Finite Group Databasebeta

← #16945006

Subgroups — #16945006

Order 4352.

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

normal subgroups only

This group has 758 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#14528864C211yesmaximal, characteristic, complemented
2#14527114C211yesmaximal, characteristic, complemented
3#14526787C214yesmaximal, complemented
4C2×C17×#394C211yesmaximal, characteristic
5#11762773—21no—
6#11762773—21no—
7#11762578—21no—
8#11762578C2212yescomplemented
9#11762535—21no—
10#11762535C2212yescomplemented
11C23×D4×C17C411yescharacteristic
12C22×C4⋊C4×C17C2211yescharacteristic
13C22×C22⋊C4×C17—21no—
14C22×C22⋊C4×C17—21no—
15C17×#394—21no—
16C17×#394—21no—
17C17×#394C412yes—
18C17×#394C2212yes—
19C2×C17×#95C411yescharacteristic
20C25×C17—21no—
21C22×D4×C17—21no—
22C22×D4×C17—41no—
23C22×D4×C17—41no—
24C22×D4×C17—44no—
25C22×D4×C17C2×C412yes—
26C23×C4×C17—21no—
27C23×C4×C17C2×C411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C2×C4⋊C4×C17—41no—
31C2×C4⋊C4×C17—41no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17C2311yescharacteristic
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17—22no—
40C2×C22⋊C4×C17—44no—
41C2×C22⋊C4×C17—44no—
42C2×#63223—22no—
43C2×#63223—41no—
44C17×#95—21no—
45C17×#95C2×C412yes—
46C22⋊#3622—22no—
47C22⋊#3622—41no—
48C22⋊#3622—41no—
49C22⋊#3622—41no—
50C24×C17—24no—
51C24×C17—22no—
52C24×C17—42no—
53C24×C17—42no—
54C24×C17—42no—
55C24×C17—42no—
56C24×C17—42no—
57C24×C17—41no—
58C24×C17—41no—
59C24×C17C22⋊C411yescharacteristic
60C2×D4×C17—42no—
61C2×D4×C17—42no—
62C2×D4×C17—44no—
63C2×D4×C17—44no—
64C2×D4×C17—48no—
65C2×D4×C17—44no—
66C2×D4×C17—44no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—41no—
76C22×C4×C17—42no—
77C22×C4×C17—42no—
78C22×C4×C17—42no—
79C22×C4×C17—42no—
80C22×C4×C17—42no—
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22⋊C411yescharacteristic
83C22×C4×C17C22⋊C411yescharacteristic
84C22×C4×C17C22×C411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C22⋊C4×C17—44no—
92C22⋊C4×C17—44no—
93C22⋊C4×C17—44no—
94C22⋊C4×C17—44no—
95#63223—42no—
96#63223—42no—
97#63223—82no—
98C2×#3622—42no—
99C2×#3622—42no—
100#11802—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—22no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—24no—
106C23×C17—42no—
107C23×C17—44no—
108C23×C17—48no—
109C23×C17—48no—
110C23×C17—42no—
111C23×C17—42no—
112C23×C17—44no—
113C23×C17—42no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17—84no—
117C23×C17—84no—
118C23×C17—82no—
119C23×C17—82no—
120C23×C17—82no—
121C23×C17—82no—
122C23×C17#9911yescharacteristic
123C23×C17#9911yescharacteristic
124C23×C17C2×C22⋊C411yescharacteristic
125D4×C17—44no—
126D4×C17—44no—
127D4×C17—48no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—42no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—42no—
147C2×C4×C17—42no—
148C2×C4×C17—42no—
149C2×C4×C17—81no—
150C2×C4×C17—84no—
151C2×C4×C17AGL(1,ℤ/8)12yes—
152#3622—82no—
153#3622—82no—
154C2×#394D1711yescharacteristic
155#1821—171no—
156#1463—171no—
157#1345—174no—
158C22×C17—24no—
159C22×C17—24no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—44no—
167C22×C17—44no—
168C22×C17—44no—
169C22×C17—82no—
170C22×C17—82no—
171C22×C17—88no—
172C22×C17—88no—
173C22×C17—82no—
174C22×C17—82no—
175C22×C17AΣL(1,𝔽16)12yes—
176C22×C17#35512yes—
177C22×C17C2×#9911yescharacteristic
178C22×C17#41811yescharacteristic
179C22×C17#42111yescharacteristic
180C4×C17—22no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17—41no—
186C4×C17—82no—
187C23×D4Dic1711yescharacteristic
188C22×C4⋊C4C2×D1711yescharacteristic
189C22×C22⋊C4—21no—
190C22×C22⋊C4—21no—
191#407—341no—
192#407—341no—
193#394—21no—
194#394—21no—
195#394Dic1712yes—
196#394C2×D1712yes—
197C2×#95Dic1711yescharacteristic
198#343—172no—
199#343—341no—
200#324—172no—
201#324—341no—
202C2×C17—21no—
203C2×C17—21no—
204C2×C17—22no—
205C2×C17—22no—
206C2×C17—84no—
207C2×C17#151011yescharacteristic
208C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
209C2×C17C2×#35511yescharacteristic
210C25—21no—
211C22×D4—21no—
212C22×D4—41no—
213C22×D4—41no—
214C22×D4—44no—
215C22×D4C2×Dic1712yes—
216C23×C4—21no—
217C23×C4C2×Dic1711yescharacteristic
218C23×C4C22⋊D1711yescharacteristic
219C23×C4C22⋊D1711yescharacteristic
220C2×M16—342no—
221C2×M16—681no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4C2×Dic1711yescharacteristic
225C2×C4⋊C4C22⋊D1711yescharacteristic
226C2×C4⋊C4C22⋊D1711yescharacteristic
227C2×C4⋊C4C22×D1711yescharacteristic
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C22⋊C8—342no—
235C22⋊C8—681no—
236C22⋊C8—681no—
237C22⋊C8—681no—
238#95—21no—
239#95C2×Dic1712yes—
240C17#1180211yescharacteristic, complemented, Sylow
241C24—24no—
242C24—22no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24C22⋊Dic1711yescharacteristic
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—44no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C22⋊Dic1711yescharacteristic
273C22×C4C22⋊Dic1711yescharacteristic
274C22×C4C22⋊Dic1711yescharacteristic
275C22×C4C22×Dic1711yescharacteristic
276C22×C4C2×C22⋊D1711yescharacteristic
277C22×C4C2×C22⋊D1711yescharacteristic
278M16—682no—
279M16—682no—
280M16—1362no—
281C2×C8—682no—
282C2×C8—682no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—24no—
296C23—42no—
297C23—44no—
298C23—48no—
299C23—48no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—44no—
306C23—84no—
307C23—84no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23#1058750511yescharacteristic
313C23#1058750511yescharacteristic
314C23C2×C22⋊Dic1711yescharacteristic
315D4—44no—
316D4—44no—
317D4—48no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—84no—
341C2×C4#1058761812yes—
342C8—1362no—
343C8—1362no—
344C22—24no—
345C22—24no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—82no—
356C22—82no—
357C22—88no—
358C22—88no—
359C22—82no—
360C22—82no—
361C22#1176259612yes—
362C22#1176260612yes—
363C22C2×#1058750511yescharacteristic
364C22#1176281411yescharacteristic
365C22#1176282611yescharacteristic
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—82no—
373C2—21no—
374C2—21no—
375C2—22no—
376C2—22no—
377C2—84no—
378C2#1452731311yescharacteristic
379C2C2×#1176259611yescharacteristic
380C2C2×#1176260611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC2×C17×#394C2
socleC22×C17C2×#99
prime radicalC23×D4×C17C4
cosocleC2×C4⋊C4×C17C23
hypercenterC2×#394D17
hypocenterC17#11802
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. #14528864C2semidirect
2. #14527114C2semidirect
3. #14526787C2semidirect
8. #11762578C22semidirect
10. #11762535C22semidirect
240. C17#11802semidirect

Maximal subgroups

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

subgroupindex
#145288642
#145271142
#145267872
C2×C17×#3942
#1180217