Finite Group Databasebeta

← #24994154

Subgroups — #24994154

Order 11008.

508 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 95 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#19145052C211yesmaximal, characteristic
2#19137776C211yesmaximal, characteristic, complemented
3#19133143C212yesmaximal
4#19132461C212yesmaximal, complemented
5C43×#1558C211yesmaximal, characteristic, complemented
6C2×#13063356—22no—
7#14583449C2211yescharacteristic
8#14583291—22no—
9#14583278C412yescomplemented
10#14583273C412yescomplemented
11C4×#10621877C2211yescharacteristic
12#14582806—22no—
13#14582785—24no—
14#14582767C2212yes—
15#14582634—24no—
16C2×C4⋊Q8×C43C2211yescharacteristic
17C43×#432C2212yes—
18C43×#333—22no—
19C2×#10621898—21no—
20#13063356—44no—
21#13063356—44no—
22C2×#10621885—24no—
23C2×#10621884—24no—
24C2×#10621877—21no—
25#13063325C2×C411yescharacteristic
26#13063322C2×C411yescharacteristic
27#13063313—21no—
28#13063299—22no—
29C4×D172—22no—
30C2×C4⋊Dic43—22no—
31C2×C4×Dic43—22no—
32#13063236—21no—
33C4×C43⋊C8C2×C412yes—
34C43⋊#95—24no—
35C43⋊#95—22no—
36C43⋊#95—22no—
37C43⋊#95D414yes—
38C22×Q8×C43D412yes—
39C4⋊Q8×C43—22no—
40C4⋊Q8×C43C2×C412yes—
41C4⋊Q8×C43D412yes—
42C2×C4⋊C4×C43—22no—
43C2×C42×C43C2311yescharacteristic
44C4⋊C8×C43—22no—
45C22⋊C8×C43—22no—
46#97×C43—22no—
47C2×Q8×C43—22no—
48C2×Q8×C43—42no—
49C2×Q8×C43—42no—
50C2×Q8×C43C22⋊C414yes—
51C22×C4×C43—22no—
52C22×C4×C43C2×D411yescharacteristic
53C22×C4×C43C2×D412yes—
54C2×C22⋊D43—41no—
55C22×Dic43—24no—
56#10621898—42no—
57#10621898—42no—
58C2×D172—21no—
59C2×D172—22no—
60C2×C4×D43—41no—
61C2×Dic86—21no—
62C2×C8×C43—42no—
63C4⋊C4×C43—24no—
64C4⋊C4×C43—24no—
65C42×C43C22×C411yescharacteristic
66C42×C43C2×D411yescharacteristic
67C42×C43#4112yes—
68#10621885—48no—
69#10621884—48no—
70(C2×D43)⋊C4—42no—
71C4⋊Dic43—22no—
72C4⋊Dic43—82no—
73C4×Dic43—82no—
74#10621877—24no—
75C2×C43⋊C8—22no—
76C23×C43#12011yescharacteristic
77C22×D43—41no—
78Q8×C43—44no—
79Q8×C43—82no—
80C2×C4×C43—22no—
81C2×C4×C43—24no—
82C2×C4×C43—42no—
83C2×C4×C43—42no—
84C2×C4×C43—42no—
85C2×C4×C43C2×C22⋊C411yescharacteristic
86C2×C4×C43C4×D412yes—
87C2×C4×C43#12112yes—
88C2×C4×C43#12311yescharacteristic
89C22⋊D43—82no—
90C2×Dic43—48no—
91C2×Dic43—41no—
92C2×Dic43—44no—
93D172—42no—
94D172—82no—
95C4×D43—81no—
96C4×D43—81no—
97Dic86—42no—
98C8×C43—82no—
99C43⋊C8—24no—
100#9429—431nomaximal, Sylow
101C22×C43—22no—
102C22×C43#38611yescharacteristic
103C22×C43#45312yes—
104C2×D43—81no—
105C2×D43—82no—
106C4×C43—21no—
107C4×C43—21no—
108C4×C43—22no—
109C4×C43—22no—
110C4×C43—44no—
111Dic43—84no—
112Dic43—81no—
113#3152—431no—
114#2381—431no—
115#1558D4311yescharacteristic, complemented
116#1514—432no—
117#1288—432no—
118C2×C43—22no—
119C2×C43#188711yescharacteristic
120C2×C43#190512yes—
121D43—161no—
122C2×#137—862no—
123#532—431no—
124C2×C4⋊Q8C2×D4311yescharacteristic
125#497—862no—
126#494—432no—
127#492—432no—
128#432C2×D4312yes—
129C4×M16—431no—
130#391—862no—
131#387—864no—
132#383—432no—
133#333—22no—
134#327—864no—
135C43#942911yescharacteristic, complemented, Sylow
136C2×#41—861no—
137C22×Q8D17212yes—
138#137—1724no—
139#137—1724no—
140C2×Q16—864no—
141C2×SD16—864no—
142C2×M16—861no—
143C4⋊Q8—22no—
144C4⋊Q8—431no—
145C4⋊Q8C4×D4312yes—
146C4⋊Q8C22⋊D4312yes—
147#127—431no—
148#124—861no—
149#121—862no—
150C4×D4—862no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—862no—
153C2×C42—862no—
154C2×C42C22×D4311yescharacteristic
155C4⋊C8—22no—
156C22⋊C8—22no—
157#97—22no—
158#97—861no—
159C4×C8—432no—
160#95—434no—
161#95—864no—
162#95—862no—
163#95—862no—
164#41—1722no—
165#41—1722no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—861no—
170C2×Q8(C2×D43)⋊C414yes—
171C2×D4—861no—
172C2×D4—862no—
173C2×D4—1721no—
174C22×C4—22no—
175C22×C4—864no—
176C22×C4—1721no—
177C22×C4C2×D17211yescharacteristic
178C22×C4D4×D4312yes—
179Q16—1728no—
180SD16—1728no—
181M16—864no—
182C2×C8—42no—
183C2×C8—862no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—862no—
187C4⋊C4—3442no—
188C22⋊C4—1722no—
189C42—3442no—
190C42C2×C4×D4311yescharacteristic
191C42#1062190012yes—
192C42C2×C22⋊D4311yescharacteristic
193C23—1721no—
194C23#1306329511yescharacteristic
195Q8—44no—
196Q8—82no—
197Q8—1722no—
198D4—3442no—
199D4—3442no—
200D4—1722no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—3441no—
207C2×C4—3441no—
208C2×C4—1728no—
209C2×C4—1724no—
210C2×C4—1721no—
211C2×C4C2×(C2×D43)⋊C411yescharacteristic
212C2×C4(C22⋊D43)⋊C412yes—
213C2×C4#1306329712yes—
214C2×C4#1306330911yescharacteristic
215C8—82no—
216C8—864no—
217C22—22no—
218C22—3441no—
219C22—3442no—
220C22#1458277611yescharacteristic
221C22#1458306912yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—3444no—
228C4—3441no—
229C2—22no—
230C2—6881no—
231C2#1913499111yescharacteristic
232C2#1913508712yes—
subgroupG/H
centerC22#14582776
commutatorC42×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1558C2
socleC22×C43#386
prime radical#14583449C22
cosocleC2×C42×C43C23
hypercenter#1558D43
hypocenterC43#9429
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #19137776C2semidirect
4. #19132461C2semidirect
5. C43×#1558C2semidirect
9. #14583278C4semidirect
10. #14583273C4semidirect
115. #1558D43semidirect
135. C43#9429semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#191450522
#191377762
#191331432
#191324612
C43×#15582
#942943