Finite Group Databasebeta

← #19137309

Subgroups — #19137309

Order 5504.

612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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
2C2×Q8×Dic43C211yesmaximal, characteristic, complemented
3C2×#13063287C211yesmaximal, characteristic
4#14582801C211yesmaximal, characteristic
5#14582788C214yesmaximal, complemented
6#14582773C214yesmaximal, complemented
7#14582768C211yesmaximal, characteristic, complemented
8#14582745C211yesmaximal, characteristic, complemented
9C43×#391C211yesmaximal, characteristic, complemented
10C22×D172C2211yescharacteristic
11C22×C4×D43C2212yes—
12#13063314C418yescomplemented
13Q8×Dic43—24no—
14#13063287—24no—
15C2×C4⋊Dic43C2211yescharacteristic
16C2×C4⋊Dic43C2212yes—
17C2×Dic43⋊C4C2214yes—
18C2×(C2×D43)⋊C4C2212yes—
19C2×(C2×D43)⋊C4C2214yes—
20C2×(C2×D43)⋊C4C2214yes—
21C2×C4×Dic43C2212yes—
22C2×C4×Dic43C2211yescharacteristic
23C2×C4×Dic43C2211yescharacteristic
24C43⋊#95C2212yes—
25C43⋊#95C2212yes—
26C22×Q8×C43C2211yescharacteristic
27C2×C4⋊C4×C43C2211yescharacteristic
28C2×C42×C43C2211yescharacteristic
29#95×C43C2214yes—
30C23×D43C2312yes—
31C2×Q8×C43—22no—
32C2×Q8×C43—22no—
33C2×Q8×C43C2×C414yescomplemented
34C22×C4×C43C2312yes—
35C22×C4×C43C2312yes—
36C22×C4×C43C2312yes—
37C22×C4×C43C2311yescharacteristic
38C22×Dic43C2312yes—
39C22×Dic43C2312yes—
40C22×Dic43C2311yescharacteristic
41C22×Dic43C2311yescharacteristic
42C2×D172—22no—
43C2×D172—22no—
44C2×D172C2×C414yes—
45C2×C4×D43—24no—
46C2×C4×D43—44no—
47C4⋊C4×C43—22no—
48C42×C43—22no—
49(C2×D43)⋊C4—24no—
50(C2×D43)⋊C4—44no—
51(C2×D43)⋊C4C2×C4116yescomplemented
52C4⋊Dic43—22no—
53C4⋊Dic43—24no—
54Dic43⋊C4—44no—
55C4×Dic43—24no—
56C4×Dic43C2×C414yescomplemented
57C4×Dic43D414yes—
58C23×C43C2411yescharacteristic
59C22×D43—22no—
60C22×D43—22no—
61C22×D43—24no—
62C22×D43—24no—
63C22×D43C22×C414yes—
64Q8×C43—28no—
65C2×C4×C43—22no—
66C2×C4×C43—22no—
67C2×C4×C43—22no—
68C2×C4×C43—22no—
69C2×C4×C43—24no—
70C2×C4×C43—22no—
71C2×C4×C43—22no—
72C2×C4×C43C22×C414yes—
73C2×C4×C43C22×C412yes—
74C2×C4×C43C2×D412yes—
75C2×C4×C43#4112yes—
76C2×Dic43—22no—
77C2×Dic43—22no—
78C2×Dic43—22no—
79C2×Dic43—22no—
80C2×Dic43—22no—
81C2×Dic43—24no—
82C2×Dic43C22×C414yes—
83C2×Dic43C2×D412yes—
84C2×Dic43C2×D412yes—
85D172—28no—
86C4×D43—48no—
87C22×C43C23×C411yescharacteristic
88C22×C43C22×D411yescharacteristic
89C22×C43C2×#4111yescharacteristic
90C22×C43H(5,𝔽2)12yes—
91C22×C432−1+412yes—
92C2×D43—24no—
93C2×D43—28no—
94C2×D43—42no—
95C2×D43—42no—
96C2×D43—44no—
97C4×C43—24no—
98C4×C43—42no—
99C4×C43—42no—
100C4×C43C4×D414yes—
101Dic43—24no—
102Dic43—22no—
103Dic43—22no—
104Dic43—42no—
105#2326—431nomaximal, Sylow
106C2×C43C2×C4×D411yescharacteristic
107C2×C43#52012yes—
108C2×C43#53511yescharacteristic
109C2×C43#53611yescharacteristic
110C2×C43#54212yes—
111D43—44no—
112C2×#124—431no—
113C2×C4×Q8—431no—
114C2×C4×D4—431no—
115#391D4311yescharacteristic, complemented
116#390—431no—
117#388—434no—
118#385—434no—
119#384—431no—
120#376—431no—
121C43#232611yescharacteristic, complemented, Sylow
122C22×Q8C2×D4311yescharacteristic
123C22×D4—431no—
124C23×C4—432no—
125#124—438no—
126C4×Q8—864no—
127C4×D4—864no—
128C2×C4⋊C4—431no—
129C2×C4⋊C4—432no—
130C2×C4⋊C4—434no—
131C2×C4⋊C4C2×D4311yescharacteristic
132C2×C22⋊C4—434no—
133C2×C22⋊C4—432no—
134C2×C22⋊C4—434no—
135C2×C42—431no—
136C2×C42—431no—
137C2×C42—432no—
138C2×C42C2×D4311yescharacteristic
139#95—432no—
140#95—432no—
141#95C2×D4314yes—
142C24—432no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8C4×D4314yescomplemented
146C2×D4—434no—
147C2×D4—862no—
148C2×D4—862no—
149C22×C4—431no—
150C22×C4—432no—
151C22×C4—431no—
152C22×C4—432no—
153C22×C4—864no—
154C22×C4—1724no—
155C22×C4C22×D4312yes—
156C22×C4C22×D4312yes—
157C22×C4C22×D4312yes—
158C22×C4C22×D4311yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—864no—
161C4⋊C4—862no—
162C4⋊C4—1724no—
163C22⋊C4—4316no—
164C22⋊C4—864no—
165C22⋊C4—1724no—
166C42—22no—
167C42—434no—
168C42—434no—
169C42—864no—
170C23—434no—
171C23—862no—
172C23—862no—
173C23—864no—
174C23—864no—
175C23C23×D4311yescharacteristic
176Q8—28no—
177D4—868no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—434no—
186C2×C4—432no—
187C2×C4—432no—
188C2×C4—862no—
189C2×C4—862no—
190C2×C4—862no—
191C2×C4—862no—
192C2×C4—862no—
193C2×C4—864no—
194C2×C4—1728no—
195C2×C4C2×C4×D4314yes—
196C2×C4C2×C4×D4312yes—
197C2×C4D4×D4312yes—
198C2×C4#1062190012yes—
199C22—864no—
200C22—868no—
201C22—1722no—
202C22—1722no—
203C22—1724no—
204C22C22×C4×D4311yescharacteristic
205C22C2×D4×D4311yescharacteristic
206C22C2×#1062190011yescharacteristic
207C22#1306337112yes—
208C22#1306337312yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—864no—
213C4—862no—
214C4—862no—
215C4—1722no—
216C4(C22⋊D43)⋊C414yes—
217C2—1724no—
218C2C2×(C22⋊D43)⋊C411yescharacteristic
219C2#1458338112yes—
220C2#1458346711yescharacteristic
221C2#1458347411yescharacteristic
222C2#1458352212yes—
subgroupG/H
center, FrattiniC23C23×D43
commutatorC22×C43C23×C4
FittingC43×#391C2
socleC23×C43C24
hypercenter#391D43
hypocenterC43#2326
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic43C2semidirect
5. #14582788C2semidirect
6. #14582773C2semidirect
7. #14582768C2semidirect
8. #14582745C2semidirect
9. C43×#391C2semidirect
12. #13063314C4semidirect
33. C2×Q8×C43C2×C4semidirect
51. (C2×D43)⋊C4C2×C4semidirect
56. C4×Dic43C2×C4semidirect
115. #391D43semidirect
121. C43#2326semidirect
145. C2×Q8C4×D43semidirect