Finite Group Databasebeta

← #14539857

Subgroups — #14539857

Order 2176.

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

normal subgroups only

This group has 396 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#11763422C211yesmaximal, characteristic, complemented
2#11763372C211yesmaximal, characteristic
3#11763203C211yesmaximal, characteristic, complemented
4#11763181C211yesmaximal, characteristic, complemented
5#11763166C211yesmaximal, characteristic, complemented
6#11763030C212yesmaximal, complemented
7#11763008C212yesmaximal, complemented
8#11762972C212yesmaximal, complemented
9#11762961C212yesmaximal, 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×#63229C2212yes—
21C2×#63229C2212yescomplemented
22C2×Dic68C2212yescomplemented
23C2×#63219C2212yes—
24#10587584—22no—
25#10587583C2212yes—
26#10587576—22no—
27#10587573C2212yescomplemented
28#10587561C2211yescharacteristic
29#10587558C2211yescharacteristic
30#10587555C2211yescharacteristic, complemented
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×D68C2311yescharacteristic
50C2×C4×D17C2311yescharacteristic
51C2×Dic34C2311yescharacteristic
52C2×Dic34C2312yes—
53C2×C8×C17C2312yes—
54C4⋊C4×C17—22no—
55C4⋊C4×C17C2312yes—
56C42×C17C2311yescharacteristic
57C22⋊Dic17—22no—
58#63229—42no—
59#63229—42no—
60#63227—22no—
61#63227—22no—
62#63227D412yes—
63C4⋊Dic17—22no—
64C4⋊Dic17—22no—
65#63225—22no—
66#63225D412yes—
67C4×Dic17—22no—
68C4×Dic17C2311yescharacteristic
69C2×#3622C2312yes—
70Dic68—42no—
71#63219—42no—
72C23×C17—21no—
73C23×C17—21no—
74C22×D17—21no—
75C22×D17—42no—
76C22×D17—82no—
77C22×D17C2×D411yescharacteristic
78D4×C17—22no—
79D4×C17—22no—
80D4×C17—82no—
81C2×C4×C17—22no—
82C2×C4×C17C2×D412yes—
83C2×C4×C17C2411yescharacteristic
84C22⋊D17—42no—
85C22⋊D17—42no—
86C22⋊D17—82no—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—42no—
90C2×Dic17C2×D412yes—
91C2×Dic17C2×D411yescharacteristic
92D68—22no—
93C4×D17—22no—
94Dic34—22no—
95Dic34—22no—
96Dic34—22no—
97C8×C17—22no—
98#3622—22no—
99#3222—171nomaximal, Sylow
100C22×C17—41no—
101C22×C17—41no—
102C22×C17—42no—
103C22×C17—42no—
104C22×C17C22×D411yescharacteristic
105C2×D17—21no—
106C2×D17—22no—
107C2×D17—41no—
108C2×D17—42no—
109C2×D17—84no—
110C4×C17—22no—
111C4×C17—42no—
112C4×C17H(5,𝔽2)12yes—
113Dic17—21no—
114Dic17—22no—
115Dic17—42no—
116Dic17—41no—
117#541—171no—
118#535—171no—
119#497—171no—
120#492—171no—
121#490D1711yescharacteristic, complemented
122#488—171no—
123#464—172no—
124#460—172no—
125#452—172no—
126#450—172no—
127#433—171no—
128C2×C17—81no—
129C2×C17—81no—
130C2×C17#53511yescharacteristic
131C2×C17#57712yes—
132D17—41no—
133D17—81no—
134C2×#41—171no—
135C22×D4—171no—
136C2×Q16—172no—
137C2×SD16—172no—
138C2×SD16—172no—
139C2×SD16—172no—
140C4⋊Q8—171no—
141#127—171no—
142#127C2×D1711yescharacteristic
143#126—342no—
144#125—342no—
145#125C2×D1711yescharacteristic, complemented
146#124—171no—
147#124—171no—
148C22⋊Q8—172no—
149#121—342no—
150#120—342no—
151C4×Q8—172no—
152#117—171no—
153C4⋊C8—172no—
154#103—172no—
155#102—172no—
156#102C2×D1712yes—
157#102C2×D1712yescomplemented
158C22⋊C8—172no—
159#97C2×D1711yescharacteristic, complemented
160C4×C8—171no—
161C17#322211yescharacteristic, complemented, Sylow
162C24—341no—
163#41—682no—
164C2×Q8—171no—
165C2×Q8—172no—
166C2×D4—22no—
167C2×D4—171no—
168C2×D4—341no—
169C2×D4—342no—
170C2×D4—341no—
171C2×D4—682no—
172C2×D4C22×D1711yescharacteristic
173C2×D4C22×D1711yescharacteristic
174C22×C4—171no—
175C22×C4—341no—
176Q16—682no—
177SD16—682no—
178SD16—682no—
179SD16—682no—
180C2×C8—172no—
181C2×C8C22×D1712yes—
182C4⋊C4—22no—
183C4⋊C4—172no—
184C4⋊C4—342no—
185C4⋊C4—342no—
186C4⋊C4—342no—
187C4⋊C4C22×D1712yes—
188C22⋊C4—172no—
189C22⋊C4—342no—
190C22⋊C4—342no—
191C22⋊C4—342no—
192C42—171no—
193C42—342no—
194C42C22×D1711yescharacteristic
195C23—21no—
196C23—21no—
197C23—171no—
198C23—341no—
199C23—682no—
200C23—1362no—
201Q8—342no—
202Q8—342no—
203Q8—342no—
204D4—22no—
205D4—22no—
206D4—82no—
207D4—342no—
208D4—682no—
209D4—682no—
210D4—1362no—
211C2×C4—22no—
212C2×C4—172no—
213C2×C4—171no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—341no—
217C2×C4—682no—
218C2×C4D4×D1712yes—
219C2×C4C23×D1711yescharacteristic
220C8—22no—
221C8—342no—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—342no—
227C22—341no—
228C22—682no—
229C22—681no—
230C22—1364no—
231C22C2×D4×D1711yescharacteristic
232C4—22no—
233C4—42no—
234C4—342no—
235C4—341no—
236C4—681no—
237C4—682no—
238C4#1058769612yes—
239C2—81no—
240C2—81no—
241C2—681no—
242C2—1361no—
243C2#1176337711yescharacteristic
244C2#1176361912yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#490C2
socleC22×C17C22×D4
prime radical#11763372C2
hypercenter#490D17
hypocenterC17#3222
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. #11763422C2semidirect
3. #11763203C2semidirect
4. #11763181C2semidirect
5. #11763166C2semidirect
6. #11763030C2semidirect
7. #11763008C2semidirect
8. #11762972C2semidirect
9. #11762961C2semidirect
10. #11762871C2semidirect
11. C17×#490C2semidirect
15. C17×#125C22semidirect
21. C2×#63229C22semidirect
22. C2×Dic68C22semidirect
27. #10587573C22semidirect
30. #10587555C22semidirect
32. C17×#102C22semidirect
33. C17×#97C22semidirect
35. #10587485C22semidirect
36. #10587481C22semidirect
38. C4×#3622C22semidirect
121. #490D17semidirect
145. #125C2×D17semidirect
157. #102C2×D17semidirect
159. #97C2×D17semidirect
161. C17#3222semidirect

Maximal subgroups

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

subgroupindex
#117634222
#117633722
#117632032
#117631812
#117631662
#117630302
#117630082
#117629722
#117629612
#117628712
C17×#4902
#322217