Finite Group Databasebeta

← #17953103

Subgroups — #17953103

Order 4352.

1134 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 108 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#14544618C212yesmaximal, complemented
2#14544034C212yesmaximal, complemented
3#14544031C212yesmaximal
4C22×#10587582C211yesmaximal, characteristic
5C2×#10587545—24no—
6#11763779C2214yes—
7#11763655—24no—
8#11763653—24no—
9C23×Dic34C2211yescharacteristic
10C23×C4×D17C411yescharacteristic
11C2×#10587582—24no—
12C2×#10587582—24no—
13C2×#10587582—24no—
14C22×#63225C411yescharacteristic
15C22×#63225C2211yescharacteristic
16C22×#63227C411yescharacteristic
17C22×#63227C2211yescharacteristic
18C22×C4⋊C4×C17C411yescharacteristic
19C24×D17C2×C411yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17C2×C411yescharacteristic
23C23×Dic17C2×C411yescharacteristic
24C23×Dic17C2×C411yescharacteristic
25C23×Dic17C2311yescharacteristic
26C22×C4×D17—22no—
27C22×C4×D17—22no—
28C22×C4×D17—22no—
29C22×C4×D17—44no—
30C22×Dic34—22no—
31C22×Dic34—22no—
32C22×Dic34—44no—
33C22×Dic34D414yes—
34C2×#63247—24no—
35C2×C4⋊C4×C17—22no—
36C2×C4⋊C4×C17—22no—
37C2×C4⋊C4×C17—22no—
38C2×#63227—22no—
39C2×#63227—22no—
40C2×#63227—22no—
41C2×#63227—22no—
42C2×#63227—22no—
43C2×#63227D414yes—
44C2×#63225—22no—
45C2×#63225—22no—
46C2×#63225—22no—
47C2×#63225—22no—
48C2×#63225—22no—
49C2×#63225D414yes—
50#10587582—232no—
51C22⋊#3624—24no—
52C22⋊#3624—24no—
53#10587545—48no—
54C24×C17C22×C411yescharacteristic
55C23×D17—21no—
56C23×D17—21no—
57C23×D17—24no—
58C23×D17—42no—
59C23×D17—42no—
60C23×D17C22⋊C412yes—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—42no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17C22⋊C412yes—
71C22×C4×C17C22⋊C412yes—
72C22×C4×C17C22⋊C412yes—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—24no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—42no—
79C22×Dic17—42no—
80C22×Dic17C22⋊C412yes—
81C22×Dic17C22⋊C412yes—
82C22×Dic17C2×D412yes—
83C22×Dic17C2×D412yes—
84C22×Dic17C2×D412yes—
85C2×C4×D17—28no—
86C2×C4×D17—48no—
87C2×C4×D17—48no—
88C2×C4×D17—44no—
89C2×C4×D17—44no—
90C2×Dic34—28no—
91C2×Dic34—28no—
92C2×Dic34—44no—
93C2×Dic34—48no—
94C2×Dic34—48no—
95#63247—48no—
96C2×#3624—44no—
97C4⋊C4×C17—28no—
98#63227—28no—
99#63227—28no—
100#63225—28no—
101#63225—28no—
102#8550—171nomaximal, Sylow
103C23×C17—22no—
104C23×C17—22no—
105C23×C17—22no—
106C23×C17C2×C22⋊C411yescharacteristic
107C23×C17C2×C22⋊C411yescharacteristic
108C23×C17C2×C22⋊C411yescharacteristic
109C22×D17—24no—
110C22×D17—24no—
111C22×D17—22no—
112C22×D17—22no—
113C22×D17—42no—
114C22×D17—416no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—42no—
120C22×D17#10114yescomplemented
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—22no—
124C2×C4×C17—24no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×Dic17—24no—
134C2×Dic17—24no—
135C2×Dic17—22no—
136C2×Dic17—24no—
137C2×Dic17—22no—
138C2×Dic17—22no—
139C2×Dic17—24no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17#9914yes—
145C2×Dic17#12014yes—
146C4×D17—416no—
147Dic34—416no—
148#3624—84no—
149C22×C22⋊Q8—171no—
150#1784—172no—
151#1463—172no—
152#1462—172no—
153C22×C17—28no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17#37911yescharacteristic
161C22×C17C2×#9912yes—
162C22×C17C2×#10112yes—
163C22×C17#41312yes—
164C2×D17—24no—
165C2×D17—48no—
166C2×D17—44no—
167C2×D17—48no—
168C2×D17—42no—
169C2×D17—42no—
170C2×D17—44no—
171C4×C17—42no—
172C4×C17—42no—
173C4×C17—42no—
174Dic17—24no—
175Dic17—24no—
176Dic17—42no—
177C23×Q8—171no—
178C24×C4—171no—
179C2×C22⋊Q8—344no—
180C2×C22⋊Q8—344no—
181C2×C22⋊Q8—344no—
182C22×C4⋊C4—171no—
183C22×C4⋊C4—171no—
184C22×C4⋊C4#59511yescharacteristic
185C22×C22⋊C4—171no—
186C22×C22⋊C4—171no—
187C2×#101—344no—
188#407—174no—
189#324—344no—
190#323—344no—
191C2×C17—22no—
192C2×C17—22no—
193C2×C17—22no—
194C2×C17#178111yescharacteristic
195C2×C17#178411yescharacteristic
196C2×C17#178511yescharacteristic
197D17—44no—
198C25—171no—
199C22×Q8—174no—
200C22×Q8—342no—
201C22×Q8—342no—
202C22×Q8—684no—
203C23×C4—171no—
204C23×C4—171no—
205C23×C4—171no—
206C23×C4—342no—
207C23×C4—342no—
208C23×C4—342no—
209C23×C4—684no—
210C23×C4C2×#59511yescharacteristic
211C23×C4C2×#59511yescharacteristic
212C23×C4C2×#59511yescharacteristic
213C2×M16—344no—
214C22⋊Q8—3432no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—174no—
219C2×C4⋊C4—342no—
220C2×C4⋊C4—342no—
221C2×C4⋊C4—342no—
222C2×C4⋊C4—342no—
223C2×C4⋊C4—342no—
224C2×C22⋊C4—174no—
225C2×C22⋊C4—342no—
226C2×C22⋊C4—342no—
227C2×C22⋊C4—342no—
228C2×C22⋊C4—342no—
229C2×C22⋊C4—342no—
230#101—688no—
231C22⋊C8—344no—
232C22⋊C8—344no—
233C17#855011yescharacteristic, complemented, Sylow
234C24—172no—
235C24—344no—
236C24—341no—
237C24—341no—
238C24—682no—
239C24—682no—
240C24C22×#59511yescharacteristic
241C2×Q8—348no—
242C2×Q8—348no—
243C2×Q8—684no—
244C2×Q8—688no—
245C2×Q8—688no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—172no—
256C22×C4—172no—
257C22×C4—172no—
258C22×C4—172no—
259C22×C4—172no—
260C22×C4—348no—
261C22×C4—344no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—684no—
267C22×C4—688no—
268C22×C4—684no—
269C22×C4—688no—
270C22×C4—682no—
271C22×C4—682no—
272C22×C4C22⋊#59512yes—
273C22×C4C22⋊#59512yes—
274C22×C4C22⋊#59512yes—
275M16—688no—
276C2×C8—684no—
277C4⋊C4—28no—
278C4⋊C4—348no—
279C4⋊C4—348no—
280C22⋊C4—348no—
281C22⋊C4—348no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—174no—
286C23—344no—
287C23—344no—
288C23—342no—
289C23—342no—
290C23—682no—
291C23—682no—
292C23—682no—
293C23—6816no—
294C23—682no—
295C23—682no—
296C23—682no—
297C23C2×C22⋊#59511yescharacteristic
298C23C2×C22⋊#59511yescharacteristic
299C23C2×C22⋊#59511yescharacteristic
300Q8—6816no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—174no—
314C2×C4—174no—
315C2×C4—344no—
316C2×C4—344no—
317C2×C4—342no—
318C2×C4—344no—
319C2×C4—342no—
320C2×C4—342no—
321C2×C4—344no—
322C2×C4—6816no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—682no—
327C8—1364no—
328C22—28no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—344no—
336C22—684no—
337C22—688no—
338C22—684no—
339C22—688no—
340C22—682no—
341C22—682no—
342C22#1176375011yescharacteristic
343C22D17⋊#9912yes—
344C22D17⋊#10112yes—
345C22#1176379512yes—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—344no—
350C4—344no—
351C4—682no—
352C2—22no—
353C2—22no—
354C2—22no—
355C2—684no—
356C2#1454461111yescharacteristic
357C2#1454461911yescharacteristic
358C2#1454462011yescharacteristic
subgroupG/H
centerC22#11763750
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC22×C17#379
prime radicalC24×D17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8550
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #14544618C2semidirect
2. #14544034C2semidirect
120. C22×D17#101semidirect
233. C17#8550semidirect

Maximal subgroups

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

subgroupindex
#145446182
#145440342
#145440312
C22×#105875822
#855017