Finite Group Databasebeta

← #14539172

Subgroups — #14539172

Order 2176.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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×#10587633C211yesmaximal, characteristic
3#11763170C214yesmaximal, complemented
4#11763165C214yesmaximal, complemented
5#11762818C211yesmaximal, characteristic, complemented
6#11762815C211yesmaximal, characteristic
7#11762813C211yesmaximal, characteristic
8M16×Dic17C211yesmaximal, characteristic, complemented
9C17×#419C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D68C2211yescharacteristic
12C2×M16×C17C2211yescharacteristic
13C17×#117C2211yescharacteristic
14#10587633—22no—
15#10587633C2212yes—
16#10587633C2212yes—
17C2×#63227—22no—
18#10587612C2211yescharacteristic
19C2×C4×Dic17C2211yescharacteristic
20C2×#63223C2211yescharacteristic
21#10587576C2212yescomplemented
22#10587574C2212yescomplemented
23#10587564—22no—
24C4×Dic34—22no—
25C17×#103C2214yescomplemented
26#10587506C2214yescomplemented
27#10587492C2214yes—
28#10587486C2212yescomplemented
29C8×Dic17C2212yescomplemented
30#10587480C2214yes—
31C23×D17—21no—
32C2×Q8×C17—21no—
33C2×Q8×C17—22no—
34C2×Q8×C17—22no—
35C2×Q8×C17C2312yes—
36C22×C4×C17—21no—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17D412yes—
39C2×D68—22no—
40C2×D68—22no—
41C2×D68—21no—
42C2×D68C2312yes—
43C2×Dic34—22no—
44M16×C17—22no—
45C2×C8×C17C2312yes—
46C4⋊C4×C17C2312yes—
47C22⋊C4×C17—21no—
48C42×C17—21no—
49C22⋊Dic17—21no—
50#63227—24no—
51#63227—24no—
52C4⋊Dic17—22no—
53C4⋊Dic17C2312yes—
54#63225—22no—
55#63225—22no—
56C4×Dic17—21no—
57C4×Dic17D412yes—
58C4×Dic17C2311yescharacteristic
59C4×Dic17C2311yescharacteristic
60#63223—22no—
61C2×#3622C2312yes—
62C23×C17C2×D411yescharacteristic
63C22×D17—22no—
64C22×D17—44no—
65C22×D17—42no—
66Q8×C17—22no—
67Q8×C17—22no—
68Q8×C17—42no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—42no—
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17#4112yes—
74C2×C4×C17#4112yes—
75C2×C4×C17C2411yescharacteristic
76C2×Dic17—22no—
77C2×Dic17—42no—
78C2×Dic17C2×D412yes—
79C2×Dic17C2×D412yes—
80D68—22no—
81D68—22no—
82D68—42no—
83Dic34—44no—
84C8×C17—22no—
85#3622—22no—
86#3140—171nomaximal, Sylow
87C22×C17—22no—
88C22×C17#12412yes—
89C22×C17C22×D411yescharacteristic
90C2×D17—42no—
91C2×D17—44no—
92C2×D17—84no—
93C4×C17—42no—
94C4×C17—42no—
95C4×C17#12412yes—
96C4×C17C2×#4111yescharacteristic
97C4×C17C2×#4111yescharacteristic
98Dic17—22no—
99Dic17—22no—
100Dic17—42no—
101#533—171no—
102C2×#124—171no—
103#489—174no—
104#488—174no—
105#419—171no—
106#419D1711yescharacteristic, complemented
107#418—171no—
108#418—171no—
109C4×M16—171no—
110C2×C17—22no—
111C2×C17C2×#12411yescharacteristic
112C2×C17#57512yes—
113D17—82no—
114C22×Q8C2×D1711yescharacteristic
115C22×D4—171no—
116C2×M16—171no—
117C2×M16C2×D1711yescharacteristic
118C4⋊Q8—172no—
119#125—172no—
120#124—172no—
121#124—172no—
122#124—342no—
123C22⋊Q8—342no—
124C4×Q8—342no—
125#117—171no—
126#117C2×D1711yescharacteristic
127C2×C22⋊C4—342no—
128C2×C42—171no—
129#103—174no—
130#103C2×D1714yescomplemented
131#102—174no—
132#102—174no—
133#97—172no—
134C4×C8—172no—
135C17#314011yescharacteristic, complemented, Sylow
136C24—341no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—342no—
141C2×Q8C22×D1712yes—
142C2×D4—172no—
143C2×D4—342no—
144C2×D4—341no—
145C2×D4—342no—
146C22×C4—21no—
147C22×C4—172no—
148C22×C4C22×D1711yescharacteristic
149M16—22no—
150M16—342no—
151C2×C8—172no—
152C2×C8C22×D1712yes—
153C4⋊C4—172no—
154C4⋊C4—342no—
155C4⋊C4—342no—
156C4⋊C4—342no—
157C4⋊C4C22×D1712yes—
158C22⋊C4—21no—
159C22⋊C4—344no—
160C22⋊C4—344no—
161C22⋊C4—341no—
162C42—21no—
163C42—172no—
164C42—171no—
165C42—171no—
166C42—341no—
167C23—342no—
168C23—684no—
169C23—682no—
170C23D4×D1711yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—42no—
174Q8—684no—
175D4—342no—
176D4—342no—
177D4—682no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—172no—
182C2×C4—172no—
183C2×C4—342no—
184C2×C4—682no—
185C2×C4#6325212yes—
186C2×C4D4×D1711yescharacteristic
187C2×C4#6325412yes—
188C2×C4C23×D1711yescharacteristic
189C8—22no—
190C8—342no—
191C22—22no—
192C22—682no—
193C22—684no—
194C22—1364no—
195C22#1058757112yes—
196C22C2×D4×D1711yescharacteristic
197C4—42no—
198C4—42no—
199C4—342no—
200C4—342no—
201C4—682no—
202C4#1058757112yes—
203C4C2×#6325211yescharacteristic
204C4C2×#6325411yescharacteristic
205C2—22no—
206C2—1362no—
207C2C2×#1058757111yescharacteristic
208C2#1176360812yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#419C2
socleC22×C17C22×D4
prime radicalC22×D68C22
hypercenter#419D17
hypocenterC17#3140
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #11763362C2semidirect
3. #11763170C2semidirect
4. #11763165C2semidirect
5. #11762818C2semidirect
8. M16×Dic17C2semidirect
9. C17×#419C2semidirect
21. #10587576C22semidirect
22. #10587574C22semidirect
25. C17×#103C22semidirect
26. #10587506C22semidirect
28. #10587486C22semidirect
29. C8×Dic17C22semidirect
106. #419D17semidirect
130. #103C2×D17semidirect
135. C17#3140semidirect

Maximal subgroups

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

subgroupindex
#117633622
C2×#105876332
#117631702
#117631652
#117628182
#117628152
#117628132
M16×Dic172
C17×#4192
#314017