Finite Group Databasebeta

← #16929263

Subgroups — #16929263

Order 4352.

578 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 115 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×#11763204C211yesmaximal, characteristic
2#14529756C212yesmaximal, complemented
3C2×#11762549C211yesmaximal, characteristic
4#14526665C212yesmaximal, complemented
5C17×#2379C211yesmaximal, characteristic
6C22×#63231—21no—
7C22×#63229—21no—
8C2×#10587555C2211yescharacteristic
9#11763204—22no—
10#11763204C414yescomplemented
11C2×#10587474C2212yes—
12C2×#10587474C2211yescharacteristic
13#11762571—22no—
14#11762571—22no—
15#11762549—22no—
16C2×C17×#124C2211yescharacteristic
17C17×#391—21no—
18C17×#388—22no—
19C17×#386—21no—
20C17×#385—22no—
21C17×#382C2212yes—
22C22×Q8×C17D411yescharacteristic
23C22×D4×C17D411yescharacteristic
24C23×C4×C17—21no—
25C22×Dic34—21no—
26C17×#124—22no—
27C17×#124C2×C412yes—
28C17×#124D412yes—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17D412yes—
31C2×C4⋊C4×C17D412yes—
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17—21no—
35C2×C42×C17—21no—
36C2×C42×C17C2311yescharacteristic
37C2×#63231—24no—
38C2×#63231—42no—
39C2×#63231—42no—
40C2×#63229—24no—
41C2×#63229—42no—
42C2×#63229—42no—
43C2×C4⋊Dic17—22no—
44C22×#3622—21no—
45C22×#3622—22no—
46#10587555—21no—
47#10587555—22no—
48#10587555C2×C412yescomplemented
49C17×#95—22no—
50C17×#95—22no—
51C17×#95—22no—
52#10587474—21no—
53#10587474—22no—
54#10587474—22no—
55#10587474C2×C412yescomplemented
56C24×C17—21no—
57C2×Q8×C17—21no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C2×Q8×C17C22⋊C412yes—
61C2×D4×C17—21no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C2×D4×C17C22⋊C412yes—
65C22×C4×C17—21no—
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—81no—
69C22×C4×C17—82no—
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D412yes—
72C22×Dic17—41no—
73C2×Dic34—22no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×Dic34—41no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—81no—
82C22⋊C4×C17—24no—
83C22⋊C4×C17—24no—
84C22⋊C4×C17—81no—
85C22⋊C4×C17—82no—
86C42×C17—81no—
87C42×C17C22×C411yescharacteristic
88C42×C17C2×D411yescharacteristic
89C42×C17#4112yes—
90#63231—48no—
91#63229—48no—
92C4⋊Dic17—24no—
93C4⋊Dic17—42no—
94C2×#3622—22no—
95C2×#3622—22no—
96C2×#3622—22no—
97C2×#3622—24no—
98C2×#3622—42no—
99C2×#3622—42no—
100#9003—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—42no—
105C23×C17—42no—
106C23×C17#12011yescharacteristic
107Q8×C17—42no—
108Q8×C17—81no—
109D4×C17—42no—
110D4×C17—81no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—84no—
125C2×C4×C17C2×C22⋊C411yescharacteristic
126C2×C4×C17C4×D412yes—
127C2×C4×C17#12112yes—
128C2×C4×C17#12311yescharacteristic
129C2×Dic17—42no—
130C2×Dic17—81no—
131C2×Dic17—81no—
132Dic34—42no—
133Dic34—42no—
134Dic34—81no—
135Dic34—81no—
136#3622—24no—
137#3622—44no—
138C2×#497—171no—
139#2379D1711yescharacteristic
140#1993—172no—
141C2×#329—171no—
142#1295—172no—
143C22×C17—44no—
144C22×C17—42no—
145C22×C17—81no—
146C22×C17—81no—
147C22×C17—82no—
148C22×C17#38611yescharacteristic
149C22×C17#45311yescharacteristic
150C22×C17#45311yescharacteristic
151C22×C17#45512yes—
152C22×C17#45612yes—
153C4×C17—22no—
154C4×C17—22no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—42no—
158C4×C17—82no—
159C4×C17—82no—
160Dic17—82no—
161C22×Q16—341no—
162C22×SD16—341no—
163C2×C4⋊Q8—171no—
164C2×#124C2×D1711yescharacteristic
165#497—174no—
166#497—342no—
167C2×#97—171no—
168C2×#97—172no—
169#391—21no—
170#388—22no—
171#386—21no—
172#385—22no—
173#382C2×D1712yes—
174#340—342no—
175#340—342no—
176#329—342no—
177C2×C17—82no—
178C2×C17#165911yescharacteristic
179C2×C17#166011yescharacteristic
180C2×C17#166112yes—
181C2×C17#188212yes—
182C2×C17#188711yescharacteristic
183C22×Q8—341no—
184C22×Q8C22⋊D1711yescharacteristic
185C22×D4C22⋊D1711yescharacteristic
186C23×C4—21no—
187C2×Q16—344no—
188C2×Q16—682no—
189C2×Q16—682no—
190C2×SD16—344no—
191C2×SD16—682no—
192C2×SD16—682no—
193C22×C8—341no—
194C22×C8—342no—
195C4⋊Q8—172no—
196C4⋊Q8—341no—
197C4⋊Q8—342no—
198#124—22no—
199#124C4×D1712yes—
200#124D6812yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4C22⋊D1712yes—
204C2×C4⋊C4C22⋊D1712yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42C22×D1711yescharacteristic
210#97—172no—
211#97—341no—
212#97—342no—
213#97—342no—
214#95—22no—
215#95—22no—
216#95—22no—
217C17#900311yescharacteristic, complemented, Sylow
218C24—21no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—342no—
223C2×Q8—681no—
224C2×Q8—681no—
225C2×Q8—681no—
226C2×Q8—681no—
227C2×Q8—681no—
228C2×Q8#6322712yes—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#6322712yes—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—681no—
239C22×C4C2×C22⋊D1711yescharacteristic
240C22×C4C2×C22⋊D1712yes—
241Q16—688no—
242SD16—688no—
243C2×C8—342no—
244C2×C8—342no—
245C2×C8—342no—
246C2×C8—344no—
247C2×C8—682no—
248C2×C8—682no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—81no—
252C4⋊C4—344no—
253C4⋊C4—682no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—81no—
257C22⋊C4—82no—
258C42—81no—
259C42C2×C4×D1711yescharacteristic
260C42C2×D6811yescharacteristic
261C42#6325212yes—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23#1058763911yescharacteristic
268Q8—42no—
269Q8—81no—
270Q8—682no—
271Q8—682no—
272Q8—1361no—
273Q8—1361no—
274D4—42no—
275D4—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4—682no—
291C2×C4—1361no—
292C2×C4—1361no—
293C2×C4C2×#6322711yescharacteristic
294C2×C4C4×C22⋊D1712yes—
295C2×C4#1058761511yescharacteristic
296C2×C4#1058761612yes—
297C8—344no—
298C8—684no—
299C22—44no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22#1176268811yescharacteristic
305C22#1176297711yescharacteristic
306C22#1176297711yescharacteristic
307C22#1176298712yes—
308C22#1176299212yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—82no—
315C4—82no—
316C4—1362no—
317C2—82no—
318C2#1452809011yescharacteristic
319C2#1452809511yescharacteristic
320C2#1452810112yes—
321C2#1452915312yes—
322C2#1452918211yescharacteristic
subgroupG/H
centerC23#10587639
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2379C2
socleC23×C17#120
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#2379D17
hypocenterC17#9003
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
2. #14529756C2semidirect
4. #14526665C2semidirect
10. #11763204C4semidirect
48. #10587555C2×C4semidirect
55. #10587474C2×C4semidirect
217. C17#9003semidirect

Maximal subgroups

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

subgroupindex
C2×#117632042
#145297562
C2×#117625492
#145266652
C17×#23792
#900317