Finite Group Databasebeta

← #46812558

Subgroups — #46812558

Order 34432.

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

normal subgroups only

This group has 238 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#31771175C211yesmaximal, characteristic
2#31770577C212yesmaximal
3#31770400C211yesmaximal, characteristic, complemented
4#31770399C212yesmaximal, complemented
5#354×C269C211yesmaximal, characteristic, complemented
6C2×#16864401C2211yescharacteristic
7C2×Q8×D269C411yescharacteristic, complemented
8#22575453C411yescharacteristic, complemented
9#22575451C411yescharacteristic, complemented
10#22575440C2211yescharacteristic
11#22575435—22no—
12#22575430—22no—
13#22575410C411yescharacteristic, complemented
14#22575409—22no—
15#22575368C2212yes—
16#22575367—22no—
17#124×C269C2211yescharacteristic
18#99×C269C2212yes—
19#16864401—42no—
20C2×Dic538—21no—
21C2×Dic538C2×C411yescharacteristic
22C2×Dic538C2×C411yescharacteristic
23Q8×D269—42no—
24C2×C22⋊D269D412yes—
25C22×Dic269D412yes—
26C2×C4×D269—21no—
27C2×C4×D269C2×C411yescharacteristic
28C4⋊Dic269—22no—
29#16864381—21no—
30#16864381—22no—
31#16864381—42no—
32C22⋊Dic269—22no—
33C22⋊Dic269—22no—
34#16864379—21no—
35C4×Dic269—42no—
36C4×Dic269C2×C411yescharacteristic
37C2×Q8×C269C2×C411yescharacteristic
38C2×D4×C269C2311yescharacteristic
39C22⋊C4×C269—22no—
40C22⋊C4×C269—22no—
41C42×C269C2×C411yescharacteristic
42C22×D269C22⋊C411yescharacteristic
43Dic538—41no—
44Dic538—42no—
45Dic538—41no—
46C22⋊D269—42no—
47C2×Dic269—22no—
48C2×Dic269—22no—
49C2×Dic269—21no—
50C2×Dic269—21no—
51C2×Dic269—42no—
52C2×Dic269C22⋊C412yes—
53C2×Dic269C22⋊C411yescharacteristic
54C4×D269—22no—
55C4×D269—42no—
56C23×C269C2×D412yes—
57Q8×C269—41no—
58D4×C269—41no—
59C2×C4×C269—21no—
60C2×C4×C269—21no—
61C2×C4×C269—42no—
62C2×C4×C269C22×C411yescharacteristic
63C22×C269—22no—
64C22×C269—42no—
65C22×C269C2×C22⋊C411yescharacteristic
66C2×D269—21no—
67C2×D269—41no—
68C4×C269—21no—
69C4×C269—41no—
70C4×C269—41no—
71C4×C269—82no—
72Dic269—22no—
73Dic269—41no—
74Dic269—41no—
75Dic269—82no—
76Dic269#9912yes—
77C2×C269—21no—
78C2×C269—42no—
79C2×C269C2×#9911yescharacteristic
80D269—41no—
81C269#212911yescharacteristic, complemented, Sylow
82#2129—2691nomaximal, Sylow
83#536—2691no—
84#410—2692no—
85#354—2691no—
86#354—2692no—
87#354D26911yescharacteristic, complemented
88C2×#41—2691no—
89C22×Q8—2691no—
90C4⋊Q8—2691no—
91C4⋊Q8—2691no—
92#124—2691no—
93#124C2×D26911yescharacteristic
94#123—5382no—
95C22⋊Q8—5382no—
96#117—2691no—
97#117—5382no—
98#99—2692no—
99#99—5382no—
100#99C2×D26912yes—
101#41—10762no—
102C2×Q8—2691no—
103C2×Q8—2691no—
104C2×Q8—5381no—
105C2×Q8—10762no—
106C2×Q8C4×D26911yescharacteristic
107C2×D4—2692no—
108C2×D4C22×D26911yescharacteristic
109C22×C4—2692no—
110C22×C4—2691no—
111C22×C4—5381no—
112C4⋊C4—5381no—
113C4⋊C4—5382no—
114C4⋊C4—5382no—
115C4⋊C4—10762no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—5381no—
119C22⋊C4—5382no—
120C22⋊C4—5382no—
121C42—2691no—
122C42—10762no—
123C42C4×D26911yescharacteristic
124C23—2691no—
125C23D4×D26912yes—
126Q8—41no—
127Q8—10761no—
128Q8—10761no—
129Q8—10762no—
130D4—41no—
131D4—10762no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—2691no—
136C2×C4—2692no—
137C2×C4—5382no—
138C2×C4—5382no—
139C2×C4—5382no—
140C2×C4—5381no—
141C2×C4—5381no—
142C2×C4—10762no—
143C2×C4—10762no—
144C2×C4C2×C4×D26911yescharacteristic
145C22—22no—
146C22—42no—
147C22—5381no—
148C22—10761no—
149C22C22⋊C4×D26911yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—2692no—
155C4—5382no—
156C4—10761no—
157C4—10761no—
158C4—21522no—
159C2—21no—
160C2—42no—
161C2—10761no—
162C2#99×D26911yescharacteristic
subgroupG/H
centerC2#99×D269
commutatorC2×C4×C269C22×C4
FrattiniC2×D4C22×D269
Fitting#354×C269C2
socleC2×C269C2×#99
prime radicalC2×#16864401C22
cosocleC2×D4×C269C23
hypercenter#354D269
hypocenterC269#2129
LCSG ▷ C2×C4×C269 ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #31770400C2semidirect
4. #31770399C2semidirect
5. #354×C269C2semidirect
7. C2×Q8×D269C4semidirect
8. #22575453C4semidirect
9. #22575451C4semidirect
13. #22575410C4semidirect
81. C269#2129semidirect
87. #354D269semidirect

Maximal subgroups

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

subgroupindex
#317711752
#317705772
#317704002
#317703992
#354×C2692
#2129269