Finite Group Databasebeta

← #14538708

Subgroups — #14538708

Order 2176.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#11763322C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
3#11763143C212yesmaximal, complemented
4#11763137C211yesmaximal, characteristic, complemented
5#11763129C212yesmaximal, complemented
6#11763123C211yesmaximal, characteristic, complemented
7#11763122C211yesmaximal, characteristic, complemented
8#11762848C212yesmaximal, complemented
9#11762809C212yesmaximal, complemented
10#11762771C211yesmaximal, characteristic, complemented
11C17×#483C211yesmaximal, characteristic, complemented
12C2×#63256C2211yescharacteristic
13C22×C4×D17—21no—
14C22×C4×D17C2211yescharacteristic
15C17×C22⋊Q8C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17#10587632C2211yescharacteristic
18#10587616C2211yescharacteristic
19C2×C4⋊Dic17C2211yescharacteristic
20C2×#63225—21no—
21C2×#63218C2212yes—
22#10587581C2211yescharacteristic
23#10587580—21no—
24#10587579—21no—
25#10587578C2211yescharacteristic
26C4⋊C4×D17—21no—
27C4⋊C4×D17—21no—
28C4⋊C4×D17C2212yes—
29C4⋊C4×D17C2212yes—
30#10587568—21no—
31#10587567—21no—
32C17×#106C2212yescomplemented
33C17×#103C2212yescomplemented
34C17×C22⋊C8C2211yescharacteristic
35#10587506C2212yescomplemented
36C22⋊#3622C2211yescharacteristic
37#10587492C2212yescomplemented
38#10587491C2212yes—
39#10587489C2212yescomplemented
40#10587480C2212yescomplemented
41#10587479C2212yescomplemented
42#10587478C2212yescomplemented
43C23×D17D411yescharacteristic
44C2×Q8×C17C2311yescharacteristic
45C22×C4×C17—21no—
46C22×C4×C17C2311yescharacteristic
47C2×C22⋊D17—21no—
48C22×Dic17—21no—
49C22×Dic17D411yescharacteristic
50#63256—24no—
51C2×D68—21no—
52C2×D68—21no—
53C2×D68C2311yescharacteristic
54C2×C4×D17—24no—
55C2×C4×D17—22no—
56C2×C4×D17—22no—
57C2×C4×D17—21no—
58C2×C4×D17—44no—
59C2×C4×D17D411yescharacteristic
60C2×C4×D17D411yescharacteristic
61C2×C4×D17C2311yescharacteristic
62C2×C4×D17C2311yescharacteristic
63C2×C8×C17C2312yes—
64C4⋊C4×C17—21no—
65C4⋊C4×C17—21no—
66C4⋊C4×C17C2311yescharacteristic
67C4⋊C4×C17C2312yes—
68C22⋊C4×C17—21no—
69#63227—21no—
70#63227—21no—
71#63227—21no—
72#63227—21no—
73C4⋊Dic17—21no—
74C4⋊Dic17C2311yescharacteristic
75C4⋊Dic17C2312yes—
76#63225—21no—
77#63225—22no—
78#63225—22no—
79C4×Dic17—21no—
80C2×#3622C2312yes—
81#63218—24no—
82C23×C17C2×D411yescharacteristic
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—21no—
87C22×D17C2×D411yescharacteristic
88C22×D17C2×D411yescharacteristic
89Q8×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—42no—
95C2×C4×C17C2×D411yescharacteristic
96C2×C4×C17C2411yescharacteristic
97C22⋊D17—42no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—21no—
101C2×Dic17—42no—
102C2×Dic17C2×D411yescharacteristic
103C2×Dic17C2×D411yescharacteristic
104D68—22no—
105D68—42no—
106D68—42no—
107C4×D17—22no—
108C4×D17—22no—
109C4×D17—22no—
110C4×D17—44no—
111C4×D17—44no—
112C4×D17—42no—
113C4×D17#4114yes—
114C8×C17—22no—
115#3622—22no—
116#3092—171nomaximal, Sylow
117C22×C17—21no—
118C22×C17—21no—
119C22×C17AGL(1,ℤ/8)12yes—
120C22×C17C22×D411yescharacteristic
121C2×D17—28no—
122C2×D17—21no—
123C2×D17—21no—
124C2×D17—21no—
125C2×D17—21no—
126C2×D17—41no—
127C2×D17—41no—
128C2×D17—41no—
129C2×D17#12312yes—
130C2×D17#13712yes—
131C4×C17—21no—
132C4×C17—41no—
133C4×C17—42no—
134C4×C17—41no—
135C4×C17C2×#4112yes—
136Dic17—21no—
137Dic17—42no—
138Dic17—41no—
139Dic17#12312yes—
140#525—171no—
141C22×C4⋊C4—171no—
142#484—172no—
143#483—171no—
144#483D1711yescharacteristic, complemented
145#481—172no—
146#480—171no—
147#480—171no—
148#428—172no—
149#417—172no—
150#407—171no—
151C2×C17—22no—
152C2×C17C2×#12311yescharacteristic
153C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C17C2×#13711yescharacteristic
155D17—22no—
156D17—22no—
157D17—81no—
158C2×#41—171no—
159C23×C4—171no—
160C23×C4—341no—
161C2×M16—172no—
162#123—341no—
163C22⋊Q8—171no—
164C22⋊Q8C2×D1711yescharacteristic
165#121—171no—
166#121—171no—
167#120—341no—
168C4×D4—341no—
169C4×D4—341no—
170#117—171no—
171C2×C4⋊C4—171no—
172C2×C4⋊C4—172no—
173C2×C4⋊C4—172no—
174C2×C4⋊C4—341no—
175C2×C4⋊C4—341no—
176C2×C4⋊C4—341no—
177C2×C4⋊C4C2×D1711yescharacteristic
178#107—172no—
179#107—172no—
180#106—172no—
181#106C2×D1712yescomplemented
182#103—172no—
183#103C2×D1712yescomplemented
184#102—172no—
185#102—172no—
186C22⋊C8—171no—
187C22⋊C8—172no—
188C22⋊C8C2×D1711yescharacteristic
189C17#309211yescharacteristic, complemented, Sylow
190C24—171no—
191#41—344no—
192C2×Q8C22×D1711yescharacteristic
193C2×D4—171no—
194C2×D4—341no—
195C2×D4—341no—
196C2×D4—341no—
197C22×C4—21no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—171no—
201C22×C4—171no—
202C22×C4—171no—
203C22×C4—344no—
204C22×C4—341no—
205C22×C4—342no—
206C22×C4—342no—
207C22×C4—341no—
208C22×C4—684no—
209C22×C4C22×D1711yescharacteristic
210M16—344no—
211C2×C8—172no—
212C2×C8C22×D1712yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—172no—
216C4⋊C4—171no—
217C4⋊C4—342no—
218C4⋊C4—342no—
219C4⋊C4—341no—
220C4⋊C4—341no—
221C4⋊C4C22×D1712yes—
222C4⋊C4C22×D1711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—341no—
225C22⋊C4—341no—
226C22⋊C4—341no—
227C22⋊C4—341no—
228C42—341no—
229C23—171no—
230C23—171no—
231C23—342no—
232C23—342no—
233C23—342no—
234C23—341no—
235C23D4×D1711yescharacteristic
236Q8—22no—
237D4—342no—
238D4—682no—
239D4—682no—
240D4—682no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—171no—
247C2×C4—174no—
248C2×C4—171no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—341no—
254C2×C4—342no—
255C2×C4—682no—
256C2×C4—684no—
257C2×C4—684no—
258C2×C4—682no—
259C2×C4D4×D1711yescharacteristic
260C2×C4C23×D1711yescharacteristic
261C8—22no—
262C8—342no—
263C22—21no—
264C22—21no—
265C22—172no—
266C22—172no—
267C22—348no—
268C22—341no—
269C22—341no—
270C22—341no—
271C22—341no—
272C22—681no—
273C22—681no—
274C22—681no—
275C22#1058760012yes—
276C22C2×D4×D1711yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—172no—
282C4—341no—
283C4—682no—
284C4—681no—
285C4#41×D1712yes—
286C2—22no—
287C2—342no—
288C2—342no—
289C2—1361no—
290C2#123×D1711yescharacteristic
291C2C2×#1058760011yescharacteristic
292C2#137×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#483C2
socleC22×C17C22×D4
prime radical#11763322C2
hypercenter#483D17
hypocenterC17#3092
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D17C2semidirect
3. #11763143C2semidirect
4. #11763137C2semidirect
5. #11763129C2semidirect
6. #11763123C2semidirect
7. #11763122C2semidirect
8. #11762848C2semidirect
9. #11762809C2semidirect
10. #11762771C2semidirect
11. C17×#483C2semidirect
32. C17×#106C22semidirect
33. C17×#103C22semidirect
35. #10587506C22semidirect
37. #10587492C22semidirect
39. #10587489C22semidirect
40. #10587480C22semidirect
41. #10587479C22semidirect
42. #10587478C22semidirect
144. #483D17semidirect
181. #106C2×D17semidirect
183. #103C2×D17semidirect
189. C17#3092semidirect

Maximal subgroups

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

subgroupindex
#117633222
C2×C4⋊C4×D172
#117631432
#117631372
#117631292
#117631232
#117631222
#117628482
#117628092
#117627712
C17×#4832
#309217