Finite Group Databasebeta

← #19141603

Subgroups — #19141603

Order 5504.

492 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13063323C211yesmaximal, characteristic, complemented
2#14582822C212yesmaximal, complemented
3#14582813C212yesmaximal
4#14582811C212yesmaximal, complemented
5#14582806C211yesmaximal, characteristic, complemented
6#14582793C212yesmaximal, complemented
7#14582787C212yesmaximal, complemented
8#14582783C212yesmaximal, complemented
9C2×#124×C43C211yesmaximal, characteristic
10C22×Dic86C2211yescharacteristic
11#13063323—24no—
12C2×C4⋊Dic43C2211yescharacteristic
13C2×Dic43⋊C4C2212yes—
14C2×Dic43⋊C4C2212yes—
15C2×Dic43⋊C4C2212yes—
16C2×C22⋊Dic43C2212yes—
17C2×C22⋊Dic43C2212yes—
18C2×C22⋊Dic43C2212yes—
19C2×C22⋊Dic43C2212yes—
20C2×C4×Dic43C2211yescharacteristic
21C2×C4×Dic43C2211yescharacteristic
22C43⋊#95C2212yes—
23C43⋊#95C2212yes—
24C43⋊#95C2212yes—
25C43⋊#95C2212yes—
26C43⋊#95C2212yes—
27C22×Q8×C43C2211yescharacteristic
28C22×D4×C43C2211yescharacteristic
29#124×C43—24no—
30C2×C22⋊C4×C43C2212yes—
31C2×C22⋊C4×C43C2212yes—
32C2×C42×C43C2211yescharacteristic
33C24×C43C2312yes—
34C2×Q8×C43—41no—
35C2×Q8×C43—41no—
36C2×Q8×C43D414yes—
37C2×D4×C43—22no—
38C2×D4×C43—41no—
39C2×D4×C43—41no—
40C22×C4×C43C2312yes—
41C22×C4×C43C2311yescharacteristic
42C22×C4×C43C2311yescharacteristic
43C22×C4×C43C2311yescharacteristic
44C22×Dic43C2312yes—
45C22×Dic43C2312yes—
46C22×Dic43C2312yes—
47C22×Dic43C2312yes—
48C2×Dic86—22no—
49C2×Dic86—41no—
50C2×Dic86—41no—
51C22⋊C4×C43—24no—
52C22⋊C4×C43—24no—
53C42×C43—22no—
54C22⋊Dic43—24no—
55C22⋊Dic43—24no—
56C22⋊Dic43—42no—
57C22⋊Dic43—42no—
58C4⋊Dic43—41no—
59Dic43⋊C4—24no—
60Dic43⋊C4—24no—
61Dic43⋊C4—24no—
62C4×Dic43—22no—
63C4×Dic43—41no—
64C23×C43—22no—
65C23×C43—22no—
66C23×C43—22no—
67C23×C43—22no—
68C23×C43—22no—
69C23×C43—22no—
70C23×C43—22no—
71C23×C43C2411yescharacteristic
72Q8×C43—44no—
73D4×C43—44no—
74C2×C4×C43—21no—
75C2×C4×C43—21no—
76C2×C4×C43—21no—
77C2×C4×C43—21no—
78C2×C4×C43—21no—
79C2×C4×C43—21no—
80C2×C4×C43—22no—
81C2×C4×C43—22no—
82C2×C4×C43—22no—
83C2×C4×C43C2×D412yes—
84C2×C4×C43C2×D412yes—
85C2×C4×C43C2×D412yes—
86C2×Dic43—22no—
87C2×Dic43—22no—
88C2×Dic43—22no—
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#4114yes—
98Dic86—44no—
99C22×C43—42no—
100C22×C43—42no—
101C22×C43—42no—
102C22×C43—42no—
103C22×C43—42no—
104C22×C43—42no—
105C22×C43—42no—
106C22×C43C22×D411yescharacteristic
107C22×C43C2×#4112yes—
108C22×C43H(5,𝔽2)11yescharacteristic
109C22×C43H(5,𝔽2)11yescharacteristic
110C22×C432−1+412yes—
111C4×C43—22no—
112C4×C43—22no—
113C4×C43—22no—
114C4×C43—42no—
115Dic43—24no—
116Dic43—42no—
117Dic43—42no—
118Dic43—42no—
119#2731—431nomaximal, Sylow
120C2×C43—82no—
121C2×C43#53411yescharacteristic
122C2×C43#54212yes—
123C2×C43#54812yes—
124C2×C43#55511yescharacteristic
125C2×C43#56311yescharacteristic
126C2×C4⋊Q8—431no—
127C2×#124D4311yescharacteristic
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×Q8—431no—
137C22×Q8C2×D4311yescharacteristic
138C22×D4C2×D4311yescharacteristic
139C4⋊Q8—864no—
140#124—24no—
141C2×C4⋊C4—432no—
142C2×C4⋊C4—432no—
143C2×C4⋊C4—432no—
144C2×C4⋊C4—431no—
145C2×C22⋊C4—432no—
146C2×C22⋊C4—432no—
147C2×C22⋊C4—432no—
148C2×C22⋊C4—432no—
149C2×C22⋊C4C2×D4312yes—
150C2×C22⋊C4C2×D4312yes—
151C2×C42—431no—
152C2×C42—431no—
153C2×C42C2×D4311yescharacteristic
154#95—432no—
155#95—432no—
156#95—432no—
157#95—432no—
158#95—432no—
159C24C22×D4312yes—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—862no—
163C2×Q8—1721no—
164C2×Q8—1721no—
165C2×Q8C22⋊D4314yes—
166C2×D4—22no—
167C2×D4—41no—
168C2×D4—41no—
169C22×C4—432no—
170C22×C4—432no—
171C22×C4—432no—
172C22×C4—432no—
173C22×C4C22×D4312yes—
174C22×C4C22×D4311yescharacteristic
175C22×C4C22×D4311yescharacteristic
176C22×C4C22×D4311yescharacteristic
177C4⋊C4—864no—
178C4⋊C4—864no—
179C4⋊C4—864no—
180C4⋊C4—1721no—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—864no—
184C22⋊C4—864no—
185C22⋊C4—1722no—
186C22⋊C4—1722no—
187C42—22no—
188C42—862no—
189C42—1721no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23C23×D4311yescharacteristic
198Q8—44no—
199Q8—1724no—
200D4—44no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—434no—
211C2×C4—862no—
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—862no—
222C2×C4C2×C22⋊D4312yes—
223C2×C4C2×C22⋊D4312yes—
224C2×C4C2×C22⋊D4312yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22C22×C22⋊D4311yescharacteristic
233C22#41×D4312yes—
234C22#1306337011yescharacteristic
235C22#1306337011yescharacteristic
236C22#1306337212yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—864no—
242C4—1722no—
243C4—1722no—
244C4—1722no—
245C2—82no—
246C2#1458345811yescharacteristic
247C2#1458351312yes—
248C2#1458355612yes—
249C2#1458361111yescharacteristic
250C2#1458364911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC2×#124×C43C2
hypercenterC2×#124D43
hypocenterC43#2731
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#13063323C2semidirect
2. #14582822C2semidirect
4. #14582811C2semidirect
5. #14582806C2semidirect
6. #14582793C2semidirect
7. #14582787C2semidirect
8. #14582783C2semidirect
135. C43#2731semidirect