Finite Group Databasebeta

← #14532440

Subgroups — #14532440

Order 2176.

852 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×Dic34C211yesmaximal, characteristic, complemented
2C2×#10587571C212yesmaximal, complemented
3C2×#10587625C211yesmaximal, characteristic
4#11762722C212yesmaximal, complemented
5#11762721C212yesmaximal, complemented
6#11762713C212yesmaximal, complemented
7#11762689C211yesmaximal, characteristic, complemented
8#11762688C211yesmaximal, characteristic, complemented
9#11762686C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic17C211yesmaximal, characteristic, complemented
11C17×#386C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C23×Dic17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C22×Dic34—24no—
19C22×Dic34—28no—
20C22×Dic34C2212yes—
21C22×Dic34C2212yes—
22C2×C22⋊C4×C17C2212yes—
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2212yes—
26C2×C22⋊Dic17C2211yescharacteristic
27#10587625—24no—
28C2×#63227C2212yes—
29C2×#63227C2211yescharacteristic
30C2×#63227C2212yes—
31C2×#63225C2211yescharacteristic
32C2×C4×Dic17C2212yes—
33#10587571—28no—
34C17×#95C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2212yes—
38#10587502C2212yes—
39C24×C17C2311yescharacteristic
40C24×C17C2311yescharacteristic
41C23×D17C2311yescharacteristic
42C2×D4×C17—22no—
43C2×D4×C17—22no—
44C2×D4×C17—22no—
45C22×C4×C17—22no—
46C22×C4×C17—24no—
47C22×C4×C17C2311yescharacteristic
48C22×C4×C17C2312yes—
49C22×C4×C17C2312yes—
50C2×C22⋊D17—22no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—22no—
53C2×C22⋊D17—24no—
54C2×C22⋊D17—22no—
55C22×Dic17—22no—
56C22×Dic17—24no—
57C22×Dic17—24no—
58C22×Dic17D414yes—
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2312yes—
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2312yes—
63C22×Dic17C2311yescharacteristic
64C2×Dic34—232no—
65C2×Dic34—24no—
66C2×Dic34—24no—
67C2×Dic34—24no—
68C2×Dic34—24no—
69C2×Dic34—24no—
70C2×Dic34D418yescomplemented
71C22⋊C4×C17—22no—
72C22⋊C4×C17—24no—
73C22⋊Dic17—22no—
74C22⋊Dic17—24no—
75C22⋊Dic17—22no—
76#63227—24no—
77#63227—24no—
78#63227—22no—
79#63225—22no—
80C4×Dic17—24no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—21no—
85C23×C17—21no—
86C23×C17—21no—
87C23×C17—22no—
88C23×C17C2×D412yes—
89C23×C17#4112yes—
90C23×C17#4112yes—
91C23×C17C2411yescharacteristic
92C22×D17—21no—
93C22×D17—22no—
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—22no—
97D4×C17—44no—
98C2×C4×C17—28no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17C2×D414yes—
108C22⋊D17—44no—
109C22⋊D17—44no—
110C2×Dic17—28no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—28no—
115C2×Dic17—21no—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—22no—
121C2×Dic17C2×D412yes—
122C2×Dic17C2×D414yes—
123C2×Dic17C2×D412yes—
124C2×Dic17C2×D414yes—
125Dic34—216no—
126Dic34—216no—
127#2436—171nomaximal, Sylow
128C22×C17—22no—
129C22×C17—22no—
130C22×C17—24no—
131C22×C17—24no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17—42no—
136C22×C17—42no—
137C22×C17#12414yescomplemented
138C22×C17C22×D411yescharacteristic
139C22×C17C22×D412yes—
140C22×C17C2×#4111yescharacteristic
141C22×C17C2×#4111yescharacteristic
142C22×C172−1+412yes—
143C2×D17—41no—
144C2×D17—41no—
145C2×D17—41no—
146C2×D17—42no—
147C2×D17—42no—
148C4×C17—24no—
149C4×C17—41no—
150C4×C17—42no—
151Dic17—22no—
152Dic17—24no—
153Dic17—41no—
154Dic17—42no—
155Dic17#12014yes—
156C23×Q8—171no—
157C2×#124—172no—
158C2×#121—171no—
159#393—172no—
160#393—172no—
161#391—172no—
162#386—171no—
163#386—171no—
164#386—171no—
165#386D1711yescharacteristic, complemented
166C4×C22⋊C4—171no—
167C2×C17—24no—
168C2×C17—81no—
169C2×C17C2×#12011yescharacteristic
170C2×C17C2×#12411yescharacteristic
171C2×C17#53611yescharacteristic
172C2×C17#54812yes—
173C2×C17#54812yes—
174D17—81no—
175C22×Q8—172no—
176C22×Q8—172no—
177C22×Q8—344no—
178C22×Q8—348no—
179C22×D4—171no—
180C22×D4—171no—
181C22×D4C2×D1711yescharacteristic
182C23×C4—171no—
183C23×C4—171no—
184C23×C4C2×D1711yescharacteristic
185#124—348no—
186#121—344no—
187C2×C4⋊C4—171no—
188C2×C22⋊C4—171no—
189C2×C22⋊C4—172no—
190C2×C22⋊C4—171no—
191C2×C22⋊C4—171no—
192C2×C22⋊C4—172no—
193C2×C22⋊C4—172no—
194C2×C22⋊C4C2×D1712yes—
195C2×C22⋊C4C2×D1711yescharacteristic
196C2×C42—172no—
197#95—172no—
198#95—172no—
199#95—172no—
200#95—172no—
201#95C2×D1712yes—
202C17#243611yescharacteristic, complemented, Sylow
203C24—171no—
204C24C22×D1711yescharacteristic
205C24C22×D1711yescharacteristic
206C2×Q8—178no—
207C2×Q8—344no—
208C2×Q8—344no—
209C2×Q8—344no—
210C2×Q8—344no—
211C2×Q8—3432no—
212C2×Q8—344no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—342no—
217C2×D4—344no—
218C2×D4—342no—
219C2×D4—342no—
220C2×D4—342no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—171no—
224C22×C4—174no—
225C22×C4—171no—
226C22×C4—172no—
227C22×C4—171no—
228C22×C4—172no—
229C22×C4—344no—
230C22×C4—344no—
231C22×C4—342no—
232C22×C4C22×D1711yescharacteristic
233C22×C4C22×D1712yes—
234C22×C4C22×D1712yes—
235C4⋊C4—342no—
236C22⋊C4—22no—
237C22⋊C4—24no—
238C22⋊C4—342no—
239C22⋊C4—344no—
240C22⋊C4—342no—
241C22⋊C4—344no—
242C22⋊C4—342no—
243C22⋊C4—344no—
244C42—344no—
245C23—22no—
246C23—22no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—341no—
253C23—342no—
254C23—341no—
255C23—341no—
256C23—342no—
257C23#6325212yes—
258C23D4×D1712yes—
259C23#6325412yes—
260C23C23×D1711yescharacteristic
261Q8—3416no—
262Q8—3416no—
263D4—44no—
264D4—684no—
265D4—684no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—174no—
276C2×C4—172no—
277C2×C4—172no—
278C2×C4—174no—
279C2×C4—348no—
280C2×C4—348no—
281C2×C4—342no—
282C2×C4—342no—
283C2×C4—342no—
284C2×C4—342no—
285C2×C4—341no—
286C2×C4—342no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—342no—
290C2×C4D4×D1714yes—
291C22—24no—
292C22—24no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—682no—
301C22—682no—
302C22—681no—
303C22—681no—
304C22—681no—
305C22#1058757114yescomplemented
306C22C2×#6325211yescharacteristic
307C22C2×D4×D1712yes—
308C22C2×D4×D1711yescharacteristic
309C22C2×#6325411yescharacteristic
310C22#1058769712yes—
311C4—24no—
312C4—42no—
313C4—41no—
314C4—174no—
315C4—342no—
316C4—344no—
317C4—682no—
318C4—681no—
319C2—24no—
320C2—81no—
321C2—1361no—
322C2#120×D1711yescharacteristic
323C2C2×#1058757111yescharacteristic
324C2#1176338111yescharacteristic
325C2#1176346512yes—
326C2#1176346812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC2×#10587625C2
hypercenter#386D17
hypocenterC17#2436
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C23×Dic34C2semidirect
2. C2×#10587571C2semidirect
4. #11762722C2semidirect
5. #11762721C2semidirect
6. #11762713C2semidirect
7. #11762689C2semidirect
8. #11762688C2semidirect
9. #11762686C2semidirect
10. C22⋊C4×Dic17C2semidirect
11. C17×#386C2semidirect
70. C2×Dic34D4semidirect
137. C22×C17#124semidirect
165. #386D17semidirect
202. C17#2436semidirect
305. C22#10587571semidirect

Maximal subgroups

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

subgroupindex
C23×Dic342
C2×#105875712
C2×#105876252
#117627222
#117627212
#117627132
#117626892
#117626882
#117626862
C22⋊C4×Dic172
C17×#3862
#243617