Finite Group Databasebeta

← #14531996

Subgroups — #14531996

Order 2176.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 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×#63254C211yesmaximal, characteristic, complemented
2C2×D4×Dic17C212yesmaximal, complemented
3C2×#10587567C211yesmaximal, characteristic, complemented
4#11762730C211yesmaximal, characteristic, complemented
5#11762725C212yesmaximal, complemented
6#11762723C211yesmaximal, characteristic, complemented
7#11762721C211yesmaximal, characteristic, complemented
8#11762717C212yesmaximal, complemented
9C22⋊C4×Dic17C212yesmaximal, complemented
10#11762656C211yesmaximal, characteristic, complemented
11C17×#392C211yesmaximal, characteristic, complemented
12C22×D4×C17C2212yes—
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C22×C22⋊D17C2212yes—
16C23×Dic17C2211yescharacteristic
17C23×Dic17C2212yes—
18C2×#63254—28no—
19C2×#63254—24no—
20C22×C4×D17C2211yescharacteristic
21C22×Dic34C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2212yes—
24C2×C22⋊Dic17C2212yes—
25C2×C22⋊Dic17C2212yes—
26C2×C22⋊Dic17C2212yes—
27D4×Dic17—28no—
28C2×#63227C2212yes—
29C2×#63227C2211yescharacteristic
30C2×C4⋊Dic17C2212yes—
31C2×#63225C2211yescharacteristic
32C2×C4×Dic17C2211yescharacteristic
33C2×C4×Dic17C2212yes—
34#10587567—24no—
35C17×#95C2211yescharacteristic
36#10587502C2212yes—
37#10587502C2211yescharacteristic
38#10587502C2212yes—
39C24×C17C2312yes—
40C24×C17C2311yescharacteristic
41C23×D17C2311yescharacteristic
42C2×D4×C17—24no—
43C2×D4×C17—24no—
44C2×D4×C17—24no—
45C2×D4×C17—22no—
46C2×D4×C17—24no—
47C22×C4×C17C2311yescharacteristic
48C22×C4×C17C2311yescharacteristic
49C22×C4×C17C2312yes—
50C2×C22⋊D17—24no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—24no—
53C2×C22⋊D17—22no—
54C2×C22⋊D17—24no—
55C2×C22⋊D17—22no—
56C22×Dic17—24no—
57C22×Dic17—24no—
58C22×Dic17—22no—
59C22×Dic17—24no—
60C22×Dic17D418yescomplemented
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2312yes—
63C22×Dic17C2312yes—
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66#63254—232no—
67C2×C4×D17—24no—
68C2×C4×D17D414yescomplemented
69C2×Dic34—22no—
70C2×Dic34—24no—
71C22⋊C4×C17—22no—
72C22⋊C4×C17—24no—
73C22⋊Dic17—24no—
74C22⋊Dic17—24no—
75C22⋊Dic17—24no—
76#63227—22no—
77#63227—24no—
78C4⋊Dic17—24no—
79#63225—22no—
80C4×Dic17—22no—
81C4×Dic17—24no—
82C23×C17—22no—
83C23×C17—21no—
84C23×C17—22no—
85C23×C17—22no—
86C23×C17—22no—
87C23×C17—22no—
88C23×C17—21no—
89C23×C17—22no—
90C23×C17—22no—
91C23×C17—22no—
92C23×C17C2×D414yes—
93C23×C17#4112yescomplemented
94C23×C17C2411yescharacteristic
95C22×D17—22no—
96C22×D17—21no—
97C22×D17—21no—
98C22×D17—22no—
99C22×D17C2×D412yes—
100D4×C17—28no—
101D4×C17—216no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17C2×D412yes—
108C2×C4×C17#4114yescomplemented
109C22⋊D17—216no—
110C22⋊D17—28no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—216no—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—21no—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—44no—
122C2×Dic17C2×D414yes—
123C2×Dic17C2×D414yes—
124C2×Dic17C2×D412yes—
125C2×Dic17#4112yescomplemented
126C4×D17—28no—
127Dic34—28no—
128#2388—171nomaximal, Sylow
129C22×C17—24no—
130C22×C17—22no—
131C22×C17—24no—
132C22×C17—24no—
133C22×C17—24no—
134C22×C17—41no—
135C22×C17—42no—
136C22×C17—42no—
137C22×C17—42no—
138C22×C17—42no—
139C22×C17—41no—
140C22×C17—42no—
141C22×C17C22×D411yescharacteristic
142C22×C17C22×D412yes—
143C22×C17C2×#4111yescharacteristic
144C22×C17C2×#4112yes—
145C22×C17C2×#4111yescharacteristic
146C2×D17—22no—
147C2×D17—22no—
148C2×D17—22no—
149C2×D17—42no—
150C2×D17—42no—
151C4×C17—22no—
152C4×C17—24no—
153C4×C17—41no—
154Dic17—24no—
155Dic17—22no—
156Dic17—41no—
157Dic17—42no—
158Dic17#12014yes—
159C22×#41—171no—
160C2×C4×D4—172no—
161C2×C4×D4—171no—
162#395—171no—
163#394—171no—
164#394—172no—
165#393—171no—
166#392—172no—
167#392D1711yescharacteristic, complemented
168C4×C22⋊C4—172no—
169C4×C22⋊C4—171no—
170C2×C17—44no—
171C2×C17—42no—
172C2×C17C2×#12011yescharacteristic
173C2×C17#52512yes—
174C2×C17#52511yescharacteristic
175C2×C17#53212yes—
176C2×C17#53211yescharacteristic
177D17—42no—
178C2×#41—348no—
179C2×#41—344no—
180C22×Q8—171no—
181C22×D4—171no—
182C22×D4—172no—
183C22×D4C2×D1712yes—
184C22×D4C2×D1711yescharacteristic
185C23×C4—172no—
186C23×C4—171no—
187C23×C4—171no—
188C4×D4—344no—
189C4×D4—348no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—172no—
192C2×C22⋊C4—172no—
193C2×C22⋊C4—172no—
194C2×C22⋊C4—172no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—172no—
197C2×C22⋊C4C2×D1711yescharacteristic
198C2×C22⋊C4C2×D1712yes—
199C2×C42—171no—
200C2×C42—172no—
201#95—172no—
202#95—172no—
203#95—171no—
204#95C2×D1711yescharacteristic
205C17#238811yescharacteristic, complemented, Sylow
206C24—171no—
207C24C22×D1712yes—
208C24C22×D1711yescharacteristic
209#41—3432no—
210C2×Q8—344no—
211C2×Q8—342no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—342no—
218C2×D4—344no—
219C2×D4—342no—
220C2×D4—344no—
221C2×D4—344no—
222C2×D4—342no—
223C22×C4—171no—
224C22×C4—178no—
225C22×C4—171no—
226C22×C4—172no—
227C22×C4—174no—
228C22×C4—172no—
229C22×C4—171no—
230C22×C4—344no—
231C22×C4—344no—
232C22×C4—344no—
233C22×C4—342no—
234C22×C4—344no—
235C22×C4C22×D1711yescharacteristic
236C22×C4C22×D1711yescharacteristic
237C22×C4C22×D1712yes—
238C4⋊C4—344no—
239C4⋊C4—342no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—342no—
243C22⋊C4—344no—
244C22⋊C4—344no—
245C22⋊C4—344no—
246C22⋊C4—344no—
247C42—344no—
248C42—342no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—172no—
260C23—342no—
261C23—342no—
262C23—341no—
263C23—341no—
264C23D4×D1714yes—
265C23#6325412yescomplemented
266C23C23×D1711yescharacteristic
267Q8—348no—
268D4—28no—
269D4—216no—
270D4—3416no—
271D4—348no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—174no—
278C2×C4—172no—
279C2×C4—174no—
280C2×C4—172no—
281C2×C4—348no—
282C2×C4—3416no—
283C2×C4—341no—
284C2×C4—342no—
285C2×C4—342no—
286C2×C4—341no—
287C2×C4—342no—
288C2×C4—342no—
289C2×C4—341no—
290C2×C4—342no—
291C2×C4—342no—
292C2×C4—684no—
293C2×C4D4×D1712yes—
294C2×C4#6325414yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—342no—
308C22—342no—
309C22—342no—
310C22—682no—
311C22—682no—
312C22C2×D4×D1712yes—
313C22C2×D4×D1711yescharacteristic
314C22C2×#6325411yescharacteristic
315C22C2×#6325412yes—
316C22#41×D1711yescharacteristic
317C4—22no—
318C4—24no—
319C4—41no—
320C4—174no—
321C4—344no—
322C4—342no—
323C4—681no—
324C4—682no—
325C2—44no—
326C2—42no—
327C2—682no—
328C2#120×D1711yescharacteristic
329C2#1176331512yes—
330C2#1176331911yescharacteristic
331C2#1176335412yes—
332C2#1176335711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2388
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#63254C2semidirect
2. C2×D4×Dic17C2semidirect
3. C2×#10587567C2semidirect
4. #11762730C2semidirect
5. #11762725C2semidirect
6. #11762723C2semidirect
7. #11762721C2semidirect
8. #11762717C2semidirect
9. C22⋊C4×Dic17C2semidirect
10. #11762656C2semidirect
11. C17×#392C2semidirect
60. C22×Dic17D4semidirect
68. C2×C4×D17D4semidirect
93. C23×C17#41semidirect
108. C2×C4×C17#41semidirect
125. C2×Dic17#41semidirect
167. #392D17semidirect
205. C17#2388semidirect
265. C23#63254semidirect
294. C2×C4#63254semidirect

Maximal subgroups

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

subgroupindex
C22×#632542
C2×D4×Dic172
C2×#105875672
#117627302
#117627252
#117627232
#117627212
#117627172
C22⋊C4×Dic172
#117626562
C17×#3922
#238817