Finite Group Databasebeta

← #19141601

Subgroups — #19141601

Order 5504.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13063314C211yesmaximal, characteristic
2#14582821C212yesmaximal, complemented
3#14582816C212yesmaximal
4#14582810C212yesmaximal, complemented
5#14582806C211yesmaximal, characteristic, complemented
6#14582790C212yesmaximal, complemented
7#14582789C212yesmaximal, complemented
8#14582780C212yesmaximal, complemented
9C2×C4⋊Q8×C43C211yesmaximal, characteristic, complemented
10C22×D172C2211yescharacteristic
11#13063314—24no—
12C2×C4⋊Dic43C2211yescharacteristic
13C2×Dic43⋊C4C2212yes—
14C2×(C2×D43)⋊C4C2212yes—
15C2×(C2×D43)⋊C4C2212yes—
16C2×(C2×D43)⋊C4C2212yes—
17C2×(C2×D43)⋊C4C2212yes—
18C2×(C2×D43)⋊C4C2212yes—
19C2×(C2×D43)⋊C4C2212yes—
20C2×C4×Dic43C2211yescharacteristic
21C2×C4×Dic43C2211yescharacteristic
22C43⋊#95C2212yes—
23C43⋊#95C2212yes—
24C43⋊#95C2212yes—
25C43⋊#95C2212yes—
26C43⋊#95C2212yes—
27C22×Q8×C43C2211yescharacteristic
28C22×Q8×C43C2211yescharacteristic
29C4⋊Q8×C43—24no—
30C2×C4⋊C4×C43C2212yes—
31C2×C4⋊C4×C43C2212yes—
32C2×C42×C43C2211yescharacteristic
33C23×D43C2312yes—
34C2×Q8×C43—22no—
35C2×Q8×C43—41no—
36C2×Q8×C43—41no—
37C2×Q8×C43—41no—
38C2×Q8×C43—41no—
39C2×Q8×C43D414yes—
40C22×C4×C43C2312yes—
41C22×C4×C43C2311yescharacteristic
42C22×C4×C43C2311yescharacteristic
43C22×C4×C43C2311yescharacteristic
44C22×C4×C43C2312yes—
45C22×Dic43C2312yes—
46C22×Dic43C2312yes—
47C22×Dic43C2312yes—
48C2×D172—22no—
49C2×D172—41no—
50C2×D172—41no—
51C4⋊C4×C43—24no—
52C4⋊C4×C43—24no—
53C42×C43—22no—
54(C2×D43)⋊C4—24no—
55(C2×D43)⋊C4—24no—
56(C2×D43)⋊C4—24no—
57(C2×D43)⋊C4—24no—
58(C2×D43)⋊C4—42no—
59(C2×D43)⋊C4—42no—
60C4⋊Dic43—41no—
61Dic43⋊C4—24no—
62C4×Dic43—22no—
63C4×Dic43—41no—
64C23×C43C2411yescharacteristic
65C22×D43—22no—
66C22×D43—22no—
67C22×D43—22no—
68C22×D43—22no—
69C22×D43—22no—
70C22×D43—22no—
71C22×D43—22no—
72Q8×C43—44no—
73Q8×C43—44no—
74C2×C4×C43—22no—
75C2×C4×C43—21no—
76C2×C4×C43—21no—
77C2×C4×C43—21no—
78C2×C4×C43—21no—
79C2×C4×C43—21no—
80C2×C4×C43—21no—
81C2×C4×C43—22no—
82C2×C4×C43—22no—
83C2×C4×C43—22no—
84C2×C4×C43—22no—
85C2×C4×C43C2×D412yes—
86C2×C4×C43C2×D412yes—
87C2×C4×C43C2×D412yes—
88C2×C4×C43#4114yes—
89C2×Dic43—22no—
90C2×Dic43—22no—
91C2×Dic43—22no—
92C2×Dic43—22no—
93C2×Dic43—22no—
94C2×Dic43—22no—
95C2×Dic43—22no—
96C2×Dic43—22no—
97C2×Dic43—22no—
98D172—44no—
99C22×C43C22×D411yescharacteristic
100C22×C43C2×#4112yes—
101C22×C43H(5,𝔽2)11yescharacteristic
102C22×C43H(5,𝔽2)11yescharacteristic
103C22×C432−1+412yes—
104C2×D43—42no—
105C2×D43—42no—
106C2×D43—42no—
107C2×D43—42no—
108C2×D43—42no—
109C2×D43—42no—
110C2×D43—42no—
111C4×C43—22no—
112C4×C43—22no—
113C4×C43—22no—
114C4×C43—24no—
115C4×C43—42no—
116Dic43—42no—
117Dic43—42no—
118Dic43—42no—
119#2731—431nomaximal, Sylow
120C2×C43#53411yescharacteristic
121C2×C43#54212yes—
122C2×C43#54812yes—
123C2×C43#55511yescharacteristic
124C2×C43#56311yescharacteristic
125D43—82no—
126C2×C4⋊Q8D4311yescharacteristic, complemented
127C2×#124—431no—
128#396—432no—
129#394—432no—
130#393—432no—
131#391—431no—
132#388—432no—
133#388—432no—
134#387—432no—
135C43#273111yescharacteristic, complemented, Sylow
136C22×Q8C2×D4311yescharacteristic
137C22×Q8C2×D4311yescharacteristic
138C22×D4—431no—
139C4⋊Q8—24no—
140#124—864no—
141C2×C4⋊C4—432no—
142C2×C4⋊C4—431no—
143C2×C4⋊C4C2×D4312yes—
144C2×C4⋊C4C2×D4312yes—
145C2×C22⋊C4—432no—
146C2×C22⋊C4—432no—
147C2×C22⋊C4—432no—
148C2×C22⋊C4—432no—
149C2×C22⋊C4—432no—
150C2×C22⋊C4—432no—
151C2×C42—431no—
152C2×C42—431no—
153C2×C42C2×D4311yescharacteristic
154#95—432no—
155#95—432no—
156#95—432no—
157#95—432no—
158#95—432no—
159C24—432no—
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8C22⋊D4314yes—
166C2×D4—862no—
167C2×D4—1721no—
168C2×D4—1721no—
169C22×C4—432no—
170C22×C4—432no—
171C22×C4—432no—
172C22×C4C22×D4312yes—
173C22×C4C22×D4312yes—
174C22×C4C22×D4311yescharacteristic
175C22×C4C22×D4311yescharacteristic
176C22×C4C22×D4311yescharacteristic
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—864no—
180C4⋊C4—1721no—
181C22⋊C4—864no—
182C22⋊C4—864no—
183C22⋊C4—864no—
184C22⋊C4—864no—
185C22⋊C4—1722no—
186C22⋊C4—1722no—
187C42—22no—
188C42—862no—
189C42—1721no—
190C23—862no—
191C23—862no—
192C23—862no—
193C23—862no—
194C23—862no—
195C23—862no—
196C23—862no—
197C23C23×D4311yescharacteristic
198Q8—44no—
199Q8—44no—
200D4—1724no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—862no—
213C2×C4—862no—
214C2×C4—862no—
215C2×C4—862no—
216C2×C4—862no—
217C2×C4—862no—
218C2×C4—862no—
219C2×C4—862no—
220C2×C4—862no—
221C2×C4#1062190214yes—
222C2×C4C2×C22⋊D4312yes—
223C2×C4C2×C22⋊D4312yes—
224C2×C4C2×C22⋊D4312yes—
225C22—1722no—
226C22—1722no—
227C22—1722no—
228C22—1722no—
229C22—1722no—
230C22—1722no—
231C22—1722no—
232C22C22×C22⋊D4311yescharacteristic
233C22C2×#1062190212yes—
234C22#1306337011yescharacteristic
235C22#1306337011yescharacteristic
236C22#1306337312yes—
237C4—24no—
238C4—22no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—1722no—
243C4—1722no—
244C4—1722no—
245C2—3442no—
246C2#1458345811yescharacteristic
247C2#1458352312yes—
248C2#1458356012yes—
249C2#1458361511yescharacteristic
250C2#1458364811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC2×C4⋊Q8×C43C2
hypercenterC2×C4⋊Q8D43
hypocenterC43#2731
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #14582821C2semidirect
4. #14582810C2semidirect
5. #14582806C2semidirect
6. #14582790C2semidirect
7. #14582789C2semidirect
8. #14582780C2semidirect
9. C2×C4⋊Q8×C43C2semidirect
126. C2×C4⋊Q8D43semidirect
135. C43#2731semidirect