Finite Group Databasebeta

← #14534505

Subgroups — #14534505

Order 2176.

782 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C23×Dic34C211yesmaximal, characteristic, complemented
2C2×#10587621C212yesmaximal
3#11762729C214yesmaximal, complemented
4#11762721C214yesmaximal, complemented
5#11762672C211yesmaximal, characteristic, complemented
6#11762670C212yesmaximal, complemented
7C2×C17×#123C211yesmaximal, characteristic
8C22×D4×C17C2211yescharacteristic
9C23×C4×C17C2211yescharacteristic
10C23×Dic17C2212yes—
11C22×Dic34—24no—
12C22×Dic34—28no—
13C22×Dic34C2214yes—
14C17×#123—24no—
15C2×C4⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2214yes—
19C2×C22⋊Dic17C2212yes—
20#10587621—28no—
21C2×#63225C2214yes—
22#10587502C2218yes—
23#10587502C2214yes—
24C24×C17C2311yescharacteristic
25C24×C17C2311yescharacteristic
26C2×D4×C17—22no—
27C2×D4×C17—24no—
28C22×C4×C17—24no—
29C22×C4×C17D414yes—
30C22×C4×C17C2311yescharacteristic
31C22×C4×C17C2312yes—
32C22×C4×C17C2312yes—
33C22×Dic17—24no—
34C22×Dic17—24no—
35C22×Dic17D418yes—
36C22×Dic17C2314yes—
37C22×Dic17C2314yes—
38C2×Dic34—232no—
39C2×Dic34—28no—
40C2×Dic34—216no—
41C4⋊C4×C17—24no—
42C22⋊C4×C17—22no—
43C22⋊C4×C17—24no—
44C22⋊Dic17—24no—
45C22⋊Dic17—28no—
46#63225—28no—
47C23×C17—24no—
48C23×C17—22no—
49C23×C17—21no—
50C23×C17—24no—
51C23×C17C2×D414yes—
52C23×C17C2×D412yes—
53C23×C17C2411yescharacteristic
54D4×C17—44no—
55C2×C4×C17—28no—
56C2×C4×C17—24no—
57C2×C4×C17—22no—
58C2×C4×C17—21no—
59C2×C4×C17—22no—
60C2×C4×C17—24no—
61C2×C4×C17C2×D414yes—
62C2×Dic17—216no—
63C2×Dic17—24no—
64C2×Dic17—24no—
65C2×Dic17—24no—
66C2×Dic17—28no—
67C2×Dic17C2×D418yes—
68Dic34—232no—
69#2614—171nomaximal, Sylow
70C22×C17—24no—
71C22×C17—28no—
72C22×C17—41no—
73C22×C17—42no—
74C22×C17—44no—
75C22×C17#12014yes—
76C22×C17C22×D412yes—
77C22×C17C22×D411yescharacteristic
78C22×C172−1+414yes—
79C4×C17—24no—
80C4×C17—41no—
81C4×C17—42no—
82Dic17—28no—
83Dic17—44no—
84C23×Q8—171no—
85C2×#123—172no—
86C2×#123D1711yescharacteristic
87#395—174no—
88#393—174no—
89#381—171no—
90#381—172no—
91C2×C17—24no—
92C2×C17—81no—
93C2×C17C2×#12011yescharacteristic
94C2×C17#53612yes—
95C2×C17#53614yes—
96C22×Q8—174no—
97C22×Q8—348no—
98C22×Q8—344no—
99C22×D4C2×D1711yescharacteristic
100C23×C4—172no—
101C23×C4C2×D1711yescharacteristic
102#123—24no—
103#123—348no—
104C2×C4⋊C4—174no—
105C2×C4⋊C4C2×D1712yes—
106C2×C22⋊C4—172no—
107C2×C22⋊C4—174no—
108C2×C22⋊C4C2×D1711yescharacteristic
109C2×C22⋊C4C2×D1712yes—
110#95—178no—
111#95—174no—
112C17#261411yescharacteristic, complemented, Sylow
113C24C22×D1711yescharacteristic
114C24C22×D1711yescharacteristic
115C2×Q8—3416no—
116C2×Q8—3432no—
117C2×Q8—348no—
118C2×D4—24no—
119C2×D4—22no—
120C22×C4—24no—
121C22×C4—178no—
122C22×C4—174no—
123C22×C4—174no—
124C22×C4—344no—
125C22×C4—344no—
126C22×C4C22⋊D1714yes—
127C22×C4C22×D1712yes—
128C22×C4C22×D1711yescharacteristic
129C22×C4C22×D1712yes—
130C4⋊C4—24no—
131C4⋊C4—348no—
132C22⋊C4—22no—
133C22⋊C4—24no—
134C22⋊C4—348no—
135C22⋊C4—344no—
136C23—24no—
137C23—22no—
138C23—24no—
139C23—21no—
140C23D4×D1714yes—
141C23C2×C22⋊D1712yes—
142C23C23×D1711yescharacteristic
143Q8—3432no—
144D4—44no—
145C2×C4—24no—
146C2×C4—28no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—178no—
152C2×C4—344no—
153C2×C4—344no—
154C2×C4—3416no—
155C2×C4—344no—
156C2×C4—348no—
157C2×C4C2×C22⋊D1714yes—
158C22—24no—
159C22—28no—
160C22—41no—
161C22—42no—
162C22—44no—
163C22#1058762314yes—
164C22C2×D4×D1712yes—
165C22#1058769714yes—
166C22C22×C22⋊D1711yescharacteristic
167C4—24no—
168C4—41no—
169C4—42no—
170C4—348no—
171C4—684no—
172C2—24no—
173C2—81no—
174C2C2×#1058762311yescharacteristic
175C2#1176338012yes—
176C2#1176338114yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
prime radicalC22×D4×C17C22
hypercenterC2×#123D17
hypocenterC17#2614
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×Dic34C2semidirect
3. #11762729C2semidirect
4. #11762721C2semidirect
5. #11762672C2semidirect
6. #11762670C2semidirect
112. C17#2614semidirect

Maximal subgroups

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

subgroupindex
C23×Dic342
C2×#105876212
#117627292
#117627212
#117626722
#117626702
C2×C17×#1232
#261417