Finite Group Databasebeta

← #14538734

Subgroups — #14538734

Order 2176.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#10587583C211yesmaximal, characteristic
2#117×D17C211yesmaximal, characteristic
3#11763153C212yesmaximal, complemented
4#11763152C212yesmaximal, complemented
5#11763146C212yesmaximal, complemented
6#11763141C212yesmaximal, complemented
7#11762849C211yesmaximal, characteristic, complemented
8#11762819C211yesmaximal, characteristic, complemented
9#11762817C211yesmaximal, characteristic, complemented
10#11762772C211yesmaximal, characteristic
11C17×#428C211yesmaximal, characteristic, complemented
12C22×C4×D17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C2×M16×C17C2211yescharacteristic
15C17×#117C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×#63227—21no—
18#10587612C2211yescharacteristic
19C2×C4⋊Dic17—21no—
20C2×#63223C2211yescharacteristic
21#10587583—21no—
22#10587583—21no—
23#10587583C2212yes—
24#10587583C2212yes—
25#10587578C2211yescharacteristic
26#10587578C2211yescharacteristic
27C4⋊C4×D17C2211yescharacteristic
28C4⋊C4×D17C2211yescharacteristic
29C22⋊C4×D17—21no—
30#10587563—21no—
31#10587558—21no—
32C42×D17—21no—
33C17×#107C2212yescomplemented
34C17×#106C2212yescomplemented
35#10587491C2212yes—
36#10587491C2212yes—
37#10587487C2212yescomplemented
38#10587487C2212yescomplemented
39#10587481C2212yescomplemented
40#10587481C2212yescomplemented
41#10587479C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17D411yescharacteristic
44C22×C4×C17—21no—
45C22×C4×C17C2311yescharacteristic
46C22×Dic17—21no—
47C22×Dic17D411yescharacteristic
48C2×C4×D17—22no—
49C2×C4×D17—22no—
50C2×C4×D17—21no—
51C2×C4×D17—21no—
52C2×C4×D17D411yescharacteristic
53C2×C4×D17D411yescharacteristic
54C2×C4×D17C2311yescharacteristic
55C2×C4×D17C2311yescharacteristic
56C2×Dic34—22no—
57C2×Dic34—22no—
58C2×Dic34—21no—
59C2×Dic34C2312yes—
60M16×C17—22no—
61C2×C8×C17C2312yes—
62C4⋊C4×C17—21no—
63C4⋊C4×C17C2312yes—
64C4⋊C4×C17C2311yescharacteristic
65C4⋊C4×C17C2311yescharacteristic
66C22⋊C4×C17—21no—
67C42×C17—21no—
68C22⋊Dic17—21no—
69#63227—21no—
70#63227—21no—
71#63227—22no—
72#63227—22no—
73C4⋊Dic17—22no—
74C4⋊Dic17—22no—
75C4⋊Dic17C2311yescharacteristic
76C4⋊Dic17C2311yescharacteristic
77#63225—21no—
78#63225—21no—
79C4×Dic17—21no—
80C4×Dic17—21no—
81C4×Dic17—21no—
82#63223—22no—
83C2×#3622C2312yes—
84C23×C17C2×D411yescharacteristic
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—41no—
88C22×D17C2×D411yescharacteristic
89C22×D17C2×D411yescharacteristic
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—42no—
94C2×C4×C17C2×D411yescharacteristic
95C2×C4×C17#4112yes—
96C2×C4×C17#4112yes—
97C2×C4×C17C2411yescharacteristic
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—22no—
102C2×Dic17—42no—
103C2×Dic17C2×D411yescharacteristic
104C2×Dic17C2×D411yescharacteristic
105C4×D17—22no—
106C4×D17—22no—
107C4×D17—21no—
108C4×D17—21no—
109C4×D17—21no—
110C4×D17—21no—
111C4×D17—42no—
112C4×D17—42no—
113Dic34—22no—
114Dic34—22no—
115Dic34—42no—
116C8×C17—22no—
117#3622—22no—
118#3095—171nomaximal, Sylow
119C22×C17—21no—
120C22×C17—21no—
121C22×C17#12312yes—
122C22×C17C22×D411yescharacteristic
123C2×D17—21no—
124C2×D17—21no—
125C2×D17—21no—
126C2×D17—21no—
127C2×D17—21no—
128C2×D17—44no—
129C2×D17#13712yes—
130C4×C17—41no—
131C4×C17—41no—
132C4×C17—42no—
133C4×C17#12312yes—
134C4×C17C2×#4111yescharacteristic
135C4×C17C2×#4111yescharacteristic
136Dic17—21no—
137Dic17—21no—
138Dic17—42no—
139Dic17—41no—
140Dic17—41no—
141C2×C22⋊Q8—171no—
142C2×#117—171no—
143#485—172no—
144#485—172no—
145#484—172no—
146#483—172no—
147#428—171no—
148#428D1711yescharacteristic, complemented
149#419—171no—
150#419—171no—
151#407—171no—
152C2×C17—22no—
153C2×C17C2×#12311yescharacteristic
154C2×C17C2×#13711yescharacteristic
155C2×C17#57511yescharacteristic
156D17—22no—
157D17—41no—
158C22×Q8—171no—
159C23×C4—171no—
160C2×M16—171no—
161C2×M16C2×D1711yescharacteristic
162C22⋊Q8—172no—
163C22⋊Q8—172no—
164C22⋊Q8—341no—
165C22⋊Q8—341no—
166#117—171no—
167#117—171no—
168#117—171no—
169#117—341no—
170#117—341no—
171#117C2×D1711yescharacteristic
172C2×C4⋊C4—171no—
173C2×C4⋊C4—171no—
174C2×C4⋊C4—341no—
175C2×C4⋊C4C2×D1711yescharacteristic
176C2×C22⋊C4—341no—
177C2×C22⋊C4—341no—
178C2×C42—341no—
179#107—172no—
180#107C2×D1712yescomplemented
181#106—172no—
182#106C2×D1712yescomplemented
183#103—172no—
184#103—172no—
185#103—172no—
186#103—172no—
187C22⋊C8—172no—
188C22⋊C8—172no—
189C17#309511yescharacteristic, complemented, Sylow
190C24—171no—
191C2×Q8—172no—
192C2×Q8—341no—
193C2×Q8—342no—
194C2×Q8—342no—
195C22×C4—21no—
196C22×C4—171no—
197C22×C4—171no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—171no—
201C22×C4—341no—
202C22×C4—341no—
203C22×C4—342no—
204C22×C4—342no—
205C22×C4—341no—
206C22×C4C22×D1711yescharacteristic
207M16—22no—
208M16—342no—
209C2×C8—172no—
210C2×C8C22×D1712yes—
211C4⋊C4—21no—
212C4⋊C4—171no—
213C4⋊C4—171no—
214C4⋊C4—341no—
215C4⋊C4—341no—
216C4⋊C4—342no—
217C4⋊C4—342no—
218C4⋊C4C22×D1711yescharacteristic
219C4⋊C4C22×D1711yescharacteristic
220C4⋊C4C22×D1712yes—
221C22⋊C4—21no—
222C22⋊C4—341no—
223C22⋊C4—341no—
224C22⋊C4—342no—
225C22⋊C4—342no—
226C22⋊C4—341no—
227C42—21no—
228C42—341no—
229C42—341no—
230C42—341no—
231C23—171no—
232C23—171no—
233C23—342no—
234C23—342no—
235C23—681no—
236C23D4×D1711yescharacteristic
237Q8—342no—
238Q8—342no—
239Q8—682no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—171no—
245C2×C4—171no—
246C2×C4—342no—
247C2×C4—341no—
248C2×C4—341no—
249C2×C4—341no—
250C2×C4—341no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—341no—
254C2×C4—341no—
255C2×C4—342no—
256C2×C4—682no—
257C2×C4—682no—
258C2×C4—682no—
259C2×C4#6325212yes—
260C2×C4D4×D1711yescharacteristic
261C2×C4#6325612yes—
262C2×C4C23×D1711yescharacteristic
263C8—22no—
264C8—342no—
265C22—21no—
266C22—21no—
267C22—172no—
268C22—341no—
269C22—341no—
270C22—341no—
271C22—341no—
272C22—341no—
273C22—684no—
274C22#1058758012yes—
275C22C2×D4×D1711yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—341no—
280C4—341no—
281C4—682no—
282C4—681no—
283C4—681no—
284C4#1058758012yes—
285C4C2×#6325211yescharacteristic
286C4C2×#6325611yescharacteristic
287C2—22no—
288C2—342no—
289C2—681no—
290C2C2×#1058758011yescharacteristic
291C2#137×D1711yescharacteristic
292C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#428C2
socleC22×C17C22×D4
prime radicalC23×D17D4
hypercenter#428D17
hypocenterC17#3095
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #11763153C2semidirect
4. #11763152C2semidirect
5. #11763146C2semidirect
6. #11763141C2semidirect
7. #11762849C2semidirect
8. #11762819C2semidirect
9. #11762817C2semidirect
11. C17×#428C2semidirect
33. C17×#107C22semidirect
34. C17×#106C22semidirect
37. #10587487C22semidirect
38. #10587487C22semidirect
39. #10587481C22semidirect
40. #10587481C22semidirect
41. #10587479C22semidirect
42. #10587478C22semidirect
148. #428D17semidirect
180. #107C2×D17semidirect
182. #106C2×D17semidirect
189. C17#3095semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
#117×D172
#117631532
#117631522
#117631462
#117631412
#117628492
#117628192
#117628172
#117627722
C17×#4282
#309517