Finite Group Databasebeta

← #14539342

Subgroups — #14539342

Order 2176.

412 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 129 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#11763364C211yesmaximal, characteristic, complemented
2#11763357C211yesmaximal, characteristic
3#11763205C211yesmaximal, characteristic, complemented
4#11763181C211yesmaximal, characteristic, complemented
5#11763160C211yesmaximal, characteristic, complemented
6#11763037C214yesmaximal, complemented
7#11762900C214yesmaximal, complemented
8C4×C8×D17C211yesmaximal, characteristic, complemented
9C17×#486C211yesmaximal, characteristic, complemented
10C2×#63254—21no—
11C17×C4⋊Q8C2211yescharacteristic
12C17×#127C2211yescharacteristic
13#10587632—21no—
14Q8×Dic17—21no—
15#10587624—21no—
16#10587624C2212yes—
17#10587622—21no—
18D4×Dic17—21no—
19D4×Dic17C2212yes—
20C2×#63229C2214yes—
21C2×C8×D17C2212yes—
22#10587583C2212yes—
23#10587578—21no—
24#10587576—21no—
25#10587573C2212yes—
26C42×D17C2211yescharacteristic
27#10587555C2211yescharacteristic
28#10587555C2211yescharacteristic
29C17×#102C2214yes—
30C4×C8×C17C2211yescharacteristic
31#10587487C2214yes—
32C8×Dic17C2212yes—
33#10587479C2214yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×D4×C17—21no—
37C2×D4×C17—21no—
38C2×D4×C17C2312yes—
39C2×C22⋊D17—22no—
40C22×Dic17—22no—
41#63254—44no—
42C2×C4×D17D411yescharacteristic
43C2×C4×D17D411yescharacteristic
44C2×C4×D17C2311yescharacteristic
45C2×Dic34—21no—
46C2×Dic34C2312yes—
47C2×C8×C17C2312yes—
48C4⋊C4×C17—21no—
49C4⋊C4×C17C2312yes—
50C42×C17C2311yescharacteristic
51C22⋊Dic17—22no—
52C22⋊Dic17—22no—
53#63229—28no—
54#63227—22no—
55C4⋊Dic17—21no—
56C4⋊Dic17—21no—
57C4⋊Dic17—21no—
58C4⋊Dic17—22no—
59C4⋊Dic17C2312yes—
60#63225—22no—
61C4×Dic17—22no—
62C4×Dic17D411yescharacteristic
63C4×Dic17D411yescharacteristic
64C4×Dic17C2311yescharacteristic
65C2×#3622C2312yes—
66C8×D17—24no—
67C23×C17—22no—
68C22×D17C2×D411yescharacteristic
69Q8×C17—42no—
70D4×C17—24no—
71D4×C17—42no—
72D4×C17—42no—
73C2×C4×C17—22no—
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2×D411yescharacteristic
76C2×C4×C17C2411yescharacteristic
77C22⋊D17—44no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—44no—
81C2×Dic17C2×D411yescharacteristic
82C2×Dic17C2×D411yescharacteristic
83C2×Dic17C2×D411yescharacteristic
84C4×D17—22no—
85C4×D17—22no—
86C4×D17SD1614yescomplemented
87Dic34—24no—
88Dic34—42no—
89C8×C17—22no—
90#3622#4114yescomplemented
91#3167—171nomaximal, Sylow
92C22×C17—42no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17C22×D411yescharacteristic
96C2×D17—21no—
97C2×D17—21no—
98C2×D17C2×SD1612yes—
99C4×C17—42no—
100C4×C17C2×SD1612yes—
101C4×C17#13512yes—
102C4×C17C2×#4112yes—
103Dic17—21no—
104Dic17—21no—
105Dic17—42no—
106Dic17—42no—
107Dic17C2×SD1612yes—
108#533—171no—
109#532—171no—
110#498—171no—
111#492—171no—
112#487—171no—
113#486D1711yescharacteristic, complemented
114#465—174no—
115C4×SD16—174no—
116C2×C4×C8—171no—
117C2×C17—82no—
118C2×C17#53211yescharacteristic
119C2×C17C22×SD1611yescharacteristic
120C2×C17C2×#13511yescharacteristic
121D17—22no—
122C2×#41—341no—
123C2×SD16—174no—
124C22×C8—172no—
125C4⋊Q8—171no—
126C4⋊Q8—171no—
127C4⋊Q8C2×D1711yescharacteristic
128#127C2×D1711yescharacteristic
129#125—341no—
130#124—341no—
131C22⋊Q8—172no—
132C22⋊Q8—341no—
133#121—172no—
134#121—341no—
135C4×Q8—172no—
136C4×Q8—341no—
137C4×D4—172no—
138C4×D4—341no—
139#117—341no—
140C2×C42—171no—
141#106—174no—
142#103—174no—
143#102C2×D1714yes—
144C4×C8—172no—
145C4×C8—171no—
146C4×C8C2×D1711yescharacteristic
147C17#316711yescharacteristic, complemented, Sylow
148#41—684no—
149C2×Q8—21no—
150C2×Q8—172no—
151C2×Q8—341no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—342no—
155C2×D4C22×D1712yes—
156C22×C4—171no—
157C22×C4—171no—
158C22×C4—171no—
159C22×C4—342no—
160SD16—348no—
161C2×C8—172no—
162C2×C8—344no—
163C2×C8C22×D1712yes—
164C4⋊C4—21no—
165C4⋊C4—172no—
166C4⋊C4—342no—
167C4⋊C4—342no—
168C4⋊C4—341no—
169C4⋊C4—341no—
170C4⋊C4—341no—
171C4⋊C4C22×D1712yes—
172C22⋊C4—342no—
173C22⋊C4—342no—
174C22⋊C4—342no—
175C42—171no—
176C42—171no—
177C42—171no—
178C42—342no—
179C42C22×D1711yescharacteristic
180C23—22no—
181C23—171no—
182Q8—42no—
183Q8—344no—
184Q8—682no—
185D4—24no—
186D4—42no—
187D4—42no—
188D4—684no—
189C2×C4—22no—
190C2×C4—171no—
191C2×C4—174no—
192C2×C4—171no—
193C2×C4—171no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—684no—
199C2×C4D4×D1711yescharacteristic
200C2×C4D4×D1711yescharacteristic
201C2×C4C23×D1711yescharacteristic
202C8—22no—
203C8—174no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—172no—
208C22—341no—
209C22—341no—
210C22C2×D4×D1711yescharacteristic
211C4—42no—
212C4—172no—
213C4—341no—
214C4—341no—
215C4—682no—
216C4—682no—
217C4#1058759812yes—
218C4SD16×D1712yes—
219C4#41×D1712yes—
220C2—82no—
221C2—342no—
222C2#1176335811yescharacteristic
223C2C2×SD16×D1711yescharacteristic
224C2C2×#1058759811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#486C2
socleC22×C17C22×D4
prime radical#11763357C2
hypercenter#486D17
hypocenterC17#3167
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #11763364C2semidirect
3. #11763205C2semidirect
4. #11763181C2semidirect
5. #11763160C2semidirect
6. #11763037C2semidirect
7. #11762900C2semidirect
8. C4×C8×D17C2semidirect
9. C17×#486C2semidirect
86. C4×D17SD16semidirect
90. #3622#41semidirect
113. #486D17semidirect
147. C17#3167semidirect

Maximal subgroups

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

subgroupindex
#117633642
#117633572
#117632052
#117631812
#117631602
#117630372
#117629002
C4×C8×D172
C17×#4862
#316717