Finite Group Databasebeta

← #17043776

Subgroups — #17043776

Order 4352.

1130 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C2×#11763435C211yesmaximal, characteristic
2#14531380C212yesmaximal
3#14529767C218yesmaximal, complemented
4#14528858C211yesmaximal, characteristic
5#14528856C212yesmaximal
6C17×#1819C211yesmaximal, characteristic
7#11763435—24no—
8#11763435—24no—
9C2×#10587610C414yescomplemented
10C2×#10587564C414yescomplemented
11C2×#10587564C2214yes—
12C22×C22⋊Dic17C2211yescharacteristic
13C22×C22⋊Dic17C2211yescharacteristic
14#11762703—24no—
15#11762697—24no—
16#11762667C2214yes—
17#11762665C2212yes—
18C22⋊C4×Dic17C2214yes—
19#11762576C2214yes—
20#11762575C2218yes—
21C22×C22⋊C4×C17C2211yescharacteristic
22C17×#379C2212yes—
23C17×#342C2214yes—
24C25×C17C2311yescharacteristic
25C23×C4×C17C2×C412yes—
26C23×Dic17C2×C412yes—
27C23×Dic17C2312yes—
28C22×Dic34C2×C414yes—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17C2×C412yes—
34C2×C22⋊C4×C17C2314yes—
35C2×C22⋊C4×C17C2312yes—
36C2×C22⋊Dic17—22no—
37C2×C22⋊Dic17—22no—
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17—22no—
40C2×C22⋊Dic17C2×C412yes—
41C2×C22⋊Dic17C2×C412yes—
42C2×C22⋊Dic17C2×C412yes—
43C2×C22⋊Dic17D414yes—
44C2×C22⋊Dic17D414yes—
45C2×C22⋊Dic17C2314yes—
46C2×C22⋊Dic17C2312yes—
47C2×C4⋊Dic17C2×C414yes—
48#10587610—28no—
49#10587610—28no—
50C2×#63225—24no—
51C2×#63225C2×C418yes—
52C2×C4×Dic17—24no—
53#10587564—28no—
54#10587564—28no—
55#10587564—28no—
56#10587564—28no—
57#10587502—24no—
58#10587502—24no—
59C24×C17—24no—
60C24×C17—82no—
61C24×C17C22×C411yescharacteristic
62C24×C17C22×C411yescharacteristic
63C24×C17C2×D412yes—
64C24×C17C2×D412yes—
65C24×C17C2411yescharacteristic
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—24no—
69C22×C4×C17—82no—
70C22×C4×C17C22×C414yes—
71C22×Dic17—22no—
72C22×Dic17—22no—
73C22×Dic17—24no—
74C22×Dic17—82no—
75C22×Dic17—82no—
76C22×Dic17C22×C414yes—
77C22×Dic17C22×C414yes—
78C22×Dic17C2×D414yes—
79C22×Dic17C2×D414yes—
80C2×Dic34—28no—
81C2×Dic34—84no—
82C22⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—24no—
85C22⋊C4×C17—28no—
86C22⋊C4×C17—82no—
87C22⋊C4×C17—82no—
88C22⋊Dic17—24no—
89C22⋊Dic17—24no—
90C22⋊Dic17—28no—
91C22⋊Dic17—82no—
92C22⋊Dic17—82no—
93C22⋊Dic17C22⋊C414yes—
94C22⋊Dic17C22⋊C414yes—
95C22⋊Dic17C22⋊C418yes—
96C4⋊Dic17—28no—
97#63225—216no—
98#63225—84no—
99C4×Dic17—84no—
100#21827—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—22no—
103C23×C17—24no—
104C23×C17—24no—
105C23×C17—42no—
106C23×C17—42no—
107C23×C17—42no—
108C23×C17—42no—
109C23×C17—42no—
110C23×C17—42no—
111C23×C17—42no—
112C23×C17—88no—
113C23×C17C2×C22⋊C412yes—
114C23×C17C2×C22⋊C411yescharacteristic
115C23×C17C2×C22⋊C411yescharacteristic
116C23×C17C23×C411yescharacteristic
117C23×C17C22×D412yes—
118C23×C172−1+414yes—
119C2×C4×C17—24no—
120C2×C4×C17—44no—
121C2×C4×C17—44no—
122C2×C4×C17—44no—
123C2×C4×C17—44no—
124C2×C4×C17—82no—
125C2×C4×C17—82no—
126C2×Dic17—24no—
127C2×Dic17—44no—
128C2×Dic17—44no—
129C2×Dic17—44no—
130C2×Dic17—44no—
131C2×Dic17—44no—
132C2×Dic17—44no—
133C2×Dic17—44no—
134C2×Dic17—82no—
135C2×Dic17—82no—
136C2×Dic17C2×C22⋊C418yes—
137Dic34—88no—
138C2×#543—171no—
139#2312—172no—
140#1996—178no—
141#1819—172no—
142#1819—171no—
143#1819D1711yescharacteristic
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—22no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—22no—
153C22×C17—22no—
154C22×C17—44no—
155C22×C17—44no—
156C22×C17—44no—
157C22×C17—84no—
158C22×C17—84no—
159C22×C17C22×C22⋊C411yescharacteristic
160C22×C17#51912yes—
161C22×C17#53614yes—
162C4×C17—44no—
163C4×C17—84no—
164Dic17—28no—
165Dic17—44no—
166Dic17—84no—
167#543—344no—
168#543—344no—
169C2×C22⋊Q8—174no—
170C2×C22⋊Q8—174no—
171C2×C22⋊Q8—174no—
172C22×C22⋊C4—171no—
173C22×C22⋊C4—171no—
174C22×C22⋊C4C2×D1711yescharacteristic
175#388—344no—
176#388—344no—
177#380—174no—
178#379—172no—
179#379C2×D1712yes—
180C4×C22⋊C4—174no—
181#342—178no—
182#342—174no—
183#342C2×D1714yes—
184C2×C17—24no—
185C2×C17—22no—
186C2×C17—41no—
187C2×C17—41no—
188C2×C17—42no—
189C2×C17#231011yescharacteristic
190C2×C17#288312yes—
191C25C22×D1711yescharacteristic
192C22×Q8—174no—
193C23×C4—172no—
194C23×C4—172no—
195C23×C4C4×D1712yes—
196C22⋊Q8—348no—
197C22⋊Q8—348no—
198C22⋊Q8—348no—
199C22⋊Q8—348no—
200C22⋊Q8—348no—
201C22⋊Q8—348no—
202C2×C4⋊C4—178no—
203C2×C4⋊C4—174no—
204C2×C4⋊C4—344no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—172no—
210C2×C22⋊C4—172no—
211C2×C22⋊C4—174no—
212C2×C22⋊C4—174no—
213C2×C22⋊C4—172no—
214C2×C22⋊C4—172no—
215C2×C22⋊C4—174no—
216C2×C22⋊C4—342no—
217C2×C22⋊C4—342no—
218C2×C22⋊C4—342no—
219C2×C22⋊C4—342no—
220C2×C22⋊C4C4×D1712yes—
221C2×C22⋊C4C22×D1714yes—
222C2×C22⋊C4C22×D1712yes—
223C2×C42—344no—
224#95—344no—
225#95—344no—
226C17#2182711yescharacteristic, complemented, Sylow
227C24—24no—
228C24—82no—
229C24C2×C4×D1711yescharacteristic
230C24C2×C4×D1711yescharacteristic
231C24D4×D1712yes—
232C24D4×D1712yes—
233C24C23×D1711yescharacteristic
234C2×Q8—348no—
235C2×Q8—1364no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—82no—
240C22×C4—174no—
241C22×C4—174no—
242C22×C4—174no—
243C22×C4—174no—
244C22×C4—342no—
245C22×C4—342no—
246C22×C4—344no—
247C22×C4—1362no—
248C22×C4—1362no—
249C22×C4C2×C4×D1714yes—
250C4⋊C4—3416no—
251C4⋊C4—348no—
252C4⋊C4—1364no—
253C22⋊C4—24no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—28no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—174no—
260C22⋊C4—174no—
261C22⋊C4—178no—
262C22⋊C4—344no—
263C22⋊C4—344no—
264C22⋊C4—348no—
265C22⋊C4—1362no—
266C22⋊C4—1362no—
267C42—1364no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—88no—
280C23C22⋊C4×D1711yescharacteristic
281C23C22⋊C4×D1711yescharacteristic
282C23C22⋊C4×D1712yes—
283C23C22×C4×D1711yescharacteristic
284C23C2×D4×D1712yes—
285C23#1058769714yes—
286Q8—1368no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—178no—
295C2×C4—344no—
296C2×C4—684no—
297C2×C4—684no—
298C2×C4—684no—
299C2×C4—684no—
300C2×C4—684no—
301C2×C4—684no—
302C2×C4—684no—
303C2×C4—1362no—
304C2×C4—1362no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—84no—
319C22—84no—
320C22C2×C22⋊C4×D1711yescharacteristic
321C22#1176328612yes—
322C22#1176338114yes—
323C4—44no—
324C4—84no—
325C4—348no—
326C4—684no—
327C4—1364no—
328C2—24no—
329C2—22no—
330C2—42no—
331C2—41no—
332C2—41no—
333C2#1453136911yescharacteristic
334C2#1453724212yes—
subgroupG/H
centerC22C2×C22⋊C4×D17
commutatorC23×C17C23×C4
FrattiniC24C23×D17
FittingC17×#1819C2
socleC22×C17C22×C22⋊C4
prime radicalC25×C17C23
cosocleC24×C17C24
hypercenter#1819D17
hypocenterC17#21827
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #14529767C2semidirect
9. C2×#10587610C4semidirect
10. C2×#10587564C4semidirect
226. C17#21827semidirect

Maximal subgroups

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

subgroupindex
C2×#117634352
#145313802
#145297672
#145288582
#145288562
C17×#18192
#2182717