Finite Group Databasebeta

← #16926965

Subgroups — #16926965

Order 4352.

1166 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 147 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#14528864C212yesmaximal, complemented
2#14527111C212yesmaximal, complemented
3#14527111C212yesmaximal, complemented
4C22×C17×#123C211yesmaximal, characteristic
5#11762773C2214yes—
6#11762578—24no—
7#11762533—24no—
8#11762533—24no—
9C23×D4×C17C411yescharacteristic
10C24×C4×C17C411yescharacteristic
11C2×C17×#123—24no—
12C2×C17×#123—24no—
13C2×C17×#123—24no—
14C22×C4⋊C4×C17C411yescharacteristic
15C22×C4⋊C4×C17C2211yescharacteristic
16C22×C22⋊C4×C17C411yescharacteristic
17C22×C22⋊C4×C17C2211yescharacteristic
18C22×C22⋊C4×C17C2211yescharacteristic
19C25×C17C2×C411yescharacteristic
20C25×C17C2×C411yescharacteristic
21C22×D4×C17—22no—
22C22×D4×C17—22no—
23C22×D4×C17—22no—
24C22×D4×C17—44no—
25C23×C4×C17—22no—
26C23×C4×C17—22no—
27C23×C4×C17—22no—
28C23×C4×C17—44no—
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17C2×C411yescharacteristic
31C23×C4×C17C2×C411yescharacteristic
32C23×C4×C17C2×C411yescharacteristic
33C23×C4×C17C2311yescharacteristic
34C17×#123—232no—
35C2×C4⋊C4×C17—22no—
36C2×C4⋊C4×C17—22no—
37C2×C4⋊C4×C17—22no—
38C2×C4⋊C4×C17—22no—
39C2×C4⋊C4×C17—22no—
40C2×C4⋊C4×C17D414yes—
41C2×C22⋊C4×C17—22no—
42C2×C22⋊C4×C17—22no—
43C2×C22⋊C4×C17—22no—
44C2×C22⋊C4×C17—22no—
45C2×C22⋊C4×C17—22no—
46C2×C22⋊C4×C17—22no—
47C2×C22⋊C4×C17—22no—
48C2×C22⋊C4×C17D414yes—
49C2×C22⋊C4×C17D414yes—
50C2×#63223—24no—
51C22⋊#3622—24no—
52C22⋊#3622—24no—
53C24×C17—24no—
54C24×C17—21no—
55C24×C17—21no—
56C24×C17—21no—
57C24×C17—21no—
58C24×C17—42no—
59C24×C17—42no—
60C24×C17—42no—
61C24×C17—42no—
62C24×C17—42no—
63C24×C17C22⋊C412yes—
64C24×C17C22⋊C412yes—
65C24×C17C22×C411yescharacteristic
66C2×D4×C17—28no—
67C2×D4×C17—48no—
68C2×D4×C17—44no—
69C2×D4×C17—48no—
70C2×D4×C17—44no—
71C22×C4×C17—28no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—42no—
81C22×C4×C17—48no—
82C22×C4×C17—48no—
83C22×C4×C17—44no—
84C22×C4×C17—44no—
85C22×C4×C17—42no—
86C22×C4×C17—42no—
87C22×C4×C17—42no—
88C22×C4×C17—42no—
89C22×C4×C17C22⋊C412yes—
90C22×C4×C17C22⋊C412yes—
91C22×C4×C17C22⋊C412yes—
92C22×C4×C17C22⋊C412yes—
93C22×C4×C17C2×D412yes—
94C22×C4×C17C2×D412yes—
95C22×C4×C17C2×D412yes—
96C4⋊C4×C17—28no—
97C4⋊C4×C17—28no—
98C22⋊C4×C17—28no—
99C22⋊C4×C17—28no—
100C22⋊C4×C17—28no—
101#63223—48no—
102C2×#3622—44no—
103#8553—171nomaximal, Sylow
104C23×C17—22no—
105C23×C17—22no—
106C23×C17—22no—
107C23×C17—24no—
108C23×C17—24no—
109C23×C17—22no—
110C23×C17—22no—
111C23×C17—22no—
112C23×C17—24no—
113C23×C17—42no—
114C23×C17—42no—
115C23×C17—416no—
116C23×C17—42no—
117C23×C17—42no—
118C23×C17—42no—
119C23×C17—42no—
120C23×C17—42no—
121C23×C17—42no—
122C23×C17—42no—
123C23×C17—42no—
124C23×C17—42no—
125C23×C17—42no—
126C23×C17—42no—
127C23×C17—42no—
128C23×C17—88no—
129C23×C17#10014yes—
130C23×C17C2×C22⋊C411yescharacteristic
131C23×C17C2×C22⋊C411yescharacteristic
132C23×C17C2×C22⋊C411yescharacteristic
133D4×C17—416no—
134C2×C4×C17—24no—
135C2×C4×C17—22no—
136C2×C4×C17—22no—
137C2×C4×C17—22no—
138C2×C4×C17—22no—
139C2×C4×C17—42no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17—416no—
145C2×C4×C17—42no—
146C2×C4×C17—42no—
147C2×C4×C17—42no—
148C2×C4×C17—42no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17#12014yes—
155#3622—84no—
156C22×#123D1711yescharacteristic
157#1821—172no—
158#1462—172no—
159#1462—172no—
160C22×C17—28no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—24no—
168C22×C17—44no—
169C22×C17—48no—
170C22×C17—44no—
171C22×C17—48no—
172C22×C17—42no—
173C22×C17—42no—
174C22×C17—42no—
175C22×C17—42no—
176C22×C17—42no—
177C22×C17—84no—
178C22×C17—84no—
179C22×C17—84no—
180C22×C17#37911yescharacteristic
181C22×C17#41012yes—
182C22×C17C2×#10012yes—
183C22×C17#41312yes—
184C4×C17—24no—
185C4×C17—42no—
186C4×C17—42no—
187C4×C17—42no—
188C4×C17—42no—
189C23×D4Dic1711yescharacteristic
190C24×C4Dic1711yescharacteristic
191C2×#123—24no—
192C2×#123—24no—
193C2×#123—24no—
194C22×C4⋊C4Dic1711yescharacteristic
195C22×C4⋊C4C2×D1711yescharacteristic
196C22×C22⋊C4Dic1711yescharacteristic
197C22×C22⋊C4C2×D1711yescharacteristic
198C22×C22⋊C4C2×D1711yescharacteristic
199#407—174no—
200#343—344no—
201#323—344no—
202#323—344no—
203C2×C17—22no—
204C2×C17—22no—
205C2×C17—22no—
206C2×C17—44no—
207C2×C17—82no—
208C2×C17#178211yescharacteristic
209C2×C17#178311yescharacteristic
210C2×C17#178511yescharacteristic
211C25C2×Dic1711yescharacteristic
212C25C2×Dic1711yescharacteristic
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—44no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—44no—
221C23×C4C2×Dic1711yescharacteristic
222C23×C4C2×Dic1711yescharacteristic
223C23×C4C2×Dic1711yescharacteristic
224C23×C4C2×Dic1711yescharacteristic
225C23×C4C22×D1711yescharacteristic
226C2×M16—344no—
227#123—232no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4C22⋊D1714yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4C22⋊D1714yes—
242C2×C22⋊C4C22⋊D1714yes—
243C22⋊C8—344no—
244C22⋊C8—344no—
245C17#855311yescharacteristic, complemented, Sylow
246C24—24no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C22⋊Dic1712yes—
257C24C22⋊Dic1712yes—
258C24C22×Dic1711yescharacteristic
259C2×D4—28no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C22×C4—28no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C22⋊Dic1712yes—
283C22×C4C22⋊Dic1712yes—
284C22×C4C22⋊Dic1712yes—
285C22×C4C22⋊Dic1712yes—
286C22×C4C2×C22⋊D1712yes—
287C22×C4C2×C22⋊D1712yes—
288C22×C4C2×C22⋊D1712yes—
289M16—688no—
290C2×C8—684no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—416no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—88no—
321C23#1058750414yes—
322C23C2×C22⋊Dic1711yescharacteristic
323C23C2×C22⋊Dic1711yescharacteristic
324C23C2×C22⋊Dic1711yescharacteristic
325D4—416no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—416no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4#1058763914yes—
347C8—1364no—
348C22—28no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—24no—
356C22—44no—
357C22—48no—
358C22—44no—
359C22—48no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22#1176266511yescharacteristic
369C22#1176278312yes—
370C22C2×#1058750412yes—
371C22#1176279312yes—
372C4—24no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—44no—
381C2—82no—
382C2#1452870211yescharacteristic
383C2#1452870511yescharacteristic
384C2#1452871211yescharacteristic
subgroupG/H
centerC22#11762665
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC22×C17×#123C2
socleC22×C17#379
prime radicalC23×D4×C17C4
cosocleC23×C4×C17C23
hypercenterC22×#123D17
hypocenterC17#8553
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. #14528864C2semidirect
2. #14527111C2semidirect
3. #14527111C2semidirect
245. C17#8553semidirect

Maximal subgroups

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

subgroupindex
#145288642
#145271112
C22×C17×#1232
#855317