Finite Group Databasebeta

← #25010010

Subgroups — #25010010

Order 11008.

384 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 67 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#19136671C212yesmaximal, complemented
2#19134259C211yesmaximal, characteristic, complemented
3#19132636C211yesmaximal, characteristic, complemented
4#19132571C212yesmaximal, complemented
5C43×#3291C211yesmaximal, characteristic, complemented
6#14583334—22no—
7#14583276C2211yescharacteristic, complemented
8#14583275—21no—
9#14583247—21no—
10#14582720C2212yescomplemented
11#14582711—22no—
12#14582620C2211yescharacteristic, complemented
13C43×#577—22no—
14C43×#492C2211yescharacteristic
15C43×#471—22no—
16C43×#456—21no—
17C43×#453—21no—
18C43×#443C2212yes—
19C2×D344—22no—
20C2×#10621882—42no—
21#13063322—21no—
22#13063268—42no—
23#13063266—42no—
24#13063265—22no—
25#13063245—42no—
26C4×C43⋊C8—21no—
272−1+4×C43—21no—
28H(5,𝔽2)×C43—21no—
29#137×C43—22no—
30#137×C43—42no—
31AGL(1,ℤ/8)×C43—22no—
32AGL(1,ℤ/8)×C43—42no—
33#135×C43—42no—
34#135×C43D412yes—
35C2×SD16×C43—22no—
36C2×SD16×C43—42no—
37#131×C43—22no—
38C4⋊Q8×C43D411yescharacteristic
39#127×C43D411yescharacteristic
40#108×C43D412yes—
41#104×C43—22no—
42#101×C43—21no—
43#100×C43—21no—
44C4×C8×C43C2311yescharacteristic
45#41×C43—22no—
46#41×C43—42no—
47#41×C43—42no—
48#41×C43—81no—
49C2×Q8×C43—21no—
50C2×Q8×C43—42no—
51C2×D4×C43—21no—
52C2×D4×C43—41no—
53C2×D4×C43—42no—
54C2×D4×C43—81no—
55C2×D172—41no—
56C2×D172—41no—
57C2×D172—41no—
58Q16×C43—42no—
59Q16×C43D812yescomplemented
60SD16×C43—22no—
61SD16×C43—42no—
62SD16×C43—42no—
63SD16×C43—42no—
64SD16×C43—42no—
65D8×C43—42no—
66D8×C43D812yescomplemented
67M16×C43—22no—
68M16×C43—42no—
69C2×C8×C43—42no—
70C2×C8×C43C2×D412yes—
71C4⋊C4×C43—41no—
72C42×C43C2×D411yescharacteristic
73#10621882—84no—
74C2×C43⋊C8—22no—
75D344—42no—
76D344—42no—
77C43⋊C16—42no—
78C23×C43—41no—
79C23×C43—81no—
80C22×D43—81no—
81Q8×C43—22no—
82Q8×C43—42no—
83Q8×C43—81no—
84D4×C43—22no—
85D4×C43—42no—
86D4×C43—42no—
87D4×C43—82no—
88D4×C43—82no—
89D4×C43—81no—
90C2×C4×C43—21no—
91C2×C4×C43—42no—
92C2×C4×C43—41no—
93C2×C4×C43—81no—
94C2×C4×C43—82no—
95C2×C4×C43#12011yescharacteristic
96D172—82no—
97D172—82no—
98D172—82no—
99C8×C43—42no—
100C8×C43C2×D812yes—
101C8×C43AGL(1,ℤ/8)12yes—
102C43⋊C8—42no—
103#12314—431nomaximal, Sylow
104C22×C43—42no—
105C22×C43—81no—
106C22×C43—161no—
107C22×C43—321no—
108C22×C43#45311yescharacteristic
109C2×D43—161no—
110C2×D43—161no—
111C2×D43—161no—
112C4×C43—22no—
113C4×C43—42no—
114C4×C43—81no—
115C4×C43#44712yes—
116#3291D4311yescharacteristic, complemented
117#2213—432no—
118#1721—431no—
119#1359—431no—
120#1336—432no—
121C2×C43—21no—
122C2×C43—82no—
123C2×C43—161no—
124C2×C43#161911yescharacteristic
125D43—321no—
126#577—22no—
127#509—862no—
128#493—431no—
129#493—861no—
130#492C2×D4311yescharacteristic
131#486—861no—
132#471—22no—
133#456—21no—
134#453—21no—
135#443C2×D4312yes—
136#364—432no—
137#361—862no—
138#322—431no—
139C43#1231411yescharacteristic, complemented, Sylow
1402−1+4—21no—
141H(5,𝔽2)—21no—
142#137—22no—
143#137—42no—
144AGL(1,ℤ/8)—22no—
145AGL(1,ℤ/8)—42no—
146#135—42no—
147#135C22⋊D4312yes—
148C2×SD16—22no—
149C2×SD16—42no—
150C2×D8—862no—
151C2×D8—1722no—
152#131—22no—
153C4⋊Q8C22⋊D4311yescharacteristic
154#127—861no—
155#127C22⋊D4311yescharacteristic
156SD32—1722no—
157D16—1722no—
158M32—862no—
159#108C22⋊D4312yes—
160#104—22no—
161#102—1722no—
162#101—21no—
163#100—21no—
164C4×C8—861no—
165C4×C8C22×D4311yescharacteristic
166#41—22no—
167#41—42no—
168#41—42no—
169#41—81no—
170C2×Q8—21no—
171C2×Q8—42no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—81no—
176C2×D4—1721no—
177C2×D4—1721no—
178C2×D4—1721no—
179Q16—42no—
180Q16#1062188212yescomplemented
181SD16—22no—
182SD16—42no—
183SD16—42no—
184SD16—42no—
185SD16—42no—
186D8—42no—
187D8—3444no—
188D8—1722no—
189D8—1722no—
190D8#1062188212yescomplemented
191M16—22no—
192M16—42no—
193C2×C8—42no—
194C2×C8—862no—
195C2×C8C2×C22⋊D4312yes—
196C4⋊C4—41no—
197C42C2×C22⋊D4311yescharacteristic
198C16—1722no—
199C23—41no—
200C23—81no—
201C23—3441no—
202Q8—22no—
203Q8—42no—
204Q8—81no—
205D4—22no—
206D4—42no—
207D4—42no—
208D4—82no—
209D4—82no—
210D4—81no—
211D4—3442no—
212D4—3442no—
213D4—3442no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4#1306329411yescharacteristic
220C8—42no—
221C8—1722no—
222C8C2×#1062188212yes—
223C8#1306335112yes—
224C22—42no—
225C22—81no—
226C22—161no—
227C22—321no—
228C22—6881no—
229C22—6881no—
230C22—6881no—
231C22#1458306711yescharacteristic
232C4—22no—
233C4—42no—
234C4—81no—
235C4#1458303312yes—
236C2—21no—
237C2—82no—
238C2—161no—
239C2—13761no—
240C2#1913364911yescharacteristic
subgroupG/H
centerC2#19133649
commutator, cosocleC4×C8×C43C23
FrattiniC4×C8C22×D43
FittingC43×#3291C2
socleC2×C43#1619
hypercenter#3291D43
hypocenterC43#12314
LCSG ▷ C4×C8×C43 ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3291

Complements

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

NH ≅ G/Ntype
1. #19136671C2semidirect
2. #19134259C2semidirect
3. #19132636C2semidirect
4. #19132571C2semidirect
5. C43×#3291C2semidirect
7. #14583276C22semidirect
10. #14582720C22semidirect
12. #14582620C22semidirect
59. Q16×C43D8semidirect
66. D8×C43D8semidirect
116. #3291D43semidirect
139. C43#12314semidirect
180. Q16#10621882semidirect
190. D8#10621882semidirect

Maximal subgroups

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

subgroupindex
#191366712
#191342592
#191326362
#191325712
C43×#32912
#1231443