Finite Group Databasebeta

← #34102156

Subgroups — #34102156

Order 20096.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#23758023C211yesmaximal, characteristic, complemented
2#23757456C212yesmaximal, complemented
3#23757071C212yesmaximal, complemented
4#23757070C211yesmaximal, characteristic, complemented
5#454×C157C211yesmaximal, characteristic
6#19097801—22no—
7#19097769C2212yes—
8#19097764—22no—
9#19097763—21no—
10#19097752—21no—
11#19097748—22no—
12#19097698—22no—
13#19097690C2212yes—
142−1+4×C157C2211yescharacteristic
15AGL(1,ℤ/8)×C157—21no—
16#124×C157C2211yescharacteristic
17#104×C157—21no—
18#101×C157C2211yescharacteristic
19C2×Dic314D412yes—
20C22×Dic157—21no—
21#14578944—42no—
22#14578942—42no—
23#14578938—22no—
24C4⋊Dic157—21no—
25C4⋊Dic157—21no—
26#14578933—22no—
27#14578933—22no—
28#14578933—22no—
29C22⋊Dic157—22no—
30C22⋊Dic157—21no—
31C4×Dic157D412yes—
32#41×C157—21no—
33#41×C157—22no—
34#41×C157—41no—
35C2×Q8×C157—41no—
36C2×Q8×C157C2311yescharacteristic
37C2×D4×C157D411yescharacteristic
38SD16×C157—41no—
39D8×C157—41no—
40M16×C157—21no—
41C22⋊C4×C157—22no—
42C42×C157D411yescharacteristic
43Dic314—42no—
44C2×Dic157—22no—
45C2×Dic157—22no—
46C2×Dic157—81no—
47#11929434—42no—
48C23×C157—21no—
49Q8×C157—22no—
50Q8×C157—21no—
51Q8×C157—41no—
52D4×C157—21no—
53D4×C157—22no—
54D4×C157—41no—
55D4×C157—41no—
56C2×C4×C157—21no—
57C2×C4×C157—41no—
58C2×C4×C157—42no—
59C2×C4×C157C2×D412yes—
60C2×C4×C157C2×D411yescharacteristic
61C8×C157—41no—
62C22×C157—42no—
63C22×C157—41no—
64C22×C157—41no—
65C22×C157#12011yescharacteristic
66C4×C157—21no—
67C4×C157—22no—
68C4×C157—41no—
69C4×C157—41no—
70Dic157—42no—
71Dic157—42no—
72C2×C157—21no—
73C2×C157—81no—
74C2×C157—81no—
75C2×C157B(4,𝔽2)11yescharacteristic
76C157#220311yescharacteristic, complemented, Sylow
77#2203—1571nomaximal, Sylow
78#563—1571no—
79#456—1572no—
80#454D15711yescharacteristic
81#356—1572no—
82#355—1571no—
832−1+4C2×D15711yescharacteristic
84#137—3142no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—1572no—
87#126—3142no—
88#125—3141no—
89#124C2×D15711yescharacteristic
90#123—3141no—
91C22⋊Q8—3142no—
92#104—21no—
93#104—3142no—
94#101—1572no—
95#101C2×D15711yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—1572no—
101C2×Q8C22×D15711yescharacteristic
102C2×D4C22⋊D15711yescharacteristic
103C22×C4—3141no—
104Q16—6282no—
105SD16—41no—
106SD16—6282no—
107D8—41no—
108M16—21no—
109M16—3142no—
110C4⋊C4—3141no—
111C4⋊C4—3142no—
112C4⋊C4—3141no—
113C4⋊C4—3142no—
114C4⋊C4—3142no—
115C22⋊C4—22no—
116C22⋊C4—3142no—
117C22⋊C4—3141no—
118C42—1572no—
119C42C22⋊D15711yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—6282no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—3142no—
133C2×C4—3142no—
134C2×C4—12561no—
135C2×C4D4×D15712yes—
136C2×C4C2×C22⋊D15711yescharacteristic
137C8—41no—
138C8—6282no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#1909773911yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—6282no—
148C4—6282no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#2375745911yescharacteristic
subgroupG/H
centerC2#23757459
commutator, cosocleC2×Q8×C157C23
FrattiniC2×Q8C22×D157
Fitting, prime radical#454×C157C2
socleC2×C157B(4,𝔽2)
hypercenter#454D157
hypocenterC157#2203
LCSG ▷ C2×Q8×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #23758023C2semidirect
2. #23757456C2semidirect
3. #23757071C2semidirect
4. #23757070C2semidirect
76. C157#2203semidirect

Maximal subgroups

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

subgroupindex
#237580232
#237574562
#237570712
#237570702
#454×C1572
#2203157