Finite Group Databasebeta

← #17953122

Subgroups — #17953122

Order 4352.

842 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544683C212yesmaximal, complemented
2#14544034C212yesmaximal, complemented
3#14544031C212yesmaximal
4C22×#10587562C211yesmaximal, characteristic
5#11763779C2214yes—
6#11763689—24no—
7#11763655—24no—
8#11763653—24no—
9C2×#10587562—24no—
10C2×#10587562—24no—
11C2×#10587562—24no—
12C22×#63225C411yescharacteristic
13C22×#63225C2211yescharacteristic
14C22×#63225C2211yescharacteristic
15C22×#63227C411yescharacteristic
16C22×#63227C411yescharacteristic
17C22×#63227C2211yescharacteristic
18C22×C42×C17C411yescharacteristic
19C24×D17C2×C411yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17C2×C411yescharacteristic
23C23×Dic17C2×C411yescharacteristic
24C23×Dic17C2×C411yescharacteristic
25C23×Dic17C2311yescharacteristic
26C2×#63247—24no—
27C2×C42×C17—22no—
28C2×C42×C17—22no—
29C2×C42×C17—22no—
30C2×#63227—22no—
31C2×#63227—22no—
32C2×#63227—22no—
33C2×#63227—22no—
34C2×#63227—22no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227—22no—
38C2×#63227D414yes—
39C2×#63225—22no—
40C2×#63225—22no—
41C2×#63225—22no—
42C2×#63225—22no—
43C2×#63225—22no—
44C2×#63225—22no—
45C2×#63225—22no—
46C2×#63225D414yes—
47C2×#63225D414yes—
48#10587562—232no—
49C22⋊#3624—24no—
50C22⋊#3624—24no—
51C24×C17C22×C411yescharacteristic
52C23×D17—21no—
53C23×D17—21no—
54C23×D17—42no—
55C23×D17—42no—
56C23×D17—42no—
57C23×D17C22⋊C412yes—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C412yes—
68C22×C4×C17C22⋊C412yes—
69C22×C4×C17C22⋊C412yes—
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—42no—
75C22×Dic17—42no—
76C22×Dic17—42no—
77C22×Dic17C22⋊C412yes—
78C22×Dic17C22⋊C412yes—
79C22×Dic17C2×D412yes—
80C22×Dic17C2×D412yes—
81C22×Dic17C2×D412yes—
82#63247—48no—
83C2×#3624—44no—
84C42×C17—28no—
85#63227—28no—
86#63227—28no—
87#63227—28no—
88#63225—28no—
89#63225—28no—
90#63225—28no—
91#8556—171nomaximal, Sylow
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17C2×C22⋊C411yescharacteristic
96C23×C17C2×C22⋊C411yescharacteristic
97C23×C17C2×C22⋊C411yescharacteristic
98C22×D17—22no—
99C22×D17—24no—
100C22×D17—42no—
101C22×D17—42no—
102C22×D17—42no—
103C22×D17—42no—
104C22×D17—42no—
105C22×D17—42no—
106C22×D17—42no—
107C22×D17—42no—
108C22×D17—88no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×Dic17—24no—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—24no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17#12014yes—
133#3624—84no—
134C22×#126—171no—
135#1821—172no—
136#1463—172no—
137#1462—172no—
138C22×C17—28no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17#37911yescharacteristic
146C22×C17#41012yes—
147C22×C17#41312yes—
148C22×C17#41312yes—
149C2×D17—42no—
150C2×D17—42no—
151C2×D17—42no—
152C2×D17—84no—
153C2×D17—84no—
154C2×D17—84no—
155C4×C17—42no—
156C4×C17—42no—
157C4×C17—42no—
158Dic17—24no—
159Dic17—42no—
160Dic17—42no—
161C2×#126—344no—
162C2×#126—344no—
163C2×#126—344no—
164C22×C4⋊C4—171no—
165C22×C4⋊C4—171no—
166C22×C4⋊C4—171no—
167C22×C22⋊C4—171no—
168C22×C22⋊C4—171no—
169C22×C22⋊C4—171no—
170C22×C42#59511yescharacteristic
171#407—174no—
172#343—344no—
173#324—344no—
174#323—344no—
175C2×C17—22no—
176C2×C17—22no—
177C2×C17—22no—
178C2×C17#178211yescharacteristic
179C2×C17#178511yescharacteristic
180C2×C17#178511yescharacteristic
181D17—82no—
182C25—171no—
183C23×C4—171no—
184C23×C4—171no—
185C23×C4—171no—
186C23×C4C2×#59511yescharacteristic
187C23×C4C2×#59511yescharacteristic
188C23×C4C2×#59511yescharacteristic
189C2×M16—344no—
190#126—3432no—
191C2×C4⋊C4—174no—
192C2×C4⋊C4—174no—
193C2×C4⋊C4—342no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—342no—
198C2×C4⋊C4—342no—
199C2×C4⋊C4—342no—
200C2×C22⋊C4—174no—
201C2×C22⋊C4—342no—
202C2×C22⋊C4—342no—
203C2×C22⋊C4—342no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4—342no—
207C2×C22⋊C4—342no—
208C2×C22⋊C4—342no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C22⋊C8—344no—
213C22⋊C8—344no—
214C17#855611yescharacteristic, complemented, Sylow
215C24—172no—
216C24—341no—
217C24—341no—
218C24—682no—
219C24—682no—
220C24—682no—
221C24C22×#59511yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—172no—
232C22×C4—172no—
233C22×C4—172no—
234C22×C4—172no—
235C22×C4—172no—
236C22×C4—341no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4—682no—
241C22×C4—682no—
242C22×C4—682no—
243C22×C4C22⋊#59512yes—
244C22×C4C22⋊#59512yes—
245C22×C4C22⋊#59512yes—
246M16—688no—
247C2×C8—684no—
248C4⋊C4—348no—
249C4⋊C4—348no—
250C4⋊C4—348no—
251C22⋊C4—348no—
252C22⋊C4—348no—
253C22⋊C4—348no—
254C42—28no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—342no—
259C23—344no—
260C23—682no—
261C23—682no—
262C23—682no—
263C23—682no—
264C23—682no—
265C23—682no—
266C23—682no—
267C23—682no—
268C23—1368no—
269C23C2×C22⋊#59511yescharacteristic
270C23C2×C22⋊#59511yescharacteristic
271C23C2×C22⋊#59511yescharacteristic
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—174no—
285C2×C4—344no—
286C2×C4—342no—
287C2×C4—342no—
288C2×C4—344no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—682no—
295C2×C4—682no—
296C8—1364no—
297C22—28no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—682no—
305C22—682no—
306C22—682no—
307C22—1364no—
308C22—1364no—
309C22—1364no—
310C22#1176375011yescharacteristic
311C22#1176378812yes—
312C22#1176379512yes—
313C22#1176379512yes—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—344no—
318C4—682no—
319C4—682no—
320C2—22no—
321C2—22no—
322C2—22no—
323C2—1362no—
324C2#1454461311yescharacteristic
325C2#1454462011yescharacteristic
326C2#1454462011yescharacteristic
subgroupG/H
centerC22#11763750
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C42×C17C4
socleC22×C17#379
prime radicalC24×D17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C42#595
hypocenterC17#8556
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. #14544683C2semidirect
2. #14544034C2semidirect
214. C17#8556semidirect

Maximal subgroups

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

subgroupindex
#145446832
#145440342
#145440312
C22×#105875622
#855617