Finite Group Databasebeta

← #16944907

Subgroups — #16944907

Order 4352.

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

normal subgroups only

This group has 768 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
1C4×#10587504C211yesmaximal, characteristic, complemented
2#14527122C211yesmaximal, characteristic, complemented
3#14526787C214yesmaximal, complemented
4C17×#2408C211yesmaximal, characteristic
5C2×#10587504—21no—
6#11762773—21no—
7C4×#63223—21no—
8#11762578C2212yescomplemented
9#11762535C2212yescomplemented
10#11762533—21no—
11C23×D4×C17C411yescharacteristic
12C2×C17×#121—21no—
13C2×C17×#120—21no—
14C2×C4×D4×C17C2211yescharacteristic
15C17×#394C2212yes—
16C17×#392C412yes—
17C17×#390—21no—
18C17×#380C411yescharacteristic
19C17×#379—21no—
20C25×C17—21no—
21C22×D4×C17—22no—
22C22×D4×C17—21no—
23C22×D4×C17—41no—
24C22×D4×C17—41no—
25C22×D4×C17—44no—
26C22×D4×C17C2×C412yes—
27C22×D4×C17D411yescharacteristic
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×C17C2311yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17—22no—
40C2×C22⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊C4×C17D411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×#63223—22no—
44C2×#63223—41no—
45C17×#95C2×C412yes—
46#10587504—24no—
47C22⋊#3622—22no—
48C22⋊#3622—41no—
49#10587474—41no—
50C4×#3622—41no—
51C24×C17—22no—
52C24×C17—42no—
53C24×C17—44no—
54C24×C17—42no—
55C24×C17—42no—
56C24×C17—82no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C22⋊C411yescharacteristic
59C2×D4×C17—24no—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—41no—
63C2×D4×C17—42no—
64C2×D4×C17—42no—
65C2×D4×C17—48no—
66C2×D4×C17—44no—
67C2×D4×C17—44no—
68C2×D4×C17—44no—
69C2×D4×C17—44no—
70C2×D4×C17—44no—
71C2×D4×C17—41no—
72C2×D4×C17—41no—
73C2×D4×C17—82no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—22no—
79C22×C4×C17—42no—
80C22×C4×C17—42no—
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22⋊C411yescharacteristic
83C22×C4×C17C22×C411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C4⋊C4×C17—41no—
87C4⋊C4×C17—42no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—41no—
90C22⋊C4×C17—42no—
91C22⋊C4×C17—44no—
92C22⋊C4×C17—44no—
93C22⋊C4×C17—82no—
94C42×C17—21no—
95C42×C17—21no—
96#63223—24no—
97#63223—42no—
98#63223—82no—
99C2×#3622—42no—
100C2×#3622—42no—
101#11779—171nomaximal, Sylow
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—24no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—42no—
113C23×C17—42no—
114C23×C17—42no—
115C23×C17—48no—
116C23×C17—42no—
117C23×C17—42no—
118C23×C17—42no—
119C23×C17—84no—
120C23×C17—88no—
121C23×C17—88no—
122C23×C17—84no—
123C23×C17#10412yes—
124C23×C17C2×C22⋊C411yescharacteristic
125D4×C17—44no—
126D4×C17—44no—
127D4×C17—44no—
128D4×C17—44no—
129D4×C17—44no—
130D4×C17—88no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—22no—
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—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—81no—
147C2×C4×C17#9911yescharacteristic
148C2×C4×C17#9911yescharacteristic
149C2×C4×C17AGL(1,ℤ/8)12yes—
150#3622—44no—
151#3622—82no—
152#2408D1711yescharacteristic
153C4×#100—171no—
154#1465—171no—
155#1345—174no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—22no—
159C22×C17—42no—
160C22×C17—42no—
161C22×C17—42no—
162C22×C17—44no—
163C22×C17—44no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—44no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—81no—
171C22×C17—82no—
172C22×C17—82no—
173C22×C17—816no—
174C22×C17—82no—
175C22×C17—82no—
176C22×C17C2×#9911yescharacteristic
177C22×C17#41811yescharacteristic
178C22×C17C2×#10411yescharacteristic
179C4×C17—21no—
180C4×C17—22no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—82no—
185C4×C17AΣL(1,𝔽16)12yes—
186C23×D4Dic1711yescharacteristic
187C2×#121—21no—
188C2×#120—21no—
189C2×C4×D4C2×D1711yescharacteristic
190C2×#100—341no—
191#407—341no—
192C4×M16—341no—
193#394C2×D1712yes—
194#392Dic1712yes—
195#390—21no—
196#380Dic1711yescharacteristic
197#379—21no—
198#343—172no—
199#324—172no—
200#323—341no—
201C2×C17—21no—
202C2×C17—21no—
203C2×C17—42no—
204C2×C17—81no—
205C2×C17—84no—
206C2×C17#150711yescharacteristic
207C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
208C2×C17#213411yescharacteristic
209C25—21no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—41no—
213C22×D4—41no—
214C22×D4—44no—
215C22×D4C2×Dic1712yes—
216C22×D4C22⋊D1711yescharacteristic
217C23×C4C2×Dic1711yescharacteristic
218C23×C4C22⋊D1711yescharacteristic
219C2×M16—342no—
220C2×M16—681no—
221#121—44no—
222#120—44no—
223C4×D4—41no—
224C4×D4—41no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22×D1711yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4C2×Dic1711yescharacteristic
232C2×C22⋊C4C22⋊D1711yescharacteristic
233C2×C42C22⋊D1711yescharacteristic
234#100—344no—
235C22⋊C8—342no—
236C22⋊C8—681no—
237#97—681no—
238C4×C8—681no—
239#95C2×Dic1712yes—
240C17#1177911yescharacteristic, complemented, Sylow
241C24—22no—
242C24—42no—
243C24—44no—
244C24—42no—
245C24—42no—
246C24—82no—
247C24C22⋊Dic1711yescharacteristic
248C24C22⋊Dic1711yescharacteristic
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—82no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C22⋊Dic1711yescharacteristic
272C22×C4C22⋊Dic1711yescharacteristic
273C22×C4C22×Dic1711yescharacteristic
274C22×C4C2×C22⋊D1711yescharacteristic
275C22×C4C2×C22⋊D1711yescharacteristic
276M16—344no—
277M16—682no—
278M16—1362no—
279C2×C8—682no—
280C2×C8—682no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—82no—
289C42—21no—
290C42—21no—
291C23—21no—
292C23—21no—
293C23—24no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—48no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—84no—
309C23—88no—
310C23—88no—
311C23—84no—
312C23#1058750812yes—
313C23C2×C22⋊Dic1711yescharacteristic
314D4—44no—
315D4—44no—
316D4—44no—
317D4—44no—
318D4—44no—
319D4—88no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4#1058750511yescharacteristic
337C2×C4#1058750511yescharacteristic
338C2×C4#1058761812yes—
339C8—684no—
340C8—1362no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22—44no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—41no—
355C22—81no—
356C22—82no—
357C22—82no—
358C22—816no—
359C22—82no—
360C22—82no—
361C22C2×#1058750511yescharacteristic
362C22#1176281411yescharacteristic
363C22C2×#1058750811yescharacteristic
364C4—22no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—82no—
370C4#1176259612yes—
371C2—21no—
372C2—21no—
373C2—42no—
374C2—81no—
375C2—84no—
376C2#1452730011yescharacteristic
377C2C2×#1176259611yescharacteristic
378C2#1453047611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
Fitting, prime radicalC17×#2408C2
socleC22×C17C2×#99
cosocleC2×C4⋊C4×C17C23
hypercenter#2408D17
hypocenterC17#11779
LCSG ▷ C22×C4×C17 ▷ C23×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. C4×#10587504C2semidirect
2. #14527122C2semidirect
3. #14526787C2semidirect
8. #11762578C22semidirect
9. #11762535C22semidirect
240. C17#11779semidirect

Maximal subgroups

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

subgroupindex
C4×#105875042
#145271222
#145267872
C17×#24082
#1177917