Finite Group Databasebeta

← #14534496

Subgroups — #14534496

Order 2176.

1134 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C2×#10587615C211yesmaximal, characteristic, complemented
2C2×#10587639C212yesmaximal, complemented
3#11762723C214yesmaximal, complemented
4#11762717C214yesmaximal, complemented
5#11762672C211yesmaximal, characteristic, complemented
6#11762665C212yesmaximal, complemented
7C23×D4×C17C211yesmaximal, characteristic, complemented
8C25×C17C2212yes—
9C22×D4×C17—28no—
10C22×D4×C17—24no—
11C22×D4×C17C2214yes—
12C23×C4×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×C22⋊D17C2214yescomplemented
15#10587639—28no—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2211yescharacteristic
18C2×C22⋊Dic17C2214yescomplemented
19C2×C22⋊Dic17C2218yescomplemented
20#10587615—24no—
21C2×#63227C2212yescomplemented
22C2×#63225C2212yescomplemented
23#10587502C2214yescomplemented
24C24×C17—24no—
25C24×C17—24no—
26C24×C17—48no—
27C24×C17D418yescomplemented
28C24×C17C2314yes—
29C24×C17C2311yescharacteristic
30C23×D17C2311yescharacteristic
31C2×D4×C17—232no—
32C2×D4×C17—216no—
33C2×D4×C17—28no—
34C22×C4×C17—24no—
35C22×C4×C17D414yescomplemented
36C22×C4×C17C2312yes—
37C2×C22⋊D17—22no—
38C2×C22⋊D17—24no—
39C2×C22⋊D17—28no—
40C2×C22⋊D17—48no—
41C22×Dic17C2311yescharacteristic
42C22×Dic17C2314yes—
43C22×Dic17C2312yes—
44C22⋊Dic17—22no—
45C22⋊Dic17—24no—
46C22⋊Dic17—28no—
47C22⋊Dic17—48no—
48#63227—24no—
49#63225—24no—
50C23×C17—24no—
51C23×C17—216no—
52C23×C17—24no—
53C23×C17—24no—
54C23×C17—28no—
55C23×C17—28no—
56C23×C17—416no—
57C23×C17—416no—
58C23×C17—416no—
59C23×C17C2×D418yes—
60C23×C17C2×D414yes—
61C23×C17C2×D412yes—
62C23×C17C2411yescharacteristic
63C22×D17—22no—
64C22×D17—21no—
65C22×D17—24no—
66D4×C17—232no—
67C2×C4×C17—28no—
68C2×C4×C17—24no—
69C2×C4×C17C2×D414yes—
70C22⋊D17—416no—
71C22⋊D17—44no—
72C2×Dic17—22no—
73C2×Dic17—24no—
74C2×Dic17—22no—
75C2×Dic17—21no—
76C2×Dic17—24no—
77C2×Dic17—28no—
78#2613—171nomaximal, Sylow
79C22×C17—28no—
80C22×C17—24no—
81C22×C17—28no—
82C22×C17—28no—
83C22×C17—28no—
84C22×C17—432no—
85C22×C17—48no—
86C22×C17—48no—
87C22×C17#12014yes—
88C22×C17C22×D412yes—
89C22×C17C22×D411yescharacteristic
90C22×C17H(5,𝔽2)14yes—
91C2×D17—41no—
92C2×D17—44no—
93C2×D17—42no—
94C4×C17—24no—
95Dic17—41no—
96Dic17—44no—
97Dic17—42no—
98C23×D4D1711yescharacteristic, complemented
99C2×#123—171no—
100C2×#120—172no—
101#394—174no—
102#392—174no—
103#381—171no—
104#379—172no—
105C2×C17—24no—
106C2×C17—48no—
107C2×C17C2×#12011yescharacteristic
108C2×C17#53414yes—
109C2×C17#53512yes—
110D17—81no—
111C25C2×D1712yes—
112C22×D4—28no—
113C22×D4—24no—
114C22×D4—171no—
115C22×D4—174no—
116C22×D4C2×D1714yes—
117C23×C4C2×D1711yescharacteristic
118#123—344no—
119#120—348no—
120C2×C4⋊C4—172no—
121C2×C22⋊C4—172no—
122C2×C22⋊C4—174no—
123C2×C22⋊C4—171no—
124C2×C22⋊C4—172no—
125C2×C22⋊C4—178no—
126#95—174no—
127C17#261311yescharacteristic, complemented, Sylow
128C24—24no—
129C24—24no—
130C24—48no—
131C24—171no—
132C24C22⋊D1718yescomplemented
133C24C22×D1714yes—
134C24C22×D1711yescharacteristic
135C2×D4—232no—
136C2×D4—216no—
137C2×D4—28no—
138C2×D4—348no—
139C2×D4—344no—
140C2×D4—342no—
141C2×D4—688no—
142C22×C4—24no—
143C22×C4—174no—
144C22×C4—172no—
145C22×C4—171no—
146C22×C4C22⋊D1714yescomplemented
147C22×C4C22×D1712yes—
148C4⋊C4—344no—
149C22⋊C4—348no—
150C22⋊C4—344no—
151C22⋊C4—342no—
152C22⋊C4—344no—
153C22⋊C4—688no—
154C23—24no—
155C23—28no—
156C23—28no—
157C23—24no—
158C23—24no—
159C23—216no—
160C23—416no—
161C23—416no—
162C23—416no—
163C23—344no—
164C23—341no—
165C23—342no—
166C23C2×C22⋊D1718yes—
167C23C2×C22⋊D1714yes—
168C23C2×C22⋊D1712yes—
169C23C23×D1711yescharacteristic
170D4—232no—
171D4—6816no—
172D4—684no—
173C2×C4—24no—
174C2×C4—28no—
175C2×C4—344no—
176C2×C4—348no—
177C2×C4—342no—
178C2×C4—342no—
179C2×C4—344no—
180C2×C4—341no—
181C2×C4C2×C22⋊D1714yes—
182C22—28no—
183C22—24no—
184C22—28no—
185C22—28no—
186C22—28no—
187C22—432no—
188C22—48no—
189C22—48no—
190C22—681no—
191C22—684no—
192C22—682no—
193C22#1058763914yes—
194C22#1058769114yes—
195C22C22×C22⋊D1712yes—
196C22C22×C22⋊D1711yescharacteristic
197C4—24no—
198C4—681no—
199C4—684no—
200C4—682no—
201C2—24no—
202C2—48no—
203C2—1361no—
204C2C2×#1058763911yescharacteristic
205C2#1176336814yes—
206C2#1176337612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC23×D4×C17C2
hypercenterC23×D4D17
hypocenterC17#2613
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
2. C2×#10587639C2semidirect
3. #11762723C2semidirect
4. #11762717C2semidirect
5. #11762672C2semidirect
6. #11762665C2semidirect
7. C23×D4×C17C2semidirect
14. C22×C22⋊D17C22semidirect
18. C2×C22⋊Dic17C22semidirect
19. C2×C22⋊Dic17C22semidirect
21. C2×#63227C22semidirect
22. C2×#63225C22semidirect
23. #10587502C22semidirect
27. C24×C17D4semidirect
35. C22×C4×C17D4semidirect
98. C23×D4D17semidirect
127. C17#2613semidirect
132. C24C22⋊D17semidirect
146. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876392
#117627232
#117627172
#117626722
#117626652
C23×D4×C172
#261317