Finite Group Databasebeta

← #19135004

Subgroups — #19135004

Order 5504.

578 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 97 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#14583374C211yesmaximal, characteristic, complemented
2C2×#13063240C211yesmaximal, characteristic, complemented
3#14582755C212yesmaximal, complemented
4#14582665C212yesmaximal, complemented
5C2×#120×C43C211yesmaximal, characteristic
6D4×Dic43—24no—
7#13063284—21no—
8C2×C22⋊Dic43—21no—
9C2×C22⋊Dic43—24no—
10C2×C22⋊Dic43C2212yes—
11C2×C22⋊Dic43C2212yes—
12#13063240—22no—
13C25×C43C2211yescharacteristic
14C22×D4×C43—21no—
15C22×D4×C43C2211yescharacteristic
16#120×C43—22no—
17#120×C43C414yescomplemented
18C2×C22⋊C4×C43—21no—
19C2×C22⋊C4×C43C2211yescharacteristic
20C24×C43—21no—
21C24×C43—22no—
22C24×C43—21no—
23C24×C43—22no—
24C24×C43—44no—
25C24×C43C2×C412yescomplemented
26C24×C43D411yescharacteristic
27C24×C43C2311yescharacteristic
28C2×D4×C43—24no—
29C2×D4×C43—22no—
30C2×D4×C43—22no—
31C2×D4×C43—44no—
32C2×D4×C43C2×C412yes—
33C2×D4×C43D412yes—
34C22×C4×C43—21no—
35C22×C4×C43D411yescharacteristic
36C22×Dic43—22no—
37C22×Dic43—22no—
38C22×Dic43D414yescomplemented
39C22⋊C4×C43—24no—
40C22⋊C4×C43—21no—
41C22⋊C4×C43C2×C412yescomplemented
42C22⋊Dic43—22no—
43C22⋊Dic43—22no—
44C22⋊Dic43—22no—
45C22⋊Dic43—22no—
46C22⋊Dic43—22no—
47C22⋊Dic43—48no—
48C4⋊Dic43—22no—
49C4×Dic43—22no—
50C23×C43—28no—
51C23×C43—24no—
52C23×C43—22no—
53C23×C43—24no—
54C23×C43—21no—
55C23×C43—21no—
56C23×C43—44no—
57C23×C43—42no—
58C23×C43—42no—
59C23×C43—44no—
60C23×C43—48no—
61C23×C43—44no—
62C23×C43—42no—
63C23×C43—41no—
64C23×C43—42no—
65C23×C43—41no—
66C23×C43C22⋊C412yescomplemented
67C23×C43C22×C411yescharacteristic
68C23×C43C2×D412yes—
69C23×C43C2×D411yescharacteristic
70C23×C43C2×D411yescharacteristic
71C23×C43#4112yes—
72D4×C43—24no—
73D4×C43—24no—
74D4×C43—48no—
75C2×C4×C43—21no—
76C2×C4×C43—21no—
77C2×C4×C43—22no—
78C2×C4×C43—42no—
79C2×C4×C43C22⋊C412yescomplemented
80C2×Dic43—24no—
81C2×Dic43—44no—
82C2×Dic43—44no—
83C2×Dic43—42no—
84C2×Dic43—82no—
85C22×C43—22no—
86C22×C43—24no—
87C22×C43—28no—
88C22×C43—24no—
89C22×C43—22no—
90C22×C43—41no—
91C22×C43—41no—
92C22×C43—42no—
93C22×C43—42no—
94C22×C43—48no—
95C22×C43—42no—
96C22×C43—44no—
97C22×C43—44no—
98C22×C43—42no—
99C22×C43—44no—
100C22×C43—42no—
101C22×C43—81no—
102C22×C43C2×C22⋊C411yescharacteristic
103C22×C43C4×D412yes—
104C22×C43#12011yescharacteristic
105C22×C43#12112yes—
106C22×C43#12311yescharacteristic
107C4×C43—22no—
108C4×C43—42no—
109Dic43—44no—
110Dic43—82no—
111#1889—431nomaximal, Sylow
112C2×C43—22no—
113C2×C43—21no—
114C2×C43—22no—
115C2×C43—21no—
116C2×C43—44no—
117C2×C43—42no—
118C2×C43#38611yescharacteristic
119C2×C43B(4,𝔽2)12yes—
120C2×#120D4311yescharacteristic
121#518—431no—
122C2×#99—431no—
123#379—432no—
124#342—432no—
125C43#188911yescharacteristic, complemented, Sylow
126C25C2×D4311yescharacteristic
127C22×D4—21no—
128C22×D4C2×D4311yescharacteristic
129#120—22no—
130#120Dic4314yescomplemented
131C4×D4—864no—
132#117—861no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—432no—
135C2×C22⋊C4—432no—
136C2×C22⋊C4—861no—
137C2×C22⋊C4—864no—
138C2×C22⋊C4C2×D4311yescharacteristic
139#99—862no—
140C24—22no—
141C24—21no—
142C24—22no—
143C24—21no—
144C24—44no—
145C24C2×Dic4312yescomplemented
146C24C22⋊D4311yescharacteristic
147C24C22×D4311yescharacteristic
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—24no—
151C2×D4—44no—
152C2×D4C2×Dic4312yes—
153C2×D4C22⋊D4312yes—
154C22×C4—21no—
155C22×C4—434no—
156C22×C4—862no—
157C22×C4—862no—
158C22×C4C22⋊D4311yescharacteristic
159C4⋊C4—862no—
160C22⋊C4—21no—
161C22⋊C4—24no—
162C22⋊C4—862no—
163C22⋊C4—862no—
164C22⋊C4—862no—
165C22⋊C4—862no—
166C22⋊C4—862no—
167C22⋊C4—1728no—
168C22⋊C4C2×Dic4312yescomplemented
169C42—862no—
170C23—28no—
171C23—24no—
172C23—22no—
173C23—24no—
174C23—21no—
175C23—21no—
176C23—42no—
177C23—44no—
178C23—42no—
179C23—44no—
180C23—44no—
181C23—48no—
182C23—42no—
183C23—41no—
184C23—42no—
185C23—41no—
186C23C22⋊Dic4312yescomplemented
187C23D4×D4312yes—
188C23#1062190012yes—
189C23C22×Dic4311yescharacteristic
190C23C2×C22⋊D4311yescharacteristic
191C23C2×C22⋊D4311yescharacteristic
192D4—24no—
193D4—24no—
194D4—48no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—864no—
200C2×C4—3442no—
201C2×C4—1724no—
202C2×C4—1722no—
203C2×C4—1724no—
204C2×C4C22⋊Dic4312yescomplemented
205C22—22no—
206C22—24no—
207C22—24no—
208C22—28no—
209C22—22no—
210C22—42no—
211C22—41no—
212C22—41no—
213C22—42no—
214C22—42no—
215C22—48no—
216C22—44no—
217C22—44no—
218C22—42no—
219C22—44no—
220C22—42no—
221C22—81no—
222C22C2×C22⋊Dic4311yescharacteristic
223C22D4×Dic4312yes—
224C22#1306329311yescharacteristic
225C22#1306329712yes—
226C22#1306330611yescharacteristic
227C4—22no—
228C4—42no—
229C4—3442no—
230C4—1724no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2—21no—
235C2—44no—
236C2—42no—
237C2#1458277911yescharacteristic
238C2#1458308512yes—
subgroupG/H
centerC22#13063293
commutatorC23×C43C22×C4
FrattiniC24C22×D43
FittingC2×#120×C43C2
socleC22×C43#120
hypercenterC2×#120D43
hypocenterC43#1889
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #14583374C2semidirect
2. C2×#13063240C2semidirect
3. #14582755C2semidirect
4. #14582665C2semidirect
17. #120×C43C4semidirect
25. C24×C43C2×C4semidirect
38. C22×Dic43D4semidirect
41. C22⋊C4×C43C2×C4semidirect
66. C23×C43C22⋊C4semidirect
79. C2×C4×C43C22⋊C4semidirect
125. C43#1889semidirect
130. #120Dic43semidirect
145. C24C2×Dic43semidirect
168. C22⋊C4C2×Dic43semidirect
186. C23C22⋊Dic43semidirect
204. C2×C4C22⋊Dic43semidirect