Finite Group Databasebeta

← #46809892

Subgroups — #46809892

Order 34432.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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#31771236C211yesmaximal, characteristic
2#31770935C212yesmaximal
3#31770341C212yesmaximal
4#31770331C211yesmaximal, characteristic
5#398×C269C211yesmaximal, characteristic
6#22575452—22no—
7#22575444—22no—
8#22575432—21no—
9#22575432C2212yes—
10#22575412—21no—
11#22575384—22no—
12#22575376—22no—
13C22⋊#14524130C2212yes—
14C2×C4⋊C4×C269—21no—
15C2×C4⋊C4×C269C2211yescharacteristic
16C2×C4⋊C4×C269C2211yescharacteristic
17#95×C269—21no—
18#95×C269C2211yescharacteristic
19C2×Dic538—21no—
20C2×Dic538D412yes—
21C2×#14524130—22no—
22C4⋊Dic269—21no—
23C4⋊Dic269—21no—
24C4⋊Dic269—21no—
25C4⋊Dic269D412yes—
26#16864381—22no—
27#16864381—22no—
28C22⋊Dic269—22no—
29C4×Dic269—22no—
30C22×C4×C269—21no—
31C22×C4×C269—21no—
32C22×C4×C269D411yescharacteristic
33C22×C4×C269D411yescharacteristic
34C22×C4×C269C2311yescharacteristic
35C4⋊C4×C269—22no—
36C4⋊C4×C269—22no—
37C4⋊C4×C269—42no—
38Dic538—42no—
39Dic538—81no—
40C2×Dic269—22no—
41C2×Dic269—22no—
42#14524130—42no—
43C23×C269C2×D411yescharacteristic
44C2×C4×C269—21no—
45C2×C4×C269—21no—
46C2×C4×C269—22no—
47C2×C4×C269—21no—
48C2×C4×C269—41no—
49C2×C4×C269—41no—
50C2×C4×C269—41no—
51C2×C4×C269—41no—
52C2×C4×C269—41no—
53C2×C4×C269—41no—
54C2×C4×C269—41no—
55C2×C4×C269SD1612yes—
56C2×C4×C269C2×D412yes—
57C22×C269—21no—
58C22×C269—21no—
59C22×C269#12011yescharacteristic
60C22×C269C2×SD1611yescharacteristic
61C22×C269#13711yescharacteristic
62C4×C269—22no—
63C4×C269—22no—
64C4×C269—42no—
65C4×C269—41no—
66C4×C269—81no—
67Dic269—42no—
68Dic269—42no—
69C2×C269—21no—
70C2×C269—21no—
71C2×C269#44811yescharacteristic
72C2×C269#45611yescharacteristic
73C2×C269#45811yescharacteristic
74C269#161111yescharacteristic, complemented, Sylow
75#1611—2691nomaximal, Sylow
76#544—2691no—
77#483—2692no—
78#398D26911yescharacteristic
79#328—2692no—
80#324—2691no—
81C4⋊Q8—5382no—
82#125—5382no—
83C22⋊Q8—2692no—
84C22⋊Q8—5381no—
85#117—5381no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2×D26911yescharacteristic
88C2×C4⋊C4C2×D26911yescharacteristic
89#106—5382no—
90#103—5382no—
91C22⋊C8—2692no—
92#95—21no—
93#95C2×D26911yescharacteristic
94C2×Q8—2692no—
95C2×Q8—5381no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D26911yescharacteristic
99C22×C4C22⋊D26911yescharacteristic
100C22×C4C22×D26911yescharacteristic
101C2×C8—5382no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4—2692no—
106C4⋊C4—5381no—
107C4⋊C4—5381no—
108C4⋊C4—5381no—
109C4⋊C4—5382no—
110C4⋊C4—5382no—
111C22⋊C4—5382no—
112C42—5382no—
113C23C2×C22⋊D26911yescharacteristic
114Q8—10762no—
115Q8—21521no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—5382no—
128C2×C4—5382no—
129C2×C4#1686439012yes—
130C2×C4D4×D26912yes—
131C8—10762no—
132C22—21no—
133C22—21no—
134C22#2257542111yescharacteristic
135C22C2×#1686439011yescharacteristic
136C22#2257548311yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—81no—
142C4—10762no—
143C4—10762no—
144C2—21no—
145C2—21no—
146C2#3177074311yescharacteristic
147C2#3177078711yescharacteristic
148C2#3177079411yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC22×C4×C269C23
FrattiniC22×C4C22×D269
Fitting#398×C269C2
socleC22×C269#120
prime radicalC23×C269C2×D4
hypercenter#398D269
hypocenterC269#1611
LCSG ▷ C22×C4×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
74. C269#1611semidirect

Maximal subgroups

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

subgroupindex
#317712362
#317709352
#317703412
#317703312
#398×C2692
#1611269