Finite Group Databasebeta

← #27542240

Subgroups — #27542240

Order 13568.

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#20287853C212yesmaximal, complemented
2#20277029C214yesmaximal, complemented
3#20277027C212yesmaximal, complemented
4#20277025C212yesmaximal, complemented
5C2×D1696C211yesmaximal, characteristic, complemented
6D32×D53C212yesmaximal, complemented
7#20277008C211yesmaximal, characteristic, complemented
8C53×#2262C211yesmaximal, characteristic, complemented
9#16811425—22no—
10#16811060—24no—
11#16811058—22no—
12#16811056C2214yescomplemented
13C2×D848C2212yescomplemented
14D16×D53C2214yescomplemented
15#16811039C2211yescharacteristic
16#16810460C2214yescomplemented
17D1696C2212yescomplemented
18D1696C2212yescomplemented
19#16810458C2214yescomplemented
20#16810457C2211yescharacteristic, complemented
21#16810456C2212yescomplemented
22C32×D53C2212yescomplemented
23C53×#508C2212yescomplemented
24C53×Q64C2211yescharacteristic, complemented
25C53×SD64C2212yescomplemented
26C53×D32C2211yescharacteristic, complemented
27C2×C32×C53C2211yescharacteristic, complemented
28#13133263—42no—
29#13133244—44no—
30#13133242—42no—
31#13133240—24no—
32C2×D424—22no—
33D8×D53—24no—
34#13133223D411yescharacteristic
35#13133160—24no—
36D848—22no—
37D848C2314yes—
38#13133158—24no—
39#13133157D411yescharacteristic
40#13133156D412yes—
41C16×D53C2312yes—
42#13133123C2312yes—
43#135×C53—22no—
44Q32×C53C2312yes—
45SD32×C53—22no—
46D16×C53C2312yes—
47C2×C16×C53C2311yescharacteristic
48C32×C53C2311yescharacteristic
49C32×C53C2311yescharacteristic
50#41×C53—42no—
51#11719949—44no—
52D4×D53—44no—
53D4×D53—84no—
54#11719945—82no—
55#11719945D811yescharacteristic
56C2×D212—42no—
57C2×D212—82no—
58Q16×C53—22no—
59SD16×C53—42no—
60D8×C53—22no—
61C2×C8×C53C2×D411yescharacteristic
62#11719924—44no—
63#11719922—44no—
64#11719917D811yescharacteristic
65D424—24no—
66D424—42no—
67#11719912D812yes—
68C8×D53C2×D412yes—
69C16×C53C2×D411yescharacteristic
70C16×C53C2411yescharacteristic
71#11719908C2×D412yes—
72C22×D53—82no—
73C22×D53—84no—
74Q8×C53—42no—
75D4×C53—42no—
76D4×C53—82no—
77C2×C4×C53—82no—
78C2×C4×C53C2×D811yescharacteristic
79C22⋊D53—84no—
80C22⋊D53D1612yes—
81D212—44no—
82D212—84no—
83D212—82no—
84D212D1611yescharacteristic
85C4×D53—84no—
86C4×D53C2×D812yes—
87Dic106D1611yescharacteristic
88C8×C53C2×D811yescharacteristic
89C8×C53C22×D411yescharacteristic
90#88322C2×D812yes—
91#33982—531nomaximal, Sylow
92C22×C53—82no—
93C22×C53C2×D1611yescharacteristic
94C2×D53—84no—
95C2×D53—324no—
96C2×D53—322no—
97C2×D53C2×D1612yes—
98C4×C53—82no—
99C4×C53C2×D1611yescharacteristic
100C4×C53C22×D811yescharacteristic
101Dic53C2×D1612yes—
102#3415—532no—
103#2263—532no—
104#2263—534no—
105#2262—532no—
106#2262D5311yescharacteristic, complemented
107C2×D32—532no—
108C2×D32—531no—
109#2258—531no—
110C2×C53—21no—
111C2×C53—162no—
112C2×C53C22×D1611yescharacteristic
113D53—22no—
114D53—164no—
115#576—1062no—
116#509—1064no—
117#509—1062no—
118#508—534no—
119#508C2×D5312yescomplemented
120C2×D16—534no—
121C2×D16—532no—
122#504—531no—
123Q64C2×D5311yescharacteristic, complemented
124SD64—534no—
125SD64C2×D5312yescomplemented
126D32—534no—
127D32—532no—
128D32—532no—
129D32C2×D5311yescharacteristic, complemented
130M64—531no—
131M64—532no—
132C2×C32—532no—
133C2×C32C2×D5311yescharacteristic, complemented
134C53#3398211yescharacteristic, complemented, Sylow
135H(5,𝔽2)—2122no—
136AGL(1,ℤ/8)—2124no—
137AGL(1,ℤ/8)—2122no—
138#135—22no—
139#135—1064no—
140C2×D8—1064no—
141C2×D8—1062no—
142#131—531no—
143Q32C22×D5312yes—
144SD32—22no—
145SD32—1064no—
146D16—534no—
147D16—1064no—
148D16—1062no—
149D16C22×D5312yes—
150M32—531no—
151M32—532no—
152C2×C16—532no—
153C2×C16C22×D5311yescharacteristic
154C32—532no—
155C32C22×D5311yescharacteristic
156C32C22×D5311yescharacteristic
157#41—42no—
158#41—531no—
159#41—4242no—
160#41—2124no—
161C2×D4—4242no—
162C2×D4—4244no—
163C2×D4—2124no—
164C2×D4—2122no—
165Q16—22no—
166SD16—42no—
167SD16—2124no—
168D8—22no—
169D8—1064no—
170D8—2124no—
171D8—2122no—
172M16—531no—
173M16—532no—
174C2×C8—532no—
175C2×C8D4×D5311yescharacteristic
176C16—532no—
177C16D4×D5311yescharacteristic
178C16C23×D5311yescharacteristic
179C23—4244no—
180C23—4242no—
181Q8—42no—
182Q8—531no—
183D4—42no—
184D4—82no—
185D4—532no—
186D4—531no—
187D4—4244no—
188D4—4244no—
189D4—4242no—
190D4—2124no—
191C2×C4—82no—
192C2×C4—532no—
193C2×C4—4244no—
194C2×C4D8×D5311yescharacteristic
195C8—532no—
196C8D8×D5311yescharacteristic
197C8C2×D4×D5311yescharacteristic
198C22—82no—
199C22—532no—
200C22—16962no—
201C22—16964no—
202C22—4244no—
203C22D16×D5311yescharacteristic
204C4—82no—
205C4—532no—
206C4D16×D5311yescharacteristic
207C4C2×D8×D5311yescharacteristic
208C2—21no—
209C2—162no—
210C2—8484no—
211C2—1062no—
212C2C2×D16×D5311yescharacteristic
subgroupG/H
centerC2C2×D16×D53
commutator, cosocleC16×C53C24
FrattiniC16C23×D53
FittingC53×#2262C2
socleC2×C53C22×D16
hypercenter#2262D53
hypocenterC53#33982
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #20287853C2semidirect
2. #20277029C2semidirect
3. #20277027C2semidirect
4. #20277025C2semidirect
5. C2×D1696C2semidirect
6. D32×D53C2semidirect
7. #20277008C2semidirect
8. C53×#2262C2semidirect
12. #16811056C22semidirect
13. C2×D848C22semidirect
14. D16×D53C22semidirect
16. #16810460C22semidirect
17. D1696C22semidirect
18. D1696C22semidirect
19. #16810458C22semidirect
20. #16810457C22semidirect
21. #16810456C22semidirect
22. C32×D53C22semidirect
23. C53×#508C22semidirect
24. C53×Q64C22semidirect
25. C53×SD64C22semidirect
26. C53×D32C22semidirect
27. C2×C32×C53C22semidirect
106. #2262D53semidirect
119. #508C2×D53semidirect
123. Q64C2×D53semidirect
125. SD64C2×D53semidirect
129. D32C2×D53semidirect
133. C2×C32C2×D53semidirect
134. C53#33982semidirect

Maximal subgroups

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

subgroupindex
#202878532
#202770292
#202770272
#202770252
C2×D16962
D32×D532
#202770082
C53×#22622
#3398253