Finite Group Databasebeta

← #17956701

Subgroups — #17956701

Order 4352.

1678 conjugacy classes of subgroups; only the 244 normal subgroups are shown.

normal subgroups only

This group has 1678 conjugacy classes of subgroups — too many to draw. Only its 244 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×#11763781C212yesmaximal
2C2×#11763689C214yesmaximal, complemented
3C22×C4⋊C4×D17C211yesmaximal, characteristic
4C22×#63247C2214yes—
5#11763781—24no—
6#11763781C418yescomplemented
7C2×#10587543—24no—
8#11763689—28no—
9C23×C4×D17C412yes—
10C23×C4×D17C2211yescharacteristic
11C22×C4⋊Dic17C411yescharacteristic
12C22×#63225C2212yes—
13C2×C4⋊C4×D17—28no—
14C2×C4⋊C4×D17—24no—
15C22×C4⋊C4×C17C411yescharacteristic
16C24×D17C2×C411yescharacteristic
17C23×C4×C17C2×C411yescharacteristic
18C23×C4×C17C2×C412yes—
19C23×Dic17C2×C412yes—
20C23×Dic17C2311yescharacteristic
21C22×C4×D17—24no—
22C22×C4×D17—24no—
23C22×C4×D17—24no—
24C22×C4×D17—24no—
25C22×C4×D17—24no—
26C22×C4×D17—48no—
27C22×C4×D17C2×C414yes—
28C22×C4×D17D412yes—
29C22×C4×D17D414yes—
30C22×C4×D17Q812yes—
31C2×#63247—24no—
32C2×#63247—28no—
33C2×#63247—28no—
34C2×#63247C2×C418yescomplemented
35C22×#3624—24no—
36C2×C4⋊C4×C17—24no—
37C2×C4⋊C4×C17—22no—
38C2×C4⋊Dic17—22no—
39C2×C4⋊Dic17—24no—
40C2×#63225—24no—
41C2×#63225—24no—
42C2×#63225D418yes—
43C4⋊C4×D17—232no—
44#10587543—216no—
45C24×C17C22×C411yescharacteristic
46C23×D17—22no—
47C23×D17—22no—
48C23×D17—24no—
49C23×D17—24no—
50C23×D17C4212yes—
51C23×D17C22⋊C412yes—
52C23×D17C4⋊C412yes—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—44no—
58C22×C4×C17C4212yes—
59C22×C4×C17C22⋊C412yes—
60C22×C4×C17C22⋊C414yes—
61C22×C4×C17C4⋊C412yes—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—44no—
65C22×Dic17C22⋊C414yes—
66C22×Dic17C22×C412yes—
67C22×Dic17C2×D411yescharacteristic
68C22×Dic17C2×D412yes—
69C22×Dic17C2×D414yes—
70C22×Dic17C2×Q811yescharacteristic
71C22×Dic17#4114yes—
72C2×C4×D17—216no—
73C2×C4×D17—28no—
74C2×C4×D17—28no—
75C2×C4×D17—28no—
76C2×C4×D17—28no—
77C2×C4×D17—216no—
78C2×C4×D17—416no—
79C2×C4×D17—48no—
80C2×C4×D17—416no—
81C2×C4×D17—48no—
82C2×C4×D17C22⋊C418yes—
83C2×C4×D17C4⋊C418yes—
84#63247—216no—
85#63247—48no—
86C2×#3624—28no—
87C2×#3624—44no—
88C2×#3624—44no—
89C4⋊C4×C17—28no—
90C4⋊Dic17—28no—
91#63225—216no—
92#10727—171nomaximal, Sylow
93C23×C17—22no—
94C23×C17—22no—
95C23×C17C2×C4211yescharacteristic
96C23×C17C2×C22⋊C412yes—
97C23×C17C2×C22⋊C411yescharacteristic
98C23×C17C2×C4⋊C411yescharacteristic
99C23×C17#11712yes—
100C22×D17—28no—
101C22×D17—216no—
102C22×D17—28no—
103C22×D17—24no—
104C22×D17—22no—
105C22×D17—24no—
106C22×D17—22no—
107C22×D17—22no—
108C22×D17—22no—
109C22×D17—28no—
110C22×D17—28no—
111C22×D17#9514yes—
112C22×D17#10014yes—
113C22×D17#10114yes—
114C2×C4×C17—24no—
115C2×C4×C17—22no—
116C2×C4×C17—24no—
117C2×C4×C17—22no—
118C2×C4×C17—24no—
119C2×C4×C17—44no—
120C2×C4×C17—44no—
121C2×C4×C17—44no—
122C2×C4×C17#9514yes—
123C2×Dic17—24no—
124C2×Dic17—44no—
125C2×Dic17—44no—
126C2×Dic17—44no—
127C2×Dic17C2×C22⋊C412yes—
128C2×Dic17C2×C4⋊C412yes—
129C2×Dic17C4×D418yes—
130C2×Dic17#12014yes—
131C2×Dic17#12114yes—
132C2×Dic17C22⋊Q814yes—
133C2×Dic17#12314yes—
134C4×D17—216no—
135C4×D17—216no—
136C4×D17—432no—
137#3624—48no—
138C23×C4⋊C4—171no—
139C2×#407—172no—
140C2×#343—174no—
141C22×C17—24no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17C2×#9511yescharacteristic
147C22×C17C4×C22⋊C412yes—
148C22×C17#37911yescharacteristic
149C22×C17#38012yes—
150C22×C17#38111yescharacteristic
151C22×C17C2×#10012yes—
152C22×C17C2×#10112yes—
153C2×D17—216no—
154C2×D17—216no—
155C2×D17—24no—
156C2×D17—24no—
157C2×D17—24no—
158C2×D17—24no—
159C2×D17—24no—
160C2×D17—24no—
161C2×D17#34318yescomplemented
162C4×C17—24no—
163C4×C17—44no—
164Dic17—44no—
165Dic17#38514yes—
166Dic17#38614yes—
167C24×C4—171no—
168C24×C4—172no—
169C22×M16—174no—
170C22×C4⋊C4—172no—
171C22×C4⋊C4—171no—
172C22×C4⋊C4—348no—
173C22×C4⋊C4—344no—
174C22×C4⋊C4#59511yescharacteristic
175#407—178no—
176#407—344no—
177C2×C22⋊C8—344no—
178#343—348no—
179C2×C17—22no—
180C2×C17—22no—
181C2×C17C22⋊#9511yescharacteristic
182C2×C17C2×#34312yes—
183C2×C17#178311yescharacteristic
184C2×C17#178411yescharacteristic
185C2×C17#182112yes—
186D17—28no—
187C25—171no—
188C23×C4—171no—
189C23×C4—174no—
190C23×C4—172no—
191C23×C4—172no—
192C23×C4—174no—
193C23×C4—172no—
194C23×C4—344no—
195C23×C4—344no—
196C23×C4—344no—
197C23×C4—344no—
198C23×C4—344no—
199C23×C4—688no—
200C23×C4C2×#59512yes—
201C23×C4C2×#59511yescharacteristic
202C2×M16—178no—
203C2×M16—344no—
204C2×M16—348no—
205C2×M16—348no—
206C22×C8—344no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—178no—
210C2×C4⋊C4—3432no—
211C2×C4⋊C4—344no—
212C2×C4⋊C4—344no—
213C2×C4⋊C4—344no—
214C2×C4⋊C4—342no—
215C22⋊C8—3416no—
216C17#1072711yescharacteristic, complemented, Sylow
217C24—172no—
218C24—172no—
219C24—172no—
220C24—344no—
221C24—344no—
222C24—342no—
223C24—342no—
224C24C22×#59511yescharacteristic
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—172no—
231C22×C4—171no—
232C22×C4—171no—
233C22×C4—174no—
234C22×C4—174no—
235C22×C4—172no—
236C22×C4—178no—
237C22×C4—178no—
238C22×C4—174no—
239C22×C4—348no—
240C22×C4—3416no—
241C22×C4—348no—
242C22×C4—348no—
243C22×C4—348no—
244C22×C4—3416no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—688no—
248C22×C4—6816no—
249C22×C4—688no—
250C22×C4—6816no—
251C22×C4—684no—
252C22×C4C4×#59512yes—
253C22×C4C4⋊#59512yes—
254C22×C4C22⋊#59512yes—
255C22×C4C22⋊#59514yes—
256M16—3416no—
257M16—688no—
258C2×C8—348no—
259C2×C8—684no—
260C2×C8—684no—
261C4⋊C4—28no—
262C4⋊C4—3416no—
263C4⋊C4—348no—
264C23—22no—
265C23—22no—
266C23—174no—
267C23—174no—
268C23—174no—
269C23—348no—
270C23—348no—
271C23—348no—
272C23—3416no—
273C23—348no—
274C23—344no—
275C23—342no—
276C23—344no—
277C23—342no—
278C23—342no—
279C23—342no—
280C23C2×C4×#59511yescharacteristic
281C23C2×C4⋊#59511yescharacteristic
282C23#1058767712yes—
283C23C2×C22⋊#59512yes—
284C23C2×C22⋊#59511yescharacteristic
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—172no—
294C2×C4—172no—
295C2×C4—178no—
296C2×C4—174no—
297C2×C4—174no—
298C2×C4—174no—
299C2×C4—174no—
300C2×C4—3416no—
301C2×C4—3416no—
302C2×C4—344no—
303C2×C4—6832no—
304C2×C4—684no—
305C2×C4—684no—
306C2×C4—684no—
307C2×C4(C2×#595)⋊C414yes—
308C8—688no—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—178no—
315C22—3416no—
316C22—3416no—
317C22—344no—
318C22—344no—
319C22—344no—
320C22—344no—
321C22—344no—
322C22—344no—
323C22C2×(C2×#595)⋊C411yescharacteristic
324C22C4×C22⋊#59512yes—
325C22#1176375011yescharacteristic
326C22#1176375112yes—
327C22#1176375211yescharacteristic
328C22D17⋊#10012yes—
329C22D17⋊#10112yes—
330C4—24no—
331C4—44no—
332C4—174no—
333C4—174no—
334C4—684no—
335C2—22no—
336C2—22no—
337C2—348no—
338C2#1454395811yescharacteristic
339C2#1454448712yes—
340C2#1454461711yescharacteristic
341C2#1454461911yescharacteristic
342C2#1454468412yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC23×C17C2×C42
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×D17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#10727
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#11763689C2semidirect
6. #11763781C4semidirect
34. C2×#63247C2×C4semidirect
161. C2×D17#343semidirect
216. C17#10727semidirect

Maximal subgroups

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

subgroupindex
C2×#117637812
C2×#117636892
C22×C4⋊C4×D172
#1072717