Finite Group Databasebeta

← #14539514

Subgroups — #14539514

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#11763345C212yesmaximal, complemented
2#11763209C211yesmaximal, characteristic, complemented
3#11763194C211yesmaximal, characteristic, complemented
4#11763165C212yesmaximal, complemented
5#11763062C211yesmaximal, characteristic
6#11763057C212yesmaximal, complemented
7#11762920C211yesmaximal, characteristic, complemented
8#11762916C211yesmaximal, characteristic, complemented
9#11762912C212yesmaximal, complemented
10M16×Dic17C211yesmaximal, characteristic, complemented
11C17×#470C211yesmaximal, characteristic, complemented
12C2×Q16×C17C2211yescharacteristic
13C2×M16×C17C2211yescharacteristic
14C17×C22⋊Q8C2212yes—
15#10587633C2212yes—
16Q8×Dic17C2212yes—
17C2×#63230C2212yes—
18#10587616C2212yes—
19C2×C4×Dic17C2211yescharacteristic
20C2×#63223C2211yescharacteristic
21C2×D136C2211yescharacteristic
22#10587584—22no—
23#10587579C2212yes—
24#10587578—22no—
25#10587576C2212yes—
26#10587572—22no—
27#10587567—22no—
28C17×#106C2211yescharacteristic
29C17×#103C2212yes—
30#10587506C2212yes—
31#10587492C2212yes—
32#10587488C2211yescharacteristic
33#10587486C2211yescharacteristic
34#10587486C2211yescharacteristic
35C8×Dic17C2211yescharacteristic
36C8×Dic17C2211yescharacteristic
37#10587480C2212yes—
38#10587478C2212yes—
39C2×Q8×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C2×C22⋊D17—22no—
42C22×Dic17D412yes—
43C2×D68C2312yes—
44C2×C4×D17—22no—
45Q16×C17—22no—
46M16×C17—22no—
47C2×C8×C17C2311yescharacteristic
48C2×C8×C17C2311yescharacteristic
49C4⋊C4×C17—22no—
50C4⋊C4×C17C2312yes—
51C22⋊C4×C17—22no—
52#63230—24no—
53#63227—22no—
54#63227—22no—
55#63227—22no—
56#63227—22no—
57C4⋊Dic17—22no—
58C4⋊Dic17—22no—
59C4⋊Dic17C2312yes—
60#63225—22no—
61C4×Dic17—22no—
62C4×Dic17D412yes—
63C4×Dic17C2311yescharacteristic
64C4×Dic17C2311yescharacteristic
65#63223—22no—
66C2×#3622C2311yescharacteristic
67C2×#3622C2311yescharacteristic
68D136—22no—
69C23×C17C2×D411yescharacteristic
70C22×D17—22no—
71Q8×C17—22no—
72Q8×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2411yescharacteristic
79C22⋊D17—44no—
80C2×Dic17—22no—
81C2×Dic17—42no—
82C2×Dic17C2×D412yes—
83C2×Dic17C2×D412yes—
84D68—22no—
85D68—22no—
86C4×D17—44no—
87C8×C17—21no—
88C8×C17#4112yes—
89#3622—21no—
90#3622#4112yes—
91#3183—171nomaximal, Sylow
92C22×C17—21no—
93C22×C17—22no—
94C22×C17C22×D411yescharacteristic
95C2×D17—42no—
96C2×D17—42no—
97C2×D17—42no—
98C4×C17—21no—
99C4×C17—42no—
100C4×C17—42no—
101C4×C17C2×#4111yescharacteristic
102C4×C17C2×#4111yescharacteristic
103Dic17—22no—
104Dic17—22no—
105Dic17—42no—
106#529—172no—
107#499—171no—
108#495—171no—
109#488—172no—
110#470D1711yescharacteristic, complemented
111#469—171no—
112#468—172no—
113#442—171no—
114#441—171no—
115#440—172no—
116C4×M16—171no—
117C2×C17—41no—
118C2×C17#53211yescharacteristic
119C2×C17#57512yes—
120D17—82no—
121C2×Q16C2×D1711yescharacteristic
122C2×SD16—172no—
123C2×D8—171no—
124C2×M16—171no—
125C2×M16C2×D1711yescharacteristic
126#126—342no—
127#125—172no—
128#124—172no—
129#123—342no—
130C22⋊Q8C2×D1712yes—
131#121—172no—
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#103C2×D1712yes—
142#102—172no—
143#102—172no—
144#97—171no—
145#97—171no—
146C4×C8—171no—
147C4×C8—171no—
148C17#318311yescharacteristic, complemented, Sylow
149C2×Q8C22×D1712yes—
150C2×D4—172no—
151C2×D4—342no—
152C22×C4—172no—
153C22×C4—342no—
154C22×C4C22×D1711yescharacteristic
155Q16—22no—
156SD16—344no—
157D8—342no—
158M16—22no—
159M16—342no—
160C2×C8—171no—
161C2×C8—171no—
162C2×C8C22×D1711yescharacteristic
163C2×C8C22×D1711yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—172no—
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—342no—
180C23D4×D1711yescharacteristic
181Q8—22no—
182Q8—22no—
183D4—342no—
184D4—342no—
185D4—684no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—172no—
191C2×C4—172no—
192C2×C4—342no—
193C2×C4—684no—
194C2×C4—682no—
195C2×C4D4×D1711yescharacteristic
196C2×C4C23×D1711yescharacteristic
197C8—21no—
198C8—172no—
199C8—341no—
200C8#6325412yes—
201C22—22no—
202C22—21no—
203C22—682no—
204C22—682no—
205C22—682no—
206C22C2×D4×D1711yescharacteristic
207C4—21no—
208C4—42no—
209C4—42no—
210C4—342no—
211C4—342no—
212C4—682no—
213C4C2×#6325411yescharacteristic
214C4#41×D1711yescharacteristic
215C2—41no—
216C2—1362no—
217C2#1176335411yescharacteristic
218C2#1176360812yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#470C2
socleC22×C17C22×D4
prime radical#11763062C2
hypercenter#470D17
hypocenterC17#3183
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #11763345C2semidirect
2. #11763209C2semidirect
3. #11763194C2semidirect
4. #11763165C2semidirect
6. #11763057C2semidirect
7. #11762920C2semidirect
8. #11762916C2semidirect
9. #11762912C2semidirect
10. M16×Dic17C2semidirect
11. C17×#470C2semidirect
110. #470D17semidirect
148. C17#3183semidirect

Maximal subgroups

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

subgroupindex
#117633452
#117632092
#117631942
#117631652
#117630622
#117630572
#117629202
#117629162
#117629122
M16×Dic172
C17×#4702
#318317