Finite Group Databasebeta

← #16929318

Subgroups — #16929318

Order 4352.

756 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#11762729C211yesmaximal, characteristic
2#14528718C212yesmaximal
3C2×#11762535C211yesmaximal, characteristic
4#14526662C212yesmaximal
5C17×#2313C211yesmaximal, characteristic
6C23×Dic34C2211yescharacteristic
7C22×#63225—21no—
8C2×#10587481—24no—
9C2×C22⋊#3622C2212yes—
10#11762729—22no—
11#11762729C414yescomplemented
12C2×#10587502—21no—
13C2×#10587502C2211yescharacteristic
14#11762535—22no—
15C22×C4⋊C4×C17C2211yescharacteristic
16C17×#387—24no—
17C17×#381—21no—
18C17×#380C2212yes—
19C4×C22⋊C4×C17—21no—
20C23×C4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17C2311yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
25C22×Dic34—22no—
26C22×Dic34—22no—
27C22×Dic34—21no—
28C22×Dic34—44no—
29C22×Dic34C2×C412yes—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17C2×C412yes—
33C2×C4⋊C4×C17D412yes—
34C2×C4⋊C4×C17D414yes—
35C2×C22⋊C4×C17—22no—
36C2×C42×C17—22no—
37C2×#63225—24no—
38C2×#63225—44no—
39C22×#3622—22no—
40C17×#95—22no—
41C17×#95—24no—
42C17×#95—22no—
43#10587502—24no—
44#10587502—21no—
45#10587502C2×C412yes—
46C22⋊#3622—22no—
47C22⋊#3622—22no—
48#10587481—48no—
49C24×C17C2×D411yescharacteristic
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—21no—
56C22×C4×C17—24no—
57C22×C4×C17—42no—
58C22×C4×C17—42no—
59C22×C4×C17C22⋊C412yes—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22×C411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D412yes—
64C22×C4×C17#4112yes—
65C22×Dic17—24no—
66C22×Dic17—24no—
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—42no—
70C22×Dic17—42no—
71C22×Dic17—41no—
72C22×Dic17—41no—
73C22×Dic17—42no—
74C2×Dic34—28no—
75C2×Dic34—24no—
76C2×Dic34—44no—
77C2×Dic34—48no—
78C2×Dic34—44no—
79C2×Dic34—44no—
80C2×Dic34—42no—
81C4⋊C4×C17—24no—
82C4⋊C4×C17—24no—
83C4⋊C4×C17—44no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—42no—
86C42×C17—82no—
87#63225—48no—
88C2×#3622—44no—
89C2×#3622—42no—
90#9014—171nomaximal, Sylow
91C23×C17—22no—
92C23×C17—22no—
93C23×C17C2×C22⋊C411yescharacteristic
94C23×C17#12011yescharacteristic
95C23×C17#12311yescharacteristic
96C23×C17#13512yes—
97C23×C17#13712yes—
98C2×C4×C17—24no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—44no—
111C2×C4×C17—44no—
112C2×C4×C17—44no—
113C2×C4×C17—81no—
114C2×C4×C17C4×D412yes—
115C2×C4×C17#12112yes—
116C2×Dic17—24no—
117C2×Dic17—24no—
118C2×Dic17—42no—
119C2×Dic17—44no—
120C2×Dic17—48no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—44no—
125Dic34—28no—
126Dic34—48no—
127Dic34—44no—
128#3622—82no—
129C2×#395—171no—
130#2313D1711yescharacteristic
131#1787—172no—
132C2×#324—171no—
133#1294—172no—
134C22×C17—24no—
135C22×C17—24no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17#38611yescharacteristic
141C22×C17#41611yescharacteristic
142C22×C17#41911yescharacteristic
143C22×C17#45212yes—
144C22×C17#45612yes—
145C22×C17B(4,𝔽2)12yes—
146C22×C17#48512yes—
147C4×C17—22no—
148C4×C17—22no—
149C4×C17—42no—
150C4×C17—42no—
151C4×C17—84no—
152Dic17—44no—
153Dic17—44no—
154C23×Q8—171no—
155C22×C4⋊C4—341no—
156C22×C4⋊C4C2×D1711yescharacteristic
157C2×#103—344no—
158C2×C22⋊C8—172no—
159#395—174no—
160#395—342no—
161#387—24no—
162#381—21no—
163#380C2×D1712yes—
164C4×C22⋊C4—21no—
165C2×#95—171no—
166C2×#95—341no—
167#324—342no—
168C2×C17—21no—
169C2×C17—21no—
170C2×C17—22no—
171C2×C17#160314yes—
172C2×C17#187311yescharacteristic
173C2×C17#188611yescharacteristic
174C2×C17#188911yescharacteristic
175C22×Q8—172no—
176C22×Q8—341no—
177C22×Q8—342no—
178C22×Q8—342no—
179C22×Q8—684no—
180C23×C4—341no—
181C23×C4—341no—
182C23×C4C22⋊D1711yescharacteristic
183C23×C4C22⋊D1711yescharacteristic
184C23×C4C22×D1711yescharacteristic
185C22×C8—342no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—344no—
189C2×C4⋊C4—684no—
190C2×C4⋊C4C4×D1712yes—
191C2×C4⋊C4D6812yes—
192C2×C4⋊C4C22⋊D1714yes—
193C2×C22⋊C4—22no—
194C2×C42—22no—
195#103—688no—
196C22⋊C8—342no—
197C22⋊C8—342no—
198#95—22no—
199#95—24no—
200#95—22no—
201#95—172no—
202#95—341no—
203#95—344no—
204C17#901411yescharacteristic, complemented, Sylow
205C24C2×C22⋊D1711yescharacteristic
206C2×Q8—348no—
207C2×Q8—344no—
208C2×Q8—684no—
209C2×Q8—684no—
210C2×Q8—684no—
211C2×Q8—688no—
212C2×Q8—682no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—344no—
223C22×C4—344no—
224C22×C4—342no—
225C22×C4—342no—
226C22×C4—682no—
227C22×C4—682no—
228C22×C4—681no—
229C22×C4—682no—
230C22×C4—681no—
231C22×C4#6322712yes—
232C22×C4#6322712yes—
233C22×C4C2×C4×D1711yescharacteristic
234C22×C4C2×D6811yescharacteristic
235C22×C4#6325212yes—
236C22×C4C2×C22⋊D1712yes—
237C2×C8—684no—
238C2×C8—682no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—688no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C42—82no—
246C23—22no—
247C23—22no—
248C23C2×#6322711yescharacteristic
249C23#1058761511yescharacteristic
250C23#1058763612yes—
251C23#1058763712yes—
252C23#1058763911yescharacteristic
253Q8—348no—
254Q8—688no—
255Q8—684no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—344no—
273C2×C4—344no—
274C2×C4—682no—
275C2×C4—682no—
276C2×C4—684no—
277C2×C4—688no—
278C2×C4—682no—
279C2×C4—684no—
280C2×C4—682no—
281C2×C4C4×C22⋊D1712yes—
282C2×C4#1058761612yes—
283C8—1362no—
284C22—24no—
285C22—24no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22#1176268811yescharacteristic
291C22#1176280311yescharacteristic
292C22#1176281711yescharacteristic
293C22#1176297112yes—
294C22#1176299212yes—
295C22#1176299612yes—
296C22#1176315012yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—84no—
302C4—684no—
303C4—684no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2#1452774714yes—
308C2#1452909611yescharacteristic
309C2#1452917711yescharacteristic
310C2#1452919211yescharacteristic
subgroupG/H
centerC23#10587639
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2313C2
socleC23×C17#120
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#2313D17
hypocenterC17#9014
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
11. #11762729C4semidirect
204. C17#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#117627292
#145287182
C2×#117625352
#145266622
C17×#23132
#901417