Finite Group Databasebeta

← #16943937

Subgroups — #16943937

Order 4352.

748 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 95 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#14527122C211yesmaximal, characteristic, complemented
2#14527105C211yesmaximal, characteristic, complemented
3#14526712C214yesmaximal, complemented
4C17×#2408C211yesmaximal, characteristic
5#11762862—21no—
6#11762773—21no—
7#11762535—21no—
8#11762533—21no—
9#11762533C2212yes—
10#11762533C2212yes—
11C23×D4×C17C411yescharacteristic
12C2×C17×#121—21no—
13C2×C17×#120C412yes—
14C2×C4×D4×C17C2211yescharacteristic
15C17×#394—21no—
16C17×#392—21no—
17C17×#390—21no—
18C17×#380C411yescharacteristic
19C17×#379C2212yes—
20C25×C17C2×C412yes—
21C22×D4×C17—22no—
22C22×D4×C17—21no—
23C22×D4×C17—21no—
24C22×D4×C17—41no—
25C22×D4×C17—41no—
26C22×D4×C17—44no—
27C22×D4×C17C2×C411yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C17×#121—44no—
31C17×#120—44no—
32C4×D4×C17—41no—
33C4×D4×C17—41no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17C2×C412yes—
40C2×C22⋊C4×C17D411yescharacteristic
41C2×C22⋊C4×C17C2311yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×#63223—41no—
44C17×#95—21no—
45C22⋊#3622—22no—
46C22⋊#3622—22no—
47C22⋊#3622—41no—
48C4⋊#3622—41no—
49C4×#3622—41no—
50C24×C17—22no—
51C24×C17—24no—
52C24×C17—42no—
53C24×C17—44no—
54C24×C17—42no—
55C24×C17—82no—
56C24×C17C22⋊C411yescharacteristic
57C24×C17C22×C411yescharacteristic
58C2×D4×C17—24no—
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—41no—
62C2×D4×C17—42no—
63C2×D4×C17—42no—
64C2×D4×C17—44no—
65C2×D4×C17—48no—
66C2×D4×C17—44no—
67C2×D4×C17—44no—
68C2×D4×C17—44no—
69C2×D4×C17—44no—
70C2×D4×C17—41no—
71C2×D4×C17—41no—
72C2×D4×C17—82no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—22no—
78C22×C4×C17—42no—
79C22×C4×C17—42no—
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22⋊C411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C4⋊C4×C17—41no—
86C4⋊C4×C17—42no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—44no—
90C22⋊C4×C17—44no—
91C22⋊C4×C17—42no—
92C22⋊C4×C17—82no—
93C42×C17—21no—
94C42×C17—21no—
95#63223—82no—
96C2×#3622—42no—
97C2×#3622—42no—
98#11561—171nomaximal, Sylow
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—24no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—42no—
110C23×C17—42no—
111C23×C17—42no—
112C23×C17—44no—
113C23×C17—44no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17—42no—
117C23×C17—84no—
118C23×C17—88no—
119C23×C17—88no—
120C23×C17—84no—
121C23×C17C2×C22⋊C411yescharacteristic
122C23×C17#13112yes—
123D4×C17—44no—
124D4×C17—44no—
125D4×C17—48no—
126D4×C17—44no—
127D4×C17—88no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—22no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—42no—
135C2×C4×C17—42no—
136C2×C4×C17—42no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—81no—
144C2×C4×C17#9911yescharacteristic
145C2×C4×C17#9911yescharacteristic
146C2×C4×C17#10012yes—
147#3622—82no—
148#3622—82no—
149#2408D1711yescharacteristic
150#1465—171no—
151#1460—171no—
152#1316—174no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—22no—
156C22×C17—42no—
157C22×C17—42no—
158C22×C17—42no—
159C22×C17—44no—
160C22×C17—44no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—44no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—81no—
168C22×C17—82no—
169C22×C17—82no—
170C22×C17—88no—
171C22×C17—88no—
172C22×C17—82no—
173C22×C17—82no—
174C22×C17#40811yescharacteristic
175C22×C17C2×#9911yescharacteristic
176C22×C17C2×#10011yescharacteristic
177C4×C17—21no—
178C4×C17—22no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—82no—
183C4×C17AΣL(1,𝔽16)12yes—
184C23×D4Dic1711yescharacteristic
185C2×#121—21no—
186C2×#120Dic1712yes—
187C2×C4×D4C2×D1711yescharacteristic
188#431—341no—
189#407—341no—
190#394—21no—
191#392—21no—
192#390—21no—
193#380Dic1711yescharacteristic
194#379C2×D1712yes—
195#324—341no—
196#323—172no—
197#323—172no—
198#323—341no—
199C2×C17—21no—
200C2×C17—21no—
201C2×C17—42no—
202C2×C17—81no—
203C2×C17—84no—
204C2×C17#147111yescharacteristic
205C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
206C2×C17#212211yescharacteristic
207C25C2×Dic1712yes—
208C22×D4—22no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—41no—
212C22×D4—41no—
213C22×D4—44no—
214C22×D4C2×Dic1711yescharacteristic
215C23×C4C2×Dic1711yescharacteristic
216C23×C4C22⋊D1711yescharacteristic
217C2×M16—681no—
218#121—44no—
219#120—44no—
220C4×D4—41no—
221C4×D4—41no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22⋊D1711yescharacteristic
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4C2×Dic1712yes—
228C2×C22⋊C4C22⋊D1711yescharacteristic
229C2×C22⋊C4C22×D1711yescharacteristic
230C2×C42C22⋊D1711yescharacteristic
231C4⋊C8—681no—
232C22⋊C8—342no—
233C22⋊C8—342no—
234C22⋊C8—681no—
235C4×C8—681no—
236#95—21no—
237C17#1156111yescharacteristic, complemented, Sylow
238C24—22no—
239C24—24no—
240C24—42no—
241C24—44no—
242C24—42no—
243C24—82no—
244C24C22⋊Dic1711yescharacteristic
245C24C22×Dic1711yescharacteristic
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—48no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—82no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4C22⋊Dic1711yescharacteristic
269C22×C4C22⋊Dic1711yescharacteristic
270C22×C4C22⋊Dic1711yescharacteristic
271C22×C4C2×C22⋊D1711yescharacteristic
272C22×C4C2×C22⋊D1711yescharacteristic
273M16—1362no—
274C2×C8—682no—
275C2×C8—682no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C42—21no—
285C42—21no—
286C23—21no—
287C23—21no—
288C23—24no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—44no—
300C23—44no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—84no—
305C23—88no—
306C23—88no—
307C23—84no—
308C23#1058763412yes—
309C23C2×C22⋊Dic1711yescharacteristic
310D4—44no—
311D4—44no—
312D4—48no—
313D4—44no—
314D4—88no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4#1058750412yes—
332C2×C4#1058750511yescharacteristic
333C2×C4#1058750511yescharacteristic
334C8—1362no—
335C8—1362no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—44no—
343C22—44no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—81no—
351C22—82no—
352C22—82no—
353C22—88no—
354C22—88no—
355C22—82no—
356C22—82no—
357C22#1176277611yescharacteristic
358C22C2×#1058750511yescharacteristic
359C22C2×#1058750411yescharacteristic
360C4—21no—
361C4—22no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—82no—
366C4#1176259612yes—
367C2—21no—
368C2—21no—
369C2—42no—
370C2—81no—
371C2—84no—
372C2#1452714411yescharacteristic
373C2C2×#1176259611yescharacteristic
374C2#1453043411yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
Fitting, prime radicalC17×#2408C2
socleC22×C17C2×#99
cosocleC2×C22⋊C4×C17C23
hypercenter#2408D17
hypocenterC17#11561
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #14527122C2semidirect
2. #14527105C2semidirect
3. #14526712C2semidirect
237. C17#11561semidirect

Maximal subgroups

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

subgroupindex
#145271222
#145271052
#145267122
C17×#24082
#1156117