Finite Group Databasebeta

← #46812555

Subgroups — #46812555

Order 34432.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#31771168C211yesmaximal, characteristic
2#31770577C212yesmaximal, complemented
3#31770398C211yesmaximal, characteristic, complemented
4#31770397C212yesmaximal, complemented
5#353×C269C211yesmaximal, characteristic, complemented
6C2×#16864401C2211yescharacteristic
7C2×D4×D269C411yescharacteristic, complemented
8#22575449C2211yescharacteristic
9#22575441C411yescharacteristic, complemented
10#22575440C411yescharacteristic, complemented
11#22575425—22no—
12#22575421—22no—
13#22575410C411yescharacteristic, complemented
14#22575409—22no—
15#22575368C2212yescomplemented
16#22575367—22no—
17#127×C269C2211yescharacteristic
18#99×C269C2212yescomplemented
19C23×D269—21no—
20#16864401—42no—
21C2×Dic538C2×C411yescharacteristic
22C2×D1076C2×C411yescharacteristic
23C2×C22⋊D269—21no—
24C2×C22⋊D269—22no—
25C2×C22⋊D269D412yes—
26D4×D269—42no—
27C22×Dic269D412yes—
28C2×C4×D269C2×C411yescharacteristic
29#16864381—21no—
30#16864381—42no—
31C22⋊Dic269—22no—
32C22⋊Dic269—22no—
33#16864379—21no—
34#16864379—22no—
35C4×Dic269—42no—
36C4×Dic269C2×C411yescharacteristic
37C2×D4×C269—22no—
38C2×D4×C269C2×C411yescharacteristic
39C2×D4×C269C2311yescharacteristic
40C22⋊C4×C269—22no—
41C42×C269C2×C411yescharacteristic
42C22×D269—21no—
43C22×D269—42no—
44C22×D269—82no—
45C22×D269C22⋊C411yescharacteristic
46Dic538—41no—
47D1076—41no—
48C22⋊D269—42no—
49C22⋊D269—42no—
50C22⋊D269—82no—
51C2×Dic269—22no—
52C2×Dic269—22no—
53C2×Dic269—21no—
54C2×Dic269—42no—
55C2×Dic269C22⋊C412yes—
56C2×Dic269C22⋊C411yescharacteristic
57C4×D269—22no—
58C23×C269—21no—
59C23×C269C2×D412yes—
60D4×C269—41no—
61D4×C269—41no—
62D4×C269—82no—
63C2×C4×C269—21no—
64C2×C4×C269—42no—
65C2×C4×C269C22×C411yescharacteristic
66C22×C269—22no—
67C22×C269—42no—
68C22×C269—42no—
69C22×C269—41no—
70C22×C269C2×C22⋊C411yescharacteristic
71C2×D269—21no—
72C2×D269—41no—
73C2×D269—42no—
74C2×D269—41no—
75C2×D269—84no—
76C4×C269—21no—
77C4×C269—41no—
78C4×C269—82no—
79Dic269—22no—
80Dic269—41no—
81Dic269—82no—
82Dic269#9912yes—
83C2×C269—21no—
84C2×C269—42no—
85C2×C269—81no—
86C2×C269C2×#9911yescharacteristic
87D269—41no—
88D269—81no—
89C269#212811yescharacteristic, complemented, Sylow
90#2128—2691nomaximal, Sylow
91#535—2691no—
92#410—2692no—
93#353—2691no—
94#353—2692no—
95#353D26911yescharacteristic, complemented
96C2×#41—2691no—
97C22×D4—2691no—
98#127—2691no—
99#127C2×D26911yescharacteristic
100#124—2691no—
101#124—2691no—
102#121—5382no—
103#120—5382no—
104#117—2691no—
105#117—5382no—
106#99—2692no—
107#99—5382no—
108#99C2×D26912yescomplemented
109C24—5381no—
110#41—10762no—
111C2×Q8—2691no—
112C2×D4—22no—
113C2×D4—2692no—
114C2×D4—2691no—
115C2×D4—5381no—
116C2×D4—5382no—
117C2×D4—10762no—
118C2×D4C4×D26911yescharacteristic
119C2×D4C22×D26911yescharacteristic
120C22×C4—2692no—
121C22×C4—2691no—
122C4⋊C4—5381no—
123C4⋊C4—10762no—
124C22⋊C4—22no—
125C22⋊C4—5382no—
126C22⋊C4—5381no—
127C22⋊C4—5382no—
128C22⋊C4—5382no—
129C42—2691no—
130C42—10762no—
131C42C4×D26911yescharacteristic
132C23—21no—
133C23—2691no—
134C23—5381no—
135C23—10762no—
136C23—21522no—
137C23D4×D26912yes—
138Q8—10761no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—10761no—
143D4—10762no—
144D4—10762no—
145D4—21522no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—2691no—
149C2×C4—2692no—
150C2×C4—5382no—
151C2×C4—5382no—
152C2×C4—5382no—
153C2×C4—5381no—
154C2×C4—10762no—
155C2×C4C2×C4×D26911yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—5381no—
161C22—10761no—
162C22—10762no—
163C22—10761no—
164C22—21524no—
165C22C22⋊C4×D26911yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—2692no—
170C4—5382no—
171C4—10761no—
172C4—21522no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—10761no—
177C2—21521no—
178C2#99×D26911yescharacteristic
subgroupG/H
centerC2#99×D269
commutatorC2×C4×C269C22×C4
FrattiniC2×D4C22×D269
Fitting#353×C269C2
socleC2×C269C2×#99
prime radical#31771168C2
cosocleC2×D4×C269C23
hypercenter#353D269
hypocenterC269#2128
LCSG ▷ C2×C4×C269 ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #31770577C2semidirect
3. #31770398C2semidirect
4. #31770397C2semidirect
5. #353×C269C2semidirect
7. C2×D4×D269C4semidirect
9. #22575441C4semidirect
10. #22575440C4semidirect
13. #22575410C4semidirect
15. #22575368C22semidirect
18. #99×C269C22semidirect
89. C269#2128semidirect
95. #353D269semidirect
108. #99C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317711682
#317705772
#317703982
#317703972
#353×C2692
#2128269