Finite Group Databasebeta

← #14539749

Subgroups — #14539749

Order 2176.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#11763343C211yesmaximal, characteristic, complemented
2#11763191C212yesmaximal, complemented
3#11763163C212yesmaximal, complemented
4#11763028C212yesmaximal, complemented
5#11763021C212yesmaximal, complemented
6#11762965C211yesmaximal, characteristic, complemented
7#11762960C211yesmaximal, characteristic, complemented
8#11762948C211yesmaximal, characteristic, complemented
9#11762943C211yesmaximal, characteristic
10#11762862C211yesmaximal, characteristic, complemented
11C2×C17×#124C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C17×#124—21no—
15C17×#124—21no—
16C17×#124C2212yes—
17C17×#124C2212yes—
18C2×C22⋊C4×C17—21no—
19C2×C22⋊C4×C17—21no—
20C2×C42×C17C2211yescharacteristic
21C2×#63231C2212yescomplemented
22C2×#63230C2212yescomplemented
23C2×#63229C2212yes—
24C2×#63228C2212yes—
25#10587616C2211yescharacteristic
26#10587615—21no—
27C4×C22⋊D17—21no—
28#10587610C2211yescharacteristic
29#10587562—21no—
30#10587561C2211yescharacteristic, complemented
31C4×D68C2211yescharacteristic, complemented
32#10587558—21no—
33#10587556C2211yescharacteristic, complemented
34C4×Dic34C2211yescharacteristic, complemented
35#10587506C2212yescomplemented
36#10587506C2212yescomplemented
37#10587503C2212yes—
38#10587503C2212yes—
39C22⋊#3622C2211yescharacteristic
40C22⋊#3622C2211yescharacteristic
41C4⋊#3622C2212yescomplemented
42C4×#3622C2212yescomplemented
43C24×C17—21no—
44C2×Q8×C17—21no—
45C2×Q8×C17—42no—
46C2×Q8×C17C2312yes—
47C2×D4×C17—21no—
48C2×D4×C17—42no—
49C2×D4×C17C2312yes—
50C22×C4×C17—21no—
51C22×C4×C17D411yescharacteristic
52C22×C4×C17D411yescharacteristic
53C22×C4×C17C2311yescharacteristic
54C2×C22⋊D17—21no—
55C2×D68C2311yescharacteristic
56C2×C4×D17—21no—
57C2×Dic34C2311yescharacteristic
58C22⋊C4×C17—22no—
59C22⋊C4×C17—22no—
60C22⋊C4×C17—22no—
61C22⋊C4×C17—22no—
62C42×C17D411yescharacteristic
63C42×C17D411yescharacteristic
64C42×C17C2311yescharacteristic
65C42×C17C2311yescharacteristic
66C22⋊Dic17—21no—
67#63231—42no—
68#63230—42no—
69#63229—42no—
70#63228—42no—
71#63227—21no—
72#63227—21no—
73#63227—21no—
74#63227—21no—
75C4⋊Dic17C2311yescharacteristic
76C4⋊Dic17C2311yescharacteristic
77#63225—21no—
78#63225—21no—
79#63225—21no—
80#63225—21no—
81C4×Dic17—21no—
82C2×#3622C2312yes—
83C2×#3622C2312yes—
84C23×C17—22no—
85C23×C17—42no—
86C23×C17—42no—
87C23×C17—42no—
88C23×C17C2×D411yescharacteristic
89C22×D17—21no—
90Q8×C17—22no—
91Q8×C17—22no—
92Q8×C17—42no—
93D4×C17—22no—
94D4×C17—22no—
95D4×C17—42no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—41no—
100C2×C4×C17—42no—
101C2×C4×C17C2×D411yescharacteristic
102C2×C4×C17C2×D411yescharacteristic
103C2×C4×C17C2×D411yescharacteristic
104C2×C4×C17C2×D411yescharacteristic
105C2×C4×C17C2×D411yescharacteristic
106C2×C4×C17C2411yescharacteristic
107C22⋊D17—42no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111D68—22no—
112C4×D17—42no—
113Dic34—22no—
114#3622—22no—
115#3622—22no—
116#3211—171nomaximal, Sylow
117C22×C17—21no—
118C22×C17—21no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—84no—
123C22×C17#13512yes—
124C22×C17C22×D411yescharacteristic
125C2×D17—41no—
126C2×D17—41no—
127C2×D17—41no—
128C4×C17—21no—
129C4×C17—21no—
130C4×C17—21no—
131C4×C17—21no—
132C4×C17—21no—
133C4×C17—42no—
134C4×C17H(5,𝔽2)12yes—
135Dic17—41no—
136Dic17—41no—
137Dic17—41no—
138#529—171no—
139C2×#124D1711yescharacteristic, complemented
140#495—172no—
141#488—172no—
142#464—172no—
143#463—172no—
144#451—171no—
145#450—171no—
146#448—171no—
147#447—171no—
148#431—171no—
149C2×C17—22no—
150C2×C17—82no—
151C2×C17#53511yescharacteristic
152C2×C17C2×#13511yescharacteristic
153C2×C17#57511yescharacteristic
154D17—81no—
155C22×Q8C2×D1711yescharacteristic
156C22×D4C2×D1711yescharacteristic
157C2×Q16—172no—
158C2×SD16—172no—
159C2×SD16—172no—
160C2×D8—172no—
161#126—341no—
162#125—171no—
163#124—21no—
164#124—21no—
165#124—171no—
166#124C2×D1712yes—
167#124C2×D1712yes—
168#123—341no—
169C22⋊Q8—171no—
170#121—171no—
171C4×Q8—171no—
172C4×D4—171no—
173C4×D4—341no—
174#117—341no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C42C2×D1711yescharacteristic
178C4⋊C8—172no—
179#103—172no—
180#103—172no—
181#102—172no—
182#102—172no—
183C22⋊C8—171no—
184C22⋊C8—171no—
185C4×C8—172no—
186C17#321111yescharacteristic, complemented, Sylow
187C24—21no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8—171no—
191C2×Q8C22×D1712yes—
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—171no—
195C2×D4—341no—
196C2×D4C22×D1712yes—
197C22×C4—21no—
198C22×C4—341no—
199C22×C4C22⋊D1711yescharacteristic
200C22×C4C22⋊D1711yescharacteristic
201C22×C4C22×D1711yescharacteristic
202Q16—682no—
203SD16—682no—
204SD16—682no—
205D8—682no—
206C2×C8—172no—
207C2×C8—172no—
208C4⋊C4—171no—
209C4⋊C4—171no—
210C4⋊C4—341no—
211C4⋊C4—341no—
212C4⋊C4—341no—
213C4⋊C4—341no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—341no—
219C22⋊C4—341no—
220C22⋊C4—341no—
221C22⋊C4—341no—
222C22⋊C4—341no—
223C42—341no—
224C42C22⋊D1711yescharacteristic
225C42C22⋊D1711yescharacteristic
226C42C22×D1711yescharacteristic
227C42C22×D1711yescharacteristic
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—341no—
233C23C2×C22⋊D1711yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—42no—
237Q8—342no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—342no—
242D4—682no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—341no—
249C2×C4—341no—
250C2×C4—341no—
251C2×C4—682no—
252C2×C4C2×C22⋊D1711yescharacteristic
253C2×C4C2×C22⋊D1711yescharacteristic
254C2×C4C2×C22⋊D1711yescharacteristic
255C2×C4C2×C22⋊D1711yescharacteristic
256C2×C4C2×C22⋊D1711yescharacteristic
257C2×C4C23×D1711yescharacteristic
258C8—342no—
259C8—342no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—681no—
267C22—681no—
268C22—681no—
269C22#1058763612yes—
270C22C22×C22⋊D1711yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—42no—
277C4—681no—
278C4—681no—
279C4—681no—
280C4#1058769112yes—
281C2—22no—
282C2—82no—
283C2—1361no—
284C2#1176337611yescharacteristic
285C2C2×#1058763611yescharacteristic
286C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#124C2
socleC22×C17C22×D4
prime radical#11762943C2
hypercenterC2×#124D17
hypocenterC17#3211
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #11763343C2semidirect
2. #11763191C2semidirect
3. #11763163C2semidirect
4. #11763028C2semidirect
5. #11763021C2semidirect
6. #11762965C2semidirect
7. #11762960C2semidirect
8. #11762948C2semidirect
10. #11762862C2semidirect
11. C2×C17×#124C2semidirect
21. C2×#63231C22semidirect
22. C2×#63230C22semidirect
30. #10587561C22semidirect
31. C4×D68C22semidirect
33. #10587556C22semidirect
34. C4×Dic34C22semidirect
35. #10587506C22semidirect
36. #10587506C22semidirect
41. C4⋊#3622C22semidirect
42. C4×#3622C22semidirect
139. C2×#124D17semidirect
186. C17#3211semidirect

Maximal subgroups

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

subgroupindex
#117633432
#117631912
#117631632
#117630282
#117630212
#117629652
#117629602
#117629482
#117629432
#117628622
C2×C17×#1242
#321117