Finite Group Databasebeta

← #14536856

Subgroups — #14536856

Order 2176.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587562C211yesmaximal, characteristic, complemented
2#11762742C213yesmaximal, complemented
3#11762732C213yesmaximal, complemented
4#11762731C213yesmaximal, complemented
5#11762725C213yesmaximal
6#11762679C211yesmaximal, characteristic, complemented
7C2×C17×#126C211yesmaximal, characteristic, complemented
8C22×C22⋊D17C2211yescharacteristic
9C17×#126—24no—
10C2×C4⋊C4×C17C2213yes—
11C2×C22⋊C4×C17C2213yes—
12C2×C42×C17C2211yescharacteristic
13C2×C22⋊Dic17C2213yes—
14C2×#63227C2213yes—
15C2×#63227C2213yes—
16C2×C4⋊Dic17C2213yes—
17C2×C4⋊Dic17C2213yes—
18C2×#63225C2213yes—
19C2×#63225C2213yes—
20#10587562—24no—
21#10587502C2213yes—
22#10587502C2213yes—
23C24×C17C2311yescharacteristic
24C23×D17C2311yescharacteristic
25C22×C4×C17C2313yes—
26C22×C4×C17C2313yes—
27C2×C22⋊D17—43no—
28C22×Dic17C2311yescharacteristic
29C22×Dic17C2313yes—
30C22×Dic17C2313yes—
31C4⋊C4×C17—26no—
32C22⋊C4×C17—26no—
33C42×C17D414yescomplemented
34C22⋊Dic17—43no—
35#63227—26no—
36#63227—43no—
37C4⋊Dic17—43no—
38C4⋊Dic17—43no—
39#63225—26no—
40#63225—43no—
41C23×C17—21no—
42C23×C17—23no—
43C23×C17—23no—
44C23×C17C2411yescharacteristic
45C22×D17—21no—
46C22×D17—23no—
47C22×D17—23no—
48C2×C4×C17—23no—
49C2×C4×C17—23no—
50C2×C4×C17—23no—
51C2×C4×C17—23no—
52C2×C4×C17—23no—
53C2×C4×C17C2×D416yes—
54C22⋊D17—44no—
55C2×Dic17—23no—
56C2×Dic17—23no—
57C2×Dic17—23no—
58C2×Dic17—23no—
59C2×Dic17—23no—
60C2×Dic17—23no—
61C2×Dic17—23no—
62#2823—171nomaximal, Sylow
63C22×C17—43no—
64C22×C17—41no—
65C22×C17—43no—
66C22×C17C22×D411yescharacteristic
67C22×C17H(5,𝔽2)13yes—
68C22×C172−1+413yes—
69C2×D17—43no—
70C2×D17—41no—
71C2×D17—43no—
72C4×C17—26no—
73C4×C17—43no—
74Dic17—43no—
75Dic17—43no—
76Dic17—41no—
77C2×#126—171no—
78C2×#126D1711yescharacteristic, complemented
79#398—173no—
80#396—173no—
81#396—173no—
82#394—173no—
83#384—171no—
84C2×C17—81no—
85C2×C17#53713yes—
86C2×C17#56011yescharacteristic
87C2×C17#56313yes—
88D17—81no—
89C22×D4—171no—
90#126—24no—
91#126—344no—
92C2×C4⋊C4—173no—
93C2×C4⋊C4—173no—
94C2×C4⋊C4—173no—
95C2×C4⋊C4—173no—
96C2×C4⋊C4C2×D1713yes—
97C2×C22⋊C4—173no—
98C2×C22⋊C4—173no—
99C2×C22⋊C4—173no—
100C2×C22⋊C4C2×D1713yes—
101C2×C42C2×D1711yescharacteristic
102#95—173no—
103#95—173no—
104C17#282311yescharacteristic, complemented, Sylow
105C24—171no—
106C24C22×D1711yescharacteristic
107C2×D4—683no—
108C22×C4—171no—
109C22×C4—173no—
110C22×C4—173no—
111C22×C4C22×D1713yes—
112C22×C4C22×D1713yes—
113C4⋊C4—26no—
114C4⋊C4—346no—
115C4⋊C4—683no—
116C4⋊C4—683no—
117C4⋊C4—683no—
118C22⋊C4—26no—
119C22⋊C4—346no—
120C22⋊C4—683no—
121C22⋊C4—683no—
122C42C22⋊D1714yescomplemented
123C23—23no—
124C23—23no—
125C23—21no—
126C23—343no—
127C23—343no—
128C23—341no—
129C23C23×D1711yescharacteristic
130D4—684no—
131C2×C4—23no—
132C2×C4—23no—
133C2×C4—23no—
134C2×C4—23no—
135C2×C4—23no—
136C2×C4—343no—
137C2×C4—343no—
138C2×C4—343no—
139C2×C4—343no—
140C2×C4—343no—
141C2×C4—343no—
142C2×C4—343no—
143C2×C4C2×C22⋊D1716yes—
144C22—41no—
145C22—43no—
146C22—43no—
147C22—681no—
148C22—683no—
149C22—683no—
150C22#1058769613yes—
151C22#1058769713yes—
152C22C22×C22⋊D1711yescharacteristic
153C4—26no—
154C4—43no—
155C4—681no—
156C4—683no—
157C4—683no—
158C2—81no—
159C2—1361no—
160C2#1176338713yes—
161C2#1176355011yescharacteristic
162C2#1176356113yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#126C2
prime radicalC22×C22⋊D17C22
hypercenterC2×#126D17
hypocenterC17#2823
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#10587562C2semidirect
2. #11762742C2semidirect
3. #11762732C2semidirect
4. #11762731C2semidirect
6. #11762679C2semidirect
7. C2×C17×#126C2semidirect
33. C42×C17D4semidirect
78. C2×#126D17semidirect
104. C17#2823semidirect
122. C42C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875622
#117627422
#117627322
#117627312
#117627252
#117626792
C2×C17×#1262
#282317