Finite Group Databasebeta

← #34099110

Subgroups — #34099110

Order 20096.

238 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 57 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#23757428C212yesmaximal
2#23757193C211yesmaximal, characteristic
3#23757010C212yesmaximal
4#23756999C211yesmaximal, characteristic
5#544×C157C211yesmaximal, characteristic
6C22×Dic314C2211yescharacteristic
7C2×#14578944—22no—
8C2×#14578933—21no—
9#19097694—22no—
10C22⋊#11929434C2212yes—
11#19097678—21no—
12#19097678C2211yescharacteristic
13C4⋊Q8×C157—22no—
14#125×C157—22no—
15C22⋊Q8×C157—21no—
16C22⋊Q8×C157C2212yes—
17#117×C157—21no—
18C2×C4⋊C4×C157C2211yescharacteristic
19C2×Dic314—22no—
20C2×Dic314—42no—
21C22×Dic157—21no—
22C22×Dic157—21no—
23#14578944—44no—
24C2×#11929434—22no—
25#14578933—42no—
26C2×Q8×C157—21no—
27C2×Q8×C157D412yes—
28C22×C4×C157D411yescharacteristic
29C22×C4×C157D411yescharacteristic
30C22×C4×C157C2311yescharacteristic
31C4⋊C4×C157—21no—
32C4⋊C4×C157—21no—
33C4⋊C4×C157—21no—
34C4⋊C4×C157—22no—
35C4⋊C4×C157—22no—
36C4⋊C4×C157—22no—
37C4⋊C4×C157D412yes—
38C22⋊C4×C157—22no—
39C42×C157—22no—
40Dic314—24no—
41Dic314—42no—
42C2×Dic157—22no—
43C2×Dic157—22no—
44C2×Dic157—41no—
45C2×Dic157—41no—
46C2×Dic157—41no—
47C2×Dic157—41no—
48#11929434—42no—
49C23×C157C2×D411yescharacteristic
50Q8×C157—42no—
51Q8×C157—81no—
52C2×C4×C157—21no—
53C2×C4×C157—21no—
54C2×C4×C157—22no—
55C2×C4×C157—22no—
56C2×C4×C157—21no—
57C2×C4×C157—41no—
58C2×C4×C157—41no—
59C2×C4×C157Q1612yescomplemented
60C2×C4×C157C2×D412yes—
61C22×C157—21no—
62C22×C157—21no—
63C22×C157#12011yescharacteristic
64C22×C157C2×Q1611yescharacteristic
65C22×C157#13711yescharacteristic
66C4×C157—22no—
67C4×C157—22no—
68C4×C157—42no—
69C4×C157—41no—
70C4×C157—42no—
71Dic157—42no—
72Dic157—42no—
73C2×C157—21no—
74C2×C157—21no—
75C2×C157#45111yescharacteristic
76C2×C157#45611yescharacteristic
77C2×C157B(4,𝔽2)11yescharacteristic
78C157#160511yescharacteristic, complemented, Sylow
79#1605—1571nomaximal, Sylow
80#544D15711yescharacteristic
81#451—1572no—
82#395—1571no—
83#328—1572no—
84#324—1571no—
85C22×Q8—1571no—
86C2×Q16—3142no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C2×D15712yes—
91#117—21no—
92C2×C4⋊C4—3141no—
93C2×C4⋊C4C2×D15711yescharacteristic
94#103—3142no—
95C22⋊C8—1572no—
96#95—3141no—
97#95—1571no—
98C2×Q8—21no—
99C2×Q8—3142no—
100C2×Q8—6282no—
101C2×Q8C22⋊D15712yes—
102C22×C4—3141no—
103C22×C4—3141no—
104C22×C4C22⋊D15711yescharacteristic
105C22×C4C22⋊D15711yescharacteristic
106C22×C4C22×D15711yescharacteristic
107Q16—6284no—
108C2×C8—3142no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—6282no—
116C4⋊C4C22⋊D15712yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×C22⋊D15711yescharacteristic
120Q8—42no—
121Q8—81no—
122Q8—3144no—
123Q8—6282no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—3142no—
132C2×C4—3142no—
133C2×C4—6281no—
134C2×C4—6281no—
135C2×C4—6281no—
136C2×C4—6281no—
137C2×C4#1457894412yescomplemented
138C2×C4C2×C22⋊D15712yes—
139C8—6282no—
140C22—21no—
141C22—21no—
142C22#1909774011yescharacteristic
143C22C2×#1457894411yescharacteristic
144C22#1909780211yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—6282no—
151C4—6282no—
152C2—21no—
153C2—21no—
154C2#2375742911yescharacteristic
155C2#2375745611yescharacteristic
156C2#2375746011yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#544×C157C2
socleC22×C157#120
prime radicalC23×C157C2×D4
hypercenter#544D157
hypocenterC157#1605
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
59. C2×C4×C157Q16semidirect
78. C157#1605semidirect
137. C2×C4#14578944semidirect

Maximal subgroups

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

subgroupindex
#237574282
#237571932
#237570102
#237569992
#544×C1572
#1605157