Finite Group Databasebeta

← #34116396

Subgroups — #34116396

Order 20096.

266 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 52 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#23758260C211yesmaximal, characteristic, complemented
2#23758253C211yesmaximal, characteristic, complemented
3#23758174C212yesmaximal, complemented
4#23758173C212yesmaximal, complemented
5#23757838C211yesmaximal, characteristic
6C2×#14578963—21no—
7#19097850—21no—
8#19097832—21no—
9#19097831C2212yes—
10#19097829—21no—
11#19097829C2212yes—
12C2×#14578953C2211yescharacteristic
13C22×C22⋊D157C411yescharacteristic, complemented
14#19097767—21no—
15#19097757—21no—
16#19097744—21no—
17#19097743—21no—
18#19097741C2211yescharacteristic
19#19097740C2211yescharacteristic
20#19097739C411yescharacteristic, complemented
21#19097727C411yescharacteristic, complemented
22#120×C157C411yescharacteristic, complemented
23#14578963—22no—
24#14578963—41no—
25C2×#11929441—41no—
26C4⋊#10597845—41no—
27C22⋊#10597845—22no—
28C4×#10597845—41no—
29C23×D157C2×C411yescharacteristic
30#14578953—41no—
31#14578953—41no—
32C2×Dic314D411yescharacteristic
33C2×D628—21no—
34C2×C22⋊D157—21no—
35C2×C22⋊D157—21no—
36C2×C22⋊D157—21no—
37C2×C22⋊D157—21no—
38C2×C22⋊D157—41no—
39C2×C22⋊D157—41no—
40C2×C22⋊D157D411yescharacteristic
41C2×C22⋊D157C2311yescharacteristic
42C22×Dic157C2×C411yescharacteristic
43C22×Dic157D411yescharacteristic
44C2×C4×D157D411yescharacteristic
45#14578933—21no—
46C22⋊Dic157—21no—
47C22⋊Dic157C2×C411yescharacteristic
48#14578931—21no—
49#14578931—21no—
50C4×Dic157—21no—
51C24×C157C2×C411yescharacteristic
52C2×D4×C157—21no—
53C2×D4×C157C2×C411yescharacteristic
54C22⋊C4×C157—21no—
55C22⋊C4×C157C2×C411yescharacteristic
56C2×#10597845—42no—
57#11929441—42no—
58C22×D157—21no—
59C22×D157—41no—
60C22×D157—81no—
61C22×D157C2×D411yescharacteristic
62Dic314—41no—
63D628—81no—
64C22⋊D157—41no—
65C22⋊D157—41no—
66C22⋊D157—42no—
67C22⋊D157—41no—
68C22⋊D157—41no—
69C22⋊D157—81no—
70C22⋊D157—81no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—21no—
74C2×Dic157—21no—
75C2×Dic157—21no—
76C2×Dic157C22⋊C411yescharacteristic
77C2×Dic157C22⋊C411yescharacteristic
78C2×Dic157C2×D411yescharacteristic
79C4×D157—21no—
80C4×D157—21no—
81C23×C157—21no—
82C23×C157—41no—
83C23×C157—81no—
84C23×C157C22⋊C411yescharacteristic
85C23×C157C22×C411yescharacteristic
86D4×C157—41no—
87D4×C157—81no—
88C2×C4×C157—21no—
89C2×C4×C157C22⋊C411yescharacteristic
90C22×C157—21no—
91C22×C157—21no—
92C22×C157—41no—
93C22×C157—41no—
94C22×C157—41no—
95C22×C157—41no—
96C22×C157—41no—
97C22×C157—82no—
98C22×C157C2×C22⋊C411yescharacteristic
99C2×D157—21no—
100C2×D157—41no—
101C2×D157—41no—
102C2×D157—41no—
103C2×D157—41no—
104C2×D157—82no—
105#10597845—82no—
106C4×C157—21no—
107C4×C157—41no—
108Dic157—21no—
109Dic157—21no—
110Dic157—41no—
111Dic157#9911yescharacteristic
112Dic157#9911yescharacteristic
113C2×C157—21no—
114C2×C157—41no—
115C2×C157—41no—
116C2×C157—81no—
117C2×C157C2×#9911yescharacteristic
118D157—41no—
119D157—81no—
120C157#212211yescharacteristic, complemented, Sylow
121#2122—1571nomaximal, Sylow
122#535—1571no—
123#413—1571no—
124#410—1571no—
125AΣL(1,𝔽16)—1572no—
126AΣL(1,𝔽16)—1572no—
127C2×#41—1571no—
128C22×D4—1571no—
129C2×M16—3141no—
130#127—3141no—
131#124—3141no—
132#121—3141no—
133#121—3141no—
134#120—1571no—
135#120—1571no—
136#120—1571no—
137#120#1059784511yescharacteristic, complemented
138#117—3141no—
139#117—1571no—
140#101—3141no—
141#100—1572no—
142#99—3141no—
143#99—1572no—
144C24—1571no—
145C24C2×#1059784511yescharacteristic
146#41—6281no—
147#41—6281no—
148C2×Q8—1571no—
149C2×D4—21no—
150C2×D4—3141no—
151C2×D4—3141no—
152C2×D4—3141no—
153C2×D4—3141no—
154C2×D4—3141no—
155C2×D4—6281no—
156C2×D4—6281no—
157C2×D4—1571no—
158C2×D4—1571no—
159C2×D4C2×#1059784511yescharacteristic
160C22×C4—1571no—
161C22×C4—1571no—
162C22×C4—1571no—
163M16—3142no—
164M16—6281no—
165C2×C8—6281no—
166C4⋊C4—3141no—
167C4⋊C4—6281no—
168C22⋊C4—21no—
169C22⋊C4—3141no—
170C22⋊C4—3141no—
171C22⋊C4—3141no—
172C22⋊C4—3142no—
173C22⋊C4—1571no—
174C22⋊C4C2×#1059784511yescharacteristic
175C42—3141no—
176C42—6281no—
177C23—21no—
178C23—41no—
179C23—81no—
180C23—3141no—
181C23—6281no—
182C23—1571no—
183C23—12561no—
184C23C22⋊#1059784511yescharacteristic
185C23C22×#1059784511yescharacteristic
186Q8—6281no—
187D4—41no—
188D4—81no—
189D4—6281no—
190D4—6281no—
191D4—6282no—
192D4—6281no—
193D4—6281no—
194D4—12561no—
195D4—12561no—
196D4—12561no—
197C2×C4—21no—
198C2×C4—3141no—
199C2×C4—3141no—
200C2×C4—3141no—
201C2×C4—3141no—
202C2×C4—3141no—
203C2×C4—3141no—
204C2×C4—3141no—
205C2×C4—6282no—
206C2×C4—1571no—
207C2×C4—1571no—
208C2×C4—1571no—
209C2×C4C22⋊#1059784511yescharacteristic
210C8—6282no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—82no—
219C22—3141no—
220C22—6281no—
221C22—6281no—
222C22—6281no—
223C22—6281no—
224C22—12562no—
225C22C2×C22⋊#1059784511yescharacteristic
226C4—21no—
227C4—41no—
228C4—3141no—
229C4—3141no—
230C4—6281no—
231C4—1571no—
232C4—1571no—
233C4—12562no—
234C2—21no—
235C2—41no—
236C2—41no—
237C2—81no—
238C2—6281no—
239C2—12561no—
240C2D157⋊#9911yescharacteristic
subgroupG/H
centerC2D157⋊#99
commutatorC23×C157C22×C4
FrattiniC23C22×#10597845
Fitting#120×C157C4
socleC2×C157C2×#99
prime radical#23757838C2
cosocleC2×C22⋊D157C23
hypercenter#120#10597845
hypocenterC157#2122
LCSG ▷ C23×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #23758260C2semidirect
2. #23758253C2semidirect
3. #23758174C2semidirect
4. #23758173C2semidirect
13. C22×C22⋊D157C4semidirect
20. #19097739C4semidirect
21. #19097727C4semidirect
22. #120×C157C4semidirect
120. C157#2122semidirect
137. #120#10597845semidirect

Maximal subgroups

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

subgroupindex
#237582602
#237582532
#237581742
#237581732
#237578382
#2122157