Finite Group Databasebeta

← #14534561

Subgroups — #14534561

Order 2176.

702 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587623C211yesmaximal, characteristic
2C22×#63227C211yesmaximal, characteristic, complemented
3#11762738C212yesmaximal, complemented
4#11762736C212yesmaximal, complemented
5#11762725C212yesmaximal, complemented
6#11762724C212yesmaximal, complemented
7#11762689C211yesmaximal, characteristic, complemented
8#11762684C211yesmaximal, characteristic, complemented
9#11762682C211yesmaximal, characteristic, complemented
10#11762664C211yesmaximal, characteristic, complemented
11C17×#386C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C22×C22⋊D17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2212yescomplemented
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2212yescomplemented
23#10587623—24no—
24C2×#63227—24no—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2212yes—
30C2×#63227C2212yes—
31C2×C4⋊Dic17C2212yescomplemented
32C2×#63225C2212yescomplemented
33C17×#95C2212yescomplemented
34#10587502C2212yescomplemented
35#10587502C2212yescomplemented
36#10587502C2212yescomplemented
37C24×C17C2311yescharacteristic
38C24×C17C2311yescharacteristic
39C23×D17—22no—
40C23×D17—22no—
41C23×D17—42no—
42C23×D17—42no—
43C23×D17D414yes—
44C23×D17C2311yescharacteristic
45C23×D17C2311yescharacteristic
46C2×D4×C17—22no—
47C2×D4×C17—22no—
48C2×D4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—42no—
51C22×C4×C17C2312yes—
52C22×C4×C17C2311yescharacteristic
53C22×C4×C17C2312yes—
54C2×C22⋊D17—22no—
55C2×C22⋊D17—22no—
56C2×C22⋊D17—42no—
57C2×C22⋊D17—42no—
58C22×Dic17—22no—
59C22×Dic17—42no—
60C22×Dic17C2312yes—
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2312yes—
64C22⋊C4×C17—22no—
65C22⋊C4×C17—42no—
66C22⋊Dic17—22no—
67C22⋊Dic17—22no—
68C22⋊Dic17—42no—
69#63227—24no—
70#63227—24no—
71#63227—22no—
72#63227—22no—
73#63227—42no—
74#63227—42no—
75C4⋊Dic17—42no—
76#63225—42no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—21no—
80C23×C17—21no—
81C23×C17—21no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17C2×D412yes—
85C23×C17#4112yes—
86C23×C17#4112yes—
87C23×C17C2411yescharacteristic
88C22×D17—28no—
89C22×D17—21no—
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—22no—
94C22×D17—22no—
95C22×D17—22no—
96C22×D17—22no—
97C22×D17—44no—
98C22×D17—44no—
99C22×D17—48no—
100C22×D17—48no—
101C22×D17C2×D412yes—
102C22×D17C2×D412yes—
103D4×C17—44no—
104C2×C4×C17—22no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—44no—
114C22⋊D17—44no—
115C22⋊D17—44no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—22no—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—22no—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—44no—
127#2622—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#12314yes—
138C22×C17C22×D411yescharacteristic
139C22×C17C2×#4111yescharacteristic
140C22×C17C2×#4111yescharacteristic
141C22×C17H(5,𝔽2)12yes—
142C22×C17H(5,𝔽2)12yes—
143C2×D17—22no—
144C2×D17—22no—
145C2×D17—22no—
146C2×D17—22no—
147C2×D17—22no—
148C2×D17—22no—
149C2×D17—416no—
150C2×D17—42no—
151C2×D17—42no—
152C2×D17—42no—
153C2×D17—42no—
154C4×C17—41no—
155C4×C17—42no—
156C4×C17—42no—
157Dic17—42no—
158Dic17—41no—
159Dic17—41no—
160Dic17—42no—
161C2×#120—171no—
162C22×C22⋊C4—171no—
163#397—172no—
164#397—172no—
165#394—172no—
166#394—172no—
167#386—171no—
168#386D1711yescharacteristic, complemented
169#385—171no—
170#385—171no—
171#379—171no—
172C2×C17—24no—
173C2×C17—81no—
174C2×C17C2×#12311yescharacteristic
175C2×C17#53411yescharacteristic
176C2×C17#53411yescharacteristic
177C2×C17#53812yes—
178C2×C17#53812yes—
179D17—42no—
180D17—42no—
181C25—171no—
182C22×D4—171no—
183C22×D4—171no—
184C22×D4C2×D1711yescharacteristic
185C23×C4—171no—
186C23×C4C2×D1711yescharacteristic
187#120—344no—
188C2×C4⋊C4—172no—
189C2×C4⋊C4—172no—
190C2×C22⋊C4—171no—
191C2×C22⋊C4—171no—
192C2×C22⋊C4—172no—
193C2×C22⋊C4—172no—
194C2×C22⋊C4—171no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—172no—
197C2×C22⋊C4—172no—
198C2×C22⋊C4—172no—
199C2×C22⋊C4—344no—
200C2×C22⋊C4C2×D1711yescharacteristic
201C2×C22⋊C4C2×D1712yescomplemented
202#95—172no—
203#95—172no—
204#95—172no—
205#95C2×D1712yescomplemented
206C17#262211yescharacteristic, complemented, Sylow
207C24—171no—
208C24—171no—
209C24—174no—
210C24—342no—
211C24—342no—
212C24—682no—
213C24—682no—
214C24C22×D1711yescharacteristic
215C24C22×D1711yescharacteristic
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—342no—
220C2×D4—342no—
221C2×D4—682no—
222C2×D4—682no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—172no—
226C22×C4—172no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—342no—
230C22×C4—682no—
231C22×C4C22×D1711yescharacteristic
232C22×C4C22×D1712yes—
233C22×C4C22×D1712yes—
234C4⋊C4—682no—
235C4⋊C4—682no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—344no—
239C22⋊C4—344no—
240C22⋊C4—342no—
241C22⋊C4—342no—
242C22⋊C4—342no—
243C22⋊C4—342no—
244C22⋊C4—682no—
245C22⋊C4—682no—
246C22⋊C4—682no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—172no—
255C23—172no—
256C23—341no—
257C23—341no—
258C23—341no—
259C23—341no—
260C23—342no—
261C23—342no—
262C23—342no—
263C23—342no—
264C23—348no—
265C23—684no—
266C23—684no—
267C23—688no—
268C23—688no—
269C23#6325212yes—
270C23D4×D1712yes—
271C23#6325412yes—
272C23C23×D1711yescharacteristic
273D4—44no—
274D4—684no—
275D4—684no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—342no—
287C2×C4—342no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—342no—
291C2×C4—342no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4—342no—
296C2×C4—684no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—24no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—342no—
307C22—342no—
308C22—342no—
309C22—342no—
310C22—342no—
311C22—342no—
312C22—6816no—
313C22—682no—
314C22—682no—
315C22—682no—
316C22—682no—
317C22#1058756914yes—
318C22C2×#6325211yescharacteristic
319C22C2×D4×D1711yescharacteristic
320C22C2×#6325411yescharacteristic
321C22#1058769112yes—
322C22#1058769612yes—
323C4—41no—
324C4—42no—
325C4—42no—
326C4—681no—
327C4—682no—
328C4—681no—
329C4—682no—
330C2—24no—
331C2—81no—
332C2—682no—
333C2—682no—
334C2C2×#1058756911yescharacteristic
335C2#1176336911yescharacteristic
336C2#1176337011yescharacteristic
337C2#1176339212yes—
338C2#1176339612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC2×#10587623C2
hypercenter#386D17
hypocenterC17#2622
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×#63227C2semidirect
3. #11762738C2semidirect
4. #11762736C2semidirect
5. #11762725C2semidirect
6. #11762724C2semidirect
7. #11762689C2semidirect
8. #11762684C2semidirect
9. #11762682C2semidirect
10. #11762664C2semidirect
11. C17×#386C2semidirect
19. C2×C22⋊C4×C17C22semidirect
22. C2×C22⋊Dic17C22semidirect
31. C2×C4⋊Dic17C22semidirect
32. C2×#63225C22semidirect
33. C17×#95C22semidirect
34. #10587502C22semidirect
35. #10587502C22semidirect
36. #10587502C22semidirect
168. #386D17semidirect
201. C2×C22⋊C4C2×D17semidirect
205. #95C2×D17semidirect
206. C17#2622semidirect

Maximal subgroups

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

subgroupindex
C2×#105876232
C22×#632272
#117627382
#117627362
#117627252
#117627242
#117626892
#117626842
#117626822
#117626642
C17×#3862
#262217