Finite Group Databasebeta

← #17015627

Subgroups — #17015627

Order 4352.

808 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 171 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#14539974C212yesmaximal
2#14539187C211yesmaximal, characteristic
3#14528112C214yesmaximal, complemented
4#14528086C214yesmaximal, complemented
5#14527411C211yesmaximal, characteristic, complemented
6#14527386C212yesmaximal, complemented
7C2×C17×#564C211yesmaximal, characteristic, complemented
8#11763359C2211yescharacteristic
9#11763219C2214yes—
10#11763179C2212yes—
11#11763175C2212yes—
12#11763175C2214yes—
13#11763166—22no—
14#11763127—22no—
15#11763122—22no—
16#11763106—24no—
17#11762866C2212yes—
18#11762808—22no—
19C4×#63223C2211yescharacteristic
20#11762551C2218yes—
21#11762551C2214yes—
22C17×#564—24no—
23C17×#564—22no—
24C17×#564C2214yes—
25C2×C17×#125—22no—
26C2×C17×#125—22no—
27C2×C17×#125—22no—
28C2×C17×#125C2212yes—
29C2×C17×#125C2211yescharacteristic
30C2×#63252—21no—
31C17×#125—28no—
32C17×#125—28no—
33C17×#125—28no—
34C17×#125—28no—
35C17×#125—28no—
36C17×#125—28no—
37C17×#125—22no—
38C17×#125—22no—
39C17×#125—22no—
40C17×#125D414yes—
41C17×#125D412yes—
42C17×#125C2314yes—
43C17×#125C2312yes—
44C2×C4⋊C4×C17—22no—
45C2×C4⋊C4×C17—22no—
46C2×C4⋊C4×C17—22no—
47C2×C4⋊C4×C17—22no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17—22no—
50C2×C4⋊C4×C17D412yes—
51C2×C4⋊C4×C17D412yes—
52C2×C4⋊C4×C17D412yes—
53C2×C42×C17—22no—
54C2×C42×C17C2311yescharacteristic
55#10587616—22no—
56C2×#63223—21no—
57#10587561—21no—
58#10587560C2311yescharacteristic
59C4×D68—22no—
60#10587555C2311yescharacteristic
61C22⋊#3622—22no—
62#10587481—24no—
63#10587480—24no—
64#10587480—24no—
65#10587480—24no—
66#10587479—24no—
67#10587478—24no—
68C4⋊#3622—22no—
69#10587474C2314yes—
70#10587474C2312yes—
71C4×#3622—21no—
72C22×C4×C17—22no—
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D412yes—
77C2×C22⋊D17—41no—
78#63252—44no—
79C2×D68—21no—
80C2×D68—22no—
81C2×C4×D17—41no—
82C2×Dic34—21no—
83C4⋊C4×C17—28no—
84C4⋊C4×C17—28no—
85C4⋊C4×C17—28no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—22no—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—24no—
90C4⋊C4×C17—24no—
91C4⋊C4×C17—24no—
92C4⋊C4×C17—24no—
93C4⋊C4×C17—24no—
94C4⋊C4×C17—24no—
95C4⋊C4×C17C2×D414yes—
96C4⋊C4×C17C2×D414yes—
97C4⋊C4×C17C2×D414yes—
98C42×C17—28no—
99C42×C17C2×D411yescharacteristic
100C42×C17C2×D412yes—
101C42×C17C2411yescharacteristic
102#63227—42no—
103C4⋊Dic17—22no—
104#63223—42no—
105C2×#3622—22no—
106C2×#3622—24no—
107#19003—171nomaximal, Sylow
108C23×C17#12011yescharacteristic
109C22×D17—41no—
110C2×C4×C17—22no—
111C2×C4×C17—24no—
112C2×C4×C17—24no—
113C2×C4×C17—24no—
114C2×C4×C17—42no—
115C2×C4×C17—44no—
116C2×C4×C17—44no—
117C2×C4×C17—44no—
118C2×C4×C17#12011yescharacteristic
119C2×C4×C17#12012yes—
120C2×C4×C17C22×D411yescharacteristic
121C2×C4×C17C22×D412yes—
122C22⋊D17—82no—
123C2×Dic17—41no—
124D68—44no—
125D68—42no—
126C4×D17—82no—
127Dic34—42no—
128#3622—44no—
129#3622—42no—
130C2×#564D1711yescharacteristic, complemented
131#3234—172no—
132#3142—171no—
133#1663—174no—
134#1658—174no—
135#1534—171no—
136#1528—172no—
137C22×C17—22no—
138C22×C17#45412yes—
139C22×C17C2×#12011yescharacteristic
140C2×D17—81no—
141C2×D17—82no—
142C4×C17—21no—
143C4×C17—21no—
144C4×C17—22no—
145C4×C17—22no—
146C4×C17—44no—
147C4×C17—44no—
148C4×C17—44no—
149Dic17—81no—
150#564—24no—
151#564—22no—
152#564C2×D1714yes—
153#532—171no—
154C2×#125—22no—
155C2×#125—22no—
156C2×#125—22no—
157C2×#125C2×D1711yescharacteristic
158C2×#125C2×D1712yes—
159#501—174no—
160#491—172no—
161#490—172no—
162#490—174no—
163#488—342no—
164#481—342no—
165#480—342no—
166#478—344no—
167#432—172no—
168#417—342no—
169C4×M16—171no—
170#330—178no—
171#330—174no—
172C2×C17—22no—
173C2×C17C2×#45411yescharacteristic
174C2×C17#302012yes—
175D17—161no—
176C2×#41—341no—
177C2×M16—341no—
178C4⋊Q8—171no—
179#127—171no—
180#125—28no—
181#125—28no—
182#125—28no—
183#125—28no—
184#125—28no—
185#125—28no—
186#125—22no—
187#125—22no—
188#125—22no—
189#125C22⋊D1714yes—
190#125C22⋊D1712yes—
191#125C22×D1712yes—
192#125C22×D1714yes—
193#124—341no—
194#121—342no—
195C4×D4—342no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C22⋊D1712yes—
203C2×C4⋊C4C22⋊D1712yes—
204C2×C4⋊C4C22⋊D1712yes—
205C2×C42—22no—
206C2×C42C22×D1711yescharacteristic
207#107—344no—
208#106—344no—
209C4⋊C8—342no—
210#103—344no—
211#102—344no—
212#102—344no—
213#102—344no—
214C22⋊C8—342no—
215#97—172no—
216#97—174no—
217C4×C8—341no—
218C17#1900311yescharacteristic, complemented, Sylow
219#41—684no—
220C2×Q8—341no—
221C2×D4—342no—
222C2×D4—341no—
223C2×D4—681no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—681no—
228C22×C4C2×C22⋊D1711yescharacteristic
229C22×C4C2×C22⋊D1712yes—
230M16—682no—
231C2×C8—342no—
232C2×C8—344no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—342no—
246C4⋊C4C2×C22⋊D1714yes—
247C4⋊C4C2×C22⋊D1714yes—
248C4⋊C4C2×C22⋊D1714yes—
249C22⋊C4—682no—
250C42—28no—
251C42C2×C22⋊D1711yescharacteristic
252C42C2×C22⋊D1712yes—
253C42C23×D1711yescharacteristic
254C23—681no—
255C23#1058763911yescharacteristic
256Q8—682no—
257D4—682no—
258D4—684no—
259D4—1362no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—681no—
269C2×C4—1362no—
270C2×C4#1058763912yes—
271C2×C4#1058763911yescharacteristic
272C2×C4C22×C22⋊D1712yes—
273C2×C4C22×C22⋊D1711yescharacteristic
274C8—682no—
275C8—684no—
276C22—22no—
277C22—1361no—
278C22—1362no—
279C22#1176298212yes—
280C22C2×#1058763911yescharacteristic
281C4—21no—
282C4—21no—
283C4—22no—
284C4—22no—
285C4—44no—
286C4—44no—
287C4—44no—
288C4—1361no—
289C2—22no—
290C2—2721no—
291C2C2×#1176298211yescharacteristic
292C2#1453814812yes—
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#564C2
socleC22×C17C2×#120
prime radical#11763359C22
hypercenterC2×#564D17
hypocenterC17#19003
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#564

Complements

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

NH ≅ G/Ntype
3. #14528112C2semidirect
4. #14528086C2semidirect
5. #14527411C2semidirect
6. #14527386C2semidirect
7. C2×C17×#564C2semidirect
130. C2×#564D17semidirect
218. C17#19003semidirect

Maximal subgroups

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

subgroupindex
#145399742
#145391872
#145281122
#145280862
#145274112
#145273862
C2×C17×#5642
#1900317