Finite Group Databasebeta

← #16926558

Subgroups — #16926558

Order 4352.

834 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 139 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#14529625C212yesmaximal, complemented
2#14527522C211yesmaximal, characteristic, complemented
3#14527379C211yesmaximal, characteristic, complemented
4#14527223C212yesmaximal, complemented
5C17×#3508C211yesmaximal, characteristic
6#11762833C2212yes—
7C2×#10587504—24no—
8C4×#63223C2212yes—
9#11762555—22no—
10#11762551—22no—
11Q8⋊#3622—22no—
12D4⋊#3622—22no—
13C17×#559—22no—
14C17×#550—22no—
15C17×#546—24no—
16C17×#540—22no—
17C17×#535C412yes—
18C17×#533C2211yescharacteristic
19C17×#532—22no—
20C2×C17×#127C2211yescharacteristic
21C17×#518C412yes—
22C4×#41×C17C2211yescharacteristic
23C2×#41×C17—24no—
24C2×#41×C17C2×C411yescharacteristic
25C22×D4×C17C2×C412yes—
26C22×D4×C17D414yes—
27C17×C4⋊Q8D412yes—
28C17×#127—22no—
29C17×#127—21no—
30C17×#127—21no—
31C17×#127—22no—
32C17×#125D412yes—
33C17×#124—24no—
34C17×#124—22no—
35C17×#123—28no—
36C17×C22⋊Q8—22no—
37C17×#121—24no—
38C17×#121—28no—
39C17×#121—24no—
40C17×#120—24no—
41C4×Q8×C17—22no—
42C4×Q8×C17D412yes—
43C4×D4×C17—24no—
44C4×D4×C17—22no—
45C4×D4×C17—24no—
46C4×D4×C17—24no—
47C4×D4×C17D412yes—
48C17×#117—21no—
49C17×#117C2×C411yescharacteristic
50C17×#117C2×C412yes—
51C2×C22⋊C4×C17—24no—
52C2×C42×C17—21no—
53C2×C42×C17C2311yescharacteristic
54C2×#63223—22no—
55C2×#63223—22no—
56#10587504—48no—
57C4⋊#3622—22no—
58C4⋊#3622—22no—
59#10587474—22no—
60C4×#3622—22no—
61C24×C17C22⋊C414yes—
62#41×C17—42no—
63#41×C17—84no—
64#41×C17—84no—
65C2×Q8×C17—22no—
66C2×Q8×C17C22⋊C411yescharacteristic
67C2×D4×C17—21no—
68C2×D4×C17—28no—
69C2×D4×C17—22no—
70C2×D4×C17—22no—
71C2×D4×C17—24no—
72C2×D4×C17—24no—
73C2×D4×C17—44no—
74C2×D4×C17—44no—
75C2×D4×C17—44no—
76C2×D4×C17C22⋊C411yescharacteristic
77C22×C4×C17—28no—
78C22×C4×C17—21no—
79C22×C4×C17—22no—
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C2×D412yes—
82C4⋊C4×C17—24no—
83C4⋊C4×C17—22no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17C22⋊C414yes—
86C4⋊C4×C17C22⋊C412yes—
87C22⋊C4×C17—28no—
88C22⋊C4×C17—28no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17—22no—
92C42×C17—22no—
93C42×C17—21no—
94C42×C17—22no—
95C42×C17C2×D411yescharacteristic
96C42×C17C2×D411yescharacteristic
97C42×C17C2×D412yes—
98#63223—24no—
99#63223—42no—
100#63223—44no—
101C2×#3622—42no—
102C2×#3622—42no—
103#8472—171nomaximal, Sylow
104C23×C17—24no—
105C23×C17—21no—
106C23×C17—48no—
107C23×C17—44no—
108C23×C17C2×C22⋊C411yescharacteristic
109Q8×C17—22no—
110Q8×C17—42no—
111Q8×C17—42no—
112D4×C17—22no—
113D4×C17—44no—
114D4×C17—42no—
115D4×C17—44no—
116D4×C17—44no—
117D4×C17—44no—
118D4×C17—84no—
119D4×C17—88no—
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—21no—
124C2×C4×C17—22no—
125C2×C4×C17—24no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—88no—
130C2×C4×C17C2×C22⋊C411yescharacteristic
131C2×C4×C17C2×C22⋊C411yescharacteristic
132C2×C4×C17#12012yes—
133C2×C4×C17#12012yes—
134#3622—44no—
135#3622—82no—
136#3508D1711yescharacteristic
137#1965—172no—
138#1561—171no—
139#1526—171no—
140#1489—172no—
141C22×C17—22no—
142C22×C17—21no—
143C22×C17—42no—
144C22×C17—48no—
145C22×C17—41no—
146C22×C17—44no—
147C22×C17—164no—
148C22×C17#37911yescharacteristic
149C4×C17—22no—
150C4×C17—21no—
151C4×C17—42no—
152C4×C17—44no—
153C4×C17—41no—
154C4×C17#41312yes—
155C4×C17#45312yes—
156C4×C17#45512yes—
157#559—22no—
158#550—22no—
159#546—24no—
160#540—22no—
161#535Dic1712yes—
162#533C2×D1711yescharacteristic
163#532—22no—
164C2×#127C2×D1711yescharacteristic
165#518Dic1712yes—
166C4×#41C2×D1711yescharacteristic
167#423—172no—
168C2×#100—344no—
169C4×M16—172no—
170#332—342no—
171#330—342no—
172Q8⋊C8—342no—
173D4⋊C8—342no—
174C2×C17—41no—
175C2×C17—81no—
176C2×C17—84no—
177C2×C17#178511yescharacteristic
178C2×C17#179212yes—
179C2×#41—24no—
180C2×#41C2×Dic1711yescharacteristic
181C22×D4C2×Dic1712yes—
182C22×D4C22⋊D1714yes—
183C2×M16—342no—
184C2×M16—342no—
185C4⋊Q8C22⋊D1712yes—
186#127—22no—
187#127—21no—
188#127—21no—
189#127—22no—
190#125C22⋊D1712yes—
191#124—24no—
192#124—22no—
193#123—28no—
194C22⋊Q8—22no—
195#121—24no—
196#121—28no—
197#121—24no—
198#120—24no—
199C4×Q8—22no—
200C4×Q8C22⋊D1712yes—
201C4×D4—24no—
202C4×D4—22no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4C22⋊D1712yes—
206#117—21no—
207#117C2×Dic1711yescharacteristic
208#117C2×Dic1712yes—
209C2×C22⋊C4—24no—
210C2×C42—21no—
211C2×C42C22×D1711yescharacteristic
212C4⋊C8—342no—
213C4⋊C8—342no—
214#100—688no—
215#97—342no—
216C4×C8—342no—
217C17#847211yescharacteristic, complemented, Sylow
218C24C22⋊Dic1714yes—
219#41—42no—
220#41—84no—
221#41—84no—
222C2×Q8—22no—
223C2×Q8C22⋊Dic1711yescharacteristic
224C2×D4—21no—
225C2×D4—28no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4C22⋊Dic1711yescharacteristic
234C22×C4—28no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4C22×Dic1711yescharacteristic
238C22×C4C2×C22⋊D1712yes—
239M16—344no—
240M16—682no—
241M16—684no—
242C2×C8—682no—
243C2×C8—682no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4C22⋊Dic1712yes—
248C4⋊C4C22⋊Dic1714yes—
249C22⋊C4—28no—
250C22⋊C4—28no—
251C22⋊C4—21no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42C2×C22⋊D1712yes—
258C42C2×C22⋊D1711yescharacteristic
259C42C2×C22⋊D1711yescharacteristic
260C23—24no—
261C23—21no—
262C23—44no—
263C23—48no—
264C23C2×C22⋊Dic1711yescharacteristic
265Q8—22no—
266Q8—42no—
267Q8—42no—
268D4—22no—
269D4—44no—
270D4—42no—
271D4—44no—
272D4—44no—
273D4—44no—
274D4—84no—
275D4—88no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—88no—
286C2×C4C2×C22⋊Dic1711yescharacteristic
287C2×C4C2×C22⋊Dic1711yescharacteristic
288C2×C4#1058763912yes—
289C2×C4#1058763912yes—
290C8—684no—
291C8—1362no—
292C22—21no—
293C22—22no—
294C22—41no—
295C22—44no—
296C22—42no—
297C22—48no—
298C22—164no—
299C22#1176266511yescharacteristic
300C4—21no—
301C4—22no—
302C4—41no—
303C4—42no—
304C4—44no—
305C4#1176279312yes—
306C4#1176297712yes—
307C4#1176298712yes—
308C2—41no—
309C2—84no—
310C2—81no—
311C2#1452871211yescharacteristic
312C2#1452873812yes—
subgroupG/H
centerC22#11762665
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
Fitting, prime radicalC17×#3508C2
socleC22×C17#379
cosocleC2×C42×C17C23
hypercenter#3508D17
hypocenterC17#8472
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #14529625C2semidirect
2. #14527522C2semidirect
3. #14527379C2semidirect
4. #14527223C2semidirect
217. C17#8472semidirect

Maximal subgroups

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

subgroupindex
#145296252
#145275222
#145273792
#145272232
C17×#35082
#847217