Finite Group Databasebeta

← #46810139

Subgroups — #46810139

Order 34432.

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

normal subgroups only

This group has 200 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#31771325C211yesmaximal, characteristic
2#31771008C211yesmaximal, characteristic, complemented
3#31770887C211yesmaximal, characteristic, complemented
4#31770352C211yesmaximal, characteristic, complemented
5#31770335C211yesmaximal, characteristic, complemented
6D4⋊#14524130C211yesmaximal, characteristic
7#461×C269C211yesmaximal, characteristic
8#22575453C2211yescharacteristic
9#22575446—22no—
10#22575440—21no—
11#22575432—21no—
12C4×Dic538C2211yescharacteristic
13#22575418—21no—
14#22575412—21no—
15#22575386—22no—
16#22575384—21no—
17C4⋊#14524130C2211yescharacteristic
18#22575376—21no—
19C4×#14524130C2211yescharacteristic
20C2×SD16×C269—21no—
21C4⋊Q8×C269C2211yescharacteristic
22C4×D4×C269C2211yescharacteristic
23C4⋊C8×C269C2211yescharacteristic
24#103×C269—21no—
25C2×Dic538—21no—
26C2×Dic538D411yescharacteristic
27C2×#14524130—21no—
28C2×#14524130—21no—
29C4⋊Dic269—21no—
30C4⋊Dic269D411yescharacteristic
31C4⋊Dic269D412yes—
32#16864381—22no—
33#16864381—21no—
34C22⋊Dic269—22no—
35C22⋊Dic269—22no—
36C4×Dic269—22no—
37C4×Dic269—21no—
38C2×Q8×C269—21no—
39C2×D4×C269D411yescharacteristic
40C22×C4×C269—21no—
41SD16×C269—42no—
42C2×C8×C269—21no—
43C4⋊C4×C269—21no—
44C4⋊C4×C269—21no—
45C4⋊C4×C269D411yescharacteristic
46C22⋊C4×C269—21no—
47C42×C269C2311yescharacteristic
48Dic538—21no—
49Dic538—21no—
50Dic538—41no—
51Dic538—41no—
52C2×Dic269—21no—
53C2×Dic269—21no—
54C2×Dic269—22no—
55#14524130—22no—
56#14524130—41no—
57C23×C269—21no—
58Q8×C269—41no—
59Q8×C269—41no—
60D4×C269—21no—
61D4×C269—21no—
62C2×C4×C269—21no—
63C2×C4×C269—41no—
64C2×C4×C269—41no—
65C2×C4×C269—41no—
66C2×C4×C269C2×D411yescharacteristic
67C2×C4×C269C2×D411yescharacteristic
68C2×C4×C269C2×D411yescharacteristic
69C8×C269—41no—
70C22×C269—41no—
71C22×C269—41no—
72C22×C269—41no—
73C22×C269#12011yescharacteristic
74C4×C269—21no—
75C4×C269—41no—
76C4×C269—81no—
77C4×C269#13511yescharacteristic
78C4×C269#13511yescharacteristic
79C4×C269AGL(1,ℤ/8)11yescharacteristic
80C4×C269#13711yescharacteristic
81Dic269—42no—
82Dic269—41no—
83Dic269—41no—
84C2×C269—81no—
85C2×C269#44911yescharacteristic
86C2×C269#45211yescharacteristic
87C2×C269#45611yescharacteristic
88C269#165011yescharacteristic, complemented, Sylow
89#1650—2691nomaximal, Sylow
90#556—2691no—
91#500—2691no—
92#475—2691no—
93#461D26911yescharacteristic
94#332—2691no—
95Q8⋊C8—2691no—
96D4⋊C8—2691no—
97C2×SD16—21no—
98C4⋊Q8—2691no—
99C4⋊Q8C2×D26911yescharacteristic
100#126—5382no—
101#124—5381no—
102C22⋊Q8—5381no—
103C4×Q8—2691no—
104C4×Q8—5381no—
105C4×D4C2×D26911yescharacteristic
106#117—5381no—
107#107—5382no—
108#106—5381no—
109C4⋊C8—2691no—
110C4⋊C8C2×D26911yescharacteristic
111#103—21no—
112#103—5381no—
113C4×C8—2691no—
114C2×Q8—21no—
115C2×Q8—2691no—
116C2×Q8—5381no—
117C2×D4C22⋊D26911yescharacteristic
118C22×C4—21no—
119SD16—42no—
120C2×C8—21no—
121C2×C8—5381no—
122C2×C8—5381no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—2691no—
126C4⋊C4—2692no—
127C4⋊C4—5382no—
128C4⋊C4—5381no—
129C4⋊C4—5381no—
130C4⋊C4C22⋊D26911yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—5382no—
133C22⋊C4—5382no—
134C42—5382no—
135C42—5381no—
136C42C22×D26911yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—5381no—
141Q8—5381no—
142Q8—10761no—
143Q8—10761no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—5381no—
151C2×C4—5382no—
152C2×C4—5381no—
153C2×C4D4×D26911yescharacteristic
154C2×C4D4×D26911yescharacteristic
155C2×C4C2×C22⋊D26911yescharacteristic
156C8—41no—
157C8—5382no—
158C8—10761no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2257542111yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—10762no—
167C4—10761no—
168C4—10761no—
169C4#2257547311yescharacteristic
170C4#2257547411yescharacteristic
171C4#2257547911yescharacteristic
172C4#2257548211yescharacteristic
173C2—81no—
174C2#3177074811yescharacteristic
175C2#3177076411yescharacteristic
176C2#3177078611yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC42×C269C23
FrattiniC42C22×D269
Fitting#461×C269C2
socleC22×C269#120
prime radicalC2×D4×C269D4
hypercenter#461D269
hypocenterC269#1650
LCSG ▷ C42×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #31771008C2semidirect
3. #31770887C2semidirect
4. #31770352C2semidirect
5. #31770335C2semidirect
88. C269#1650semidirect

Maximal subgroups

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

subgroupindex
#317713252
#317710082
#317708872
#317703522
#317703352
D4⋊#145241302
#461×C2692
#1650269