Finite Group Databasebeta

← #14540458

Subgroups — #14540458

Order 2176.

620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 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×D136C211yesmaximal, characteristic, complemented
2C2×D8×D17C211yesmaximal, characteristic, complemented
3#11763443C212yesmaximal, complemented
4#11763184C211yesmaximal, characteristic, complemented
5#11763046C211yesmaximal, characteristic, complemented
6#11763003C212yesmaximal, complemented
7#11762944C212yesmaximal, complemented
8#11762942C212yesmaximal, complemented
9#11762891C211yesmaximal, characteristic, complemented
10C8×C22⋊D17C211yesmaximal, characteristic, complemented
11C17×#466C211yesmaximal, characteristic, complemented
12C2×D4×D17—22no—
13C2×D4×D17C2212yescomplemented
14C22×D68C2212yescomplemented
15C2×D8×C17C2211yescharacteristic, complemented
16C22×C8×C17C2211yescharacteristic, complemented
17C17×#121C2212yescomplemented
18#10587626C2212yescomplemented
19#10587623—22no—
20C2×#63228C2212yescomplemented
21C2×#63228C2212yescomplemented
22C4×C22⋊D17C2211yescharacteristic
23D8×D17—24no—
24C2×D136—24no—
25C2×D136C2211yescharacteristic
26C2×D136C2212yescomplemented
27C2×D136C2211yescharacteristic, complemented
28C2×C8×D17C2211yescharacteristic, complemented
29#10587581C2212yescomplemented
30#10587579C2212yescomplemented
31#10587570—22no—
32#10587568—22no—
33C17×#107C2211yescharacteristic, complemented
34C17×#102C2212yescomplemented
35C22⋊#3622C2211yescharacteristic, complemented
36#10587491C2211yescharacteristic, complemented
37#10587485C2211yescharacteristic, complemented
38C8×Dic17C2211yescharacteristic, complemented
39#10587480C2212yescomplemented
40#10587480C2212yescomplemented
41C23×D17—22no—
42C23×D17—22no—
43C2×D4×C17—22no—
44C2×D4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—22no—
47C2×C22⋊D17—22no—
48C2×C22⋊D17—22no—
49C2×C22⋊D17D411yescharacteristic
50D4×D17—24no—
51D4×D17—42no—
52D4×D17—48no—
53C2×D68—24no—
54C2×D68—22no—
55C2×D68—22no—
56C2×D68—42no—
57C2×D68C2312yes—
58C2×D68C2312yes—
59C2×C4×D17—22no—
60C2×C4×D17C2311yescharacteristic
61D8×C17—22no—
62C2×C8×C17—22no—
63C2×C8×C17C2311yescharacteristic
64C2×C8×C17C2311yescharacteristic
65C4⋊C4×C17C2312yes—
66C22⋊C4×C17—22no—
67C22⋊Dic17D411yescharacteristic
68#63228—24no—
69#63228—42no—
70#63227—22no—
71#63227—22no—
72#63227D411yescharacteristic
73#63225D411yescharacteristic
74C4×Dic17C2311yescharacteristic
75C2×#3622C2311yescharacteristic
76C2×#3622C2311yescharacteristic
77D136—24no—
78D136—22no—
79D136D414yescomplemented
80C8×D17—22no—
81C23×C17—22no—
82C23×C17C2×D411yescharacteristic
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—44no—
86C22×D17—44no—
87C22×D17—44no—
88C22×D17—44no—
89C22×D17—82no—
90C22×D17—82no—
91C22×D17C2×D411yescharacteristic
92D4×C17—22no—
93D4×C17—22no—
94D4×C17—44no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17C2×D411yescharacteristic
99C2×C4×C17C2411yescharacteristic
100C22⋊D17—44no—
101C22⋊D17—44no—
102C22⋊D17—82no—
103C22⋊D17D814yescomplemented
104C2×Dic17C2×D411yescharacteristic
105C2×Dic17C2×D411yescharacteristic
106C2×Dic17C2×D411yescharacteristic
107D68—22no—
108D68—22no—
109D68—22no—
110D68—44no—
111D68—44no—
112D68C2×D414yescomplemented
113C4×D17—21no—
114C4×D17—21no—
115C4×D17—44no—
116C8×C17—21no—
117C8×C17C2×D412yescomplemented
118#3622—21no—
119#3622—21no—
120D4×D8—171nomaximal, Sylow
121C22×C17—21no—
122C22×C17—21no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17C2×D812yescomplemented
127C22×C17C22×D411yescharacteristic
128C2×D17—21no—
129C2×D17—21no—
130C2×D17—24no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—88no—
137C2×D17—88no—
138C2×D17C2×D812yescomplemented
139C4×C17—21no—
140C4×C17—42no—
141C4×C17C22×D411yescharacteristic
142C4×C17H(5,𝔽2)11yescharacteristic
143Dic17—21no—
144Dic17—21no—
145Dic17C2×D812yescomplemented
146C22×D8—171no—
147C22×D8—171no—
148D42—172no—
149#493—171no—
150#466—171no—
151#466D1711yescharacteristic, complemented
152#459—172no—
153#447—172no—
154#447—172no—
155C4×D8—171no—
156C8×D4—171no—
157C2×C17—22no—
158C2×C17—82no—
159C2×C17D4211yescharacteristic
160C2×C17C22×D811yescharacteristic
161C2×C17#57611yescharacteristic
162D17—22no—
163D17—44no—
164D17—82no—
165C22×D4—172no—
166C22×D4—172no—
167C22×D4—342no—
168C2×D8—172no—
169C2×D8—172no—
170C2×D8—171no—
171C2×D8—172no—
172C2×D8—171no—
173C2×D8—344no—
174C2×D8—344no—
175C2×D8C2×D1711yescharacteristic, complemented
176C22×C8—171no—
177C22×C8C2×D1711yescharacteristic, complemented
178#127—172no—
179#121—172no—
180#121—342no—
181#121C2×D1712yescomplemented
182#120—342no—
183#120—342no—
184C4×D4—171no—
185C4×D4—172no—
186#107C2×D1711yescharacteristic, complemented
187C4⋊C8—171no—
188#102—172no—
189#102—172no—
190#102C2×D1712yescomplemented
191C22⋊C8—171no—
192C22⋊C8—171no—
193C4×C8—171no—
194C17D4×D811yescharacteristic, complemented, Sylow
195C24—342no—
196C24—342no—
197C2×D4—22no—
198C2×D4—171no—
199C2×D4—172no—
200C2×D4—172no—
201C2×D4—344no—
202C2×D4—342no—
203C2×D4—342no—
204C2×D4—342no—
205C2×D4—342no—
206C2×D4—344no—
207C2×D4—342no—
208C2×D4—682no—
209C2×D4—682no—
210C2×D4—688no—
211C2×D4C22×D1712yes—
212C22×C4—171no—
213C22×C4—342no—
214C22×C4C22×D1711yescharacteristic
215D8—22no—
216D8—174no—
217D8—344no—
218D8—344no—
219D8—342no—
220D8—682no—
221C2×C8—22no—
222C2×C8—171no—
223C2×C8—171no—
224C2×C8—342no—
225C2×C8C22×D1711yescharacteristic
226C2×C8C22×D1711yescharacteristic
227C4⋊C4—171no—
228C4⋊C4C22×D1712yes—
229C22⋊C4—22no—
230C22⋊C4—171no—
231C22⋊C4—171no—
232C22⋊C4—342no—
233C22⋊C4—342no—
234C42—171no—
235C23—22no—
236C23—171no—
237C23—342no—
238C23—342no—
239C23—684no—
240C23—684no—
241C23—684no—
242C23—684no—
243C23—1362no—
244C23—1362no—
245C23D4×D1711yescharacteristic
246D4—22no—
247D4—22no—
248D4—44no—
249D4—174no—
250D4—174no—
251D4—342no—
252D4—342no—
253D4—342no—
254D4—684no—
255D4—684no—
256D4—684no—
257D4—684no—
258D4—1362no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—171no—
263C2×C4—171no—
264C2×C4—171no—
265C2×C4—341no—
266C2×C4—341no—
267C2×C4—684no—
268C2×C4D4×D1711yescharacteristic
269C2×C4C23×D1711yescharacteristic
270C8—21no—
271C8—341no—
272C8—341no—
273C8D4×D1712yescomplemented
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—172no—
280C22—341no—
281C22—341no—
282C22—344no—
283C22—682no—
284C22—682no—
285C22—682no—
286C22—682no—
287C22—682no—
288C22—1368no—
289C22—1368no—
290C22D8×D1712yescomplemented
291C22C2×D4×D1711yescharacteristic
292C4—21no—
293C4—42no—
294C4—172no—
295C4—341no—
296C4—341no—
297C4C2×D4×D1711yescharacteristic
298C4#1058769611yescharacteristic
299C2—22no—
300C2—82no—
301C2—342no—
302C2—684no—
303C2—1362no—
304C2#1176344311yescharacteristic
305C2C2×D8×D1711yescharacteristic
306C2#1176361111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#466C2
socleC22×C17C22×D4
hypercenter#466D17
hypocenterC17D4×D8
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C22×D136C2semidirect
2. C2×D8×D17C2semidirect
3. #11763443C2semidirect
4. #11763184C2semidirect
5. #11763046C2semidirect
6. #11763003C2semidirect
7. #11762944C2semidirect
8. #11762942C2semidirect
9. #11762891C2semidirect
10. C8×C22⋊D17C2semidirect
11. C17×#466C2semidirect
13. C2×D4×D17C22semidirect
14. C22×D68C22semidirect
15. C2×D8×C17C22semidirect
16. C22×C8×C17C22semidirect
17. C17×#121C22semidirect
18. #10587626C22semidirect
20. C2×#63228C22semidirect
21. C2×#63228C22semidirect
26. C2×D136C22semidirect
27. C2×D136C22semidirect
28. C2×C8×D17C22semidirect
29. #10587581C22semidirect
30. #10587579C22semidirect
33. C17×#107C22semidirect
34. C17×#102C22semidirect
35. C22⋊#3622C22semidirect
36. #10587491C22semidirect
37. #10587485C22semidirect
38. C8×Dic17C22semidirect
39. #10587480C22semidirect
40. #10587480C22semidirect
79. D136D4semidirect
103. C22⋊D17D8semidirect
112. D68C2×D4semidirect
117. C8×C17C2×D4semidirect
126. C22×C17C2×D8semidirect
138. C2×D17C2×D8semidirect
145. Dic17C2×D8semidirect
151. #466D17semidirect
175. C2×D8C2×D17semidirect
177. C22×C8C2×D17semidirect
181. #121C2×D17semidirect
186. #107C2×D17semidirect
190. #102C2×D17semidirect
194. C17D4×D8semidirect
273. C8D4×D17semidirect
290. C22D8×D17semidirect

Maximal subgroups

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

subgroupindex
C22×D1362
C2×D8×D172
#117634432
#117631842
#117630462
#117630032
#117629442
#117629422
#117628912
C8×C22⋊D172
C17×#4662
D4×D817