Finite Group Databasebeta

← #34098726

Subgroups — #34098726

Order 20096.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#23757822C211yesmaximal, characteristic
2#23757225C212yesmaximal, complemented
3#23757013C211yesmaximal, characteristic, complemented
4#23757012C212yesmaximal, complemented
5#329×C157C211yesmaximal, characteristic, complemented
6C2×#14578955C411yescharacteristic
7C2×#14578953C411yescharacteristic
8C2×#14578937—22no—
9#19097769C411yescharacteristic
10#19097767C411yescharacteristic
11#19097759C2211yescharacteristic
12#19097744—22no—
13#19097731—22no—
14C42×D157C2211yescharacteristic
15#19097682C2212yes—
16C8×Dic157—22no—
17#124×C157C2211yescharacteristic
18#97×C157C2212yes—
19#14578955—42no—
20#14578953—42no—
21C2×Dic314C2×C411yescharacteristic
22C2×D628—21no—
23C2×D628C2×C411yescharacteristic
24C2×C22⋊D157—21no—
25C2×C22⋊D157—22no—
26C22×Dic157—21no—
27C2×C4×D157—21no—
28C2×C4×D157C2×C411yescharacteristic
29C2×C4×D157D412yes—
30#14578937—28no—
31C2×#11929434—22no—
32#14578933—22no—
33#14578931—22no—
34C4×Dic157C2×C411yescharacteristic
35C4×Dic157D412yes—
36C2×Q8×C157C2×C411yescharacteristic
37C2×D4×C157C2×C411yescharacteristic
38C2×C8×C157—22no—
39C22⋊C4×C157—22no—
40C42×C157C2311yescharacteristic
41C22×D157—21no—
42C22×D157C22⋊C411yescharacteristic
43Dic314—41no—
44D628—41no—
45D628—42no—
46C22⋊D157—42no—
47C22⋊D157—44no—
48C2×Dic157—21no—
49C2×Dic157—42no—
50C2×Dic157C22⋊C411yescharacteristic
51C2×Dic157C22⋊C412yes—
52C4×D157—22no—
53C4×D157—24no—
54C4×D157—42no—
55#11929434—24no—
56C23×C157—21no—
57Q8×C157—41no—
58D4×C157—41no—
59C2×C4×C157—21no—
60C2×C4×C157C22×C411yescharacteristic
61C2×C4×C157C2×D412yes—
62C8×C157—24no—
63C22×C157—41no—
64C22×C157—42no—
65C22×C157C2×C22⋊C411yescharacteristic
66C2×D157—21no—
67C2×D157—22no—
68C2×D157—41no—
69C2×D157—42no—
70C4×C157—21no—
71C4×C157—22no—
72C4×C157—41no—
73Dic157—41no—
74Dic157#10012yes—
75Dic157#10414yes—
76C2×C157—81no—
77C2×C157C2×#10011yescharacteristic
78C2×C157C2×#10412yes—
79D157—41no—
80D157—81no—
81C157#152511yescharacteristic, complemented, Sylow
82#1525—1571nomaximal, Sylow
83#532—1571no—
84C4×M16—1572no—
85#329—1571no—
86#329—1572no—
87#329D15711yescharacteristic, complemented
88C2×#41—1571no—
89C2×#41—1571no—
90C2×M16—3142no—
91C4⋊Q8—1571no—
92#127—1571no—
93#124—1571no—
94#124C2×D15711yescharacteristic
95#121—3142no—
96C4×D4—3142no—
97C2×C42—1571no—
98#97—1572no—
99#97C2×D15712yes—
100C4×C8—3142no—
101#41—6282no—
102#41—6282no—
103C2×Q8—1571no—
104C2×Q8C4×D15711yescharacteristic
105C2×D4—3141no—
106C2×D4—3141no—
107C2×D4—3142no—
108C2×D4—1571no—
109C2×D4C4×D15711yescharacteristic
110C22×C4—3141no—
111C22×C4—3141no—
112C22×C4—1571no—
113C22×C4—1572no—
114M16—3148no—
115C2×C8—22no—
116C2×C8—3142no—
117C4⋊C4—3142no—
118C22⋊C4—22no—
119C22⋊C4—3142no—
120C42—1571no—
121C42—1572no—
122C42C22×D15711yescharacteristic
123C23—21no—
124C23—3141no—
125C23—1571no—
126Q8—41no—
127Q8—6281no—
128D4—41no—
129D4—6281no—
130D4—6282no—
131D4—6282no—
132D4—6284no—
133C2×C4—21no—
134C2×C4—3144no—
135C2×C4—3142no—
136C2×C4—3141no—
137C2×C4—6282no—
138C2×C4—6282no—
139C2×C4—1572no—
140C2×C4—1571no—
141C2×C4C2×C4×D15711yescharacteristic
142C2×C4D4×D15712yes—
143C8—24no—
144C8—3144no—
145C22—41no—
146C22—42no—
147C22—3141no—
148C22—3142no—
149C22—6281no—
150C22—6282no—
151C22C22⋊C4×D15711yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—6281no—
156C4—1572no—
157C4—1574no—
158C2—81no—
159C2—6281no—
160C2—12561no—
161C2#100×D15711yescharacteristic
162C2#104×D15712yes—
subgroupG/H
centerC22C22⋊C4×D157
commutatorC2×C4×C157C22×C4
FrattiniC42C22×D157
Fitting#329×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757822C2
cosocleC42×C157C23
hypercenter#329D157
hypocenterC157#1525
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #23757225C2semidirect
3. #23757013C2semidirect
4. #23757012C2semidirect
5. #329×C157C2semidirect
81. C157#1525semidirect
87. #329D157semidirect

Maximal subgroups

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

subgroupindex
#237578222
#237572252
#237570132
#237570122
#329×C1572
#1525157