Finite Group Databasebeta

← #34099078

Subgroups — #34099078

Order 20096.

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

normal subgroups only

This group has 266 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, complemented
2#23757426C212yesmaximal
3#23757190C211yesmaximal, characteristic
4#23757011C212yesmaximal, complemented
5#324×C157C211yesmaximal, characteristic, complemented
6C22×D628C2211yescharacteristic
7C2×#14578943—22no—
8#19097770—22no—
9#19097762—22no—
10#19097750—21no—
11#19097750C2212yes—
12#19097730—21no—
13C2×#14578931—21no—
14C2×#14578931—21no—
15#19097693—22no—
16#19097678C2211yescharacteristic
17C2×C4⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yes—
19C23×D157—21no—
20C2×Dic314—21no—
21C2×Dic314D412yes—
22C2×D628—22no—
23C2×D628—42no—
24C22×Dic157—21no—
25#14578943—44no—
26C4⋊Dic157—21no—
27C4⋊Dic157—21no—
28C4⋊Dic157—21no—
29C4⋊Dic157D412yes—
30#14578933—22no—
31#14578933—22no—
32C22⋊Dic157—22no—
33#14578931—42no—
34#14578931—42no—
35C4×Dic157—22no—
36C22×C4×C157D411yescharacteristic
37C22×C4×C157D411yescharacteristic
38C22×C4×C157C2311yescharacteristic
39C2×C8×C157—22no—
40C4⋊C4×C157—22no—
41C22×D157—22no—
42C22×D157—41no—
43C22×D157—41no—
44C22×D157—41no—
45C22×D157—41no—
46C22×D157—41no—
47C22×D157—41no—
48Dic314—42no—
49Dic314—81no—
50D628—24no—
51D628—42no—
52C2×Dic157—22no—
53C2×Dic157—22no—
54C2×Dic157—41no—
55C2×Dic157—41no—
56C2×Dic157—41no—
57C23×C157C2×D411yescharacteristic
58C2×C4×C157—21no—
59C2×C4×C157—21no—
60C2×C4×C157—21no—
61C2×C4×C157—41no—
62C2×C4×C157—41no—
63C2×C4×C157SD1612yescomplemented
64C2×C4×C157C2×D412yes—
65C8×C157—42no—
66C22×C157—21no—
67C22×C157—21no—
68C22×C157#12011yescharacteristic
69C22×C157C2×SD1611yescharacteristic
70C22×C157#13711yescharacteristic
71C2×D157—42no—
72C2×D157—42no—
73C2×D157—42no—
74C2×D157—81no—
75C2×D157—81no—
76C2×D157—81no—
77C2×D157—81no—
78C4×C157—22no—
79C4×C157—22no—
80C4×C157—41no—
81Dic157—42no—
82Dic157—42no—
83Dic157—81no—
84C2×C157—21no—
85C2×C157—21no—
86C2×C157#44811yescharacteristic
87C2×C157#45411yescharacteristic
88C2×C157B(4,𝔽2)11yescharacteristic
89D157—82no—
90C157#159911yescharacteristic, complemented, Sylow
91#1599—1571nomaximal, Sylow
92#544—1571no—
93#450—1572no—
94#394—1571no—
95#328—1572no—
96#324D15711yescharacteristic, complemented
97C22×D4—1571no—
98C2×SD16—3142no—
99C4⋊Q8—3142no—
100#125—3142no—
101C22⋊Q8—3141no—
102C22⋊Q8—1572no—
103#117—3141no—
104C2×C4⋊C4C2×D15711yescharacteristic
105C2×C22⋊C4—3141no—
106C2×C22⋊C4—3141no—
107#102—3142no—
108C22⋊C8C2×D15712yes—
109#95—1571no—
110C24—3141no—
111C2×Q8—3141no—
112C2×Q8—1572no—
113C2×D4—3142no—
114C2×D4—6282no—
115C22×C4—3141no—
116C22×C4D62811yescharacteristic
117C22×C4D62811yescharacteristic
118C22×C4C22×D15711yescharacteristic
119SD16—6284no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—3141no—
123C4⋊C4—3141no—
124C4⋊C4—3141no—
125C4⋊C4—3142no—
126C4⋊C4—3142no—
127C4⋊C4—1572no—
128C22⋊C4—3142no—
129C22⋊C4—6282no—
130C22⋊C4—6282no—
131C42—3142no—
132C23—3142no—
133C23—6281no—
134C23—6281no—
135C23—6281no—
136C23—6281no—
137C23—6281no—
138C23—6281no—
139C23C2×D62811yescharacteristic
140Q8—6282no—
141Q8—12561no—
142D4—3144no—
143D4—6282no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—3142no—
150C2×C4—3142no—
151C2×C4—6281no—
152C2×C4—6281no—
153C2×C4—6281no—
154C2×C4#1457894312yescomplemented
155C2×C4D4×D15712yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—6282no—
160C22—6282no—
161C22—6282no—
162C22—12561no—
163C22—12561no—
164C22—12561no—
165C22—12561no—
166C22#1909774111yescharacteristic
167C22C2×#1457894311yescharacteristic
168C22#1909780411yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—6282no—
173C4—6282no—
174C4—12561no—
175C2—21no—
176C2—21no—
177C2—12562no—
178C2#2375741411yescharacteristic
179C2#2375744811yescharacteristic
180C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#324×C157C2
socleC22×C157#120
prime radicalC22×D628C22
hypercenter#324D157
hypocenterC157#1599
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
1. #23757904C2semidirect
4. #23757011C2semidirect
5. #324×C157C2semidirect
63. C2×C4×C157SD16semidirect
90. C157#1599semidirect
96. #324D157semidirect
154. C2×C4#14578943semidirect

Maximal subgroups

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

subgroupindex
#237579042
#237574262
#237571902
#237570112
#324×C1572
#1599157