Finite Group Databasebeta

← #14527752

Subgroups — #14527752

Order 2176.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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#11763390C211yesmaximal, characteristic
2#11762948C212yesmaximal, complemented
3#11762542C212yesmaximal, complemented
4#11762535C211yesmaximal, characteristic, complemented
5C17×#395C211yesmaximal, characteristic, complemented
6C22×Q8×C17C2211yescharacteristic
7C2×#63252—21no—
8C2×C4⋊C4×C17—21no—
9C2×C4⋊C4×C17C2211yescharacteristic
10C2×#63230—22no—
11#10587616—21no—
12#10587616C2212yes—
13#10587583—22no—
14#10587581—22no—
15C17×#95—21no—
16C17×#95C2211yescharacteristic
17#10587506—22no—
18C22⋊#3622C2212yes—
19C2×Q8×C17—22no—
20C2×Q8×C17—42no—
21C22×C4×C17—21no—
22C22×C4×C17—21no—
23C22×C4×C17D411yescharacteristic
24C22×C4×C17D411yescharacteristic
25C22×C4×C17C2311yescharacteristic
26C2×C22⋊D17—22no—
27#63252—42no—
28#63252—42no—
29C2×D68—21no—
30C2×D68—21no—
31C2×D68D412yes—
32C2×C4×D17—22no—
33C2×Dic34—21no—
34C4⋊C4×C17—22no—
35C4⋊C4×C17—42no—
36#63230—44no—
37#63227—22no—
38#63227—22no—
39C4⋊Dic17—21no—
40C4⋊Dic17D412yes—
41C2×#3622—22no—
42C23×C17C2×D411yescharacteristic
43C22×D17—22no—
44Q8×C17—24no—
45Q8×C17—42no—
46C2×C4×C17—21no—
47C2×C4×C17—21no—
48C2×C4×C17—22no—
49C2×C4×C17—22no—
50C2×C4×C17—21no—
51C2×C4×C17—41no—
52C2×C4×C17—41no—
53C2×C4×C17—41no—
54C2×C4×C17—41no—
55C2×C4×C17—41no—
56C2×C4×C17—41no—
57C2×C4×C17SD1612yescomplemented
58C2×C4×C17C2×D412yes—
59C22⋊D17—42no—
60C22⋊D17—42no—
61C2×Dic17—22no—
62D68—42no—
63D68—42no—
64D68—81no—
65C4×D17—44no—
66Dic34—42no—
67#3622—42no—
68#1604—171nomaximal, Sylow
69C22×C17—21no—
70C22×C17—21no—
71C22×C17#12011yescharacteristic
72C22×C17C2×SD1611yescharacteristic
73C22×C17AGL(1,ℤ/8)11yescharacteristic
74C2×D17—42no—
75C2×D17—42no—
76C2×D17—42no—
77C4×C17—22no—
78C4×C17—22no—
79C4×C17—42no—
80C4×C17—41no—
81C4×C17—42no—
82Dic17—42no—
83#537—171no—
84#448—172no—
85#395D1711yescharacteristic, complemented
86#327—172no—
87#324—171no—
88C2×C17—21no—
89C2×C17—21no—
90C2×C17#45011yescharacteristic
91C2×C17#45611yescharacteristic
92C2×C17B(4,𝔽2)11yescharacteristic
93D17—82no—
94C2×#41—341no—
95C22×Q8C2×D1711yescharacteristic
96C2×SD16—342no—
97C22⋊Q8—342no—
98#121—172no—
99#121—342no—
100#121—341no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2×D1711yescharacteristic
103#103—342no—
104C22⋊C8—172no—
105#95—21no—
106#95C2×D1711yescharacteristic
107C17#160411yescharacteristic, complemented, Sylow
108#41—682no—
109#41—682no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—341no—
113C2×D4—172no—
114C2×D4—341no—
115C2×D4—341no—
116C2×D4—342no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—342no—
120C22×C4C22⋊D1711yescharacteristic
121C22×C4C22⋊D1711yescharacteristic
122C22×C4C22×D1711yescharacteristic
123SD16—684no—
124C2×C8—342no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—172no—
128C4⋊C4—341no—
129C22⋊C4—342no—
130C22⋊C4—342no—
131C23—342no—
132C23C2×C22⋊D1711yescharacteristic
133Q8—24no—
134Q8—42no—
135Q8—682no—
136D4—682no—
137D4—682no—
138D4—682no—
139D4—682no—
140D4—1361no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—342no—
153C2×C4—684no—
154C2×C4#6323012yescomplemented
155C2×C4D4×D1712yes—
156C8—682no—
157C22—21no—
158C22—21no—
159C22—682no—
160C22—682no—
161C22—682no—
162C22#1058762311yescharacteristic
163C22C2×#6323011yescharacteristic
164C22#1058763511yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—42no—
169C4—41no—
170C4—682no—
171C2—21no—
172C2—21no—
173C2—1362no—
174C2#1176295911yescharacteristic
175C2#1176299111yescharacteristic
176C2#1176299511yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#395C2
socleC22×C17#120
prime radical#11763390C2
hypercenter#395D17
hypocenterC17#1604
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. #11762948C2semidirect
3. #11762542C2semidirect
4. #11762535C2semidirect
5. C17×#395C2semidirect
57. C2×C4×C17SD16semidirect
85. #395D17semidirect
107. C17#1604semidirect
154. C2×C4#63230semidirect

Maximal subgroups

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

subgroupindex
#117633902
#117629482
#117625422
#117625352
C17×#3952
#160417