Finite Group Databasebeta

← #14539156

Subgroups — #14539156

Order 2176.

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

normal subgroups only

This group has 484 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#11763362C211yesmaximal, characteristic, complemented
2C2×#10587626C211yesmaximal, characteristic
3#11763172C214yesmaximal, complemented
4#11763154C214yesmaximal, complemented
5#11762815C211yesmaximal, characteristic, complemented
6#11762814C211yesmaximal, characteristic, complemented
7#11762813C211yesmaximal, characteristic, complemented
8M16×Dic17C211yesmaximal, characteristic, complemented
9C17×#418C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17—22no—
12C22×D68C2211yescharacteristic
13C2×M16×C17C2211yescharacteristic
14C17×#117C2211yescharacteristic
15#10587626—22no—
16#10587626C2212yes—
17#10587626C2212yes—
18#10587612C2211yescharacteristic
19C2×C4×Dic17C2211yescharacteristic
20C2×#63223C2211yescharacteristic
21#10587576C2212yescomplemented
22#10587574C2212yescomplemented
23#10587564—22no—
24C4×Dic34—22no—
25C17×#102C2214yescomplemented
26#10587503C2214yescomplemented
27#10587492C2214yescomplemented
28#10587486C2212yescomplemented
29C8×Dic17C2212yescomplemented
30#10587480C2214yescomplemented
31C24×C17—21no—
32C23×D17—21no—
33C2×D4×C17—21no—
34C2×D4×C17—22no—
35C2×D4×C17—22no—
36C2×D4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C2×C22⋊D17—24no—
39C2×C22⋊D17—24no—
40C2×C22⋊D17—48no—
41C22×Dic17D412yes—
42C2×D68—22no—
43C2×D68—22no—
44C2×D68—21no—
45C2×D68C2312yes—
46C2×Dic34—22no—
47M16×C17—22no—
48C2×C8×C17C2312yes—
49C4⋊C4×C17C2312yes—
50C22⋊C4×C17—21no—
51C42×C17—21no—
52C22⋊Dic17—21no—
53C4⋊Dic17—22no—
54C4⋊Dic17C2312yes—
55#63225—22no—
56#63225—22no—
57C4×Dic17—21no—
58C4×Dic17D412yes—
59C4×Dic17C2311yescharacteristic
60C4×Dic17C2311yescharacteristic
61#63223—22no—
62C2×#3622C2312yes—
63C23×C17—22no—
64C23×C17—44no—
65C23×C17—42no—
66C23×C17C2×D411yescharacteristic
67C22×D17—22no—
68C22×D17—44no—
69C22×D17—42no—
70D4×C17—22no—
71D4×C17—22no—
72D4×C17—42no—
73C2×C4×C17—22no—
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17#4112yes—
76C2×C4×C17#4112yes—
77C2×C4×C17C2411yescharacteristic
78C22⋊D17—48no—
79C22⋊D17—48no—
80C2×Dic17—24no—
81C2×Dic17—22no—
82C2×Dic17C2×D412yes—
83C2×Dic17C2×D412yes—
84D68—22no—
85D68—22no—
86D68—42no—
87Dic34—44no—
88C8×C17—22no—
89#3622—22no—
90#3138—171nomaximal, Sylow
91C22×C17—22no—
92C22×C17—42no—
93C22×C17—44no—
94C22×C17—84no—
95C22×C17#12412yes—
96C22×C17C22×D411yescharacteristic
97C2×D17—42no—
98C2×D17—44no—
99C2×D17—84no—
100C4×C17—42no—
101C4×C17#12412yes—
102C4×C17C2×#4111yescharacteristic
103C4×C17C2×#4111yescharacteristic
104Dic17—22no—
105Dic17—42no—
106Dic17AGL(1,ℤ/8)14yes—
107#533—171no—
108C2×#127—171no—
109#490—174no—
110#486—174no—
111#418—171no—
112#418—171no—
113#418—171no—
114#418D1711yescharacteristic, complemented
115C4×M16—171no—
116C2×C17—22no—
117C2×C17—82no—
118C2×C17C2×#12411yescharacteristic
119C2×C17C2×AGL(1,ℤ/8)12yes—
120D17—82no—
121C22×D4—171no—
122C22×D4—342no—
123C22×D4C2×D1711yescharacteristic
124C2×M16—171no—
125C2×M16C2×D1711yescharacteristic
126C4⋊Q8—172no—
127#127—172no—
128#127—172no—
129#127—342no—
130#125—172no—
131C22⋊Q8—342no—
132C4×Q8—342no—
133#117—171no—
134#117C2×D1711yescharacteristic
135C2×C42—171no—
136#102—174no—
137#102—174no—
138#102—174no—
139#102C2×D1714yescomplemented
140#97—172no—
141C4×C8—172no—
142C17#313811yescharacteristic, complemented, Sylow
143C24—21no—
144C24—341no—
145C2×Q8—342no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—172no—
150C2×D4—344no—
151C2×D4—344no—
152C2×D4—342no—
153C2×D4—341no—
154C2×D4—342no—
155C2×D4—688no—
156C2×D4C22×D1712yes—
157C22×C4—172no—
158C22×C4C22×D1711yescharacteristic
159M16—22no—
160M16—342no—
161C2×C8—172no—
162C2×C8C22×D1712yes—
163C4⋊C4—172no—
164C4⋊C4—342no—
165C4⋊C4—342no—
166C4⋊C4—342no—
167C4⋊C4C22×D1712yes—
168C22⋊C4—21no—
169C22⋊C4—341no—
170C42—21no—
171C42—172no—
172C42—171no—
173C42—171no—
174C42—341no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—342no—
179C23—684no—
180C23—682no—
181C23D4×D1711yescharacteristic
182Q8—684no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—342no—
187D4—342no—
188D4—688no—
189D4—688no—
190D4—682no—
191C2×C4—22no—
192C2×C4—172no—
193C2×C4—172no—
194C2×C4—344no—
195C2×C4—342no—
196C2×C4#6325212yes—
197C2×C4D4×D1711yescharacteristic
198C2×C4#6325412yes—
199C2×C4C23×D1711yescharacteristic
200C8—22no—
201C8—342no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—682no—
207C22—684no—
208C22—1364no—
209C22#1058757112yes—
210C22C2×D4×D1711yescharacteristic
211C4—42no—
212C4—174no—
213C4—342no—
214C4—682no—
215C4#1058757112yes—
216C4C2×#6325211yescharacteristic
217C4C2×#6325411yescharacteristic
218C2—22no—
219C2—82no—
220C2—1362no—
221C2C2×#1058757111yescharacteristic
222C2AGL(1,ℤ/8)×D1712yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#418C2
socleC22×C17C22×D4
prime radicalC2×#10587626C2
hypercenter#418D17
hypocenterC17#3138
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #11763362C2semidirect
3. #11763172C2semidirect
4. #11763154C2semidirect
5. #11762815C2semidirect
6. #11762814C2semidirect
7. #11762813C2semidirect
8. M16×Dic17C2semidirect
9. C17×#418C2semidirect
21. #10587576C22semidirect
22. #10587574C22semidirect
25. C17×#102C22semidirect
26. #10587503C22semidirect
27. #10587492C22semidirect
28. #10587486C22semidirect
29. C8×Dic17C22semidirect
30. #10587480C22semidirect
114. #418D17semidirect
139. #102C2×D17semidirect
142. C17#3138semidirect

Maximal subgroups

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

subgroupindex
#117633622
C2×#105876262
#117631722
#117631542
#117628152
#117628142
#117628132
M16×Dic172
C17×#4182
#313817