Finite Group Databasebeta

← #14530780

Subgroups — #14530780

Order 2176.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#11762998C212yesmaximal, complemented
2#11762976C211yesmaximal, characteristic
3#11762602C211yesmaximal, characteristic, complemented
4#11762599C212yesmaximal, complemented
5C17×#559C211yesmaximal, characteristic, complemented
6#10587691C2211yescharacteristic
7C17×#127C2211yescharacteristic
8C17×#124—22no—
9C17×#123—21no—
10C17×#123C2212yes—
11C4×Q8×C17—21no—
12C4×D4×C17—22no—
13#10587618—21no—
14#10587615—22no—
15#10587505C2212yes—
16#10587504C2211yescharacteristic
17#10587477—22no—
18#10587476—21no—
19C2×Q8×C17—21no—
20C2×D4×C17—22no—
21C2×D4×C17D411yescharacteristic
22C2×D4×C17C2311yescharacteristic
23C22×C4×C17—21no—
24C22×C4×C17D412yes—
25C2×C22⋊D17—22no—
26#63254—41no—
27D4×D17—41no—
28#63252—21no—
29C4⋊C4×C17—21no—
30C4⋊C4×C17—22no—
31C22⋊C4×C17—21no—
32C22⋊C4×C17—22no—
33C22⋊C4×C17—22no—
34C22⋊C4×C17D412yes—
35C42×C17—22no—
36C42×C17D411yescharacteristic
37C22⋊Dic17—22no—
38#63229—41no—
39#63228—41no—
40#63227—42no—
41#63225—42no—
42#63223—21no—
43C23×C17—21no—
44C23×C17C2×D412yes—
45C22×D17—22no—
46Q8×C17—42no—
47D4×C17—41no—
48D4×C17—41no—
49D4×C17—44no—
50C2×C4×C17—21no—
51C2×C4×C17—22no—
52C2×C4×C17—22no—
53C2×C4×C17—42no—
54C2×C4×C17—81no—
55C2×C4×C17C2×D411yescharacteristic
56C22⋊D17—42no—
57C22⋊D17—41no—
58C2×Dic17—22no—
59C2×Dic17—42no—
60D68—21no—
61C4×D17—41no—
62Dic34—21no—
63#3622—41no—
64#2198—171nomaximal, Sylow
65C22×C17—22no—
66C22×C17—42no—
67C22×C17—42no—
68C22×C17—41no—
69C22×C17#12011yescharacteristic
70C2×D17—41no—
71C2×D17—82no—
72C4×C17—22no—
73C4×C17—21no—
74C4×C17—42no—
75C4×C17—42no—
76Dic17—41no—
77Dic17—82no—
78#559D1711yescharacteristic, complemented
79#458—172no—
80#453—171no—
81#353—171no—
82#352—172no—
83C2×C17—21no—
84C2×C17—42no—
85C2×C17—81no—
86C2×C17B(4,𝔽2)11yescharacteristic
87D17—81no—
88H(5,𝔽2)—171no—
89AGL(1,ℤ/8)—341no—
90#127C2×D1711yescharacteristic
91#124—22no—
92#123—21no—
93#123—342no—
94#123C2×D1712yes—
95C4×Q8—21no—
96C4×D4—22no—
97#104—341no—
98#100—171no—
99#99—172no—
100#99—342no—
101C17#219811yescharacteristic, complemented, Sylow
102#41—341no—
103#41—681no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—342no—
107C2×D4—681no—
108C2×D4C22⋊D1711yescharacteristic
109C2×D4C22×D1711yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D1712yes—
112SD16—681no—
113D8—681no—
114M16—341no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—682no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—342no—
122C22⋊C4—682no—
123C22⋊C4C22⋊D1712yes—
124C42—22no—
125C42C22⋊D1711yescharacteristic
126C23—21no—
127C23—342no—
128C23C2×C22⋊D1712yes—
129Q8—42no—
130Q8—341no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—341no—
135D4—682no—
136D4—681no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—342no—
143C2×C4—681no—
144C2×C4—682no—
145C2×C4C2×C22⋊D1711yescharacteristic
146C8—681no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—681no—
152C22—1362no—
153C22#1058763911yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—681no—
159C4—1362no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—1361no—
164C2#1176299611yescharacteristic
subgroupG/H
centerC2#11762996
commutator, cosocleC2×D4×C17C23
FrattiniC2×D4C22×D17
FittingC17×#559C2
socleC2×C17B(4,𝔽2)
prime radical#11762976C2
hypercenter#559D17
hypocenterC17#2198
LCSG ▷ C2×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #11762998C2semidirect
3. #11762602C2semidirect
4. #11762599C2semidirect
5. C17×#559C2semidirect
78. #559D17semidirect
101. C17#2198semidirect

Maximal subgroups

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

subgroupindex
#117629982
#117629762
#117626022
#117625992
C17×#5592
#219817