Finite Group Databasebeta

← #14539508

Subgroups — #14539508

Order 2176.

370 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 115 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#11763340C212yesmaximal, complemented
2#11763209C211yesmaximal, characteristic, complemented
3#11763192C211yesmaximal, characteristic, complemented
4#11763162C212yesmaximal, complemented
5#11763067C211yesmaximal, characteristic, complemented
6#11763053C212yesmaximal, complemented
7#11762921C211yesmaximal, characteristic, complemented
8#11762915C211yesmaximal, characteristic, complemented
9#11762906C212yesmaximal, complemented
10M16×Dic17C211yesmaximal, characteristic, complemented
11C17×#469C211yesmaximal, characteristic
12C2×D8×C17C2211yescharacteristic
13C2×M16×C17C2211yescharacteristic
14C17×#121C2212yes—
15#10587622C2212yes—
16#10587621—22no—
17D4×Dic17—22no—
18D4×Dic17C2212yes—
19C2×#63229C2212yes—
20#10587610C2212yes—
21C2×C4×Dic17C2211yescharacteristic
22C2×#63223C2211yescharacteristic
23C2×Dic68C2211yescharacteristic
24#10587576C2212yes—
25#10587573C2212yes—
26#10587565—22no—
27#10587563—22no—
28C17×#106C2211yescharacteristic
29C17×#102C2212yes—
30#10587503C2212yes—
31#10587488C2211yescharacteristic
32#10587487C2212yes—
33#10587486C2211yescharacteristic
34#10587486C2211yescharacteristic
35C8×Dic17C2211yescharacteristic
36C8×Dic17C2211yescharacteristic
37#10587481C2212yes—
38#10587478C2212yes—
39C2×D4×C17—22no—
40C2×D4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×Dic17—22no—
43C22×Dic17D412yes—
44C2×Dic34C2312yes—
45D8×C17—22no—
46M16×C17—22no—
47C2×C8×C17C2311yescharacteristic
48C2×C8×C17C2311yescharacteristic
49C4⋊C4×C17C2312yes—
50C22⋊C4×C17—22no—
51C22⋊Dic17—22no—
52C22⋊Dic17—22no—
53C22⋊Dic17—22no—
54C22⋊Dic17—22no—
55#63229—24no—
56C4⋊Dic17—22no—
57C4⋊Dic17C2312yes—
58#63225—22no—
59#63225—22no—
60#63225—22no—
61C4×Dic17—22no—
62C4×Dic17D412yes—
63C4×Dic17C2311yescharacteristic
64C4×Dic17C2311yescharacteristic
65#63223—22no—
66C2×#3622C2311yescharacteristic
67C2×#3622C2311yescharacteristic
68Dic68—22no—
69C23×C17—22no—
70C23×C17C2×D411yescharacteristic
71D4×C17—22no—
72D4×C17—22no—
73D4×C17—44no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2411yescharacteristic
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—42no—
82C2×Dic17—44no—
83C2×Dic17C2×D412yes—
84C2×Dic17C2×D412yes—
85Dic34—22no—
86Dic34—22no—
87C8×C17—21no—
88C8×C17#4112yes—
89#3622—21no—
90#3622#4112yes—
91#3183—171nomaximal, Sylow
92C22×C17—21no—
93C22×C17—22no—
94C22×C17—42no—
95C22×C17—42no—
96C22×C17—42no—
97C22×C17C22×D411yescharacteristic
98C4×C17—21no—
99C4×C17—42no—
100C4×C17C2×#4111yescharacteristic
101C4×C17C2×#4111yescharacteristic
102Dic17—22no—
103Dic17—22no—
104Dic17—42no—
105Dic17—42no—
106#529—172no—
107#499—171no—
108#495—171no—
109#488—172no—
110#470—171no—
111#469D1711yescharacteristic
112#468—172no—
113#442—171no—
114#441—171no—
115#440—172no—
116C4×M16—171no—
117C2×C17—41no—
118C2×C17—82no—
119C2×C17#53211yescharacteristic
120C2×C17#57512yes—
121C2×Q16—171no—
122C2×SD16—172no—
123C2×D8C2×D1711yescharacteristic
124C2×M16—171no—
125C2×M16C2×D1711yescharacteristic
126#126—342no—
127#125—172no—
128#124—172no—
129#123—342no—
130C22⋊Q8—172no—
131#121C2×D1712yes—
132C4×Q8—172no—
133C4×D4—172no—
134C4×D4—342no—
135#117—342no—
136C2×C42—171no—
137#107—172no—
138#106—171no—
139#106C2×D1711yescharacteristic
140#103—172no—
141#103—172no—
142#102—172no—
143#102C2×D1712yes—
144#97—171no—
145#97—171no—
146C4×C8—171no—
147C4×C8—171no—
148C17#318311yescharacteristic, complemented, Sylow
149C2×Q8—172no—
150C2×D4—22no—
151C2×D4C22×D1712yes—
152C22×C4—172no—
153C22×C4—342no—
154C22×C4C22×D1711yescharacteristic
155Q16—342no—
156SD16—344no—
157D8—22no—
158M16—22no—
159M16—342no—
160C2×C8—171no—
161C2×C8—171no—
162C2×C8C22×D1711yescharacteristic
163C2×C8C22×D1711yescharacteristic
164C4⋊C4—172no—
165C4⋊C4—342no—
166C4⋊C4—342no—
167C4⋊C4—342no—
168C4⋊C4—342no—
169C4⋊C4C22×D1712yes—
170C22⋊C4—22no—
171C22⋊C4—342no—
172C22⋊C4—342no—
173C22⋊C4—342no—
174C22⋊C4—342no—
175C42—172no—
176C42—171no—
177C42—171no—
178C42—342no—
179C23—22no—
180C23D4×D1711yescharacteristic
181Q8—342no—
182Q8—342no—
183D4—22no—
184D4—22no—
185D4—44no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—172no—
190C2×C4—172no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—682no—
194C2×C4—684no—
195C2×C4D4×D1711yescharacteristic
196C2×C4C23×D1711yescharacteristic
197C8—21no—
198C8—172no—
199C8—341no—
200C8#6325412yes—
201C22—22no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22C2×D4×D1711yescharacteristic
207C4—21no—
208C4—42no—
209C4—342no—
210C4—342no—
211C4—682no—
212C4—682no—
213C4C2×#6325411yescharacteristic
214C4#41×D1711yescharacteristic
215C2—41no—
216C2—82no—
217C2#1176335411yescharacteristic
218C2#1176360512yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#469C2
socleC22×C17C22×D4
hypercenter#469D17
hypocenterC17#3183
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #11763340C2semidirect
2. #11763209C2semidirect
3. #11763192C2semidirect
4. #11763162C2semidirect
5. #11763067C2semidirect
6. #11763053C2semidirect
7. #11762921C2semidirect
8. #11762915C2semidirect
9. #11762906C2semidirect
10. M16×Dic17C2semidirect
148. C17#3183semidirect

Maximal subgroups

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

subgroupindex
#117633402
#117632092
#117631922
#117631622
#117630672
#117630532
#117629212
#117629152
#117629062
M16×Dic172
C17×#4692
#318317