—
51#10587560—22no—
52#10587560C2314yes—
53C17×#103—24no—
54C17×#102—24no—
55C4×C8×C17D412yes—
56C4×C8×C17C2312yes—
57C22⋊#3622D414yes—
58#10587480—28no—
59#10587480—28no—
60C4⋊#3622D414yes—
61C4×#3622C2314yes—
62C23×D17—42no—
63#41×C17—44no—
64C2×Q8×C17—22no—
65C2×Q8×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—22no—
68C2×D4×C17—41no—
69C22×C4×C17—41no—
70C22×C4×C17C2×D412yes—
71C22×C4×C17C2×D411yescharacteristic
72C2×D68—24no—
73C2×D68—24no—
74C2×D68—24no—
75C2×D68—48no—
76C2×D68—42no—
77C2×D68—42no—
78C2×D68—42no—
79C2×D68—84no—
80C2×D68—84no—
81C2×C8×C17—24no—
82C2×C8×C17C2×D414yes—
83C4⋊C4×C17—22no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—42no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—42no—
89C42×C17—41no—
90C42×C17C2×D412yes—
91C42×C17C2×D411yescharacteristic
92C42×C17C2411yescharacteristic
93#63230—88no—
94#63228—88no—
95C2×#3622C2×D418yes—
96D136—416no—
97#19698—171nomaximal, Sylow
98C23×C17—41no—
99C23×C17#12711yescharacteristic
100C22×D17—44no—
101C22×D17—84no—
102C22×D17—84no—
103C22×D17—84no—
104Q8×C17—42no—
105Q8×C17—44no—
106D4×C17—42no—
107D4×C17—44no—
108D4×C17—82no—
109C2×C4×C17—24no—
110C2×C4×C17—42no—
111C2×C4×C17—41no—
112C2×C4×C17—82no—
113C2×C4×C17#12712yes—
114C2×C4×C17#12711yescharacteristic
115C2×C4×C17AGL(1,ℤ/8)14yes—
116C2×C4×C17C22×D412yes—
117C2×C4×C17C22×D411yescharacteristic
118D68—48no—
119D68—48no—
120D68—48no—
121D68—84no—
122D68—84no—
123D68—84no—
124C8×C17—24no—
125#3622—28no—
126#3208—172no—
127#3205—172no—
128C2×#493—171no—
129#3131D1711yescharacteristic, complemented
130#1721—174no—
131#1717—174no—
132#1452—171no—
133C22×C17—22no—
134C22×C17—81no—
135C22×C17—82no—
136C22×C17#49612yes—
137C22×C17C2×#12711yescharacteristic
138C2×D17—84no—
139C2×D17—84no—
140C2×D17—84no—
141C2×D17—168no—
142C4×C17—22no—
143C4×C17—81no—
144C4×C17—82no—
145C4×C17#49612yes—
146C4×C17C2×AGL(1,ℤ/8)12yes—
147C4×C17#57614yes—
148C22×D8—342no—
149#533C2×D1711yescharacteristic
150#532C2×D1711yescharacteristic
151C2×#127—172no—
152#493—174no—
153#493—174no—
154#493—342no—
155#492—174no—
156#488—22no—
157#487C2×D1712yescomplemented
158#486—174no—
159#486—174no—
160#486C2×D1712yescomplemented
161#460—344no—
162#459—344no—
163#450—344no—
164#447—344no—
165#431—172no—
166#416—22no—
167C2×C4×C8C2×D1711yescharacteristic
168#322—174no—
169C2×C17—22no—
170C2×C17—161no—
171C2×C17C2×#49611yescharacteristic
172C2×C17#315512yes—
173D17—164no—
174C2×#41—21no—
175C22×D4—342no—
176C22×D4—342no—
177C22×D4—342no—
178C2×SD16—348no—
179C2×D8—348no—
180C2×D8—348no—
181C2×D8—688no—
182C22×C8C22⋊D1712yes—
183C4⋊Q8C22×D1711yescharacteristic
184C4⋊Q8C22×D1712yes—
185#127—174no—
186#127—342no—
187#127—342no—
188#127—342no—
189#127C22×D1711yescharacteristic
190#125—21no—
191#124—21no—
192C22⋊Q8—22no—
193#121—22no—
194C4×Q8—22no—
195C4×D4—22no—
196#117—21no—
197C2×C42C22×D1711yescharacteristic
198C4⋊C8—174no—
199#103—24no—
200#102—24no—
201#102—348no—
202#102—348no—
203C22⋊C8—174no—
204C4×C8—174no—
205C4×C8C22⋊D1712yes—
206C4×C8C22×D1712yes—
207C17#1969811yescharacteristic, complemented, Sylow
208C24—682no—
209#41—44no—
210C2×Q8—22no—
211C2×Q8—21no—
212C2×D4—22no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—344no—
216C2×D4—344no—
217C2×D4—344no—
218C2×D4—682no—
219C2×D4—682no—
220C2×D4—682no—
221C2×D4—688no—
222C2×D4—1364no—
223C2×D4—1364no—
224C22×C4—41no—
225C22×C4D4×D1712yes—
226C22×C4C2×C22⋊D1711yescharacteristic
227SD16—1368no—
228D8—6816no—
229D8—1368no—
230C2×C8—24no—
231C2×C8—178no—
232C2×C8C2×C22⋊D1714yes—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C42—41no—
240C42D4×D1712yes—
241C42C2×C22⋊D1711yescharacteristic
242C42C23×D1711yescharacteristic
243C23—41no—
244C23—684no—
245C23—1364no—
246C23—1364no—
247C23—1364no—
248C23#1058762611yescharacteristic
249Q8—42no—
250Q8—44no—
251D4—42no—
252D4—44no—
253D4—82no—
254D4—688no—
255D4—688no—
256D4—688no—
257D4—1364no—
258D4—1364no—
259D4—1364no—
260C2×C4—24no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4#1058762612yes—
265C2×C4#1058762611yescharacteristic
266C2×C4#1058763514yes—
267C2×C4C2×D4×D1712yes—
268C2×C4C22×C22⋊D1711yescharacteristic
269C8—24no—
270C8—348no—
271C22—22no—
272C22—82no—
273C22—81no—
274C22—2728no—
275C22—1364no—
276C22—1364no—
277C22—1364no—
278C22#1176319912yes—
279C22C2×#1058762611yescharacteristic
280C4—22no—
281C4—82no—
282C4—81no—
283C4#1176319912yes—
284C4C2×#1058763512yes—
285C4#1176361114yes—
286C2—22no—
287C2—161no—
288C2—2724no—
289C2C2×#1176319911yescharacteristic
290C2#1453927112yes—
subgroupG/H
centerC22C2×#10587626
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3131C2
socleC22×C17C2×#127
hypercenter#3131D17
hypocenterC17#19698
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #14539722C2semidirect
2. #14539699C2semidirect
3. C2×#11763186C2semidirect
4. #14528452C2semidirect
5. #14528426C2semidirect
6. #14527076C2semidirect
7. C17×#3131C2semidirect
12. #11763185C22semidirect
13. #11763183C22semidirect
14. #11763157C22semidirect
15. #11763157C22semidirect
21. #11762531C22semidirect
25. C17×#487C22semidirect
26. C17×#486C22semidirect
129. #3131D17semidirect
157. #487C2×D17semidirect
160. #486C2×D17semidirect
207. C17#19698semidirect

Maximal subgroups

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

subgroupindex
#145397222
#145396992
C2×#117631862
#145284522
#145284262
#145270762
C17×#31312
#1969817