Finite Group Databasebeta

← #17953087

Subgroups — #17953087

Order 4352.

910 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544679C212yesmaximal
2#14544679C212yesmaximal
3#14544679C212yesmaximal
4C22×#10587584C211yesmaximal, characteristic
5#11763750C2214yes—
6#11763687—24no—
7#11763687—24no—
8#11763687—24no—
9C2×#10587584—24no—
10C2×#10587584—24no—
11C2×#10587584—24no—
12C22×C4⋊Dic17C411yescharacteristic, complemented
13C22×#63225C411yescharacteristic, complemented
14C22×#63227C2211yescharacteristic
15C22×#63227C2211yescharacteristic
16C22×#63227C2211yescharacteristic
17C22×C4×Dic17C411yescharacteristic, complemented
18C22×C4⋊C4×C17C411yescharacteristic, complemented
19C24×D17C2311yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17C2×C411yescharacteristic
23C23×Dic17C2×C411yescharacteristic
24C23×Dic17C2×C411yescharacteristic
25C23×Dic17C2×C411yescharacteristic
26C2×C22⋊#595—24no—
27C2×C22⋊#595—24no—
28C2×C22⋊#595—24no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—22no—
32C2×#63227—22no—
33C2×#63227—22no—
34C2×#63227—22no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227—22no—
38C2×#63227D414yes—
39C2×#63227D414yes—
40C2×#63227D414yes—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17—22no—
43C2×C4⋊Dic17—22no—
44C2×#63225—22no—
45C2×#63225—22no—
46C2×#63225—22no—
47C2×C4×Dic17—22no—
48C2×C4×Dic17—22no—
49C2×C4×Dic17—22no—
50#10587584—232no—
51C24×C17C22×C411yescharacteristic
52C23×D17—42no—
53C23×D17—42no—
54C23×D17—42no—
55C23×D17C2×D412yes—
56C23×D17C2×D412yes—
57C23×D17C2×D412yes—
58C22×#595—44no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17C22⋊C412yes—
69C22×C4×C17C22⋊C412yes—
70C22×C4×C17C22⋊C412yes—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—42no—
78C22×Dic17—42no—
79C22×Dic17—42no—
80C22×Dic17C22⋊C412yes—
81C22×Dic17C22⋊C412yes—
82C22×Dic17C22⋊C412yes—
83C22⋊#595—48no—
84C22⋊#595—48no—
85C22⋊#595—48no—
86C4⋊C4×C17—28no—
87#63227—28no—
88#63227—28no—
89#63227—28no—
90C4⋊Dic17—28no—
91#63225—28no—
92C4×Dic17—28no—
93#8545—171nomaximal, Sylow
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—22no—
97C23×C17C2×C22⋊C411yescharacteristic
98C23×C17C2×C22⋊C411yescharacteristic
99C23×C17C2×C22⋊C411yescharacteristic
100C22×D17—24no—
101C22×D17—24no—
102C22×D17—24no—
103C22×D17—42no—
104C22×D17—42no—
105C22×D17—42no—
106C22×D17—42no—
107C22×D17—42no—
108C22×D17—42no—
109C22×D17—88no—
110C22×D17#12014yes—
111C2×#595—84no—
112C2×#595—84no—
113C2×#595—84no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C22×#126—171no—
139#1819—172no—
140#1819—172no—
141#1819—172no—
142C22×C17—28no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17#37911yescharacteristic
150C22×C17#41012yes—
151C22×C17#41012yes—
152C22×C17#41012yes—
153C2×D17—24no—
154C2×D17—24no—
155C2×D17—24no—
156C2×D17—84no—
157C2×D17—84no—
158C2×D17—84no—
159#595—164no—
160C4×C17—42no—
161C4×C17—42no—
162C4×C17—42no—
163Dic17—42no—
164Dic17—42no—
165Dic17—42no—
166C2×#126—344no—
167C2×#126—344no—
168C2×#126—344no—
169C22×C4⋊C4—171no—
170C22×C4⋊C4—171no—
171C22×C4⋊C4#59511yescharacteristic, complemented
172C22×C22⋊C4—171no—
173C22×C22⋊C4—171no—
174C22×C22⋊C4—171no—
175C22×C42—171no—
176#379—174no—
177#342—344no—
178#342—344no—
179#342—344no—
180C2×C17—22no—
181C2×C17—22no—
182C2×C17—22no—
183C2×C17#178211yescharacteristic
184C2×C17#178211yescharacteristic
185C2×C17#178211yescharacteristic
186D17—44no—
187C25—171no—
188C23×C4—171no—
189C23×C4—171no—
190C23×C4—171no—
191C23×C4C2×#59511yescharacteristic
192C23×C4C2×#59511yescharacteristic
193C23×C4C2×#59511yescharacteristic
194#126—3432no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—342no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—342no—
201C2×C4⋊C4—342no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—342no—
204C2×C22⋊C4—174no—
205C2×C22⋊C4—174no—
206C2×C22⋊C4—174no—
207C2×C22⋊C4—342no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4—342no—
210C2×C22⋊C4—342no—
211C2×C22⋊C4—342no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4—344no—
214C2×C22⋊C4—344no—
215C2×C22⋊C4—344no—
216C2×C42—342no—
217C2×C42—342no—
218C2×C42—342no—
219C17#854511yescharacteristic, complemented, Sylow
220C24—172no—
221C24—172no—
222C24—172no—
223C24—682no—
224C24—682no—
225C24—682no—
226C24C22×#59511yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—172no—
237C22×C4—172no—
238C22×C4—172no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—682no—
246C22×C4—682no—
247C22×C4—682no—
248C22×C4—684no—
249C22×C4C22⋊#59512yes—
250C22×C4C22⋊#59512yes—
251C22×C4C22⋊#59512yes—
252C4⋊C4—28no—
253C4⋊C4—348no—
254C4⋊C4—348no—
255C22⋊C4—348no—
256C22⋊C4—348no—
257C22⋊C4—348no—
258C22⋊C4—688no—
259C22⋊C4—688no—
260C22⋊C4—688no—
261C42—348no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—174no—
266C23—344no—
267C23—344no—
268C23—344no—
269C23—682no—
270C23—682no—
271C23—682no—
272C23—682no—
273C23—682no—
274C23—682no—
275C23—1368no—
276C23C2×C22⋊#59511yescharacteristic
277C23C2×C22⋊#59511yescharacteristic
278C23C2×C22⋊#59511yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—342no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4—682no—
303C2×C4—1364no—
304C2×C4—1364no—
305C2×C4—1364no—
306C22—28no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—344no—
314C22—344no—
315C22—344no—
316C22—1364no—
317C22—1364no—
318C22—1364no—
319C22#1176375011yescharacteristic
320C22#1176378812yes—
321C22#1176378912yes—
322C22#1176379012yes—
323C4—42no—
324C4—42no—
325C4—42no—
326C4—2724no—
327C4—682no—
328C4—682no—
329C4—682no—
330C2—22no—
331C2—22no—
332C2—22no—
333C2—684no—
334C2#1454461311yescharacteristic
335C2#1454461411yescharacteristic
336C2#1454461511yescharacteristic
subgroupG/H
centerC22#11763750
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC22×C17#379
prime radical, cosocleC24×D17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8545
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. C22×C4⋊Dic17C4semidirect
13. C22×#63225C4semidirect
17. C22×C4×Dic17C4semidirect
18. C22×C4⋊C4×C17C4semidirect
171. C22×C4⋊C4#595semidirect
219. C17#8545semidirect

Maximal subgroups

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

subgroupindex
#145446792
C22×#105875842
#854517