Finite Group Databasebeta

← #17953121

Subgroups — #17953121

Order 4352.

1102 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 112 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#14544029C212yesmaximal
4C22×#10587564C211yesmaximal, characteristic
5#11763779C2214yes—
6#11763689—24no—
7#11763655—24no—
8#11763652—24no—
9C23×Dic34C411yescharacteristic
10C24×Dic17C411yescharacteristic
11C2×#10587564—24no—
12C2×#10587564—24no—
13C2×#10587564—24no—
14C22×C4⋊Dic17C411yescharacteristic
15C22×#63225C2211yescharacteristic
16C22×#63225C2211yescharacteristic
17C22×C22⋊Dic17C2211yescharacteristic
18C22×C22⋊C4×C17C411yescharacteristic
19C25×C17C2×C411yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17—22no—
23C23×Dic17—22no—
24C23×Dic17—22no—
25C23×Dic17—44no—
26C23×Dic17C2×C411yescharacteristic
27C23×Dic17C2×C411yescharacteristic
28C23×Dic17C2×C411yescharacteristic
29C23×Dic17C2311yescharacteristic
30C22×Dic34—22no—
31C22×Dic34—22no—
32C22×Dic34—22no—
33C22×Dic34—44no—
34C2×#63247—24no—
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17—22no—
40C2×C22⋊Dic17D414yes—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17—22no—
43C2×C4⋊Dic17—22no—
44C2×#63225—22no—
45C2×#63225—22no—
46C2×#63225—22no—
47C2×#63225—22no—
48C2×#63225D414yes—
49C2×#63225D414yes—
50#10587564—232no—
51C22⋊#3624—24no—
52C22⋊#3624—24no—
53C24×C17—24no—
54C24×C17—21no—
55C24×C17—21no—
56C24×C17—42no—
57C24×C17—42no—
58C24×C17C22⋊C412yes—
59C24×C17C22×C411yescharacteristic
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×C17C22⋊C412yes—
67C22×C4×C17C22⋊C412yes—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—24no—
71C22×Dic17—28no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—42no—
77C22×Dic17—48no—
78C22×Dic17—48no—
79C22×Dic17—44no—
80C22×Dic17—44no—
81C22×Dic17—42no—
82C22×Dic17—42no—
83C22×Dic17C22⋊C412yes—
84C22×Dic17C22⋊C412yes—
85C22×Dic17C22⋊C412yes—
86C22×Dic17C2×D412yes—
87C22×Dic17C2×D412yes—
88C22×Dic17C2×D412yes—
89C2×Dic34—28no—
90C2×Dic34—48no—
91C2×Dic34—48no—
92C2×Dic34—44no—
93C2×Dic34—44no—
94#63247—48no—
95C2×#3624—44no—
96C22⋊C4×C17—28no—
97C22⋊Dic17—28no—
98C4⋊Dic17—28no—
99#63225—28no—
100#63225—28no—
101#8555—171nomaximal, Sylow
102C23×C17—22no—
103C23×C17—22no—
104C23×C17—22no—
105C23×C17—24no—
106C23×C17—24no—
107C23×C17—22no—
108C23×C17—22no—
109C23×C17—42no—
110C23×C17—42no—
111C23×C17—42no—
112C23×C17—416no—
113C23×C17—42no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17#10114yes—
117C23×C17C2×C22⋊C411yescharacteristic
118C23×C17C2×C22⋊C411yescharacteristic
119C23×C17C2×C22⋊C411yescharacteristic
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×Dic17—24no—
129C2×Dic17—24no—
130C2×Dic17—22no—
131C2×Dic17—24no—
132C2×Dic17—22no—
133C2×Dic17—22no—
134C2×Dic17—22no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—416no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17#10014yes—
145C2×Dic17#12014yes—
146Dic34—416no—
147#3624—84no—
148C22×C22⋊Q8—171no—
149#1821—172no—
150#1463—172no—
151#1462—172no—
152C22×C17—28no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—24no—
160C22×C17—44no—
161C22×C17—48no—
162C22×C17—48no—
163C22×C17—44no—
164C22×C17—42no—
165C22×C17—42no—
166C22×C17#37911yescharacteristic
167C22×C17#41012yes—
168C22×C17C2×#10012yes—
169C22×C17C2×#10112yes—
170C4×C17—42no—
171C4×C17—42no—
172Dic17—24no—
173Dic17—24no—
174Dic17—42no—
175Dic17—42no—
176C23×Q8—171no—
177C24×C4—171no—
178C2×C22⋊Q8—344no—
179C2×C22⋊Q8—344no—
180C2×C22⋊Q8—344no—
181C22×C4⋊C4—171no—
182C22×C4⋊C4—171no—
183C22×C4⋊C4—171no—
184C22×C22⋊C4—171no—
185C22×C22⋊C4#59511yescharacteristic
186#407—174no—
187#343—344no—
188#324—344no—
189#323—344no—
190C2×C17—22no—
191C2×C17—22no—
192C2×C17—22no—
193C2×C17—44no—
194C2×C17#178211yescharacteristic
195C2×C17#178311yescharacteristic
196C2×C17#178411yescharacteristic
197C25C2×#59511yescharacteristic
198C22×Q8—342no—
199C22×Q8—342no—
200C22×Q8—342no—
201C22×Q8—684no—
202C23×C4—171no—
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
212C2×M16—344no—
213C22⋊Q8—3432no—
214C2×C4⋊C4—174no—
215C2×C4⋊C4—174no—
216C2×C4⋊C4—342no—
217C2×C4⋊C4—342no—
218C2×C4⋊C4—342no—
219C2×C4⋊C4—342no—
220C2×C4⋊C4—342no—
221C2×C4⋊C4—342no—
222C2×C4⋊C4—342no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—174no—
227C2×C22⋊C4—342no—
228C2×C22⋊C4—342no—
229C22⋊C8—344no—
230C22⋊C8—344no—
231C17#855511yescharacteristic, complemented, Sylow
232C24—21no—
233C24—21no—
234C24—24no—
235C24—42no—
236C24—42no—
237C24C22⋊#59512yes—
238C24C22×#59511yescharacteristic
239C2×Q8—348no—
240C2×Q8—684no—
241C2×Q8—688no—
242C2×Q8—684no—
243C2×Q8—688no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—172no—
251C22×C4—172no—
252C22×C4—172no—
253C22×C4—172no—
254C22×C4—172no—
255C22×C4—172no—
256C22×C4—344no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4—348no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—684no—
265C22×C4—682no—
266C22×C4—688no—
267C22×C4—688no—
268C22×C4—684no—
269C22×C4—682no—
270C22×C4—682no—
271C22×C4C22⋊#59512yes—
272C22×C4C22⋊#59512yes—
273M16—688no—
274C2×C8—684no—
275C4⋊C4—348no—
276C4⋊C4—348no—
277C4⋊C4—348no—
278C22⋊C4—28no—
279C22⋊C4—348no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—24no—
284C23—24no—
285C23—22no—
286C23—22no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—416no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23#1058754514yes—
295C23C2×C22⋊#59511yescharacteristic
296C23C2×C22⋊#59511yescharacteristic
297C23C2×C22⋊#59511yescharacteristic
298Q8—6816no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—174no—
308C2×C4—174no—
309C2×C4—344no—
310C2×C4—344no—
311C2×C4—342no—
312C2×C4—344no—
313C2×C4—342no—
314C2×C4—342no—
315C2×C4—342no—
316C2×C4—682no—
317C2×C4—682no—
318C2×C4—6816no—
319C2×C4—682no—
320C2×C4—682no—
321C2×C4—682no—
322C2×C4—682no—
323C2×C4—682no—
324C2×C4—682no—
325C8—1364no—
326C22—28no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—44no—
335C22—48no—
336C22—48no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22#1176375011yescharacteristic
341C22#1176378912yes—
342C22D17⋊#10012yes—
343C22C2×#1058754512yes—
344C4—42no—
345C4—42no—
346C4—344no—
347C4—344no—
348C4—682no—
349C4—682no—
350C2—22no—
351C2—22no—
352C2—22no—
353C2—44no—
354C2#1454461411yescharacteristic
355C2#1454461711yescharacteristic
356C2#1454461811yescharacteristic
subgroupG/H
centerC22#11763750
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C22⋊C4×C17C4
socleC22×C17#379
prime radicalC25×C17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C22⋊C4#595
hypocenterC17#8555
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #14544683C2semidirect
2. #14544034C2semidirect
231. C17#8555semidirect

Maximal subgroups

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

subgroupindex
#145446832
#145440342
#145440292
C22×#105875642
#855517