Finite Group Databasebeta

← #14545080

Subgroups — #14545080

Order 2176.

230 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 52 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
1D17⋊#101C212yesmaximal, complemented
2#11763720C212yesmaximal, complemented
3#11763717C212yesmaximal, complemented
4#11763379C211yesmaximal, characteristic
5C2×Q8×D17C2211yescharacteristic
6C2×#63254C411yescharacteristic
7D17⋊M16—22no—
8C17×#124C411yescharacteristic
9#10587630C411yescharacteristic
10#10587622C2211yescharacteristic
11#10587569—22no—
12#10587564—22no—
13#10587558C411yescharacteristic
14#10587555C2211yescharacteristic
15#10587545—22no—
16#10587545C2214yes—
17C2×Q8×C17C2×C411yescharacteristic
18C2×D4×C17C2×C411yescharacteristic
19C2×C22⋊D17—21no—
20C22×Dic17—21no—
21Q8×D17—42no—
22#63254—42no—
23C2×C4×D17C2×C411yescharacteristic
24C2×C4×D17D412yes—
25C2×Dic34C2×C411yescharacteristic
26C2×Dic34D412yes—
27C2×Dic34C2311yescharacteristic
28#63247—24no—
29#63243—42no—
30#63242—42no—
31C22⋊C4×C17—22no—
32C42×C17C2×C411yescharacteristic
33C22⋊Dic17—22no—
34#63227—21no—
35C4⋊Dic17—22no—
36#63225—22no—
37#63225—21no—
38C4×Dic17C2×C411yescharacteristic
39C23×C17—21no—
40C22×D17C22⋊C411yescharacteristic
41Q8×C17—41no—
42D4×C17—41no—
43C2×C4×C17—21no—
44C2×C4×C17C22⋊C412yes—
45C2×C4×C17C22×C411yescharacteristic
46C22⋊D17—42no—
47C2×Dic17—21no—
48C2×Dic17—21no—
49C2×Dic17—42no—
50C2×Dic17C22⋊C411yescharacteristic
51C2×Dic17C2×D412yes—
52C4×D17—22no—
53C4×D17—22no—
54C4×D17—22no—
55Dic34—42no—
56Dic34—41no—
57Dic34—41no—
58#3624—44no—
59#2133—171nomaximal, Sylow
60C22×C17—42no—
61C22×C17—41no—
62C22×C17C2×C22⋊C411yescharacteristic
63C2×D17—41no—
64C2×D17#9912yes—
65C4×C17—21no—
66C4×C17—22no—
67C4×C17—41no—
68Dic17—22no—
69Dic17—21no—
70Dic17—41no—
71Dic17—41no—
72#536—171no—
73C2×#101—172no—
74#356—172no—
75#355—172no—
76C2×C17—21no—
77C2×C17—81no—
78C2×C17C2×#9911yescharacteristic
79D17—22no—
80C2×#41—171no—
81C22×Q8—171no—
82C2×M16—342no—
83C4⋊Q8—171no—
84C4⋊Q8—171no—
85#124—171no—
86#124#59511yescharacteristic
87#123—342no—
88C22⋊Q8—342no—
89#117—171no—
90#101—174no—
91#101—342no—
92C17#213311yescharacteristic, complemented, Sylow
93#41—682no—
94C2×Q8—172no—
95C2×Q8—171no—
96C2×Q8—171no—
97C2×Q8—682no—
98C2×Q8C2×#59511yescharacteristic
99C2×D4—341no—
100C2×D4C2×#59511yescharacteristic
101C22×C4—172no—
102C22×C4—171no—
103C22×C4—341no—
104M16—344no—
105M16—682no—
106C2×C8—682no—
107C4⋊C4—341no—
108C4⋊C4—342no—
109C4⋊C4—342no—
110C22⋊C4—22no—
111C22⋊C4—341no—
112C22⋊C4—342no—
113C42—171no—
114C42C2×#59511yescharacteristic
115C23—21no—
116C23—171no—
117Q8—41no—
118Q8—682no—
119Q8—681no—
120Q8—681no—
121D4—41no—
122D4—682no—
123C2×C4—21no—
124C2×C4—172no—
125C2×C4—171no—
126C2×C4—342no—
127C2×C4—342no—
128C2×C4—342no—
129C2×C4—341no—
130C2×C4—341no—
131C2×C4—682no—
132C2×C4C22⋊#59512yes—
133C2×C4C22×#59511yescharacteristic
134C8—684no—
135C22—42no—
136C22—41no—
137C22—172no—
138C22—681no—
139C22C2×C22⋊#59511yescharacteristic
140C4—22no—
141C4—21no—
142C4—41no—
143C4—342no—
144C4—341no—
145C4—681no—
146C4—681no—
147C2—21no—
148C2—81no—
149C2—342no—
150C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC2×C17C2×#99
prime radicalC2×#63254C4
cosocleC2×Dic34C23
hypercenter#124#595
hypocenterC17#2133
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. D17⋊#101C2semidirect
2. #11763720C2semidirect
3. #11763717C2semidirect
92. C17#2133semidirect

Maximal subgroups

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

subgroupindex
D17⋊#1012
#117637202
#117637172
#117633792
#213317