Finite Group Databasebeta

← #14530767

Subgroups — #14530767

Order 2176.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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#11763444C211yesmaximal, characteristic, complemented
2#11762996C212yesmaximal, complemented
3#11762602C211yesmaximal, characteristic, complemented
4#11762597C212yesmaximal, complemented
5C17×#453C211yesmaximal, characteristic, complemented
6C17×H(5,𝔽2)C2211yescharacteristic, complemented
7C2×D4×D17—22no—
8C17×AGL(1,ℤ/8)—21no—
9C17×#127C2211yescharacteristic
10#10587624—22no—
11#10587623—21no—
12#10587623—22no—
13#10587623C2212yes—
14C4×D68—21no—
15C17×#104—21no—
16C17×#100C2211yescharacteristic, complemented
17#10587505—22no—
18#10587505C2212yescomplemented
19C23×D17—21no—
20C23×D17D412yescomplemented
21#41×C17—21no—
22#41×C17—41no—
23C2×D4×C17—22no—
24C2×D4×C17—22no—
25C2×D4×C17—41no—
26C2×D4×C17D411yescharacteristic, complemented
27C2×D4×C17C2311yescharacteristic
28C2×C22⋊D17—22no—
29C2×C22⋊D17—22no—
30C2×C22⋊D17—42no—
31D4×D17—48no—
32C2×D68—21no—
33C2×C4×D17—22no—
34SD16×C17—41no—
35D8×C17—41no—
36M16×C17—21no—
37C42×C17D411yescharacteristic, complemented
38C22⋊Dic17—21no—
39C22⋊Dic17—22no—
40C22⋊Dic17D412yescomplemented
41#63227—22no—
42#63227—42no—
43C4⋊Dic17—21no—
44C23×C17—22no—
45C23×C17—21no—
46C23×C17C2×D412yes—
47C22×D17—22no—
48C22×D17—44no—
49C22×D17—44no—
50C22×D17—42no—
51C22×D17—81no—
52Q8×C17—21no—
53D4×C17—21no—
54D4×C17—41no—
55D4×C17—41no—
56D4×C17—44no—
57D4×C17—42no—
58D4×C17—41no—
59C2×C4×C17—21no—
60C2×C4×C17—22no—
61C2×C4×C17—41no—
62C2×C4×C17C2×D411yescharacteristic
63C22⋊D17—44no—
64C22⋊D17—44no—
65C22⋊D17—84no—
66C2×Dic17—22no—
67C2×Dic17—42no—
68D68—24no—
69C4×D17—44no—
70C8×C17—41no—
71#2196—171nomaximal, Sylow
72C22×C17—22no—
73C22×C17—42no—
74C22×C17—42no—
75C22×C17—41no—
76C22×C17—41no—
77C22×C17—82no—
78C22×C17#12011yescharacteristic
79C2×D17—24no—
80C2×D17—42no—
81C2×D17—42no—
82C2×D17—44no—
83C2×D17—82no—
84C2×D17—84no—
85C4×C17—22no—
86C4×C17—21no—
87C4×C17—41no—
88Dic17—42no—
89Dic17—82no—
90D42—171no—
91B(4,𝔽2)—172no—
92#453D1711yescharacteristic, complemented
93#353—171no—
94AΣL(1,𝔽16)—172no—
95C2×C17—21no—
96C2×C17—42no—
97C2×C17—81no—
98C2×C17—81no—
99C2×C17B(4,𝔽2)11yescharacteristic
100D17—44no—
101H(5,𝔽2)C2×D1711yescharacteristic, complemented
102C22×D4—342no—
103AGL(1,ℤ/8)—21no—
104#127C2×D1711yescharacteristic
105#121—342no—
106#120—172no—
107#120—341no—
108#120—342no—
109C4×D4—341no—
110#104—21no—
111#100C2×D1711yescharacteristic, complemented
112#99—172no—
113#99—342no—
114C17#219611yescharacteristic, complemented, Sylow
115C24—172no—
116C24—341no—
117#41—21no—
118#41—41no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—41no—
122C2×D4—341no—
123C2×D4—342no—
124C2×D4—342no—
125C2×D4—688no—
126C2×D4—682no—
127C2×D4C22⋊D1711yescharacteristic, complemented
128C2×D4C22×D1711yescharacteristic
129C22×C4—342no—
130SD16—41no—
131D8—41no—
132M16—21no—
133C4⋊C4—341no—
134C22⋊C4—172no—
135C22⋊C4—341no—
136C22⋊C4—342no—
137C22⋊C4—342no—
138C22⋊C4—682no—
139C42C22⋊D1711yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—342no—
143C23—682no—
144C23—684no—
145C23—684no—
146C23—1361no—
147C23D4×D1712yes—
148Q8—21no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—42no—
154D4—41no—
155D4—344no—
156D4—684no—
157D4—684no—
158D4—1364no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—342no—
163C2×C4—684no—
164C2×C4—682no—
165C2×C4C2×C22⋊D1711yescharacteristic
166C8—41no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—82no—
173C22—344no—
174C22—682no—
175C22—684no—
176C22—682no—
177C22—1362no—
178C22—1364no—
179C22#1058762311yescharacteristic
180C4—22no—
181C4—21no—
182C4—41no—
183C4—682no—
184C4—1362no—
185C2—21no—
186C2—42no—
187C2—81no—
188C2—81no—
189C2—684no—
190C2#1176299511yescharacteristic
subgroupG/H
centerC2#11762995
commutator, cosocleC2×D4×C17C23
FrattiniC2×D4C22×D17
FittingC17×#453C2
socleC2×C17B(4,𝔽2)
hypercenter#453D17
hypocenterC17#2196
LCSG ▷ C2×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #11763444C2semidirect
2. #11762996C2semidirect
3. #11762602C2semidirect
4. #11762597C2semidirect
5. C17×#453C2semidirect
6. C17×H(5,𝔽2)C22semidirect
16. C17×#100C22semidirect
18. #10587505C22semidirect
20. C23×D17D4semidirect
26. C2×D4×C17D4semidirect
37. C42×C17D4semidirect
40. C22⋊Dic17D4semidirect
92. #453D17semidirect
101. H(5,𝔽2)C2×D17semidirect
111. #100C2×D17semidirect
114. C17#2196semidirect
127. C2×D4C22⋊D17semidirect
139. C42C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
#117634442
#117629962
#117626022
#117625972
C17×#4532
#219617