Finite Group Databasebeta

← #34108416

Subgroups — #34108416

Order 20096.

1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C24×Dic157C211yesmaximal, characteristic, complemented
2C2×#19097743C213yesmaximal, complemented
3C2×#19097740C211yesmaximal, characteristic, complemented
4#23757153C213yesmaximal, complemented
5#23757150C213yesmaximal, complemented
6#23757131C213yesmaximal, complemented
7C2×#120×C157C211yesmaximal, characteristic, complemented
8C22×C22⋊D157C2213yes—
9C22×C22⋊D157C2211yescharacteristic, complemented
10C23×Dic157—212no—
11C23×Dic157C2213yes—
12C23×Dic157C2213yes—
13#19097743—212no—
14#19097740—24no—
15C2×#14578933C2213yescomplemented
16C2×C22⋊Dic157C2213yes—
17C2×C22⋊Dic157C2213yescomplemented
18C2×#14578931C2213yescomplemented
19#19097678C2216yescomplemented
20C25×C157C2211yescharacteristic
21C22×D4×C157C2213yes—
22#120×C157—24no—
23C2×C22⋊C4×C157C2213yes—
24C23×D157C2311yescharacteristic
25C2×C22⋊D157—26no—
26C2×C22⋊D157—26no—
27C2×C22⋊D157—26no—
28C2×C22⋊D157—26no—
29C22×Dic157—232no—
30C22×Dic157—212no—
31C22×Dic157—212no—
32C22×Dic157—26no—
33C22×Dic157D4112yescomplemented
34C22×Dic157C2313yes—
35C22×Dic157C2313yes—
36C22×Dic157C2311yescharacteristic
37#14578933—26no—
38C22⋊Dic157—26no—
39C22⋊Dic157—26no—
40#14578931—26no—
41C24×C157—26no—
42C24×C157—26no—
43C24×C157D414yescomplemented
44C24×C157C2311yescharacteristic
45C24×C157C2313yes—
46C2×D4×C157—26no—
47C2×D4×C157—26no—
48C2×D4×C157—26no—
49C22×C4×C157C2313yes—
50C22⋊C4×C157—26no—
51C22×D157—21no—
52C22×D157—23no—
53C22×D157—23no—
54C22⋊D157—412no—
55C22⋊D157—44no—
56C2×Dic157—224no—
57C2×Dic157—26no—
58C2×Dic157—224no—
59C2×Dic157—23no—
60C2×Dic157—23no—
61C2×Dic157—23no—
62C2×Dic157C2×D416yes—
63C2×Dic157C2×D416yes—
64C23×C157—28no—
65C23×C157—224no—
66C23×C157—224no—
67C23×C157—26no—
68C23×C157—26no—
69C23×C157—23no—
70C23×C157—21no—
71C23×C157—23no—
72C23×C157C2×D416yes—
73C23×C157C2×D416yes—
74C23×C157#4116yes—
75C23×C157C2411yescharacteristic
76D4×C157—412no—
77C2×C4×C157—23no—
78C2×C4×C157—23no—
79C2×C4×C157—23no—
80C22×C157—232no—
81C22×C157—26no—
82C22×C157—212no—
83C22×C157—26no—
84C22×C157—212no—
85C22×C157—41no—
86C22×C157—43no—
87C22×C157—43no—
88C22×C157#121112yescomplemented
89C22×C157C22×D413yes—
90C22×C157C22×D411yescharacteristic
91C22×C157C2×#4113yes—
92C2×D157—41no—
93C2×D157—43no—
94C2×D157—43no—
95C4×C157—43no—
96Dic157—26no—
97Dic157—43no—
98Dic157#12014yescomplemented
99C2×C157—212no—
100C2×C157—81no—
101C2×C157C2×#12011yescharacteristic
102C2×C157C2×#12113yes—
103C2×C157#52513yes—
104D157—81no—
105C157#284711yescharacteristic, complemented, Sylow
106#2847—1571nomaximal, Sylow
107C24×C4—1571no—
108C2×#121—1573no—
109C2×#120—1571no—
110C2×#120D15711yescharacteristic, complemented
111#386—1573no—
112#386—1573no—
113#380—1573no—
114C25C2×D15711yescharacteristic
115C22×D4—1573no—
116C22×D4—1571no—
117C22×D4C2×D15713yes—
118C23×C4—31412no—
119C23×C4—1573no—
120C23×C4—1573no—
121#121—31412no—
122#120—24no—
123#120—3144no—
124C2×C4⋊C4—1573no—
125C2×C22⋊C4—1573no—
126C2×C22⋊C4—1573no—
127C2×C22⋊C4—1573no—
128C2×C22⋊C4C2×D15713yes—
129#95—1576no—
130C24—26no—
131C24—26no—
132C24—1571no—
133C24C22⋊D15714yescomplemented
134C24C22×D15711yescharacteristic
135C24C22×D15713yes—
136C2×D4—26no—
137C2×D4—26no—
138C2×D4—26no—
139C2×D4—3146no—
140C2×D4—3146no—
141C2×D4—3146no—
142C2×D4—3146no—
143C22×C4—31432no—
144C22×C4—3146no—
145C22×C4—31412no—
146C22×C4—31412no—
147C22×C4—1573no—
148C22×C4—15712no—
149C22×C4—1573no—
150C22×C4—1571no—
151C22×C4C22×D15713yes—
152C4⋊C4—3146no—
153C22⋊C4—26no—
154C22⋊C4—3146no—
155C22⋊C4—3146no—
156C22⋊C4—3146no—
157C23—28no—
158C23—224no—
159C23—224no—
160C23—26no—
161C23—26no—
162C23—23no—
163C23—21no—
164C23—23no—
165C23—3143no—
166C23—3141no—
167C23—3143no—
168C23D4×D15716yes—
169C23C2×C22⋊D15716yes—
170C23#1457895316yes—
171C23C23×D15711yescharacteristic
172D4—412no—
173D4—62812no—
174D4—6284no—
175C2×C4—23no—
176C2×C4—23no—
177C2×C4—23no—
178C2×C4—31424no—
179C2×C4—3146no—
180C2×C4—31424no—
181C2×C4—3143no—
182C2×C4—3143no—
183C2×C4—3143no—
184C2×C4—1576no—
185C2×C4—1576no—
186C22—232no—
187C22—26no—
188C22—212no—
189C22—26no—
190C22—212no—
191C22—41no—
192C22—43no—
193C22—43no—
194C22—6281no—
195C22—6283no—
196C22—6283no—
197C22#19097743112yescomplemented
198C22C2×D4×D15713yes—
199C22C22×C22⋊D15711yescharacteristic
200C22C2×#1457895313yes—
201C4—43no—
202C4—3146no—
203C4—6283no—
204C4—1574no—
205C2—212no—
206C2—81no—
207C2—12561no—
208C2#120×D15711yescharacteristic
209C2C2×#1909774313yes—
210C2#2375777913yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
FittingC2×#120×C157C2
hypercenterC2×#120D157
hypocenterC157#2847
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic157C2semidirect
2. C2×#19097743C2semidirect
3. C2×#19097740C2semidirect
4. #23757153C2semidirect
5. #23757150C2semidirect
6. #23757131C2semidirect
7. C2×#120×C157C2semidirect
9. C22×C22⋊D157C22semidirect
15. C2×#14578933C22semidirect
17. C2×C22⋊Dic157C22semidirect
18. C2×#14578931C22semidirect
19. #19097678C22semidirect
33. C22×Dic157D4semidirect
43. C24×C157D4semidirect
88. C22×C157#121semidirect
98. Dic157#120semidirect
105. C157#2847semidirect
110. C2×#120D157semidirect
133. C24C22⋊D157semidirect
197. C22#19097743semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1572
C2×#190977432
C2×#190977402
#237571532
#237571502
#237571312
C2×#120×C1572
#2847157