Finite Group Databasebeta

← #14539835

Subgroups — #14539835

Order 2176.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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#11763415C211yesmaximal, characteristic, complemented
2#11763372C211yesmaximal, characteristic, complemented
3#11763198C211yesmaximal, characteristic, complemented
4#11763185C211yesmaximal, characteristic, complemented
5#11763166C211yesmaximal, characteristic, complemented
6#11763023C212yesmaximal, complemented
7#11763003C212yesmaximal, complemented
8#11762955C212yesmaximal, complemented
9#11762944C212yesmaximal, complemented
10#11762871C211yesmaximal, characteristic, complemented
11C17×#490C211yesmaximal, characteristic, complemented
12C2×#63254C2211yescharacteristic
13C2×D4×D17C2211yescharacteristic
14C17×#127C2211yescharacteristic
15C17×#125C2211yescharacteristic, complemented
16#10587626C2211yescharacteristic
17#10587625—22no—
18#10587623—22no—
19#10587622C2211yescharacteristic
20C2×#63228C2212yes—
21C2×#63228C2212yescomplemented
22C2×D136C2212yes—
23C2×#63219C2212yescomplemented
24#10587581—22no—
25#10587581C2212yes—
26#10587580—22no—
27#10587579C2212yescomplemented
28#10587561C2211yescharacteristic
29#10587560C2211yescharacteristic, complemented
30#10587558C2211yescharacteristic
31C17×#102C2212yes—
32C17×#102C2212yescomplemented
33C17×#97C2211yescharacteristic, complemented
34#10587491C2212yes—
35#10587485C2212yescomplemented
36#10587481C2212yescomplemented
37#10587480C2212yes—
38C4×#3622C2211yescharacteristic, complemented
39C23×D17—21no—
40C2×D4×C17—22no—
41C2×D4×C17C2311yescharacteristic
42C2×D4×C17C2311yescharacteristic
43C2×C22⋊D17—21no—
44C2×C22⋊D17—21no—
45C2×C22⋊D17—22no—
46C22×Dic17—21no—
47#63254—42no—
48D4×D17—42no—
49C2×D68—22no—
50C2×D68—22no—
51C2×D68C2311yescharacteristic
52C2×D68C2312yes—
53C2×C4×D17—22no—
54C2×C4×D17C2311yescharacteristic
55C2×Dic34C2311yescharacteristic
56C2×C8×C17C2312yes—
57C4⋊C4×C17—22no—
58C4⋊C4×C17C2312yes—
59C42×C17C2311yescharacteristic
60C22⋊Dic17—22no—
61#63228—42no—
62#63228—42no—
63#63227—22no—
64#63227—22no—
65#63227—22no—
66#63227D412yes—
67#63225D412yes—
68C4×Dic17C2311yescharacteristic
69C2×#3622C2312yes—
70D136—42no—
71#63219—42no—
72C23×C17—21no—
73C23×C17—21no—
74C22×D17—22no—
75C22×D17—21no—
76C22×D17—42no—
77C22×D17—82no—
78C22×D17C2×D411yescharacteristic
79D4×C17—22no—
80D4×C17—22no—
81D4×C17—82no—
82C2×C4×C17—22no—
83C2×C4×C17C2×D412yes—
84C2×C4×C17C2411yescharacteristic
85C22⋊D17—42no—
86C22⋊D17—42no—
87C22⋊D17—82no—
88C2×Dic17—21no—
89C2×Dic17—42no—
90C2×Dic17C2×D412yes—
91C2×Dic17C2×D411yescharacteristic
92D68—22no—
93D68—22no—
94D68—22no—
95D68—82no—
96D68—82no—
97C4×D17—22no—
98C4×D17—82no—
99Dic34—22no—
100C8×C17—22no—
101#3622—22no—
102#3220—171nomaximal, Sylow
103C22×C17—41no—
104C22×C17—41no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17C22×D411yescharacteristic
108C2×D17—21no—
109C2×D17—22no—
110C2×D17—41no—
111C2×D17—42no—
112C2×D17—42no—
113C2×D17—42no—
114C2×D17—42no—
115C2×D17—84no—
116C4×C17—22no—
117C4×C17—42no—
118C4×C17H(5,𝔽2)12yes—
119Dic17—21no—
120Dic17—22no—
121Dic17—41no—
122#540—171no—
123#535—171no—
124#496—171no—
125#493—171no—
126#490D1711yescharacteristic, complemented
127#488—171no—
128#463—172no—
129#459—172no—
130#449—172no—
131#447—172no—
132#433—171no—
133C2×C17—81no—
134C2×C17—81no—
135C2×C17#53511yescharacteristic
136C2×C17#57612yes—
137D17—41no—
138D17—82no—
139D17—81no—
140C2×#41—171no—
141C22×D4—171no—
142C2×SD16—172no—
143C2×D8—172no—
144C2×D8—172no—
145C2×D8—172no—
146#127—171no—
147#127—171no—
148#127C2×D1711yescharacteristic
149#125C2×D1711yescharacteristic, complemented
150#124—171no—
151#124—171no—
152#123—342no—
153#121—172no—
154#121—342no—
155#121—342no—
156#120—342no—
157C4×D4—172no—
158#117—171no—
159C4⋊C8—172no—
160#103—172no—
161#102—172no—
162#102C2×D1712yes—
163#102C2×D1712yescomplemented
164C22⋊C8—172no—
165#97C2×D1711yescharacteristic, complemented
166C4×C8—171no—
167C17#322011yescharacteristic, complemented, Sylow
168C24—341no—
169#41—682no—
170C2×Q8—171no—
171C2×D4—22no—
172C2×D4—171no—
173C2×D4—172no—
174C2×D4—342no—
175C2×D4—341no—
176C2×D4—342no—
177C2×D4—341no—
178C2×D4—342no—
179C2×D4—682no—
180C2×D4C22×D1711yescharacteristic
181C2×D4C22×D1711yescharacteristic
182C22×C4—171no—
183C22×C4—342no—
184C22×C4—341no—
185SD16—682no—
186D8—682no—
187D8—682no—
188D8—682no—
189C2×C8—172no—
190C2×C8C22×D1712yes—
191C4⋊C4—22no—
192C4⋊C4—172no—
193C4⋊C4C22×D1712yes—
194C22⋊C4—172no—
195C22⋊C4—342no—
196C22⋊C4—342no—
197C22⋊C4—342no—
198C22⋊C4—342no—
199C42—171no—
200C42C22×D1711yescharacteristic
201C23—21no—
202C23—21no—
203C23—171no—
204C23—342no—
205C23—341no—
206C23—682no—
207C23—1362no—
208Q8—342no—
209D4—22no—
210D4—22no—
211D4—82no—
212D4—342no—
213D4—342no—
214D4—342no—
215D4—682no—
216D4—682no—
217D4—1362no—
218D4—1362no—
219D4—1362no—
220C2×C4—22no—
221C2×C4—172no—
222C2×C4—171no—
223C2×C4—342no—
224C2×C4—341no—
225C2×C4—682no—
226C2×C4—1362no—
227C2×C4D4×D1712yes—
228C2×C4C23×D1711yescharacteristic
229C8—22no—
230C8—342no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—342no—
236C22—341no—
237C22—682no—
238C22—682no—
239C22—682no—
240C22—681no—
241C22—682no—
242C22—1364no—
243C22C2×D4×D1711yescharacteristic
244C4—22no—
245C4—42no—
246C4—342no—
247C4—341no—
248C4—681no—
249C4#1058769612yes—
250C2—81no—
251C2—81no—
252C2—681no—
253C2—1361no—
254C2—1362no—
255C2#1176337711yescharacteristic
256C2#1176361312yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#490C2
socleC22×C17C22×D4
hypercenter#490D17
hypocenterC17#3220
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #11763415C2semidirect
2. #11763372C2semidirect
3. #11763198C2semidirect
4. #11763185C2semidirect
5. #11763166C2semidirect
6. #11763023C2semidirect
7. #11763003C2semidirect
8. #11762955C2semidirect
9. #11762944C2semidirect
10. #11762871C2semidirect
11. C17×#490C2semidirect
15. C17×#125C22semidirect
21. C2×#63228C22semidirect
23. C2×#63219C22semidirect
27. #10587579C22semidirect
29. #10587560C22semidirect
32. C17×#102C22semidirect
33. C17×#97C22semidirect
35. #10587485C22semidirect
36. #10587481C22semidirect
38. C4×#3622C22semidirect
126. #490D17semidirect
149. #125C2×D17semidirect
163. #102C2×D17semidirect
165. #97C2×D17semidirect
167. C17#3220semidirect

Maximal subgroups

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

subgroupindex
#117634152
#117633722
#117631982
#117631852
#117631662
#117630232
#117630032
#117629552
#117629442
#117628712
C17×#4902
#322017