Finite Group Databasebeta

← #34102157

Subgroups — #34102157

Order 20096.

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

normal subgroups only

This group has 216 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#23757455C212yesmaximal, complemented
2#23757445C211yesmaximal, characteristic
3#23757072C212yesmaximal, complemented
4#23757070C211yesmaximal, characteristic, complemented
5#563×C157C211yesmaximal, characteristic, complemented
6#19097819C2211yescharacteristic
7#19097802—22no—
8#19097797—21no—
9#19097699—21no—
10#19097699—22no—
11#19097690C2212yes—
12#19097690C2211yescharacteristic
13C4⋊Q8×C157C2212yes—
14#126×C157—22no—
15#125×C157—21no—
16#124×C157C2211yescharacteristic
17#123×C157—21no—
18C22⋊Q8×C157—22no—
19#14578955—41no—
20#14578954—21no—
21#14578954—22no—
22Q8×D157—41no—
23#14578944—42no—
24#14578942—41no—
25#14578941—42no—
26#14578939—41no—
27#14578938—21no—
28#14578938—22no—
29C2×Q8×C157D412yes—
30C2×Q8×C157C2311yescharacteristic
31C2×D4×C157D411yescharacteristic
32C22×C4×C157—21no—
33C4⋊C4×C157—21no—
34C4⋊C4×C157—22no—
35C4⋊C4×C157—21no—
36C4⋊C4×C157—22no—
37C4⋊C4×C157—22no—
38C22⋊C4×C157—22no—
39C22⋊C4×C157—21no—
40C22⋊C4×C157—22no—
41C42×C157D411yescharacteristic
42C42×C157D412yes—
43Dic314—22no—
44Dic314—21no—
45D628—21no—
46D628—22no—
47C22⋊D157—41no—
48C4×D157—41no—
49C4×D157—42no—
50#11929434—42no—
51#11929434—41no—
52C23×C157—21no—
53Q8×C157—41no—
54Q8×C157—42no—
55D4×C157—41no—
56C2×C4×C157—21no—
57C2×C4×C157—22no—
58C2×C4×C157—22no—
59C2×C4×C157—81no—
60C2×C4×C157C2×D412yes—
61C2×C4×C157C2×D411yescharacteristic
62C22×C157—42no—
63C22×C157—41no—
64C22×C157#12011yescharacteristic
65C2×D157—41no—
66C4×C157—21no—
67C4×C157—22no—
68C4×C157—41no—
69C4×C157—42no—
70C4×C157—42no—
71Dic157—41no—
72C2×C157—21no—
73C2×C157—81no—
74C2×C157B(4,𝔽2)11yescharacteristic
75D157—81no—
76C157#220311yescharacteristic, complemented, Sylow
77#2203—1571nomaximal, Sylow
78#563D15711yescharacteristic, complemented
79#456—1572no—
80#454—1571no—
81#356—1572no—
82#355—1571no—
832−1+4—1571no—
84#137—3142no—
85AGL(1,ℤ/8)—3141no—
86C4⋊Q8C2×D15712yes—
87#126—22no—
88#125—21no—
89#124C2×D15711yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—3141no—
93#104—3142no—
94#101—1572no—
95#101—1571no—
96#41—3141no—
97#41—3142no—
98#41—6281no—
99C2×Q8—6281no—
100C2×Q8C22⋊D15712yes—
101C2×Q8C22×D15711yescharacteristic
102C2×D4C22⋊D15711yescharacteristic
103C22×C4—21no—
104Q16—6282no—
105SD16—6281no—
106SD16—6282no—
107D8—6281no—
108M16—3141no—
109M16—3142no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D15711yescharacteristic
119C42C22⋊D15712yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—3141no—
124Q8—3142no—
125D4—41no—
126D4—3141no—
127D4—3142no—
128D4—6281no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—6281no—
134C2×C4—6282no—
135C2×C4C2×C22⋊D15712yes—
136C2×C4C2×C22⋊D15711yescharacteristic
137C8—6281no—
138C8—6282no—
139C22—42no—
140C22—41no—
141C22—6281no—
142C22#1909774011yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—6281no—
149C2—21no—
150C2—81no—
151C2—12561no—
152C2#2375746011yescharacteristic
subgroupG/H
centerC2#23757460
commutator, cosocleC2×Q8×C157C23
FrattiniC2×Q8C22×D157
Fitting#563×C157C2
socleC2×C157B(4,𝔽2)
prime radical#23757445C2
hypercenter#563D157
hypocenterC157#2203
LCSG ▷ C2×Q8×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #23757455C2semidirect
3. #23757072C2semidirect
4. #23757070C2semidirect
5. #563×C157C2semidirect
76. C157#2203semidirect
78. #563D157semidirect

Maximal subgroups

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

subgroupindex
#237574552
#237574452
#237570722
#237570702
#563×C1572
#2203157