Finite Group Databasebeta

← #17950565

Subgroups — #17950565

Order 4352.

1058 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 188 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×#11763779C212yesmaximal
2#14543622C214yesmaximal, complemented
3C2×C22⋊C4×Dic17C211yesmaximal, characteristic
4C22×#63247—22no—
5#11763779—24no—
6#11763779C418yescomplemented
7C2×C22⋊#3624—22no—
8C2×C22⋊#3624C2214yes—
9C24×Dic17C2211yescharacteristic
10C22×C22⋊Dic17C411yescharacteristic
11C22×C4×Dic17C412yes—
12C22⋊C4×Dic17—22no—
13C22⋊C4×Dic17—22no—
14C2×#10587502C2212yes—
15C22×C22⋊C4×C17C411yescharacteristic
16C25×C17C2×C411yescharacteristic
17C23×C4×C17C2×C412yes—
18C23×Dic17—44no—
19C23×Dic17C2×C411yescharacteristic
20C23×Dic17C2×C412yes—
21C23×Dic17C2×C414yes—
22C23×Dic17D412yes—
23C23×Dic17D412yes—
24C23×Dic17D412yes—
25C23×Dic17Q812yes—
26C23×Dic17C2311yescharacteristic
27C2×#63247—28no—
28C2×#63247—48no—
29C22×#3624—24no—
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17—24no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—24no—
34C2×C4×Dic17—22no—
35C2×C4×Dic17—22no—
36C2×C4×Dic17—44no—
37C22⋊#3624—28no—
38C22⋊#3624—24no—
39C22⋊#3624C2×C418yes—
40#10587502—24no—
41C24×C17—22no—
42C24×C17—22no—
43C24×C17—42no—
44C24×C17—42no—
45C24×C17C4212yes—
46C24×C17C22⋊C412yes—
47C24×C17C4⋊C412yes—
48C24×C17C22×C411yescharacteristic
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—42no—
52C22×C4×C17—42no—
53C22×C4×C17M1614yescomplemented
54C22×Dic17—28no—
55C22×Dic17—24no—
56C22×Dic17—22no—
57C22×Dic17—22no—
58C22×Dic17—48no—
59C22×Dic17—44no—
60C22×Dic17—44no—
61C22×Dic17—44no—
62C22×Dic17—42no—
63C22×Dic17—42no—
64C22×Dic17—42no—
65C22×Dic17C4212yes—
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C22⋊C414yes—
68C22×Dic17C22⋊C414yes—
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C4⋊C412yes—
71C22×Dic17C4⋊C414yes—
72C22×Dic17C4⋊C414yes—
73C22×Dic17M1614yescomplemented
74C22×Dic17C22×C412yes—
75C22×Dic17C2×D412yes—
76C22×Dic17C2×D411yescharacteristic
77C22×Dic17C2×Q811yescharacteristic
78#63247—416no—
79C2×#3624—28no—
80C2×#3624—48no—
81C22⋊C4×C17—28no—
82C22⋊Dic17—28no—
83C4×Dic17—44no—
84C4×Dic17—44no—
85#7911—171nomaximal, Sylow
86C23×C17—22no—
87C23×C17—22no—
88C23×C17—24no—
89C23×C17—22no—
90C23×C17—22no—
91C23×C17—24no—
92C23×C17—24no—
93C23×C17—24no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17—44no—
99C23×C17—44no—
100C23×C17—44no—
101C23×C17—44no—
102C23×C17#9514yes—
103C23×C17C2×C4211yescharacteristic
104C23×C17C2×C22⋊C411yescharacteristic
105C23×C17C2×C4⋊C411yescharacteristic
106C23×C17C2×M1612yes—
107C23×C17C2×M1612yes—
108C2×C4×C17—24no—
109C2×C4×C17—44no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—44no—
113C2×Dic17—24no—
114C2×Dic17—22no—
115C2×Dic17—28no—
116C2×Dic17—24no—
117C2×Dic17—24no—
118C2×Dic17—28no—
119C2×Dic17—24no—
120C2×Dic17—44no—
121C2×Dic17—44no—
122C2×Dic17—42no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—44no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—44no—
129C2×Dic17#9512yes—
130C2×Dic17#9512yes—
131C2×Dic17#9914yes—
132C2×Dic17C2×C22⋊C412yes—
133C2×Dic17C2×C4⋊C412yes—
134#3624—48no—
135C2×#407—172no—
136C2×C4×C22⋊C4—171no—
137#1280—174no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—24no—
147C22×C17—24no—
148C22×C17—24no—
149C22×C17—48no—
150C22×C17—48no—
151C22×C17—44no—
152C22×C17—44no—
153C22×C17C2×#9511yescharacteristic
154C22×C17C4×M1612yes—
155C22×C17#40711yescharacteristic
156C22×C17#40711yescharacteristic
157C22×C17C2×#9912yes—
158C22×C17#41312yes—
159C22×C17#42312yes—
160C4×C17—44no—
161Dic17—22no—
162Dic17—22no—
163Dic17—22no—
164Dic17—44no—
165Dic17#34214yes—
166C24×C4—171no—
167C22×M16—342no—
168C22×C22⋊C4—171no—
169C22×C22⋊C4#59511yescharacteristic
170C22×C42—172no—
171#407—178no—
172#407—344no—
173C2×C22⋊C8—174no—
174C2×C22⋊C8—342no—
175C4×C22⋊C4—342no—
176C4×C22⋊C4—342no—
177C2×#95—172no—
178C2×C17—22no—
179C2×C17—22no—
180C2×C17—44no—
181C2×C17#146512yes—
182C2×C17#146512yes—
183C2×C17#172811yescharacteristic
184C2×C17C2×#34211yescharacteristic
185C2×C17#174211yescharacteristic
186C25C2×#59511yescharacteristic
187C23×C4—172no—
188C23×C4—171no—
189C23×C4—174no—
190C23×C4—172no—
191C23×C4—172no—
192C23×C4—172no—
193C23×C4—171no—
194C23×C4—172no—
195C23×C4—684no—
196C23×C4C2×#59512yes—
197C2×M16—348no—
198C2×M16—688no—
199C22×C8—344no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—344no—
203C2×C22⋊C4—342no—
204C2×C42—342no—
205C2×C42—342no—
206C2×C42—684no—
207C22⋊C8—178no—
208C22⋊C8—344no—
209C22⋊C8—348no—
210#95—344no—
211C17#791111yescharacteristic, complemented, Sylow
212C24—22no—
213C24—22no—
214C24—42no—
215C24—42no—
216C24C4×#59512yes—
217C24C4⋊#59512yes—
218C24C22⋊#59512yes—
219C24C22×#59511yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—171no—
225C22×C4—172no—
226C22×C4—172no—
227C22×C4—171no—
228C22×C4—172no—
229C22×C4—174no—
230C22×C4—174no—
231C22×C4—172no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—174no—
235C22×C4—174no—
236C22×C4—174no—
237C22×C4—344no—
238C22×C4—348no—
239C22×C4—342no—
240C22×C4—342no—
241C22×C4—684no—
242C22×C4—688no—
243C22×C4—684no—
244C22×C4—684no—
245C22×C4—682no—
246C22×C4—682no—
247C22×C4—682no—
248C22×C4#6324714yescomplemented
249M16—6816no—
250C2×C8—348no—
251C2×C8—688no—
252C22⋊C4—28no—
253C22⋊C4—348no—
254C42—684no—
255C42—684no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—24no—
263C23—24no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—44no—
269C23—44no—
270C23—44no—
271C23—44no—
272C23(C2×#595)⋊C414yes—
273C23D17⋊M1612yes—
274C23C2×C4×#59511yescharacteristic
275C23C2×C4⋊#59511yescharacteristic
276C23C2×#6324712yes—
277C23C2×C22⋊#59511yescharacteristic
278C2×C4—24no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—172no—
284C2×C4—174no—
285C2×C4—172no—
286C2×C4—172no—
287C2×C4—172no—
288C2×C4—348no—
289C2×C4—344no—
290C2×C4—344no—
291C2×C4—348no—
292C2×C4—342no—
293C2×C4—344no—
294C2×C4—344no—
295C2×C4—681no—
296C2×C4—682no—
297C2×C4—684no—
298C2×C4—681no—
299C2×C4—684no—
300C2×C4—684no—
301C2×C4—682no—
302C2×C4—682no—
303C2×C4—684no—
304C8—688no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—48no—
317C22—48no—
318C22—44no—
319C22—44no—
320C22C2×(C2×#595)⋊C411yescharacteristic
321C22C4×#6324712yes—
322C22#1176377911yescharacteristic
323C22#1176378111yescharacteristic
324C22D17⋊#9912yes—
325C22#1176379612yes—
326C22#1176381212yes—
327C4—44no—
328C4—174no—
329C4—342no—
330C4—342no—
331C4—342no—
332C4—684no—
333C2—22no—
334C2—22no—
335C2—44no—
336C2#1454404012yes—
337C2#1454404212yes—
338C2#1454444711yescharacteristic
339C2#1454448111yescharacteristic
340C2#1454449011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC23×C17C2×C42
FrattiniC24C22×#595
FittingC22×C22⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC25×C17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C22⋊C4#595
hypocenterC17#7911
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543622C2semidirect
6. #11763779C4semidirect
53. C22×C4×C17M16semidirect
73. C22×Dic17M16semidirect
211. C17#7911semidirect
248. C22×C4#63247semidirect

Maximal subgroups

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

subgroupindex
C2×#117637792
#145436222
C2×C22⋊C4×Dic172
#791117