Finite Group Databasebeta

← #14529951

Subgroups — #14529951

Order 2176.

290 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 69 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#11763435C211yesmaximal, characteristic
2#11762672C212yesmaximal
3#11762576C211yesmaximal, characteristic
4#11762575C212yesmaximal
5C17×#381C211yesmaximal, characteristic
6C23×C4×C17C2211yescharacteristic
7C2×C22⋊C4×C17C2211yescharacteristic
8C2×C22⋊C4×C17C2211yescharacteristic
9C2×C22⋊Dic17C2212yes—
10C2×C22⋊Dic17C2212yes—
11#10587610—22no—
12#10587564—24no—
13C17×#95—22no—
14#10587502—24no—
15C24×C17C2311yescharacteristic
16C22×C4×C17—22no—
17C22×C4×C17—21no—
18C22×C4×C17—21no—
19C22×C4×C17—22no—
20C22×C4×C17—21no—
21C22×C4×C17D412yes—
22C22×C4×C17Q812yescomplemented
23C22×Dic17—22no—
24C22×Dic17D414yes—
25C2×Dic34—22no—
26C22⋊C4×C17—22no—
27C22⋊C4×C17—22no—
28C22⋊Dic17—22no—
29C22⋊Dic17—22no—
30C22⋊Dic17—22no—
31C22⋊Dic17—22no—
32C4⋊Dic17—22no—
33#63225—24no—
34C23×C17—42no—
35C23×C17C2×D412yes—
36C23×C17C2×D411yescharacteristic
37C23×C17C2×Q811yescharacteristic
38C23×C17#4111yescharacteristic
39C23×C17#4112yes—
40C2×C4×C17—22no—
41C2×C4×C17—24no—
42C2×C4×C17—22no—
43C2×C4×C17—41no—
44C2×C4×C17—42no—
45C2×C4×C17—41no—
46C2×C4×C17—42no—
47C2×C4×C17—42no—
48C2×C4×C17—41no—
49C2×Dic17—24no—
50C2×Dic17—44no—
51C2×Dic17—44no—
52C2×Dic17—42no—
53Dic34—82no—
54#2030—171nomaximal, Sylow
55C22×C17—24no—
56C22×C17—22no—
57C22×C17—41no—
58C22×C17—41no—
59C22×C17—42no—
60C22×C17#12011yescharacteristic
61C22×C17C22⋊Q812yes—
62C22×C17C22⋊Q811yescharacteristic
63C22×C17#12412yes—
64C22×C17#12411yescharacteristic
65C4×C17—42no—
66C4×C17—42no—
67C4×C17—81no—
68Dic17—44no—
69Dic17—82no—
70#543—171no—
71#381—172no—
72#381D1711yescharacteristic
73#342—172no—
74#342—171no—
75C2×C17—22no—
76C2×C17—21no—
77C2×C17—22no—
78C2×C17—21no—
79C2×C17#39311yescharacteristic
80C2×C17#45812yes—
81C23×C4C2×D1711yescharacteristic
82C22⋊Q8—344no—
83C22⋊Q8—342no—
84C2×C22⋊C4—172no—
85C2×C22⋊C4—172no—
86C2×C22⋊C4C2×D1711yescharacteristic
87C2×C22⋊C4C2×D1711yescharacteristic
88#95—22no—
89#95—344no—
90C17#203011yescharacteristic, complemented, Sylow
91C24C22×D1711yescharacteristic
92C2×Q8—342no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—174no—
99C22×C4—342no—
100C22×C4Dic3412yescomplemented
101C22×C4C22⋊D1712yes—
102C4⋊C4—344no—
103C4⋊C4—342no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—342no—
107C22⋊C4—342no—
108C22⋊C4—342no—
109C22⋊C4—342no—
110C23—42no—
111C23C2×Dic3411yescharacteristic
112C23#6325211yescharacteristic
113C23D4×D1712yes—
114C23#6325412yes—
115C23C2×C22⋊D1711yescharacteristic
116Q8—1362no—
117C2×C4—24no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—344no—
127C2×C4—684no—
128C2×C4—684no—
129C2×C4—682no—
130C22—22no—
131C22—24no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#1058756412yes—
136C22#1058757112yes—
137C22#1058761011yescharacteristic
138C22#1058762211yescharacteristic
139C22#1058762311yescharacteristic
140C4—42no—
141C4—42no—
142C4—81no—
143C4—684no—
144C4—1362no—
145C2—21no—
146C2—22no—
147C2—22no—
148C2—21no—
149C2#1176272111yescharacteristic
150C2#1176299812yes—
subgroupG/H
centerC22#10587623
commutator, prime radical, cosocleC24×C17C23
FrattiniC24C22×D17
FittingC17×#381C2
socleC22×C17#120
hypercenter#381D17
hypocenterC17#2030
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
22. C22×C4×C17Q8semidirect
90. C17#2030semidirect
100. C22×C4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117634352
#117626722
#117625762
#117625752
C17×#3812
#203017