Finite Group Databasebeta

← #14534653

Subgroups — #14534653

Order 2176.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×C4×D68C211yesmaximal, characteristic
2#11762744C212yesmaximal, complemented
3#11762739C212yesmaximal, complemented
4#11762735C212yesmaximal, complemented
5#11762726C212yesmaximal
6#11762680C214yesmaximal, complemented
7#11762677C211yesmaximal, characteristic, complemented
8C2×C17×#125C211yesmaximal, characteristic, complemented
9C22×D68C2211yescharacteristic
10C22×C4×D17C2212yes—
11C17×#125—24no—
12C2×C4⋊C4×C17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2212yes—
16C2×C42×C17C2211yescharacteristic
17C2×#63227C2214yes—
18C2×#63227C2212yes—
19C2×#63227C2214yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2214yes—
22C4×D68—24no—
23#10587502C2212yes—
24#10587502C2214yes—
25#10587502C2212yes—
26#10587502C2212yes—
27C23×D17C2312yes—
28C22×C4×C17C2312yes—
29C22×C4×C17C2312yes—
30C22×C4×C17C2311yescharacteristic
31C22×C4×C17C2311yescharacteristic
32C22×C4×C17C2311yescharacteristic
33C22×Dic17C2312yes—
34C22×Dic17C2314yes—
35C2×D68—24no—
36C2×D68—22no—
37C2×C4×D17—24no—
38C2×C4×D17—44no—
39C4⋊C4×C17—24no—
40C4⋊C4×C17—24no—
41C4⋊C4×C17—22no—
42C4⋊C4×C17—22no—
43C42×C17D414yes—
44#63227—28no—
45#63227—24no—
46#63227—44no—
47C4⋊Dic17—22no—
48#63225—44no—
49C23×C17C2411yescharacteristic
50C22×D17—22no—
51C22×D17—22no—
52C22×D17—24no—
53C22×D17—24no—
54C22×D17#4114yes—
55C2×C4×C17—22no—
56C2×C4×C17—22no—
57C2×C4×C17—22no—
58C2×C4×C17—22no—
59C2×C4×C17—22no—
60C2×C4×C17—24no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17C2×D412yes—
64C2×C4×C17C2×D412yes—
65C2×C4×C17C2×D412yes—
66C2×Dic17—22no—
67C2×Dic17—24no—
68C2×Dic17—24no—
69C2×Dic17—24no—
70C2×Dic17—24no—
71D68—28no—
72C4×D17—48no—
73#2633—171nomaximal, Sylow
74C22×C17C22×D411yescharacteristic
75C22×C17C2×#4112yes—
76C22×C17H(5,𝔽2)12yes—
77C22×C172−1+412yes—
78C2×D17—24no—
79C2×D17—28no—
80C2×D17—42no—
81C2×D17—42no—
82C2×D17—44no—
83C4×C17—22no—
84C4×C17—22no—
85C4×C17—22no—
86C4×C17—42no—
87C4×C17—42no—
88Dic17—42no—
89Dic17—44no—
90C2×#125D1711yescharacteristic, complemented
91C2×C4×D4—171no—
92#399—172no—
93#398—172no—
94#397—172no—
95#394—172no—
96#385—174no—
97#383—171no—
98C2×C17#52511yescharacteristic
99C2×C17#53511yescharacteristic
100C2×C17#53611yescharacteristic
101C2×C17#53914yes—
102D17—44no—
103C22×D4—171no—
104C23×C4—172no—
105#125—24no—
106C4×D4—344no—
107C2×C4⋊C4—171no—
108C2×C4⋊C4—174no—
109C2×C4⋊C4C2×D1712yes—
110C2×C4⋊C4C2×D1712yes—
111C2×C4⋊C4C2×D1711yescharacteristic
112C2×C4⋊C4C2×D1711yescharacteristic
113C2×C22⋊C4—172no—
114C2×C22⋊C4—174no—
115C2×C22⋊C4—174no—
116C2×C42C2×D1711yescharacteristic
117#95—172no—
118#95—172no—
119#95—172no—
120#95—174no—
121C17#263311yescharacteristic, complemented, Sylow
122C24—172no—
123C2×D4—344no—
124C2×D4—342no—
125C22×C4—174no—
126C22×C4—172no—
127C22×C4—344no—
128C22×C4—684no—
129C22×C4C22×D1712yes—
130C22×C4C22×D1712yes—
131C22×C4C22×D1711yescharacteristic
132C22×C4C22×D1711yescharacteristic
133C22×C4C22×D1711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—342no—
139C4⋊C4—684no—
140C22⋊C4—348no—
141C22⋊C4—344no—
142C22⋊C4—684no—
143C42C22⋊D1714yes—
144C23—174no—
145C23—344no—
146C23—342no—
147C23—344no—
148C23—342no—
149C23C23×D1711yescharacteristic
150D4—348no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—344no—
160C2×C4—344no—
161C2×C4—344no—
162C2×C4—344no—
163C2×C4—342no—
164C2×C4—688no—
165C2×C4C2×C22⋊D1712yes—
166C2×C4C2×C22⋊D1712yes—
167C2×C4C2×C22⋊D1712yes—
168C22—344no—
169C22—348no—
170C22—682no—
171C22—682no—
172C22—684no—
173C22#1058769412yes—
174C22#41×D1712yes—
175C22#1058769612yes—
176C22C22×C22⋊D1711yescharacteristic
177C4—22no—
178C4—22no—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—682no—
183C4—684no—
184C2—684no—
185C2#1176331611yescharacteristic
186C2#1176337311yescharacteristic
187C2#1176338311yescharacteristic
188C2#1176340714yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#125C2
prime radicalC22×D68C22
hypercenterC2×#125D17
hypocenterC17#2633
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #11762744C2semidirect
3. #11762739C2semidirect
4. #11762735C2semidirect
6. #11762680C2semidirect
7. #11762677C2semidirect
8. C2×C17×#125C2semidirect
90. C2×#125D17semidirect
121. C17#2633semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D682
#117627442
#117627392
#117627352
#117627262
#117626802
#117626772
C2×C17×#1252
#263317