Finite Group Databasebeta

← #16944840

Subgroups — #16944840

Order 4352.

470 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 91 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#14528523C211yesmaximal, characteristic, complemented
2#14527118C211yesmaximal, characteristic, complemented
3#14526814C212yesmaximal, complemented
4#14526787C212yesmaximal, complemented
5C17×#2610C211yesmaximal, characteristic
6C22×#63223—21no—
7#11762866—21no—
8#11762578—21no—
9#11762578C2212yes—
10#11762535C2212yes—
11#11762533—21no—
12C2×C17×#124C411yescharacteristic
13C2×C17×#117C2211yescharacteristic
14C17×#395C2211yescharacteristic
15C17×#394C2211yescharacteristic
16C17×#393C411yescharacteristic
17C17×#392C411yescharacteristic
18C17×#390—21no—
19C17×#388—21no—
20C17×#388—21no—
21C17×#386—21no—
22C17×#376C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×D4×C17—21no—
25C22×D4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C17×#124—41no—
28C17×#124—41no—
29C17×#117—41no—
30C17×#117—41no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊C4×C17D411yescharacteristic
42C2×C42×C17—21no—
43C2×C42×C17C2×C411yescharacteristic
44C2×C42×C17D411yescharacteristic
45C2×#63223—24no—
46C2×#63223—22no—
47C2×#63223—41no—
48C2×#63223—42no—
49C22×#3622—41no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95C2×C411yescharacteristic
54C17×#95C2×C411yescharacteristic
55C22⋊#3622—22no—
56C4⋊#3622—41no—
57#10587474—41no—
58C24×C17—21no—
59C24×C17C22⋊C411yescharacteristic
60C2×Q8×C17—41no—
61C2×Q8×C17—81no—
62C2×D4×C17—41no—
63C2×D4×C17—42no—
64C2×D4×C17—42no—
65C2×D4×C17—81no—
66C2×D4×C17—81no—
67C22×C4×C17—22no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22×C411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—81no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—42no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—81no—
93C22⋊C4×C17—81no—
94C42×C17—21no—
95C42×C17—21no—
96C42×C17—21no—
97C42×C17—21no—
98C42×C17—42no—
99#63223—24no—
100#63223—44no—
101#63223—42no—
102C2×#3622—44no—
103C2×#3622—42no—
104C2×#3622—42no—
105#11765—171nomaximal, Sylow
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17#10412yes—
118C23×C17C2×C22⋊C411yescharacteristic
119Q8×C17—82no—
120D4×C17—84no—
121D4×C17—82no—
122C2×C4×C17—22no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—22no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
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—21no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17#9911yescharacteristic
148C2×C4×C17#9911yescharacteristic
149C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C17#13711yescharacteristic
151#3622—44no—
152#3622—82no—
153#2610D1711yescharacteristic
154#1742—171no—
155#1464—171no—
156#1354—172no—
157#1345—172no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—22no—
161C22×C17—42no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—81no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17—81no—
170C22×C17—81no—
171C22×C17—81no—
172C22×C17C2×#9911yescharacteristic
173C22×C17#41711yescharacteristic
174C22×C17C2×#10411yescharacteristic
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—42no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—81no—
185C4×C17—81no—
186C22×M16—341no—
187C2×#124Dic1711yescharacteristic
188C2×#117C2×D1711yescharacteristic
189#432—341no—
190#395C2×D1711yescharacteristic
191#394C2×D1711yescharacteristic
192#393Dic1711yescharacteristic
193#392Dic1711yescharacteristic
194#390—21no—
195#388—21no—
196#388—21no—
197#386—21no—
198#376Dic1711yescharacteristic
199#343—172no—
200#343—341no—
201#324—172no—
202#323—341no—
203C2×C17—21no—
204C2×C17—21no—
205C2×C17—42no—
206C2×C17—161no—
207C2×C17#150411yescharacteristic
208C2×C17#212211yescharacteristic
209C2×C17#213211yescharacteristic
210C22×Q8C2×Dic1711yescharacteristic
211C22×D4—21no—
212C22×D4C2×Dic1711yescharacteristic
213C23×C4C22⋊D1711yescharacteristic
214C2×M16—344no—
215C2×M16—342no—
216C2×M16—681no—
217C2×M16—682no—
218C22×C8—681no—
219#124—41no—
220#124—41no—
221#117—41no—
222#117—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22⋊D1711yescharacteristic
225C2×C4⋊C4C22×D1711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4C2×Dic1711yescharacteristic
233C2×C22⋊C4C22⋊D1711yescharacteristic
234C2×C42—21no—
235C2×C42C2×Dic1711yescharacteristic
236C2×C42C22⋊D1711yescharacteristic
237C4⋊C8—681no—
238C22⋊C8—342no—
239#97—681no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95C2×Dic1711yescharacteristic
244#95C2×Dic1711yescharacteristic
245C17#1176511yescharacteristic, complemented, Sylow
246C24—21no—
247C24C22⋊Dic1711yescharacteristic
248C2×Q8—41no—
249C2×Q8—81no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—81no—
254C2×D4—81no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4C22⋊Dic1711yescharacteristic
264C22×C4C22⋊Dic1711yescharacteristic
265C22×C4C22⋊Dic1711yescharacteristic
266C22×C4C22×Dic1711yescharacteristic
267C22×C4C2×C22⋊D1711yescharacteristic
268C22×C4C2×C22⋊D1711yescharacteristic
269M16—344no—
270M16—684no—
271M16—682no—
272C2×C8—684no—
273C2×C8—682no—
274C2×C8—682no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—42no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#1058750812yes—
305C23C2×C22⋊Dic1711yescharacteristic
306Q8—82no—
307D4—84no—
308D4—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4#1058750511yescharacteristic
335C2×C4#1058750511yescharacteristic
336C2×C4#1058763511yescharacteristic
337C2×C4#1058763711yescharacteristic
338C8—684no—
339C8—1362no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22C2×#1058750511yescharacteristic
355C22#1176281011yescharacteristic
356C22C2×#1058750811yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C2—21no—
369C2—21no—
370C2—42no—
371C2—161no—
372C2#1452728811yescharacteristic
373C2#1453043411yescharacteristic
374C2#1453046911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2610C2
socleC22×C17C2×#99
prime radicalC17×#392C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2610D17
hypocenterC17#11765
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #14528523C2semidirect
2. #14527118C2semidirect
3. #14526814C2semidirect
4. #14526787C2semidirect
245. C17#11765semidirect

Maximal subgroups

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

subgroupindex
#145285232
#145271182
#145268142
#145267872
C17×#26102
#1176517