Finite Group Databasebeta

← #14544322

Subgroups — #14544322

Order 2176.

254 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 60 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#11763814C212yesmaximal, complemented
2#11763669C212yesmaximal, complemented
3#11763667C212yesmaximal, complemented
4#11763358C211yesmaximal, characteristic
5C2×#63256C411yescharacteristic
6C2×#63254C411yescharacteristic
7C2×#63243—22no—
8C17×#124C411yescharacteristic
9#10587630C2211yescharacteristic
10#10587626C2211yescharacteristic
11#10587570—22no—
12#10587567—22no—
13#10587561C411yescharacteristic
14C42×D17C2211yescharacteristic
15#10587544—22no—
16#10587544C2214yes—
17C2×Q8×C17C2×C411yescharacteristic
18C2×D4×C17C2×C411yescharacteristic
19C2×C22⋊D17—21no—
20C2×C22⋊D17—22no—
21C22×Dic17—21no—
22#63256—42no—
23#63254—42no—
24C2×D68—21no—
25C2×D68C2×C411yescharacteristic
26C2×C4×D17—21no—
27C2×C4×D17C2×C411yescharacteristic
28C2×C4×D17D412yes—
29C2×Dic34C2×C411yescharacteristic
30C2×#3624—24no—
31#63243—44no—
32C22⋊C4×C17—22no—
33C42×C17C2×C411yescharacteristic
34#63227—22no—
35#63225—22no—
36C4×Dic17D412yes—
37C4×Dic17C2311yescharacteristic
38C23×C17—21no—
39C22×D17—21no—
40C22×D17C22⋊C411yescharacteristic
41Q8×C17—41no—
42D4×C17—41no—
43C2×C4×C17—21no—
44C2×C4×C17C22⋊C412yes—
45C2×C4×C17C22×C411yescharacteristic
46C22⋊D17—42no—
47C22⋊D17—42no—
48C22⋊D17—42no—
49C2×Dic17—21no—
50C2×Dic17—42no—
51C2×Dic17C22⋊C411yescharacteristic
52C2×Dic17C2×D412yes—
53D68—41no—
54D68—42no—
55C4×D17—22no—
56C4×D17—22no—
57C4×D17—22no—
58C4×D17—42no—
59Dic34—41no—
60#3624—44no—
61#1560—171nomaximal, Sylow
62C22×C17—41no—
63C22×C17—42no—
64C22×C17C2×C22⋊C411yescharacteristic
65C2×D17—21no—
66C2×D17—22no—
67C2×D17—41no—
68C2×D17—42no—
69C4×C17—21no—
70C4×C17—22no—
71C4×C17—41no—
72Dic17—41no—
73Dic17#10012yes—
74Dic17AGL(1,ℤ/8)12yes—
75Dic17#13712yes—
76#532—171no—
77#423—172no—
78#333—172no—
79#331—172no—
80C2×C17—81no—
81C2×C17C2×#10011yescharacteristic
82C2×C17#41712yes—
83D17—41no—
84D17—81no—
85C2×#41—171no—
86C2×#41—171no—
87C2×M16—342no—
88C4⋊Q8—171no—
89#127—171no—
90#124—171no—
91#124#59511yescharacteristic
92#121—342no—
93C4×D4—342no—
94C2×C42—171no—
95C4⋊C8—174no—
96C4⋊C8—342no—
97C17#156011yescharacteristic, complemented, Sylow
98#41—682no—
99#41—682no—
100C2×Q8—171no—
101C2×Q8C2×#59511yescharacteristic
102C2×D4—171no—
103C2×D4—341no—
104C2×D4—342no—
105C2×D4—341no—
106C2×D4C2×#59511yescharacteristic
107C22×C4—172no—
108C22×C4—171no—
109C22×C4—341no—
110C22×C4—341no—
111M16—684no—
112C2×C8—344no—
113C4⋊C4—342no—
114C22⋊C4—22no—
115C22⋊C4—342no—
116C42—172no—
117C42—171no—
118C42C2×#59511yescharacteristic
119C23—21no—
120C23—171no—
121C23—341no—
122Q8—41no—
123Q8—681no—
124D4—41no—
125D4—681no—
126D4—682no—
127D4—682no—
128D4—682no—
129D4—682no—
130C2×C4—21no—
131C2×C4—172no—
132C2×C4—171no—
133C2×C4—342no—
134C2×C4—342no—
135C2×C4—342no—
136C2×C4—341no—
137C2×C4—682no—
138C2×C4—682no—
139C2×C4C22⋊#59512yes—
140C2×C4C22×#59511yescharacteristic
141C8—684no—
142C22—41no—
143C22—42no—
144C22—341no—
145C22—342no—
146C22—681no—
147C22—682no—
148C22C2×C22⋊#59511yescharacteristic
149C4—21no—
150C4—22no—
151C4—41no—
152C4—172no—
153C4—172no—
154C4—172no—
155C4—681no—
156C2—81no—
157C2—681no—
158C2—1361no—
159C2D17⋊#10011yescharacteristic
160C2#1176380012yes—
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC22×C17C2×C22⋊C4
prime radical#11763358C2
cosocleC4×Dic17C23
hypercenter#124#595
hypocenterC17#1560
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #11763814C2semidirect
2. #11763669C2semidirect
3. #11763667C2semidirect
97. C17#1560semidirect

Maximal subgroups

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

subgroupindex
#117638142
#117636692
#117636672
#117633582
#156017