Finite Group Databasebeta

← #34099112

Subgroups — #34099112

Order 20096.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757904C211yesmaximal, characteristic
2#23757431C212yesmaximal
3#23757193C211yesmaximal, characteristic
4#23757011C212yesmaximal
5#324×C157C211yesmaximal, characteristic
6C22×Dic314C2211yescharacteristic
7C2×Dic628—22no—
8#19097770—22no—
9#19097762—22no—
10#19097750—21no—
11#19097750C2212yes—
12#19097730—21no—
13C2×#14578933—21no—
14#19097696—22no—
15#19097678—21no—
16#19097678C2211yescharacteristic
17C2×C4⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yes—
19C2×Dic314—21no—
20C2×Dic314—22no—
21C2×Dic314—42no—
22C2×Dic314D412yes—
23C22×Dic157—21no—
24C22×Dic157—21no—
25Dic628—44no—
26C4⋊Dic157—21no—
27C4⋊Dic157—21no—
28C4⋊Dic157—21no—
29C4⋊Dic157D412yes—
30#14578933—22no—
31#14578933—22no—
32#14578933—42no—
33C22⋊Dic157—22no—
34C4×Dic157—22no—
35C22×C4×C157D411yescharacteristic
36C22×C4×C157D411yescharacteristic
37C22×C4×C157C2311yescharacteristic
38C2×C8×C157—22no—
39C4⋊C4×C157—22no—
40Dic314—24no—
41Dic314—42no—
42Dic314—42no—
43Dic314—81no—
44C2×Dic157—22no—
45C2×Dic157—22no—
46C2×Dic157—22no—
47C2×Dic157—22no—
48C2×Dic157—41no—
49C2×Dic157—41no—
50C2×Dic157—41no—
51C2×Dic157—41no—
52C23×C157C2×D411yescharacteristic
53C2×C4×C157—21no—
54C2×C4×C157—21no—
55C2×C4×C157—21no—
56C2×C4×C157—41no—
57C2×C4×C157—41no—
58C2×C4×C157Q1612yescomplemented
59C2×C4×C157C2×D412yes—
60C8×C157—42no—
61C22×C157—21no—
62C22×C157—21no—
63C22×C157#12011yescharacteristic
64C22×C157C2×Q1611yescharacteristic
65C22×C157#13711yescharacteristic
66C4×C157—22no—
67C4×C157—22no—
68C4×C157—41no—
69Dic157—42no—
70Dic157—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#544—1571no—
81#451—1572no—
82#395—1571no—
83#328—1572no—
84#324D15711yescharacteristic
85C22×Q8—1571no—
86C2×Q16—3142no—
87C4⋊Q8—3142no—
88#125—3142no—
89C22⋊Q8—3141no—
90C22⋊Q8—1572no—
91#117—3141no—
92C2×C4⋊C4—3141no—
93C2×C4⋊C4C2×D15711yescharacteristic
94#103—3142no—
95C22⋊C8C2×D15712yes—
96#95—3141no—
97#95—1571no—
98C2×Q8—3141no—
99C2×Q8—3142no—
100C2×Q8—6282no—
101C2×Q8—1572no—
102C22×C4—3141no—
103C22×C4—3141no—
104C22×C4D62811yescharacteristic
105C22×C4D62811yescharacteristic
106C22×C4C22×D15711yescharacteristic
107Q16—6284no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—3141no—
111C4⋊C4—3141no—
112C4⋊C4—3141no—
113C4⋊C4—3142no—
114C4⋊C4—3142no—
115C4⋊C4—6282no—
116C4⋊C4—1572no—
117C22⋊C4—3142no—
118C42—3142no—
119C23C2×D62811yescharacteristic
120Q8—3144no—
121Q8—6282no—
122Q8—6282no—
123Q8—12561no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—3142no—
130C2×C4—3142no—
131C2×C4—3142no—
132C2×C4—3142no—
133C2×C4—6281no—
134C2×C4—6281no—
135C2×C4—6281no—
136C2×C4—6281no—
137C2×C4Dic62812yescomplemented
138C2×C4D4×D15712yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1909774111yescharacteristic
143C22C2×Dic62811yescharacteristic
144C22#1909780411yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—6282no—
149C4—6282no—
150C4—6282no—
151C4—6282no—
152C2—21no—
153C2—21no—
154C2#2375743111yescharacteristic
155C2#2375745811yescharacteristic
156C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#324×C157C2
socleC22×C157#120
prime radicalC23×C157C2×D4
hypercenter#324D157
hypocenterC157#1605
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C2×C4×C157Q16semidirect
78. C157#1605semidirect
137. C2×C4Dic628semidirect

Maximal subgroups

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

subgroupindex
#237579042
#237574312
#237571932
#237570112
#324×C1572
#1605157