Finite Group Databasebeta

← #16926936

Subgroups — #16926936

Order 4352.

1222 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#10587622C211yesmaximal, characteristic
2#14528856C214yesmaximal, complemented
3#14528699C211yesmaximal, characteristic
4C17×#1781C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic, complemented
6C2×#10587622—28no—
7C2×#10587622—22no—
8C2×#10587622—22no—
9C22×C22⋊Dic17C412yescomplemented
10C22×C22⋊Dic17C2212yes—
11C22×C4×Dic17C411yescharacteristic, complemented
12C2×#10587505—22no—
13#11762665C2212yes—
14#11762575—28no—
15C23×D4×C17C2211yescharacteristic
16C2×C17×#99—22no—
17C17×#379C2212yes—
18C25×C17C2×C411yescharacteristic
19C25×C17C2311yescharacteristic
20C22×D4×C17—24no—
21C22×D4×C17—44no—
22C22×D4×C17D412yes—
23C22×D4×C17D412yes—
24C23×C4×C17C2×C411yescharacteristic
25C23×Dic17C2×C412yes—
26C23×Dic17C2×C412yes—
27C22×Dic34—22no—
28C22×Dic34—24no—
29C22×Dic34—44no—
30C2×C22⋊C4×C17—24no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊Dic17—24no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—24no—
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17—24no—
37C2×C22⋊Dic17—24no—
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17—24no—
40C2×C22⋊Dic17D418yes—
41#10587622—216no—
42#10587622—216no—
43C2×C4×Dic17—22no—
44C2×C4×Dic17—24no—
45C17×#99—44no—
46#10587505—44no—
47C24×C17—22no—
48C24×C17—42no—
49C24×C17—42no—
50C24×C17—44no—
51C24×C17—44no—
52C24×C17C22⋊C412yes—
53C24×C17C22×C411yescharacteristic
54C24×C17C2×D411yescharacteristic
55C24×C17C2×D414yes—
56C24×C17C2×D411yescharacteristic
57C2×D4×C17—24no—
58C2×D4×C17—24no—
59C2×D4×C17—24no—
60C2×D4×C17—24no—
61C2×D4×C17—42no—
62C2×D4×C17—42no—
63C2×D4×C17—416no—
64C22×C4×C17—22no—
65C22×C4×C17—24no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C412yes—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—22no—
72C22×Dic17—44no—
73C22×Dic17—44no—
74C22×Dic17—42no—
75C22×Dic17C22⋊C414yes—
76C22×Dic17C22⋊C414yes—
77C2×Dic34—28no—
78C2×Dic34—416no—
79C2×Dic34—44no—
80C2×Dic34—44no—
81C22⋊C4×C17—44no—
82C22⋊C4×C17—48no—
83C22⋊Dic17—216no—
84C22⋊Dic17—28no—
85C22⋊Dic17—28no—
86C22⋊Dic17—44no—
87C22⋊Dic17—48no—
88C4×Dic17—28no—
89#8544—171nomaximal, Sylow
90C23×C17—22no—
91C23×C17—24no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—28no—
95C23×C17—22no—
96C23×C17—22no—
97C23×C17—22no—
98C23×C17—44no—
99C23×C17—42no—
100C23×C17—44no—
101C23×C17—42no—
102C23×C17—44no—
103C23×C17—44no—
104C23×C17—44no—
105C23×C17—44no—
106C23×C17—88no—
107C23×C17—88no—
108C23×C17C2×C22⋊C412yes—
109C23×C17C2×C22⋊C411yescharacteristic
110C23×C17#12012yes—
111C23×C17#12012yes—
112D4×C17—48no—
113D4×C17—48no—
114C2×C4×C17—28no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—82no—
118C2×C4×C17—84no—
119C2×C4×C17#9914yes—
120C2×Dic17—24no—
121C2×Dic17—24no—
122C2×Dic17—44no—
123C2×Dic17—44no—
124C2×Dic17—44no—
125C2×Dic17—44no—
126C2×Dic17—44no—
127C2×Dic17—44no—
128C2×Dic17—82no—
129C2×Dic17—84no—
130Dic34—416no—
131C22×#124—171no—
132#1819—174no—
133#1781—171no—
134#1781D1711yescharacteristic
135C22×C17—28no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—24no—
142C22×C17—24no—
143C22×C17—42no—
144C22×C17—44no—
145C22×C17—82no—
146C22×C17—82no—
147C22×C17—88no—
148C22×C17—82no—
149C22×C17—82no—
150C22×C17—88no—
151C22×C17#37911yescharacteristic
152C22×C17C2×#9912yes—
153C22×C17#41014yes—
154C4×C17—24no—
155C4×C17—162no—
156Dic17—44no—
157Dic17—44no—
158Dic17—162no—
159C23×Q8—171no—
160C23×D4C2×D1711yescharacteristic
161C2×#124—348no—
162C2×#124—342no—
163C2×#124—342no—
164C22×C22⋊C4—172no—
165C22×C22⋊C4—172no—
166C22×C42—171no—
167C2×#99—22no—
168C2×#99—342no—
169#379—172no—
170#379C2×D1712yes—
171#342—348no—
172C2×C17—22no—
173C2×C17—24no—
174C2×C17—42no—
175C2×C17—42no—
176C2×C17—82no—
177C2×C17#178111yescharacteristic
178C2×C17#178212yes—
179C25C4×D1711yescharacteristic
180C25C22×D1711yescharacteristic
181C22×Q8—344no—
182C22×Q8—342no—
183C22×Q8—684no—
184C22×D4—24no—
185C22×D4—44no—
186C22×D4D6812yes—
187C22×D4C22⋊D1712yes—
188C23×C4—172no—
189C23×C4—172no—
190C23×C4C4×D1711yescharacteristic
191#124—3416no—
192#124—3416no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—178no—
196C2×C22⋊C4—344no—
197C2×C22⋊C4—344no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4—344no—
200C2×C22⋊C4—342no—
201C2×C22⋊C4—344no—
202C2×C22⋊C4—342no—
203C2×C22⋊C4—344no—
204C2×C42—344no—
205C2×C42—342no—
206#99—44no—
207#99—684no—
208C17#854411yescharacteristic, complemented, Sylow
209C24—22no—
210C24—42no—
211C24—42no—
212C24—44no—
213C24—44no—
214C24#6322712yes—
215C24C2×C4×D1711yescharacteristic
216C24C2×D6811yescharacteristic
217C24D4×D1714yes—
218C24C2×C22⋊D1711yescharacteristic
219C2×Q8—348no—
220C2×Q8—6816no—
221C2×Q8—684no—
222C2×Q8—684no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—416no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—174no—
234C22×C4—174no—
235C22×C4—342no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4—342no—
239C22×C4—684no—
240C22×C4—684no—
241C22×C4—682no—
242C22×C4#6322712yes—
243C22⋊C4—44no—
244C22⋊C4—48no—
245C22⋊C4—3416no—
246C22⋊C4—348no—
247C22⋊C4—348no—
248C22⋊C4—684no—
249C22⋊C4—688no—
250C42—348no—
251C23—22no—
252C23—24no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—44no—
260C23—42no—
261C23—44no—
262C23—42no—
263C23—44no—
264C23—44no—
265C23—44no—
266C23—44no—
267C23—88no—
268C23—88no—
269C23C22⋊C4×D1712yes—
270C23#1058756812yes—
271C23C2×#6322711yescharacteristic
272C23#1058762312yes—
273Q8—6816no—
274D4—48no—
275D4—48no—
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—82no—
280C2×C4—84no—
281C2×C4—344no—
282C2×C4—344no—
283C2×C4—684no—
284C2×C4—684no—
285C2×C4—684no—
286C2×C4—684no—
287C2×C4—684no—
288C2×C4—684no—
289C2×C4—1362no—
290C2×C4—1364no—
291C2×C4#1058747714yes—
292C22—28no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—24no—
299C22—24no—
300C22—42no—
301C22—44no—
302C22—82no—
303C22—82no—
304C22—88no—
305C22—82no—
306C22—82no—
307C22—88no—
308C22#1176266411yescharacteristic
309C22C2×#1058747712yes—
310C22#1176278114yes—
311C4—24no—
312C4—162no—
313C4—2722no—
314C4—684no—
315C4—684no—
316C2—22no—
317C2—24no—
318C2—42no—
319C2—42no—
320C2—82no—
321C2#1452869811yescharacteristic
322C2#1452870012yes—
subgroupG/H
centerC22#11762664
commutatorC24×C17C22×C4
FrattiniC25C22×D17
FittingC17×#1781C2
socleC22×C17#379
prime radicalC23×D4×C17C22
cosocleC25×C17C23
hypercenter#1781D17
hypocenterC17#8544
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #14528856C2semidirect
5. C23×Dic34C4semidirect
9. C22×C22⋊Dic17C4semidirect
11. C22×C4×Dic17C4semidirect
208. C17#8544semidirect

Maximal subgroups

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

subgroupindex
C22×#105876222
#145288562
#145286992
C17×#17812
#854417