Finite Group Databasebeta

← #17953102

Subgroups — #17953102

Order 4352.

1134 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1134 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#14544618C212yesmaximal, complemented
2#14544032C212yesmaximal, complemented
3#14544029C212yesmaximal
4C22×#10587564C211yesmaximal, characteristic
5C2×#10587545—24no—
6#11763779C2214yes—
7#11763654—24no—
8#11763652—24no—
9C23×Dic34C2211yescharacteristic
10C24×Dic17C411yescharacteristic
11C2×#10587564—24no—
12C2×#10587564—24no—
13C2×#10587564—24no—
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63225C411yescharacteristic
16C22×#63225C411yescharacteristic
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—44no—
33C22×Dic34D414yes—
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⋊Dic17D414yes—
44C2×#63225—22no—
45C2×#63225—22no—
46C2×#63225—22no—
47C2×#63225—22no—
48C2×#63225—22no—
49C2×#63225—22no—
50#10587564—232no—
51C22⋊#3624—24no—
52C22⋊#3624—24no—
53#10587545—48no—
54C24×C17—24no—
55C24×C17—21no—
56C24×C17—21no—
57C24×C17—42no—
58C24×C17—42no—
59C24×C17C22⋊C412yes—
60C24×C17C22×C411yescharacteristic
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C412yes—
68C22×C4×C17C22⋊C412yes—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—24no—
72C22×Dic17—28no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—42no—
78C22×Dic17—48no—
79C22×Dic17—48no—
80C22×Dic17—44no—
81C22×Dic17—44no—
82C22×Dic17—42no—
83C22×Dic17—42no—
84C22×Dic17C22⋊C412yes—
85C22×Dic17C22⋊C412yes—
86C22×Dic17C22⋊C412yes—
87C22×Dic17C2×D412yes—
88C22×Dic17C2×D412yes—
89C22×Dic17C2×D412yes—
90C2×Dic34—28no—
91C2×Dic34—28no—
92C2×Dic34—44no—
93C2×Dic34—48no—
94C2×Dic34—48no—
95#63247—48no—
96C2×#3624—44no—
97C22⋊C4×C17—28no—
98C22⋊Dic17—28no—
99C4⋊Dic17—28no—
100#63225—28no—
101#63225—28no—
102#8550—171nomaximal, Sylow
103C23×C17—22no—
104C23×C17—22no—
105C23×C17—22no—
106C23×C17—24no—
107C23×C17—24no—
108C23×C17—22no—
109C23×C17—22no—
110C23×C17—42no—
111C23×C17—42no—
112C23×C17—42no—
113C23×C17—416no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17—42no—
117C23×C17#10114yescomplemented
118C23×C17C2×C22⋊C411yescharacteristic
119C23×C17C2×C22⋊C411yescharacteristic
120C23×C17C2×C22⋊C411yescharacteristic
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—24no—
124C2×C4×C17—24no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×Dic17—24no—
130C2×Dic17—24no—
131C2×Dic17—22no—
132C2×Dic17—24no—
133C2×Dic17—22no—
134C2×Dic17—22no—
135C2×Dic17—22no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—416no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17—42no—
145C2×Dic17#9914yes—
146C2×Dic17#12014yes—
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—24no—
161C22×C17—44no—
162C22×C17—48no—
163C22×C17—44no—
164C22×C17—48no—
165C22×C17—42no—
166C22×C17—42no—
167C22×C17#37911yescharacteristic
168C22×C17C2×#9912yes—
169C22×C17C2×#10112yes—
170C22×C17#41312yes—
171C4×C17—42no—
172C4×C17—42no—
173Dic17—24no—
174Dic17—24no—
175Dic17—42no—
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—171no—
185C22×C22⋊C4—171no—
186C22×C22⋊C4#59511yescharacteristic
187C2×#101—344no—
188#407—174no—
189#324—344no—
190#323—344no—
191C2×C17—22no—
192C2×C17—22no—
193C2×C17—22no—
194C2×C17—44no—
195C2×C17#178111yescharacteristic
196C2×C17#178411yescharacteristic
197C2×C17#178511yescharacteristic
198C25C2×#59511yescharacteristic
199C22×Q8—174no—
200C22×Q8—342no—
201C22×Q8—342no—
202C22×Q8—684no—
203C23×C4—171no—
204C23×C4—171no—
205C23×C4—171no—
206C23×C4—171no—
207C23×C4—342no—
208C23×C4—342no—
209C23×C4—342no—
210C23×C4—684no—
211C23×C4C2×#59511yescharacteristic
212C23×C4C2×#59511yescharacteristic
213C2×M16—344no—
214C22⋊Q8—3432no—
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×C4⋊C4—342no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—174no—
228C2×C22⋊C4—342no—
229C2×C22⋊C4—342no—
230#101—688no—
231C22⋊C8—344no—
232C22⋊C8—344no—
233C17#855011yescharacteristic, complemented, Sylow
234C24—21no—
235C24—21no—
236C24—24no—
237C24—42no—
238C24—42no—
239C24C22⋊#59512yes—
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—42no—
251C22×C4—42no—
252C22×C4—172no—
253C22×C4—172no—
254C22×C4—172no—
255C22×C4—172no—
256C22×C4—172no—
257C22×C4—172no—
258C22×C4—344no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—348no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—684no—
267C22×C4—682no—
268C22×C4—688no—
269C22×C4—688no—
270C22×C4—684no—
271C22×C4—682no—
272C22×C4—682no—
273C22×C4C22⋊#59512yes—
274C22×C4C22⋊#59512yes—
275M16—688no—
276C2×C8—684no—
277C4⋊C4—348no—
278C4⋊C4—348no—
279C4⋊C4—348no—
280C22⋊C4—28no—
281C22⋊C4—348no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—24no—
286C23—24no—
287C23—22no—
288C23—22no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—416no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23#1058754514yescomplemented
297C23C2×C22⋊#59511yescharacteristic
298C23C2×C22⋊#59511yescharacteristic
299C23C2×C22⋊#59511yescharacteristic
300Q8—6816no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—174no—
310C2×C4—174no—
311C2×C4—344no—
312C2×C4—344no—
313C2×C4—342no—
314C2×C4—344no—
315C2×C4—342no—
316C2×C4—342no—
317C2×C4—342no—
318C2×C4—682no—
319C2×C4—682no—
320C2×C4—6816no—
321C2×C4—682no—
322C2×C4—682no—
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—24no—
336C22—44no—
337C22—48no—
338C22—44no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22#1176375011yescharacteristic
343C22D17⋊#9912yes—
344C22C2×#1058754512yes—
345C22#1176379612yes—
346C4—42no—
347C4—42no—
348C4—344no—
349C4—344no—
350C4—682no—
351C4—682no—
352C2—22no—
353C2—22no—
354C2—22no—
355C2—44no—
356C2#1454461111yescharacteristic
357C2#1454461811yescharacteristic
358C2#1454462111yescharacteristic
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#8550
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. #14544618C2semidirect
2. #14544032C2semidirect
117. C23×C17#101semidirect
233. C17#8550semidirect
296. C23#10587545semidirect

Maximal subgroups

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

subgroupindex
#145446182
#145440322
#145440292
C22×#105875642
#855017