Finite Group Databasebeta

← #17016380

Subgroups — #17016380

Order 4352.

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

normal subgroups only

This group has 936 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#14539177C212yesmaximal, complemented
2#14539166C211yesmaximal, characteristic, complemented
3#14528107C218yesmaximal, complemented
4#14527379C211yesmaximal, characteristic, complemented
5#14527376C212yesmaximal, complemented
6C17×#3548C211yesmaximal, characteristic
7C2×#10587555C2211yescharacteristic
8#11763179C2214yes—
9#11763168C2218yes—
10#11763161—22no—
11#11763159—22no—
12#11763155—22no—
13#11762817—22no—
14#11762810—24no—
15C4×#63223C2211yescharacteristic
16C4×#63223C2212yes—
17#11762551C2214yes—
18#11762549C2218yes—
19C17×#562C2214yes—
20C17×#554—22no—
21C17×#550—22no—
22C17×#547—22no—
23C17×#540—22no—
24C17×#537—24no—
25C17×#533C2211yescharacteristic
26C17×#532C2212yes—
27C2×#41×C17—24no—
28C2×#41×C17D414yes—
29C22×Dic34—21no—
30C17×C4⋊Q8D412yes—
31C17×C4⋊Q8D412yes—
32C17×#127—22no—
33C17×#127D412yes—
34C17×#125—22no—
35C17×#125C2312yes—
36C17×#124C2314yes—
37C17×#123—28no—
38C17×C22⋊Q8—24no—
39C17×C22⋊Q8—28no—
40C17×C22⋊Q8—22no—
41C17×#121—28no—
42C17×#121—24no—
43C17×#121—28no—
44C17×#121—22no—
45C17×#121—22no—
46C4×Q8×C17—22no—
47C4×D4×C17—24no—
48C4×D4×C17—22no—
49C4×D4×C17—22no—
50C17×#117D412yes—
51C2×C4⋊C4×C17—24no—
52C2×C42×C17C2311yescharacteristic
53C2×C4⋊Dic17—22no—
54C2×#63223—22no—
55C2×#63223—21no—
56#10587555—21no—
57#10587555—22no—
58#10587555C2312yes—
59#10587506—28no—
60#10587503—28no—
61#10587481—28no—
62#10587474C2312yes—
63#10587474C2314yes—
64C4×#3622—22no—
65C4×#3622—21no—
66#41×C17—48no—
67#41×C17—88no—
68C2×Q8×C17—22no—
69C2×Q8×C17C2×D414yes—
70C2×D4×C17—28no—
71C2×D4×C17—24no—
72C2×D4×C17—22no—
73C2×D4×C17—22no—
74C2×D4×C17C2×D414yes—
75C22×C4×C17—28no—
76C22×C4×C17—24no—
77C22×C4×C17C2×D412yes—
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—41no—
80C2×Dic34—22no—
81C2×Dic34—44no—
82C2×Dic34—41no—
83C4⋊C4×C17—28no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—24no—
88C4⋊C4×C17C2×D414yes—
89C22⋊C4×C17—28no—
90C22⋊C4×C17—22no—
91C22⋊C4×C17—24no—
92C22⋊C4×C17—24no—
93C42×C17—22no—
94C42×C17C2×D412yes—
95C42×C17C2×D411yescharacteristic
96C42×C17C2411yescharacteristic
97C4⋊Dic17—24no—
98C4⋊Dic17—42no—
99#63223—42no—
100#63223—44no—
101C2×#3622—22no—
102C2×#3622—24no—
103#19080—171nomaximal, Sylow
104C23×C17—24no—
105C23×C17#12011yescharacteristic
106Q8×C17—44no—
107Q8×C17—44no—
108D4×C17—44no—
109D4×C17—44no—
110D4×C17—44no—
111D4×C17—84no—
112D4×C17—88no—
113C2×C4×C17—22no—
114C2×C4×C17—24no—
115C2×C4×C17—24no—
116C2×C4×C17—24no—
117C2×C4×C17—48no—
118C2×C4×C17—88no—
119C2×C4×C17#12011yescharacteristic
120C2×C4×C17#12012yes—
121C2×C4×C17C22×D411yescharacteristic
122C2×C4×C17C22×D412yes—
123C2×Dic17—42no—
124C2×Dic17—82no—
125Dic34—44no—
126Dic34—82no—
127#3622—44no—
128#3622—42no—
129#3548D1711yescharacteristic
130#3141—172no—
131#3139—171no—
132#1662—178no—
133#1526—171no—
134#1525—172no—
135C22×C17—22no—
136C22×C17—21no—
137C22×C17—44no—
138C22×C17—44no—
139C22×C17—44no—
140C22×C17C2×#12011yescharacteristic
141C4×C17—22no—
142C4×C17—21no—
143C4×C17—44no—
144C4×C17—44no—
145C4×C17#45412yes—
146C4×C17#45514yes—
147Dic17—82no—
148#562C2×D1714yes—
149#554—22no—
150#550—22no—
151#547—22no—
152#540—22no—
153#537—24no—
154#533C2×D1711yescharacteristic
155#532C2×D1712yes—
156C2×C4⋊Q8—171no—
157#491—174no—
158#489—178no—
159#487—342no—
160#487—342no—
161#486—342no—
162#419—342no—
163#417—344no—
164C4×M16—171no—
165C4×M16—172no—
166#330—174no—
167#329—178no—
168C2×C17—41no—
169C2×C17—84no—
170C2×C17C2×#45512yes—
171C2×C17C2×#45411yescharacteristic
172C2×#41—24no—
173C2×#41C22⋊D1714yes—
174C22×Q8—341no—
175C2×M16—342no—
176C2×M16—341no—
177C4⋊Q8—172no—
178C4⋊Q8—342no—
179C4⋊Q8—341no—
180C4⋊Q8C22⋊D1712yes—
181C4⋊Q8C22⋊D1712yes—
182#127—22no—
183#127C22⋊D1712yes—
184#125—22no—
185#125C22×D1712yes—
186#124C22×D1714yes—
187#123—28no—
188C22⋊Q8—24no—
189C22⋊Q8—28no—
190C22⋊Q8—22no—
191#121—28no—
192#121—24no—
193#121—28no—
194#121—22no—
195#121—22no—
196C4×Q8—22no—
197C4×D4—24no—
198C4×D4—22no—
199C4×D4—22no—
200#117C22⋊D1712yes—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—342no—
203C2×C42C22×D1711yescharacteristic
204#103—348no—
205#103—348no—
206#102—348no—
207#97—172no—
208#97—174no—
209C4×C8—342no—
210C4×C8—341no—
211C17#1908011yescharacteristic, complemented, Sylow
212#41—48no—
213#41—88no—
214C2×Q8—22no—
215C2×Q8—342no—
216C2×Q8—684no—
217C2×Q8—681no—
218C2×Q8C2×C22⋊D1714yes—
219C2×D4—28no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4C2×C22⋊D1714yes—
224C22×C4—28no—
225C22×C4—24no—
226C22×C4—681no—
227C22×C4C2×C22⋊D1712yes—
228C22×C4C2×C22⋊D1711yescharacteristic
229M16—682no—
230M16—684no—
231C2×C8—344no—
232C2×C8—342no—
233C4⋊C4—28no—
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—344no—
239C4⋊C4—682no—
240C4⋊C4C2×C22⋊D1714yes—
241C22⋊C4—28no—
242C22⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C42—22no—
246C42C2×C22⋊D1711yescharacteristic
247C42C2×C22⋊D1712yes—
248C42C23×D1711yescharacteristic
249C23—24no—
250C23#1058763911yescharacteristic
251Q8—44no—
252Q8—44no—
253Q8—684no—
254Q8—1362no—
255D4—44no—
256D4—44no—
257D4—44no—
258D4—84no—
259D4—88no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—48no—
265C2×C4—88no—
266C2×C4—682no—
267C2×C4—1362no—
268C2×C4#1058763911yescharacteristic
269C2×C4#1058763912yes—
270C2×C4C22×C22⋊D1711yescharacteristic
271C2×C4C22×C22⋊D1712yes—
272C8—684no—
273C8—682no—
274C22—22no—
275C22—21no—
276C22—44no—
277C22—44no—
278C22—44no—
279C22C2×#1058763911yescharacteristic
280C4—22no—
281C4—21no—
282C4—44no—
283C4—44no—
284C4—1362no—
285C4#1176298212yes—
286C4#1176298714yes—
287C2—41no—
288C2—84no—
289C2C2×#1176298712yes—
290C2C2×#1176298211yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
Fitting, prime radicalC17×#3548C2
socleC22×C17C2×#120
hypercenter#3548D17
hypocenterC17#19080
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #14539177C2semidirect
2. #14539166C2semidirect
3. #14528107C2semidirect
4. #14527379C2semidirect
5. #14527376C2semidirect
211. C17#19080semidirect

Maximal subgroups

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

subgroupindex
#145391772
#145391662
#145281072
#145273792
#145273762
C17×#35482
#1908017