Finite Group Databasebeta

← #16944894

Subgroups — #16944894

Order 4352.

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

normal subgroups only

This group has 652 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×#10587505C211yesmaximal, characteristic, complemented
2#14527122C211yesmaximal, characteristic, complemented
3#14526781C214yesmaximal, complemented
4C17×#2541C211yesmaximal, characteristic
5C2×#10587505—21no—
6#11762773—21no—
7C4×C22⋊Dic17—21no—
8#11762576C2212yes—
9#11762535—21no—
10#11762533C2212yes—
11C2×C17×#127—21no—
12C2×C17×#127C411yescharacteristic, complemented
13C2×C17×#121—21no—
14C2×C17×#121—21no—
15C2×C4×D4×C17C2211yescharacteristic
16C17×#392C412yescomplemented
17C17×#392C2212yes—
18C17×#390—21no—
19C4×C4⋊C4×C17C411yescharacteristic, complemented
20C22×D4×C17—22no—
21C22×D4×C17—22no—
22C22×D4×C17—21no—
23C22×D4×C17—21no—
24C22×D4×C17C2×C412yes—
25C22×D4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C17×#127—41no—
29C17×#127—41no—
30C17×#127—44no—
31C17×#121—44no—
32C17×#121—44no—
33C4×D4×C17—41no—
34C4×D4×C17—41no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17C2×C411yescharacteristic
40C2×C22⋊C4×C17C2311yescharacteristic
41C2×C42×C17—21no—
42C2×C42×C17C2×C411yescharacteristic
43C2×C22⋊Dic17—22no—
44C2×C4×Dic17—41no—
45C2×#63223—41no—
46C17×#95C2×C412yes—
47#10587505—24no—
48#10587502—41no—
49C22⋊#3622—22no—
50C22⋊#3622—41no—
51C24×C17—22no—
52C24×C17C22⋊C411yescharacteristic
53C24×C17C2×D411yescharacteristic
54C2×D4×C17—22no—
55C2×D4×C17—22no—
56C2×D4×C17—41no—
57C2×D4×C17—42no—
58C2×D4×C17—42no—
59C2×D4×C17—42no—
60C2×D4×C17—42no—
61C2×D4×C17—42no—
62C2×D4×C17—44no—
63C2×D4×C17—44no—
64C2×D4×C17—44no—
65C2×D4×C17—44no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C2×D4×C17—42no—
69C2×D4×C17—82no—
70C2×D4×C17—82no—
71C2×D4×C17—82no—
72C22×C4×C17—22no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—42no—
78C22×C4×C17—42no—
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22×C411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×Dic17—42no—
85C4⋊C4×C17—24no—
86C4⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—44no—
90C22⋊C4×C17—44no—
91C42×C17—21no—
92C42×C17—21no—
93C42×C17—42no—
94C22⋊Dic17—24no—
95C22⋊Dic17—42no—
96C4×Dic17—82no—
97#63223—82no—
98C2×#3622—42no—
99#11776—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
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—42no—
113C23×C17—42no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17C2×C22⋊C411yescharacteristic
117C23×C17AGL(1,ℤ/8)12yes—
118D4×C17—44no—
119D4×C17—82no—
120D4×C17—84no—
121D4×C17—88no—
122D4×C17—88no—
123D4×C17—84no—
124D4×C17—82no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—22no—
128C2×C4×C17—24no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17—42no—
135C2×C4×C17—42no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—81no—
140C2×C4×C17—81no—
141C2×C4×C17#9911yescharacteristic
142C2×C4×C17#9911yescharacteristic
143C2×C4×C17#10412yes—
144C2×Dic17—44no—
145C2×Dic17—82no—
146C2×Dic17—82no—
147#3622—82no—
148#2541D1711yescharacteristic
149C4×#99—171no—
150#1465—171no—
151#1343—174no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—22no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—81no—
163C22×C17—81no—
164C22×C17—82no—
165C22×C17—82no—
166C22×C17—82no—
167C22×C17—82no—
168C22×C17—82no—
169C22×C17—82no—
170C22×C17—82no—
171C22×C17C2×#9911yescharacteristic
172C22×C17#41811yescharacteristic
173C22×C17C2×#10411yescharacteristic
174C4×C17—21no—
175C4×C17—22no—
176C4×C17—44no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17#35312yes—
181Dic17—84no—
182C2×#127—21no—
183C2×#127Dic1711yescharacteristic, complemented
184C2×#121—21no—
185C2×#121—21no—
186C2×C4×D4C2×D1711yescharacteristic
187C2×#99—341no—
188#407—341no—
189#392Dic1712yescomplemented
190#392C2×D1712yes—
191#390—21no—
192C4×C4⋊C4Dic1711yescharacteristic, complemented
193C4×C22⋊C4—341no—
194#342—172no—
195#324—341no—
196#323—172no—
197C2×C17—21no—
198C2×C17—21no—
199C2×C17—42no—
200C2×C17—81no—
201C2×C17—162no—
202C2×C17#150711yescharacteristic
203C2×C17#212211yescharacteristic
204C2×C17C2×#35311yescharacteristic
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C2×Dic1712yes—
210C22×D4C22⋊D1711yescharacteristic
211C23×C4C22⋊D1711yescharacteristic
212C23×C4C22⋊D1711yescharacteristic
213C2×M16—681no—
214#127—41no—
215#127—41no—
216#127—44no—
217#121—44no—
218#121—44no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C22⋊D1711yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—342no—
226C2×C22⋊C4C2×Dic1711yescharacteristic
227C2×C22⋊C4C22×D1711yescharacteristic
228C2×C42—21no—
229C2×C42—681no—
230C2×C42C2×Dic1711yescharacteristic
231#99—344no—
232C22⋊C8—342no—
233C22⋊C8—681no—
234#95—681no—
235#95C2×Dic1712yes—
236C17#1177611yescharacteristic, complemented, Sylow
237C24—22no—
238C24C22⋊Dic1711yescharacteristic
239C24C2×C22⋊D1711yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—682no—
266C22×C4C22⋊Dic1711yescharacteristic
267C22×C4C22⋊Dic1711yescharacteristic
268C22×C4C22⋊Dic1711yescharacteristic
269C22×C4C22×Dic1711yescharacteristic
270C22×C4C2×C22⋊D1711yescharacteristic
271M16—1362no—
272C2×C8—682no—
273C4⋊C4—24no—
274C4⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—344no—
280C22⋊C4—682no—
281C42—21no—
282C42—21no—
283C42—42no—
284C42—1362no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23#1058761812yes—
302C23C2×C22⋊Dic1711yescharacteristic
303D4—44no—
304D4—84no—
305D4—82no—
306D4—88no—
307D4—88no—
308D4—84no—
309D4—82no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—684no—
327C2×C4—1362no—
328C2×C4—1362no—
329C2×C4#1058750511yescharacteristic
330C2×C4#1058750511yescharacteristic
331C2×C4#1058750812yes—
332C8—1362no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22C2×#1058750511yescharacteristic
353C22#1176281411yescharacteristic
354C22C2×#1058750811yescharacteristic
355C4—22no—
356C4—21no—
357C4—44no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—1364no—
362C4#1176260212yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—162no—
368C2#1452730011yescharacteristic
369C2#1453043411yescharacteristic
370C2C2×#1176260211yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
Fitting, prime radicalC17×#2541C2
socleC22×C17C2×#99
cosocleC2×C22⋊C4×C17C23
hypercenter#2541D17
hypocenterC17#11776
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C4×#10587505C2semidirect
2. #14527122C2semidirect
3. #14526781C2semidirect
12. C2×C17×#127C4semidirect
16. C17×#392C4semidirect
19. C4×C4⋊C4×C17C4semidirect
183. C2×#127Dic17semidirect
189. #392Dic17semidirect
192. C4×C4⋊C4Dic17semidirect
236. C17#11776semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271222
#145267812
C17×#25412
#1177617