Finite Group Databasebeta

← #26374426

Subgroups — #26374426

Order 12032.

472 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 127 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#20256526C212yesmaximal, complemented
2#20245702C214yesmaximal, complemented
3#20245700C212yesmaximal, complemented
4#20245698C212yesmaximal, complemented
5C2×D1504C211yesmaximal, characteristic, complemented
6D32×D47C212yesmaximal, complemented
7#20245681C211yesmaximal, characteristic, complemented
8C47×#2262C211yesmaximal, characteristic, complemented
9#15689925—22no—
10#15689560—24no—
11#15689558—22no—
12#15689556C2214yescomplemented
13C2×D752C2212yescomplemented
14D16×D47C2214yescomplemented
15#15689539C2211yescharacteristic
16#15688960C2214yescomplemented
17D1504C2212yescomplemented
18D1504C2212yescomplemented
19#15688958C2214yescomplemented
20#15688957C2211yescharacteristic, complemented
21#15688956C2212yescomplemented
22C32×D47C2212yescomplemented
23C47×#508C2212yescomplemented
24C47×Q64C2211yescharacteristic, complemented
25C47×SD64C2212yescomplemented
26C47×D32C2211yescharacteristic, complemented
27C2×C32×C47C2211yescharacteristic, complemented
28#13094070—42no—
29#13094051—44no—
30#13094049—42no—
31#13094047—24no—
32C2×D376—22no—
33D8×D47—24no—
34#13094030D411yescharacteristic
35#13093967—24no—
36D752—22no—
37D752C2314yes—
38#13093965—24no—
39#13093964D411yescharacteristic
40#13093963D412yes—
41C16×D47C2312yes—
42C47⋊C32C2312yes—
43#135×C47—22no—
44Q32×C47C2312yes—
45SD32×C47—22no—
46D16×C47C2312yes—
47C2×C16×C47C2311yescharacteristic
48C32×C47C2311yescharacteristic
49C32×C47C2311yescharacteristic
50#41×C47—42no—
51#10625354—44no—
52D4×D47—44no—
53D4×D47—84no—
54#10625350—82no—
55#10625350D811yescharacteristic
56C2×D188—42no—
57C2×D188—82no—
58Q16×C47—22no—
59SD16×C47—42no—
60D8×C47—22no—
61C2×C8×C47C2×D411yescharacteristic
62#10625336—44no—
63#10625334—44no—
64#10625329D811yescharacteristic
65D376—24no—
66D376—42no—
67#10625324D812yes—
68C8×D47C2×D412yes—
69C16×C47C2×D411yescharacteristic
70C16×C47C2411yescharacteristic
71C47⋊C16C2×D412yes—
72C22×D47—82no—
73C22×D47—84no—
74Q8×C47—42no—
75D4×C47—42no—
76D4×C47—82no—
77C2×C4×C47—82no—
78C2×C4×C47C2×D811yescharacteristic
79C22⋊D47—84no—
80C22⋊D47D1612yes—
81D188—44no—
82D188—84no—
83D188—82no—
84D188D1611yescharacteristic
85C4×D47—84no—
86C4×D47C2×D812yes—
87Dic94D1611yescharacteristic
88C8×C47C2×D811yescharacteristic
89C8×C47C22×D411yescharacteristic
90C47⋊C8C2×D812yes—
91#33982—471nomaximal, Sylow
92C22×C47—82no—
93C22×C47C2×D1611yescharacteristic
94C2×D47—84no—
95C2×D47—324no—
96C2×D47—322no—
97C2×D47C2×D1612yes—
98C4×C47—82no—
99C4×C47C2×D1611yescharacteristic
100C4×C47C22×D811yescharacteristic
101Dic47C2×D1612yes—
102#3415—472no—
103#2263—472no—
104#2263—474no—
105#2262—472no—
106#2262D4711yescharacteristic, complemented
107C2×D32—472no—
108C2×D32—471no—
109#2258—471no—
110C2×C47—21no—
111C2×C47—162no—
112C2×C47C22×D1611yescharacteristic
113D47—22no—
114D47—164no—
115#576—942no—
116#509—944no—
117#509—942no—
118#508—474no—
119#508C2×D4712yescomplemented
120C2×D16—474no—
121C2×D16—472no—
122#504—471no—
123Q64C2×D4711yescharacteristic, complemented
124SD64—474no—
125SD64C2×D4712yescomplemented
126D32—474no—
127D32—472no—
128D32—472no—
129D32C2×D4711yescharacteristic, complemented
130M64—471no—
131M64—472no—
132C2×C32—472no—
133C2×C32C2×D4711yescharacteristic, complemented
134C47#3398211yescharacteristic, complemented, Sylow
135H(5,𝔽2)—1882no—
136AGL(1,ℤ/8)—1884no—
137AGL(1,ℤ/8)—1882no—
138#135—22no—
139#135—944no—
140C2×D8—944no—
141C2×D8—942no—
142#131—471no—
143Q32C22×D4712yes—
144SD32—22no—
145SD32—944no—
146D16—474no—
147D16—944no—
148D16—942no—
149D16C22×D4712yes—
150M32—471no—
151M32—472no—
152C2×C16—472no—
153C2×C16C22×D4711yescharacteristic
154C32—472no—
155C32C22×D4711yescharacteristic
156C32C22×D4711yescharacteristic
157#41—42no—
158#41—471no—
159#41—3762no—
160#41—1884no—
161C2×D4—3762no—
162C2×D4—3764no—
163C2×D4—1884no—
164C2×D4—1882no—
165Q16—22no—
166SD16—42no—
167SD16—1884no—
168D8—22no—
169D8—944no—
170D8—1884no—
171D8—1882no—
172M16—471no—
173M16—472no—
174C2×C8—472no—
175C2×C8D4×D4711yescharacteristic
176C16—472no—
177C16D4×D4711yescharacteristic
178C16C23×D4711yescharacteristic
179C23—3764no—
180C23—3762no—
181Q8—42no—
182Q8—471no—
183D4—42no—
184D4—82no—
185D4—472no—
186D4—471no—
187D4—3764no—
188D4—3764no—
189D4—3762no—
190D4—1884no—
191C2×C4—82no—
192C2×C4—472no—
193C2×C4—3764no—
194C2×C4D8×D4711yescharacteristic
195C8—472no—
196C8D8×D4711yescharacteristic
197C8C2×D4×D4711yescharacteristic
198C22—82no—
199C22—472no—
200C22—3764no—
201C22—15042no—
202C22—15044no—
203C22D16×D4711yescharacteristic
204C4—82no—
205C4—472no—
206C4D16×D4711yescharacteristic
207C4C2×D8×D4711yescharacteristic
208C2—21no—
209C2—162no—
210C2—942no—
211C2—7524no—
212C2C2×D16×D4711yescharacteristic
subgroupG/H
centerC2C2×D16×D47
commutator, cosocleC16×C47C24
FrattiniC16C23×D47
FittingC47×#2262C2
socleC2×C47C22×D16
hypercenter#2262D47
hypocenterC47#33982
LCSG ▷ C16×C47 ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #20256526C2semidirect
2. #20245702C2semidirect
3. #20245700C2semidirect
4. #20245698C2semidirect
5. C2×D1504C2semidirect
6. D32×D47C2semidirect
7. #20245681C2semidirect
8. C47×#2262C2semidirect
12. #15689556C22semidirect
13. C2×D752C22semidirect
14. D16×D47C22semidirect
16. #15688960C22semidirect
17. D1504C22semidirect
18. D1504C22semidirect
19. #15688958C22semidirect
20. #15688957C22semidirect
21. #15688956C22semidirect
22. C32×D47C22semidirect
23. C47×#508C22semidirect
24. C47×Q64C22semidirect
25. C47×SD64C22semidirect
26. C47×D32C22semidirect
27. C2×C32×C47C22semidirect
106. #2262D47semidirect
119. #508C2×D47semidirect
123. Q64C2×D47semidirect
125. SD64C2×D47semidirect
129. D32C2×D47semidirect
133. C2×C32C2×D47semidirect
134. C47#33982semidirect

Maximal subgroups

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

subgroupindex
#202565262
#202457022
#202457002
#202456982
C2×D15042
D32×D472
#202456812
C47×#22622
#3398247