Finite Group Databasebeta

← #14530784

Subgroups — #14530784

Order 2176.

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

normal subgroups only

This group has 270 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#11763551C211yesmaximal, characteristic, complemented
2#11762996C212yesmaximal, complemented
3#11762604C211yesmaximal, characteristic, complemented
4#11762597C212yesmaximal, complemented
5C17×#455C211yesmaximal, characteristic, complemented
6C17×H(5,𝔽2)C2211yescharacteristic
7C17×#137—21no—
8C17×#124C2211yescharacteristic
9#10587633—21no—
10#10587623—22no—
11#10587623C2212yes—
12#10587571—22no—
13#10587568—22no—
14#10587561—21no—
15C17×#104—21no—
16C17×#100C2211yescharacteristic
17#10587505—22no—
18#10587505C2212yes—
19C23×D17D412yescomplemented
20#41×C17—21no—
21#41×C17—41no—
22C2×Q8×C17D411yescharacteristic, complemented
23C2×D4×C17—22no—
24C2×D4×C17—41no—
25C2×D4×C17C2311yescharacteristic
26C2×C22⋊D17—22no—
27C2×C22⋊D17—42no—
28C2×D68—21no—
29C2×D68—21no—
30C2×Dic34—21no—
31Q16×C17—41no—
32SD16×C17—41no—
33M16×C17—21no—
34C22⋊C4×C17—22no—
35C42×C17D411yescharacteristic, complemented
36C22⋊Dic17—22no—
37C22⋊Dic17D412yescomplemented
38#63227—22no—
39#63227—22no—
40#63227—22no—
41#63227—42no—
42C4×Dic17—21no—
43C23×C17—22no—
44C23×C17C2×D412yes—
45C22×D17—22no—
46C22×D17—42no—
47C22×D17—82no—
48Q8×C17—21no—
49Q8×C17—41no—
50D4×C17—21no—
51D4×C17—41no—
52D4×C17—42no—
53D4×C17—41no—
54C2×C4×C17—22no—
55C2×C4×C17—21no—
56C2×C4×C17—21no—
57C2×C4×C17—41no—
58C2×C4×C17C2×D411yescharacteristic
59C22⋊D17—82no—
60C22⋊D17—84no—
61C2×Dic17—22no—
62C2×Dic17—42no—
63D68—81no—
64D68—81no—
65Dic34—81no—
66C8×C17—41no—
67#2199—171nomaximal, Sylow
68C22×C17—22no—
69C22×C17—42no—
70C22×C17—41no—
71C22×C17—82no—
72C22×C17#12011yescharacteristic
73C2×D17—42no—
74C2×D17—42no—
75C2×D17—44no—
76C2×D17—42no—
77C2×D17—82no—
78C4×C17—21no—
79C4×C17—41no—
80C4×C17—41no—
81C4×C17—41no—
82Dic17—42no—
83Dic17—82no—
84H(3,𝔽4)—171no—
85B(4,𝔽2)—172no—
86#455D1711yescharacteristic, complemented
87#354—171no—
88AΣL(1,𝔽16)—172no—
89C2×C17—21no—
90C2×C17—42no—
91C2×C17—81no—
92C2×C17B(4,𝔽2)11yescharacteristic
93D17—82no—
94H(5,𝔽2)C2×D1711yescharacteristic
95#137—21no—
96#124—342no—
97#124—341no—
98#124—341no—
99#124C2×D1711yescharacteristic
100#120—172no—
101#120—342no—
102#120—342no—
103#104—21no—
104#100C2×D1711yescharacteristic
105#99—172no—
106#99—342no—
107C17#219911yescharacteristic, complemented, Sylow
108C24—172no—
109#41—21no—
110#41—41no—
111C2×Q8—341no—
112C2×Q8C22⋊D1711yescharacteristic, complemented
113C2×D4—22no—
114C2×D4—41no—
115C2×D4—341no—
116C2×D4—341no—
117C2×D4—342no—
118C2×D4—682no—
119C2×D4C22×D1711yescharacteristic
120Q16—41no—
121SD16—41no—
122M16—21no—
123C22⋊C4—22no—
124C22⋊C4—172no—
125C22⋊C4—342no—
126C22⋊C4—342no—
127C22⋊C4—342no—
128C22⋊C4—342no—
129C22⋊C4—682no—
130C42—341no—
131C42C22⋊D1711yescharacteristic, complemented
132C23—22no—
133C23—342no—
134C23—682no—
135C23—1362no—
136C23D4×D1712yes—
137Q8—21no—
138Q8—41no—
139Q8—1361no—
140D4—21no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—1361no—
145D4—1361no—
146D4—1362no—
147D4—1364no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—342no—
153C2×C4—682no—
154C2×C4C2×C22⋊D1711yescharacteristic
155C8—41no—
156C22—22no—
157C22—42no—
158C22—41no—
159C22—82no—
160C22—682no—
161C22—682no—
162C22—684no—
163C22—682no—
164C22—1362no—
165C22#1058762311yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—682no—
171C4—1362no—
172C2—21no—
173C2—42no—
174C2—81no—
175C2—1362no—
176C2#1176299511yescharacteristic
subgroupG/H
centerC2#11762995
commutator, cosocleC2×D4×C17C23
FrattiniC2×D4C22×D17
FittingC17×#455C2
socleC2×C17B(4,𝔽2)
hypercenter#455D17
hypocenterC17#2199
LCSG ▷ C2×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #11763551C2semidirect
2. #11762996C2semidirect
3. #11762604C2semidirect
4. #11762597C2semidirect
5. C17×#455C2semidirect
19. C23×D17D4semidirect
22. C2×Q8×C17D4semidirect
35. C42×C17D4semidirect
37. C22⋊Dic17D4semidirect
86. #455D17semidirect
107. C17#2199semidirect
112. C2×Q8C22⋊D17semidirect
131. C42C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
#117635512
#117629962
#117626042
#117625972
C17×#4552
#219917