Finite Group Databasebeta

← #19135719

Subgroups — #19135719

Order 5504.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#13063293C212yesmaximal, complemented
2C2×#13063240C212yesmaximal, complemented
3#14582754C211yesmaximal, characteristic, complemented
4#14582666C211yesmaximal, characteristic, complemented
5C43×#534C211yesmaximal, characteristic, complemented
6C24×D43C2211yescharacteristic
7C22×C22⋊D43—22no—
8#13063293—24no—
9#13063293—24no—
10C2×C22⋊Dic43C2211yescharacteristic
11C2×C22⋊Dic43C2212yes—
12C2×(C2×D43)⋊C4—22no—
13C2×(C2×D43)⋊C4—22no—
14#13063240—24no—
15C22×D4×C43C2212yes—
16#123×C43—22no—
17#121×C43—22no—
18#120×C43—22no—
19C2×C22⋊C4×C43C2211yescharacteristic
20C24×C43D412yescomplemented
21C24×C43C2311yescharacteristic
22C23×D43—21no—
23C23×D43—22no—
24C23×D43—22no—
25C23×D43—21no—
26C23×D43—44no—
27C23×D43D412yescomplemented
28C2×D4×C43—22no—
29C2×D4×C43—22no—
30C2×D4×C43—22no—
31C2×D4×C43—22no—
32C2×D4×C43—22no—
33C2×D4×C43—22no—
34C2×D4×C43—22no—
35C22×C4×C43D412yescomplemented
36C22×C4×C43D412yescomplemented
37C2×C22⋊D43—28no—
38C2×C22⋊D43—48no—
39C22×Dic43—22no—
40C22×Dic43—21no—
41C4⋊C4×C43—22no—
42C22⋊C4×C43—24no—
43C22⋊C4×C43—22no—
44C22⋊Dic43—22no—
45C22⋊Dic43—22no—
46C22⋊Dic43D414yescomplemented
47(C2×D43)⋊C4—28no—
48(C2×D43)⋊C4—44no—
49C23×C43—22no—
50C23×C43—22no—
51C23×C43—22no—
52C23×C43—42no—
53C23×C43—44no—
54C23×C43C2×D411yescharacteristic
55C23×C43C2×D412yes—
56C23×C43C2×D412yes—
57C23×C43C2×D411yescharacteristic
58C23×C43#4111yescharacteristic
59C22×D43—24no—
60C22×D43—28no—
61C22×D43—24no—
62C22×D43—22no—
63C22×D43—44no—
64C22×D43—44no—
65C22×D43—48no—
66C22×D43—44no—
67C22×D43—42no—
68C22×D43—41no—
69C22×D43—42no—
70C22×D43—41no—
71D4×C43—42no—
72D4×C43—42no—
73D4×C43—42no—
74D4×C43—42no—
75D4×C43—82no—
76C2×C4×C43—22no—
77C2×C4×C43—22no—
78C2×C4×C43—22no—
79C2×C4×C43—22no—
80C2×C4×C43—42no—
81C22⋊D43—416no—
82C2×Dic43—24no—
83C2×Dic43—44no—
84C2×Dic43—42no—
85C2×Dic43—41no—
86C22×C43—24no—
87C22×C43—22no—
88C22×C43—41no—
89C22×C43—42no—
90C22×C43—41no—
91C22×C43—44no—
92C22×C43—42no—
93C22×C43—44no—
94C22×C43—82no—
95C22×C43#12012yes—
96C22×C43#12011yescharacteristic
97C22×C43#12111yescharacteristic
98C22×C43#12112yes—
99C22×C43#12711yescharacteristic
100C2×D43—24no—
101C2×D43—28no—
102C2×D43—44no—
103C2×D43—44no—
104C2×D43—42no—
105C2×D43—48no—
106C2×D43—42no—
107C2×D43—44no—
108C4×C43—42no—
109C4×C43—42no—
110Dic43—44no—
111Dic43—81no—
112#2021—431nomaximal, Sylow
113C2×C43—22no—
114C2×C43—21no—
115C2×C43—21no—
116C2×C43—22no—
117C2×C43—82no—
118C2×C43#39211yescharacteristic
119C2×C43B(4,𝔽2)12yes—
120D43—44no—
121#534D4311yescharacteristic, complemented
122C2×#120—432no—
123C2×#99—432no—
124#379—431no—
125#342—431no—
126C43#202111yescharacteristic, complemented, Sylow
127C25—431no—
128C22×D4—862no—
129C22×D4C2×D4312yes—
130#123—22no—
131#121—22no—
132#120—22no—
133#120—864no—
134#120—864no—
135C2×C22⋊C4—432no—
136C2×C22⋊C4—431no—
137C2×C22⋊C4—862no—
138C2×C22⋊C4—862no—
139C2×C22⋊C4C2×D4311yescharacteristic
140#99—864no—
141C24—432no—
142C24—861no—
143C24—862no—
144C24—862no—
145C24—861no—
146C24—1724no—
147C24C22⋊D4312yescomplemented
148C24C22×D4311yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—868no—
157C2×D4—1728no—
158C22×C4—862no—
159C22×C4—861no—
160C22×C4C22⋊D4312yescomplemented
161C22×C4C22⋊D4312yescomplemented
162C4⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—434no—
166C22⋊C4—868no—
167C22⋊C4—862no—
168C22⋊C4—862no—
169C22⋊C4—1724no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—44no—
175C23—864no—
176C23—868no—
177C23—864no—
178C23—862no—
179C23—1722no—
180C23—1724no—
181C23—1724no—
182C23—1728no—
183C23—1724no—
184C23—1721no—
185C23—1722no—
186C23—1721no—
187C23D4×D4311yescharacteristic
188C23D4×D4312yes—
189C23#1062190011yescharacteristic
190C23C2×C22⋊D4312yes—
191C23C2×C22⋊D4311yescharacteristic
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197D4—17216no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—864no—
204C2×C4—1722no—
205C2×C4—1721no—
206C2×C4—1724no—
207C22—22no—
208C22—24no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—44no—
213C22—42no—
214C22—44no—
215C22—82no—
216C22—868no—
217C22—864no—
218C22—1724no—
219C22—1724no—
220C22—1728no—
221C22—1722no—
222C22—1722no—
223C22—1724no—
224C22#1306329312yes—
225C22#1306329411yescharacteristic
226C22#1306329611yescharacteristic
227C22#1306329712yes—
228C22#1306332111yescharacteristic
229C4—42no—
230C4—42no—
231C4—3441no—
232C4—1724no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2—21no—
237C2—82no—
238C2—1724no—
239C2#1458280711yescharacteristic
240C2#1458308612yes—
subgroupG/H
centerC22#13063294
commutatorC24×C43C23
FrattiniC24C22×D43
FittingC43×#534C2
socleC22×C43#120
hypercenter#534D43
hypocenterC43#2021
LCSG ▷ C24×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#13063293C2semidirect
2. C2×#13063240C2semidirect
3. #14582754C2semidirect
4. #14582666C2semidirect
5. C43×#534C2semidirect
20. C24×C43D4semidirect
27. C23×D43D4semidirect
35. C22×C4×C43D4semidirect
36. C22×C4×C43D4semidirect
46. C22⋊Dic43D4semidirect
121. #534D43semidirect
126. C43#2021semidirect
147. C24C22⋊D43semidirect
160. C22×C4C22⋊D43semidirect
161. C22×C4C22⋊D43semidirect