Finite Group Databasebeta

← #34099061

Subgroups — #34099061

Order 20096.

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

normal subgroups only

This group has 316 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#23757899C211yesmaximal, characteristic, complemented
2#23757426C212yesmaximal, complemented
3#23757182C211yesmaximal, characteristic
4#23757011C212yesmaximal, complemented
5#323×C157C211yesmaximal, characteristic, complemented
6C22×D628C2211yescharacteristic
7C22×C22⋊D157—21no—
8C2×#14578943—22no—
9#19097750—21no—
10#19097750C2212yes—
11#19097748—22no—
12#19097748—22no—
13C2×C22⋊Dic157—21no—
14C2×#14578931—21no—
15#19097693—22no—
16#19097678C2211yescharacteristic
17C2×C22⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yes—
19C23×D157—21no—
20C2×Dic314—21no—
21C2×Dic314D412yescomplemented
22C2×D628—22no—
23C2×D628—42no—
24C2×C22⋊D157—42no—
25C2×C22⋊D157—42no—
26C2×C22⋊D157—42no—
27C22×Dic157—21no—
28C22×Dic157—21no—
29C22×Dic157—21no—
30#14578943—44no—
31C4⋊Dic157—21no—
32C4⋊Dic157D412yescomplemented
33#14578933—22no—
34#14578933—22no—
35C22⋊Dic157—22no—
36C22⋊Dic157—22no—
37#14578931—42no—
38C24×C157D411yescharacteristic
39C22×C4×C157D411yescharacteristic
40C22×C4×C157C2311yescharacteristic
41C2×C8×C157—22no—
42C22⋊C4×C157—22no—
43C22×D157—22no—
44C22×D157—41no—
45C22×D157—41no—
46C22×D157—41no—
47C22×D157—41no—
48C22×D157—41no—
49C22×D157—41no—
50Dic314—42no—
51Dic314—81no—
52D628—24no—
53D628—42no—
54C22⋊D157—48no—
55C2×Dic157—22no—
56C2×Dic157—22no—
57C2×Dic157—22no—
58C2×Dic157—41no—
59C2×Dic157—41no—
60C2×Dic157—81no—
61C2×Dic157—81no—
62C23×C157—21no—
63C23×C157—21no—
64C23×C157—41no—
65C23×C157—41no—
66C23×C157SD1612yescomplemented
67C23×C157C2×D411yescharacteristic
68C2×C4×C157—21no—
69C2×C4×C157—41no—
70C2×C4×C157—41no—
71C2×C4×C157C2×D412yes—
72C8×C157—42no—
73C22×C157—21no—
74C22×C157—21no—
75C22×C157—22no—
76C22×C157—42no—
77C22×C157—41no—
78C22×C157—41no—
79C22×C157—81no—
80C22×C157#12011yescharacteristic
81C22×C157C2×SD1611yescharacteristic
82C22×C157#13711yescharacteristic
83C2×D157—42no—
84C2×D157—42no—
85C2×D157—42no—
86C2×D157—81no—
87C2×D157—81no—
88C2×D157—81no—
89C2×D157—81no—
90C4×C157—22no—
91C4×C157—41no—
92Dic157—42no—
93Dic157—42no—
94Dic157—42no—
95C2×C157—21no—
96C2×C157—21no—
97C2×C157—42no—
98C2×C157#44811yescharacteristic
99C2×C157#45311yescharacteristic
100C2×C157B(4,𝔽2)11yescharacteristic
101D157—82no—
102C157#159611yescharacteristic, complemented, Sylow
103#1596—1571nomaximal, Sylow
104#543—1571no—
105#450—1572no—
106#392—1571no—
107#328—1572no—
108#323D15711yescharacteristic, complemented
109C22×D4—3141no—
110C22×D4—1571no—
111C2×SD16—3142no—
112C22⋊Q8—3142no—
113C22⋊Q8—3142no—
114C22⋊Q8—3141no—
115C22⋊Q8—1572no—
116C2×C22⋊C4—3141no—
117C2×C22⋊C4—3141no—
118C2×C22⋊C4C2×D15711yescharacteristic
119#102—3142no—
120C22⋊C8C2×D15712yes—
121#95—1571no—
122C24—3141no—
123C24D62811yescharacteristic
124C2×Q8—3141no—
125C2×Q8—1572no—
126C2×D4—3142no—
127C2×D4—6282no—
128C2×D4—6282no—
129C2×D4—6282no—
130C2×D4—6282no—
131C22×C4—3141no—
132C22×C4—3141no—
133C22×C4—3141no—
134C22×C4D62811yescharacteristic
135C22×C4C22×D15711yescharacteristic
136SD16—6284no—
137C2×C8—22no—
138C4⋊C4—3142no—
139C4⋊C4—3141no—
140C4⋊C4—3142no—
141C4⋊C4—1572no—
142C22⋊C4—22no—
143C22⋊C4—3142no—
144C22⋊C4—3142no—
145C22⋊C4—6282no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—3142no—
151C23—6281no—
152C23—6281no—
153C23—6281no—
154C23—6281no—
155C23—6281no—
156C23—6281no—
157C23#1457894312yescomplemented
158C23C2×D62811yescharacteristic
159Q8—6282no—
160Q8—12561no—
161D4—3144no—
162D4—6282no—
163D4—6288no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—3142no—
168C2×C4—3142no—
169C2×C4—3142no—
170C2×C4—6281no—
171C2×C4—6281no—
172C2×C4—12561no—
173C2×C4—12561no—
174C2×C4D4×D15712yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—6282no—
184C22—6282no—
185C22—6282no—
186C22—12561no—
187C22—12561no—
188C22—12561no—
189C22—12561no—
190C22#1909774111yescharacteristic
191C22C2×#1457894311yescharacteristic
192C22#1909780411yescharacteristic
193C4—22no—
194C4—41no—
195C4—6282no—
196C4—6282no—
197C4—6282no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—12562no—
202C2#2375741411yescharacteristic
203C2#2375744311yescharacteristic
204C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#323×C157C2
socleC22×C157#120
prime radical#23757182C2
hypercenter#323D157
hypocenterC157#1596
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #23757899C2semidirect
2. #23757426C2semidirect
4. #23757011C2semidirect
5. #323×C157C2semidirect
21. C2×Dic314D4semidirect
32. C4⋊Dic157D4semidirect
66. C23×C157SD16semidirect
102. C157#1596semidirect
108. #323D157semidirect
157. C23#14578943semidirect

Maximal subgroups

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

subgroupindex
#237578992
#237574262
#237571822
#237570112
#323×C1572
#1596157