Finite Group Databasebeta

← #14540591

Subgroups — #14540591

Order 2176.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D68C211yesmaximal, characteristic, complemented
2#11763157C211yesmaximal, characteristic, complemented
3#11763122C212yesmaximal, complemented
4#11763086C211yesmaximal, characteristic, complemented
5#11763043C212yesmaximal, complemented
6#11763004C211yesmaximal, characteristic, complemented
7#11762942C212yesmaximal, complemented
8C4×#63228C211yesmaximal, characteristic, complemented
9D4×#3622C211yesmaximal, characteristic, complemented
10C2×#10587480C212yesmaximal, complemented
11C17×#547C211yesmaximal, characteristic, complemented
12C2×#41×C17—21no—
13C2×D4×D17—21no—
14C22×D68C2212yes—
15C17×C4⋊Q8C2211yescharacteristic, complemented
16C17×#123—22no—
17C17×C22⋊Q8—21no—
18C17×#121C2212yescomplemented
19C4×D4×C17C2211yescharacteristic
20C4×D4×C17C2211yescharacteristic, complemented
21C2×C4⋊C4×C17C2212yescomplemented
22C2×#63228C2211yescharacteristic
23C2×#63228C2212yescomplemented
24#10587616C2212yes—
25C22×#3622C2212yescomplemented
26#10587581—21no—
27#10587568—22no—
28#10587560C2211yescharacteristic
29C4×D68C2211yescharacteristic
30#10587503C2211yescharacteristic, complemented
31C22⋊#3622C2212yescomplemented
32#10587480C2211yescharacteristic
33#10587480C2212yes—
34#10587480C2212yescomplemented
35#10587480C2212yescomplemented
36#10587480C2212yescomplemented
37#10587478C2212yescomplemented
38#10587478C2211yescharacteristic, complemented
39C4⋊#3622C2211yescharacteristic, complemented
40C4×#3622C2211yescharacteristic, complemented
41C23×D17—22no—
42#41×C17—44no—
43C2×Q8×C17—21no—
44C2×D4×C17—22no—
45C2×D4×C17D411yescharacteristic
46C2×D4×C17C2311yescharacteristic
47C22×C4×C17—22no—
48C22×C4×C17—21no—
49C22×C4×C17C2312yes—
50C2×C22⋊D17—22no—
51D4×D17—44no—
52C2×D68—24no—
53C2×D68—22no—
54C2×D68—21no—
55C2×D68—21no—
56C2×D68—42no—
57C2×D68C2312yes—
58C2×D68C2311yescharacteristic
59C2×C4×D17—21no—
60C4⋊C4×C17—22no—
61C4⋊C4×C17—21no—
62C4⋊C4×C17D411yescharacteristic
63C4⋊C4×C17C2312yes—
64C4⋊C4×C17C2311yescharacteristic
65C22⋊C4×C17—22no—
66C22⋊C4×C17—21no—
67C22⋊C4×C17D412yes—
68C42×C17C2311yescharacteristic
69#63228—22no—
70#63228—42no—
71#63227—21no—
72#63227—22no—
73C4⋊Dic17C2311yescharacteristic
74C2×#3622—42no—
75C2×#3622C2311yescharacteristic
76C2×#3622C2311yescharacteristic
77C2×#3622C2312yes—
78C23×C17—21no—
79C23×C17C2×D412yes—
80C22×D17—21no—
81C22×D17—22no—
82C22×D17—44no—
83C22×D17—44no—
84C22×D17—82no—
85Q8×C17—42no—
86D4×C17—21no—
87D4×C17—21no—
88D4×C17—44no—
89D4×C17D814yescomplemented
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—44no—
95C2×C4×C17—42no—
96C2×C4×C17C2×D411yescharacteristic
97C2×C4×C17C2×D412yes—
98C2×C4×C17C2×D411yescharacteristic
99C2×C4×C17C2411yescharacteristic
100C22⋊D17—44no—
101C2×Dic17—21no—
102D68—22no—
103D68—21no—
104D68—22no—
105D68—44no—
106D68—42no—
107D68—81no—
108D68#4112yescomplemented
109C4×D17—42no—
110#3622—21no—
111#3622—21no—
112#3622—22no—
113#3294—171nomaximal, Sylow
114C22×C17—22no—
115C22×C17—22no—
116C22×C17—41no—
117C22×C17—41no—
118C22×C17—41no—
119C22×C17C2×D814yes—
120C22×C17C22×D411yescharacteristic
121C2×D17—22no—
122C2×D17—41no—
123C2×D17—41no—
124C2×D17—42no—
125C2×D17—42no—
126C2×D17—42no—
127C2×D17—88no—
128C4×C17—22no—
129C4×C17—21no—
130C4×C17—41no—
131C4×C17—42no—
132C4×C17C2×D812yes—
133C4×C17C2×#4111yescharacteristic
134C4×C17H(5,𝔽2)11yescharacteristic
135Dic17—41no—
136#547D1711yescharacteristic, complemented
137D42—171no—
138#486—171no—
139#480—172no—
140#474—171no—
141#466—172no—
142#459—171no—
143#447—172no—
144C4×D8—171no—
145C8×D4—171no—
146C2×#102—172no—
147C2×C17—24no—
148C2×C17—81no—
149C2×C17#54611yescharacteristic
150C2×C17C22×D811yescharacteristic
151C2×C17#57711yescharacteristic
152D17—42no—
153D17—82no—
154C2×#41—21no—
155C22×D4—172no—
156C22×D4—341no—
157C2×D8—171no—
158C2×D8—172no—
159C22×C8—172no—
160C4⋊Q8C2×D1711yescharacteristic, complemented
161#127—171no—
162#123—22no—
163C22⋊Q8—21no—
164#121—172no—
165#121—341no—
166#121C2×D1712yescomplemented
167#120—342no—
168C4×D4—171no—
169C4×D4C2×D1711yescharacteristic
170C4×D4C2×D1711yescharacteristic, complemented
171C2×C4⋊C4C2×D1712yescomplemented
172#107—171no—
173#107—172no—
174C4⋊C8—171no—
175#102—171no—
176#102—171no—
177#102—172no—
178#102—172no—
179#102—172no—
180#102—172no—
181C22⋊C8—172no—
182C4×C8—171no—
183C17#329411yescharacteristic, complemented, Sylow
184C24—342no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—171no—
189C2×D4—172no—
190C2×D4—344no—
191C2×D4—342no—
192C2×D4—342no—
193C2×D4—341no—
194C2×D4—341no—
195C2×D4—682no—
196C2×D4—684no—
197C2×D4C22⋊D1711yescharacteristic
198C2×D4C22×D1711yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—341no—
202C22×C4C22×D1712yes—
203D8—342no—
204D8—682no—
205C2×C8—171no—
206C2×C8—171no—
207C2×C8—172no—
208C2×C8—682no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—171no—
212C4⋊C4C22⋊D1711yescharacteristic
213C4⋊C4C22×D1711yescharacteristic
214C4⋊C4C22×D1712yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—341no—
218C22⋊C4—342no—
219C22⋊C4C22⋊D1712yes—
220C42C22×D1711yescharacteristic
221C23—21no—
222C23—341no—
223C23—342no—
224C23—684no—
225C23—684no—
226C23—1362no—
227C23C2×C22⋊D1712yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—172no—
233D4—342no—
234D4—341no—
235D4—342no—
236D4—684no—
237D4—684no—
238D4—682no—
239D4—1361no—
240D4#6322814yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—341no—
248C2×C4—682no—
249C2×C4C2×C22⋊D1711yescharacteristic
250C2×C4C2×C22⋊D1712yes—
251C2×C4C2×C22⋊D1711yescharacteristic
252C2×C4C23×D1711yescharacteristic
253C8—341no—
254C8—341no—
255C8—342no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—342no—
262C22—681no—
263C22—681no—
264C22—682no—
265C22—682no—
266C22—682no—
267C22—1368no—
268C22C2×#6322814yes—
269C22C22×C22⋊D1711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—681no—
275C4C2×#6322812yes—
276C4#41×D1711yescharacteristic
277C4#1058769611yescharacteristic
278C2—24no—
279C2—81no—
280C2—682no—
281C2—1362no—
282C2#1176344711yescharacteristic
283C2C22×#6322811yescharacteristic
284C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#547C2
socleC22×C17C22×D4
hypercenter#547D17
hypocenterC17#3294
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D68C2semidirect
2. #11763157C2semidirect
3. #11763122C2semidirect
4. #11763086C2semidirect
5. #11763043C2semidirect
6. #11763004C2semidirect
7. #11762942C2semidirect
8. C4×#63228C2semidirect
9. D4×#3622C2semidirect
10. C2×#10587480C2semidirect
11. C17×#547C2semidirect
15. C17×C4⋊Q8C22semidirect
18. C17×#121C22semidirect
20. C4×D4×C17C22semidirect
21. C2×C4⋊C4×C17C22semidirect
23. C2×#63228C22semidirect
25. C22×#3622C22semidirect
30. #10587503C22semidirect
31. C22⋊#3622C22semidirect
34. #10587480C22semidirect
35. #10587480C22semidirect
36. #10587480C22semidirect
37. #10587478C22semidirect
38. #10587478C22semidirect
39. C4⋊#3622C22semidirect
40. C4×#3622C22semidirect
89. D4×C17D8semidirect
108. D68#41semidirect
136. #547D17semidirect
160. C4⋊Q8C2×D17semidirect
166. #121C2×D17semidirect
170. C4×D4C2×D17semidirect
171. C2×C4⋊C4C2×D17semidirect
183. C17#3294semidirect
240. D4#63228semidirect

Maximal subgroups

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

subgroupindex
D4×D682
#117631572
#117631222
#117630862
#117630432
#117630042
#117629422
C4×#632282
D4×#36222
C2×#105874802
C17×#5472
#329417