Finite Group Databasebeta

← #14533673

Subgroups — #14533673

Order 2176.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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
2C2×D4×Dic17C211yesmaximal, characteristic, complemented
3C22×#63227C211yesmaximal, characteristic, complemented
4#11762731C212yesmaximal, complemented
5#11762724C212yesmaximal, complemented
6#11762699C212yesmaximal, complemented
7#11762686C212yesmaximal, complemented
8#11762672C211yesmaximal, characteristic, complemented
9#11762667C211yesmaximal, characteristic, complemented
10#11762664C211yesmaximal, characteristic, complemented
11C2×C17×#123C211yesmaximal, characteristic, complemented
12C24×D17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2212yes—
16C23×Dic17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic, complemented
18C17×#123—24no—
19C2×C4⋊C4×C17C2212yescomplemented
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2212yescomplemented
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic, complemented
25#10587623—24no—
26D4×Dic17—24no—
27C2×#63227—24no—
28C2×#63227C2212yes—
29C2×#63227C2212yes—
30C2×#63227C2212yes—
31C2×#63227C2212yes—
32C2×#63227C2212yes—
33C2×C4⋊Dic17C2211yescharacteristic, complemented
34C2×#63225C2212yescomplemented
35C2×C4×Dic17C2211yescharacteristic, complemented
36#10587502C2212yescomplemented
37#10587502C2212yescomplemented
38#10587502C2212yescomplemented
39C24×C17C2311yescharacteristic
40C24×C17C2311yescharacteristic
41C23×D17—22no—
42C23×D17—24no—
43C23×D17—22no—
44C23×D17—22no—
45C23×D17—42no—
46C23×D17C2312yes—
47C2×D4×C17—22no—
48C2×D4×C17—22no—
49C2×D4×C17D414yes—
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17C2312yes—
54C22×C4×C17C2312yes—
55C22×C4×C17C2311yescharacteristic
56C2×C22⋊D17—24no—
57C2×C22⋊D17—24no—
58C2×C22⋊D17—42no—
59C22×Dic17—22no—
60C22×Dic17—42no—
61C22×Dic17—42no—
62C22×Dic17D414yescomplemented
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2312yes—
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C4⋊C4×C17—24no—
69C22⋊C4×C17—22no—
70C22⋊C4×C17—24no—
71C22⋊Dic17—22no—
72C22⋊Dic17—22no—
73C22⋊Dic17—22no—
74#63227—24no—
75#63227—24no—
76#63227—24no—
77#63227—24no—
78#63227—42no—
79C4⋊Dic17—22no—
80#63225—42no—
81C4×Dic17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—21no—
85C23×C17—22no—
86C23×C17—21no—
87C23×C17—22no—
88C23×C17C2×D412yes—
89C23×C17C2×D412yes—
90C23×C17C2×D412yes—
91C23×C17#4112yes—
92C23×C17C2411yescharacteristic
93C22×D17—28no—
94C22×D17—22no—
95C22×D17—22no—
96C22×D17—22no—
97C22×D17—22no—
98C22×D17—22no—
99C22×D17—22no—
100C22×D17—48no—
101C22×D17—48no—
102C22×D17—44no—
103C22×D17—44no—
104C22×D17#4114yes—
105D4×C17—28no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—44no—
115C2×C4×C17C2×D412yes—
116C22⋊D17—48no—
117C2×Dic17—22no—
118C2×Dic17—22no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—22no—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—44no—
127C2×Dic17—44no—
128C2×Dic17C2×D412yes—
129C2×Dic17C2×D412yes—
130#2534—171nomaximal, Sylow
131C22×C17—22no—
132C22×C17—22no—
133C22×C17—24no—
134C22×C17—24no—
135C22×C17—22no—
136C22×C17—24no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—42no—
140C22×C17#12114yes—
141C22×C17C22×D411yescharacteristic
142C22×C17C22×D411yescharacteristic
143C22×C17C2×#4111yescharacteristic
144C22×C17C2×#4112yes—
145C22×C17H(5,𝔽2)12yes—
146C2×D17—24no—
147C2×D17—24no—
148C2×D17—24no—
149C2×D17—416no—
150C2×D17—42no—
151C2×D17—42no—
152C2×D17—42no—
153C2×D17—42no—
154C4×C17—22no—
155C4×C17—42no—
156C4×C17—42no—
157Dic17—22no—
158Dic17—22no—
159Dic17—41no—
160Dic17—41no—
161Dic17—42no—
162C2×#123D1711yescharacteristic, complemented
163C2×#120—171no—
164C2×C4×D4—171no—
165C22×C22⋊C4—171no—
166#396—172no—
167#394—172no—
168#388—172no—
169#386—172no—
170#381—171no—
171#380—171no—
172#379—171no—
173C2×C17—24no—
174C2×C17—42no—
175C2×C17C2×#12111yescharacteristic
176C2×C17#52511yescharacteristic
177C2×C17#53411yescharacteristic
178C2×C17#54612yes—
179C2×C17#55112yes—
180D17—44no—
181C25—171no—
182C22×D4—172no—
183C22×D4C2×D1711yescharacteristic
184C23×C4—171no—
185C23×C4—171no—
186C23×C4C2×D1711yescharacteristic
187#123—24no—
188#120—344no—
189C4×D4—344no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—172no—
192C2×C4⋊C4C2×D1712yescomplemented
193C2×C22⋊C4—171no—
194C2×C22⋊C4—171no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—172no—
197C2×C22⋊C4—172no—
198C2×C22⋊C4—172no—
199C2×C22⋊C4—172no—
200C2×C22⋊C4—172no—
201C2×C22⋊C4—344no—
202C2×C22⋊C4C2×D1711yescharacteristic
203C2×C22⋊C4C2×D1712yescomplemented
204C2×C42—171no—
205#95—172no—
206#95—172no—
207#95—172no—
208C17#253411yescharacteristic, complemented, Sylow
209C24—172no—
210C24—344no—
211C24—342no—
212C24—342no—
213C24—342no—
214C24—682no—
215C24C22×D1711yescharacteristic
216C24C22×D1711yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—344no—
220C2×D4—344no—
221C2×D4—682no—
222C2×D4C22⋊D1714yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—174no—
229C22×C4—172no—
230C22×C4—171no—
231C22×C4—171no—
232C22×C4—342no—
233C22×C4—682no—
234C22×C4—682no—
235C22×C4C22×D1712yes—
236C22×C4C22×D1712yes—
237C22×C4C22×D1711yescharacteristic
238C4⋊C4—24no—
239C4⋊C4—342no—
240C4⋊C4—682no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—344no—
244C22⋊C4—344no—
245C22⋊C4—344no—
246C22⋊C4—342no—
247C22⋊C4—342no—
248C22⋊C4—342no—
249C22⋊C4—344no—
250C22⋊C4—682no—
251C42—342no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—174no—
259C23—348no—
260C23—342no—
261C23—342no—
262C23—342no—
263C23—342no—
264C23—342no—
265C23—342no—
266C23—688no—
267C23—688no—
268C23—684no—
269C23—684no—
270C23D4×D1712yes—
271C23#6325412yes—
272C23C2×C22⋊D1712yes—
273C23C2×C22⋊D1712yes—
274C23C23×D1711yescharacteristic
275D4—28no—
276D4—688no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—172no—
287C2×C4—172no—
288C2×C4—342no—
289C2×C4—342no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4—342no—
296C2×C4—342no—
297C2×C4—684no—
298C2×C4—684no—
299C2×C4C2×C22⋊D1712yes—
300C22—24no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—22no—
305C22—24no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—344no—
310C22—344no—
311C22—344no—
312C22—6816no—
313C22—682no—
314C22—682no—
315C22—682no—
316C22—682no—
317C22#1058762514yes—
318C22C2×D4×D1711yescharacteristic
319C22C2×#6325411yescharacteristic
320C22#41×D1712yes—
321C22#1058769612yes—
322C22C22×C22⋊D1711yescharacteristic
323C4—22no—
324C4—42no—
325C4—42no—
326C4—342no—
327C4—342no—
328C4—681no—
329C4—681no—
330C4—682no—
331C2—24no—
332C2—42no—
333C2—684no—
334C2C2×#1058762511yescharacteristic
335C2#1176331711yescharacteristic
336C2#1176337011yescharacteristic
337C2#1176344712yes—
338C2#1176348212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
prime radicalC2×#10587623C2
hypercenterC2×#123D17
hypocenterC17#2534
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic17C2semidirect
3. C22×#63227C2semidirect
4. #11762731C2semidirect
5. #11762724C2semidirect
6. #11762699C2semidirect
7. #11762686C2semidirect
8. #11762672C2semidirect
9. #11762667C2semidirect
10. #11762664C2semidirect
11. C2×C17×#123C2semidirect
17. C23×Dic17C22semidirect
19. C2×C4⋊C4×C17C22semidirect
21. C2×C22⋊C4×C17C22semidirect
24. C2×C22⋊Dic17C22semidirect
33. C2×C4⋊Dic17C22semidirect
34. C2×#63225C22semidirect
35. C2×C4×Dic17C22semidirect
36. #10587502C22semidirect
37. #10587502C22semidirect
38. #10587502C22semidirect
62. C22×Dic17D4semidirect
162. C2×#123D17semidirect
192. C2×C4⋊C4C2×D17semidirect
203. C2×C22⋊C4C2×D17semidirect
208. C17#2534semidirect

Maximal subgroups

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

subgroupindex
C2×#105876232
C2×D4×Dic172
C22×#632272
#117627312
#117627242
#117626992
#117626862
#117626722
#117626672
#117626642
C2×C17×#1232
#253417