Finite Group Databasebeta

← #34102215

Subgroups — #34102215

Order 20096.

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

normal subgroups only

This group has 150 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#23757713C211yesmaximal, characteristic
2#23757651C211yesmaximal, characteristic, complemented
3#23757390C211yesmaximal, characteristic
4#23757093C211yesmaximal, characteristic, complemented
5#23757090C211yesmaximal, characteristic, complemented
6#23757089C211yesmaximal, characteristic, complemented
7#510×C157C211yesmaximal, characteristic
8C2×Dic628C2211yescharacteristic
9C2×#14578944—21no—
10#19097778—21no—
11#19097769—21no—
12#19097717—21no—
13#19097715—21no—
14#19097711C2211yescharacteristic
15#19097707C2211yescharacteristic
16#19097698—21no—
17C8×Dic157C2211yescharacteristic
18#135×C157C2211yescharacteristic
19C2×Q16×C157C2211yescharacteristic
20Q32×C157—21no—
21SD32×C157—21no—
22M32×C157C2211yescharacteristic
23C2×Dic314—21no—
24Dic628—21no—
25Dic628—21no—
26#14578944—41no—
27#14578944—41no—
28#14578938—21no—
29#14578938—41no—
30C2×#11929434—41no—
31C2×#11929434D411yescharacteristic
32#14578933—42no—
33C4×Dic157D411yescharacteristic
34#14578929—21no—
35#41×C157—21no—
36C2×Q8×C157—21no—
37Q16×C157—21no—
38Q16×C157—21no—
39Q16×C157D411yescharacteristic
40SD16×C157—21no—
41D8×C157D411yescharacteristic
42C2×C8×C157C2311yescharacteristic
43C16×C157—21no—
44Dic314—41no—
45Dic314—41no—
46C2×Dic157—21no—
47C2×Dic157—41no—
48#11929434—41no—
49#11929434D811yescharacteristic
50#11929434D811yescharacteristic
51Q8×C157—21no—
52Q8×C157—41no—
53Q8×C157—41no—
54D4×C157—21no—
55D4×C157—41no—
56C2×C4×C157—41no—
57C2×C4×C157—41no—
58C2×C4×C157C2×D411yescharacteristic
59C8×C157C2×D411yescharacteristic
60C8×C157#4111yescharacteristic
61C22×C157—41no—
62C22×C157AGL(1,ℤ/8)11yescharacteristic
63C4×C157—41no—
64C4×C157—81no—
65C4×C157#12111yescharacteristic
66C4×C157C2×D811yescharacteristic
67Dic157—22no—
68Dic157—81no—
69C2×C157—21no—
70C2×C157—81no—
71C2×C157#45911yescharacteristic
72C157#221411yescharacteristic, complemented, Sylow
73#2214—1571nomaximal, Sylow
74#510—1571no—
75#510D15711yescharacteristic
76#494—1571no—
77#443—1571no—
78#364—1571no—
79#362—1571no—
80#362—1571no—
81#135C2×D15711yescharacteristic
82C2×Q16—3141no—
83C2×Q16—1571no—
84C2×Q16C2×D15711yescharacteristic
85#131—3141no—
86C4⋊Q8—3141no—
87Q32—21no—
88Q32—3141no—
89SD32—21no—
90SD32—3141no—
91M32—1571no—
92M32C2×D15711yescharacteristic
93#108—1571no—
94#104—3141no—
95C4×C8—1571no—
96#41—21no—
97C2×Q8—21no—
98C2×Q8—3141no—
99Q16—21no—
100Q16—21no—
101Q16—3141no—
102Q16—3141no—
103Q16—6281no—
104Q16—6281no—
105Q16C22⋊D15711yescharacteristic
106SD16—21no—
107D8C22⋊D15711yescharacteristic
108M16—3141no—
109M16—6281no—
110C2×C8—6281no—
111C2×C8—1571no—
112C2×C8C22×D15711yescharacteristic
113C4⋊C4—6282no—
114C42—1571no—
115C16—21no—
116C16—3141no—
117Q8—21no—
118Q8—41no—
119Q8—41no—
120Q8—6281no—
121Q8—6281no—
122D4—21no—
123D4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—3141no—
127C2×C4—6281no—
128C2×C4D4×D15711yescharacteristic
129C8—6281no—
130C8—1571no—
131C8—1571no—
132C8C2×C22⋊D15711yescharacteristic
133C8#1457895311yescharacteristic
134C22—41no—
135C22#1909779511yescharacteristic
136C4—41no—
137C4—81no—
138C4—3142no—
139C4—12561no—
140C4#1909774311yescharacteristic
141C4D8×D15711yescharacteristic
142C2—21no—
143C2—81no—
144C2#2375746611yescharacteristic
subgroupG/H
centerC2#23757466
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting#510×C157C2
socleC2×C157#459
prime radical#135×C157C22
hypercenter#510D157
hypocenterC157#2214
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #23757651C2semidirect
4. #23757093C2semidirect
5. #23757090C2semidirect
6. #23757089C2semidirect
72. C157#2214semidirect

Maximal subgroups

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

subgroupindex
#237577132
#237576512
#237573902
#237570932
#237570902
#237570892
#510×C1572
#2214157