Finite Group Databasebeta

← #34099426

Subgroups — #34099426

Order 20096.

262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 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#23758015C211yesmaximal, characteristic, complemented
2#23757663C211yesmaximal, characteristic, complemented
3#23757662C212yesmaximal, complemented
4#23757012C212yesmaximal, complemented
5#329×C157C211yesmaximal, characteristic, complemented
6C2×#14578943—22no—
7C2×#14578941—21no—
8C2×D1256—22no—
9C2×#14578939—21no—
10#19097768C2211yescharacteristic, complemented
11#19097760—21no—
12#19097759C2212yes—
13#19097757—22no—
14#19097741—22no—
15#19097739—21no—
16#19097682C2211yescharacteristic, complemented
17#124×C157C2211yescharacteristic
18#97×C157C2212yescomplemented
19C23×D157—21no—
20C2×Dic314D412yes—
21C2×D628—21no—
22C2×D628—21no—
23C2×D628—22no—
24C2×D628D412yes—
25C2×C22⋊D157—22no—
26#14578943—44no—
27#14578941—42no—
28D1256—44no—
29#14578939—42no—
30C2×#11929434—21no—
31C22⋊Dic157—21no—
32#14578931—22no—
33#14578931—22no—
34#14578931—22no—
35C4×Dic157—21no—
36C2×Q8×C157D411yescharacteristic
37C2×D4×C157D411yescharacteristic
38C2×C8×C157—22no—
39C22⋊C4×C157—22no—
40C42×C157C2311yescharacteristic
41C22×D157—22no—
42C22×D157—41no—
43C22×D157—82no—
44C22×D157—81no—
45Dic314—42no—
46D628—42no—
47D628—41no—
48D628—42no—
49D628—41no—
50D628—42no—
51D628—81no—
52C22⋊D157—82no—
53C2×Dic157—22no—
54#11929434—22no—
55C23×C157—21no—
56Q8×C157—41no—
57D4×C157—41no—
58C2×C4×C157—21no—
59C2×C4×C157C2×D412yes—
60C2×C4×C157C2×D411yescharacteristic
61C8×C157—24no—
62C22×C157—41no—
63C22×C157—42no—
64C22×C157#12011yescharacteristic
65C2×D157—42no—
66C2×D157—42no—
67C2×D157—42no—
68C2×D157—84no—
69C2×D157—82no—
70C2×D157—81no—
71C4×C157—22no—
72C4×C157—21no—
73C4×C157—41no—
74Dic157—42no—
75C2×C157—81no—
76C2×C157#45312yes—
77C2×C157#45311yescharacteristic
78D157—82no—
79D157—161no—
80C157#165511yescharacteristic, complemented, Sylow
81#1655—1571nomaximal, Sylow
82H(3,𝔽4)—1571no—
83#496—1571no—
84#496—1572no—
85#329—1572no—
86#329D15711yescharacteristic, complemented
87C2×SD16—3142no—
88C2×SD16—3141no—
89C2×D8—3142no—
90C2×D8—3141no—
91#127—1571no—
92#124—3142no—
93#124—3141no—
94#124—1572no—
95#124C2×D15711yescharacteristic
96#120—3142no—
97#120—3141no—
98#97—1571no—
99#97C2×D15712yescomplemented
100C24—3141no—
101C2×Q8—1572no—
102C2×Q8D62811yescharacteristic
103C2×D4—3141no—
104C2×D4—3142no—
105C2×D4—3142no—
106C2×D4—3141no—
107C2×D4—1572no—
108C2×D4D62811yescharacteristic
109SD16—6284no—
110SD16—6282no—
111D8—6284no—
112D8—6282no—
113C2×C8—22no—
114C2×C8—3141no—
115C22⋊C4—22no—
116C22⋊C4—3142no—
117C22⋊C4—3141no—
118C22⋊C4—3142no—
119C22⋊C4—3142no—
120C42—3141no—
121C42C22×D15711yescharacteristic
122C23—21no—
123C23—3142no—
124C23—6281no—
125C23—12562no—
126C23—12561no—
127Q8—41no—
128Q8—6282no—
129D4—41no—
130D4—6282no—
131D4—6282no—
132D4—6281no—
133D4—6282no—
134D4—6281no—
135D4—12561no—
136D4—12562no—
137C2×C4—21no—
138C2×C4—3142no—
139C2×C4D4×D15712yes—
140C2×C4C2×D62811yescharacteristic
141C8—24no—
142C8—3142no—
143C22—42no—
144C22—41no—
145C22—6282no—
146C22—6282no—
147C22—6282no—
148C22—12564no—
149C22—12562no—
150C22—12561no—
151C22#1909774111yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—6282no—
156C2—81no—
157C2—25121no—
158C2—12562no—
159C2#2375744212yes—
160C2#2375744311yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#329×C157C2
socleC22×C157#120
hypercenter#329D157
hypocenterC157#1655
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #23758015C2semidirect
2. #23757663C2semidirect
3. #23757662C2semidirect
4. #23757012C2semidirect
5. #329×C157C2semidirect
10. #19097768C22semidirect
16. #19097682C22semidirect
18. #97×C157C22semidirect
80. C157#1655semidirect
86. #329D157semidirect
99. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580152
#237576632
#237576622
#237570122
#329×C1572
#1655157