Finite Group Databasebeta

← #17959843

Subgroups — #17959843

Order 4352.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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#14545153C212yesmaximal
2#14543789C214yesmaximal, complemented
3#14539164C211yesmaximal, characteristic
4#11763732—22no—
5#11763705—22no—
6#11763701C2214yes—
7#11763366C411yescharacteristic
8#11763177C412yes—
9#11763177C2212yes—
10#11763159—22no—
11#11762818—22no—
12C4×#63223C2211yescharacteristic
13C2×C17×C4⋊Q8C411yescharacteristic
14C22×Q8×C17—21no—
15C17×C4⋊Q8—21no—
16C17×C4⋊Q8—41no—
17C17×C4⋊Q8C2×C412yes—
18C2×C4⋊C4×C17—41no—
19C2×C42×C17C2×C411yescharacteristic
20#10587612—21no—
21#10587610—41no—
22C2×#63223D412yes—
23#10587556C2×C412yes—
24#10587555—21no—
25C4×Dic34—41no—
26#10587538—24no—
27#10587506—24no—
28#10587506—24no—
29#10587474C2×C411yescharacteristic
30#10587474C2311yescharacteristic
31C4×#3622D412yes—
32C2×Q8×C17—22no—
33C2×Q8×C17—44no—
34C2×Q8×C17—41no—
35C22×C4×C17—21no—
36C22×C4×C17—41no—
37C22×C4×C17C22⋊C411yescharacteristic
38C2×Dic34—41no—
39C4⋊C4×C17—42no—
40C4⋊C4×C17—42no—
41C42×C17—21no—
42C42×C17C22⋊C411yescharacteristic
43C42×C17C22×C411yescharacteristic
44C22⋊Dic17—41no—
45C4⋊Dic17—22no—
46C4⋊Dic17—41no—
47#63225—42no—
48C4×Dic17—41no—
49#63223—42no—
50C2×#3622C22⋊C412yes—
51C2×#3622C2×D412yes—
52#63216—44no—
53#11916—171nomaximal, Sylow
54C23×C17#10011yescharacteristic
55Q8×C17—44no—
56Q8×C17—82no—
57C2×C4×C17—21no—
58C2×C4×C17—22no—
59C2×C4×C17—21no—
60C2×C4×C17—42no—
61C2×C4×C17—42no—
62C2×C4×C17—82no—
63C2×C4×C17#10011yescharacteristic
64C2×C4×C17C2×C22⋊C411yescharacteristic
65C2×Dic17—42no—
66Dic34—82no—
67#3622—22no—
68#3622—22no—
69#3139—171no—
70#2165—172no—
71#1356—174no—
72C22×C17—21no—
73C22×C17—41no—
74C22×C17C2×#10011yescharacteristic
75C4×C17—22no—
76C4×C17—21no—
77C4×C17—41no—
78C4×C17—82no—
79C4×C17#42112yes—
80Dic17—82no—
81#533—171no—
82C2×C4⋊Q8#59511yescharacteristic
83#491—172no—
84#491—172no—
85#487—342no—
86#419—342no—
87C4×M16—171no—
88#364—342no—
89#349—342no—
90#347—174no—
91C2×C17—21no—
92C2×C17—41no—
93C2×C17#153311yescharacteristic
94C22×Q8—21no—
95C2×M16—172no—
96C4⋊Q8—21no—
97C4⋊Q8—41no—
98C4⋊Q8—341no—
99C4⋊Q8C2×#59512yes—
100#125—172no—
101C22⋊Q8—681no—
102C4×Q8—681no—
103#117—341no—
104C2×C4⋊C4—41no—
105C2×C42C2×#59511yescharacteristic
106M32—344no—
107#103—344no—
108#103—344no—
109#97—171no—
110#97—171no—
111C4×C8—172no—
112C17#1191611yescharacteristic, complemented, Sylow
113C2×Q8—22no—
114C2×Q8—44no—
115C2×Q8—41no—
116C2×Q8—681no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4C22⋊#59511yescharacteristic
120M16—682no—
121C2×C8—172no—
122C2×C8—172no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—342no—
126C4⋊C4—682no—
127C4⋊C4—681no—
128C22⋊C4—681no—
129C42—21no—
130C42—681no—
131C42C22⋊#59511yescharacteristic
132C42C22×#59511yescharacteristic
133C16—684no—
134C23#1058755311yescharacteristic
135Q8—44no—
136Q8—82no—
137Q8—1362no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—82no—
144C2×C4—682no—
145C2×C4#1058755311yescharacteristic
146C2×C4C2×C22⋊#59511yescharacteristic
147C8—342no—
148C8—342no—
149C22—21no—
150C22—41no—
151C22C2×#1058755311yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—82no—
156C4—1362no—
157C4#1176380712yes—
158C2—21no—
159C2—41no—
160C2#1454426311yescharacteristic
subgroupG/H
centerC2#14544263
commutatorC42×C17C22×C4
FrattiniC42C22×#595
FittingC2×C17×C4⋊Q8C4
socleC2×C17#1533
prime radicalC23×C17#100
cosocle#10587474C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11916
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #14543789C2semidirect
112. C17#11916semidirect

Maximal subgroups

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

subgroupindex
#145451532
#145437892
#145391642
#1191617