Finite Group Databasebeta

← #17009172

Subgroups — #17009172

Order 4352.

998 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 191 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#14542594C211yesmaximal, characteristic, complemented
2#14539725C212yesmaximal
3#14527723C218yesmaximal, complemented
4#14527282C212yesmaximal, complemented
5#14527108C211yesmaximal, characteristic, complemented
6C17×#2548C211yesmaximal, characteristic
7Q8×Dic34—22no—
8#11763538—22no—
9#11763535—24no—
10#11763440—21no—
11#11763440C2214yes—
12#11763366—22no—
13#11763366C2212yes—
14#11763287—21no—
15#11763181—22no—
16#11763155—22no—
17#11763006—24no—
18#11762960C2218yes—
19#11762862C2212yes—
20#11762545C2218yes—
21#11762535C2214yes—
22C2×C17×#127C2211yescharacteristic
23C2×C17×#124—21no—
24C2×C4×Q8×C17C2211yescharacteristic
25C17×#394C2214yes—
26C17×#390—21no—
27C17×#390—22no—
28C4×C4⋊C4×C17C2211yescharacteristic
29C22×Q8×C17D411yescharacteristic
30C22×D4×C17—21no—
31C22×D4×C17—21no—
32C22×D4×C17C2312yes—
33C17×#127—22no—
34C17×#127—22no—
35C17×#124—44no—
36C4×Q8×C17—22no—
37C4×Q8×C17—22no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17D411yescharacteristic
40C2×C4⋊C4×C17C2312yes—
41C2×C22⋊C4×C17—24no—
42C2×C22⋊C4×C17—24no—
43C2×C42×C17—21no—
44C2×C42×C17D412yes—
45C2×C42×C17C2311yescharacteristic
46Q8×Dic17—24no—
47Q8×Dic17—24no—
48C2×#63229—28no—
49#10587612—24no—
50#10587612—22no—
51#10587610—22no—
52#10587610—24no—
53#10587610C2314yes—
54#10587576—24no—
55#10587576—28no—
56#10587574—24no—
57#10587574—28no—
58#10587556—24no—
59#10587556—22no—
60#10587555—22no—
61#10587555—22no—
62#10587555D412yes—
63#10587555D412yes—
64C4×Dic34—24no—
65C4×Dic34—24no—
66C4×Dic34—22no—
67C4×Dic34D414yes—
68C17×#95C2312yes—
69#10587503—28no—
70C22⋊#3622C2314yes—
71C4⋊#3622—22no—
72C4×#3622—22no—
73C24×C17—22no—
74C2×Q8×C17—22no—
75C2×Q8×C17—22no—
76C2×Q8×C17SD1614yescomplemented
77C2×D4×C17—24no—
78C2×D4×C17—44no—
79C2×D4×C17—41no—
80C2×D4×C17—42no—
81C2×D4×C17—41no—
82C2×D4×C17—42no—
83C2×D4×C17—42no—
84C2×D4×C17—82no—
85C22×C4×C17—22no—
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D412yes—
88C22×C4×C17C2×D412yes—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2411yescharacteristic
91C2×Dic34—22no—
92C2×Dic34—24no—
93C2×Dic34C2×D414yes—
94C4⋊C4×C17—22no—
95C4⋊C4×C17—24no—
96C4⋊C4×C17—42no—
97C22⋊C4×C17—48no—
98C22⋊C4×C17—84no—
99C42×C17—22no—
100C42×C17—22no—
101C42×C17—42no—
102C42×C17C2×D412yes—
103C42×C17C2×D412yes—
104C22⋊Dic17—24no—
105#63229—88no—
106C4⋊Dic17—24no—
107C4⋊Dic17—24no—
108C4⋊Dic17—24no—
109C4⋊Dic17—22no—
110C4⋊Dic17—22no—
111C4⋊Dic17—22no—
112C4⋊Dic17C2×D414yes—
113#63225—28no—
114#63225—24no—
115#63225—24no—
116C4×Dic17—28no—
117C4×Dic17—24no—
118C2×#3622—24no—
119#18403—171nomaximal, Sylow
120C23×C17—44no—
121C23×C17—44no—
122C23×C17—42no—
123C23×C17—42no—
124C23×C17—42no—
125C23×C17C22×D411yescharacteristic
126Q8×C17—28no—
127D4×C17—48no—
128D4×C17—82no—
129D4×C17—84no—
130D4×C17—82no—
131C2×C4×C17—22no—
132C2×C4×C17—22no—
133C2×C4×C17—22no—
134C2×C4×C17—22no—
135C2×C4×C17—41no—
136C2×C4×C17—42no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—42no—
140C2×C4×C17#12011yescharacteristic
141C2×C4×C17#12012yes—
142C2×C4×C17#12011yescharacteristic
143C2×C4×C17C2×SD1614yes—
144C2×C4×C17C2×SD1612yes—
145C2×C4×C17#13712yes—
146C2×C4×C17C22×D412yes—
147C2×Dic17—24no—
148C2×Dic17—24no—
149Dic34—44no—
150Dic34—44no—
151Dic34—84no—
152#3622—44no—
153#3502—171no—
154#3208—172no—
155#2548D1711yescharacteristic
156#1599—178no—
157#1503—172no—
158#1461—171no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—84no—
162C22×C17—84no—
163C22×C17—82no—
164C22×C17—82no—
165C22×C17—82no—
166C22×C17C2×#12011yescharacteristic
167C22×C17C22×SD1611yescharacteristic
168C22×C17C2×#13711yescharacteristic
169C4×C17—24no—
170C4×C17—21no—
171C4×C17—22no—
172C4×C17—42no—
173C4×C17—82no—
174C4×C17#44814yes—
175C4×C17#45412yes—
176Dic17—44no—
177Dic17—44no—
178Q82—342no—
179#557—344no—
180#557—342no—
181#544—174no—
182#544—341no—
183#533—172no—
184#533—342no—
185C2×#127C2×D1711yescharacteristic
186C2×#124—21no—
187#519—341no—
188C2×C4×Q8C2×D1711yescharacteristic
189#492—342no—
190#486—342no—
191#460—344no—
192#450—178no—
193#431—172no—
194#394C2×D1714yes—
195#390—21no—
196#390—22no—
197C4×C4⋊C4C2×D1711yescharacteristic
198#328—178no—
199#324—174no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—162no—
203C2×C17C2×#44811yescharacteristic
204C2×C17C2×#45411yescharacteristic
205C2×C17#302711yescharacteristic
206C22×Q8C22⋊D1711yescharacteristic
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C22×D1712yes—
210C2×SD16—348no—
211C4⋊Q8—172no—
212C4⋊Q8—172no—
213C4⋊Q8—342no—
214C4⋊Q8—342no—
215C4⋊Q8—348no—
216C4⋊Q8—344no—
217#127—22no—
218#127—22no—
219#125—344no—
220#125—348no—
221#125—344no—
222#125—342no—
223#124—44no—
224C22⋊Q8—174no—
225C22⋊Q8—342no—
226C22⋊Q8—344no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—174no—
230C4×Q8—344no—
231C4×Q8—344no—
232C4×Q8—344no—
233C4×Q8—344no—
234C4×Q8—342no—
235#117—342no—
236#117—344no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C22⋊D1711yescharacteristic
239C2×C4⋊C4C22×D1712yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C42—21no—
243C2×C42C22⋊D1712yes—
244C2×C42C22×D1711yescharacteristic
245C4⋊C8—342no—
246#102—348no—
247C22⋊C8—174no—
248C4×C8—342no—
249#95C22×D1712yes—
250C17#1840311yescharacteristic, complemented, Sylow
251C24—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—174no—
255C2×Q8—344no—
256C2×Q8—342no—
257C2×Q8#6322914yescomplemented
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—82no—
266C22×C4—22no—
267C22×C4C2×C22⋊D1712yes—
268C22×C4C2×C22⋊D1711yescharacteristic
269C22×C4C2×C22⋊D1712yes—
270C22×C4C2×C22⋊D1711yescharacteristic
271C22×C4C23×D1711yescharacteristic
272SD16—1368no—
273C2×C8—344no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—174no—
278C4⋊C4—344no—
279C4⋊C4—344no—
280C4⋊C4—344no—
281C4⋊C4—348no—
282C4⋊C4—344no—
283C4⋊C4—342no—
284C4⋊C4—342no—
285C4⋊C4—344no—
286C4⋊C4—342no—
287C22⋊C4—48no—
288C22⋊C4—84no—
289C22⋊C4—344no—
290C42—22no—
291C42—22no—
292C42—42no—
293C42—348no—
294C42—344no—
295C42D4×D1712yes—
296C42D4×D1712yes—
297C23—44no—
298C23—44no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23C22×C22⋊D1711yescharacteristic
303Q8—28no—
304Q8—684no—
305Q8—684no—
306Q8—1364no—
307D4—48no—
308D4—82no—
309D4—84no—
310D4—82no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—344no—
321C2×C4—344no—
322C2×C4C2×#6322914yes—
323C2×C4C2×#6322912yes—
324C2×C4#1058762311yescharacteristic
325C2×C4#1058762312yes—
326C2×C4#1058762311yescharacteristic
327C2×C4#1058763712yes—
328C2×C4C2×D4×D1712yes—
329C8—684no—
330C22—21no—
331C22—21no—
332C22—84no—
333C22—84no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22C2×#1058762311yescharacteristic
338C22C22×#6322911yescharacteristic
339C22C2×#1058763711yescharacteristic
340C4—21no—
341C4—22no—
342C4—24no—
343C4—42no—
344C4—82no—
345C4—684no—
346C4—684no—
347C4#1176294714yes—
348C4#1176298112yes—
349C2—21no—
350C2—21no—
351C2—162no—
352C2C2×#1176294711yescharacteristic
353C2C2×#1176298111yescharacteristic
354C2#1453818511yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2548C2
socleC22×C17C2×#120
prime radicalC2×C17×#127C22
hypercenter#2548D17
hypocenterC17#18403
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. #14542594C2semidirect
3. #14527723C2semidirect
4. #14527282C2semidirect
5. #14527108C2semidirect
76. C2×Q8×C17SD16semidirect
250. C17#18403semidirect
257. C2×Q8#63229semidirect

Maximal subgroups

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

subgroupindex
#145425942
#145397252
#145277232
#145272822
#145271082
C17×#25482
#1840317