Finite Group Databasebeta

← #46845274

Subgroups — #46845274

Order 34688.

458 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 217 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#31774664C212yesmaximal, complemented
2#31774019C214yesmaximal, complemented
3#31774014C214yesmaximal, complemented
4#31773946C211yesmaximal, characteristic, complemented
5C2×#22576087C212yesmaximal, complemented
6M16×Dic271C211yesmaximal, characteristic, complemented
7#575×C271C211yesmaximal, characteristic
8#22576167—24no—
9#22576167C2218yescomplemented
10C2×#16864638—24no—
11C2×#16864638C2212yes—
12C2×#16864638C2211yescharacteristic
13C2×C4×Dic271C2211yescharacteristic
14#22576096C2214yescomplemented
15#22576087C2214yescomplemented
16#22576087C2214yescomplemented
17#22576070C2212yescomplemented
18C8×Dic271C2212yescomplemented
19C2×#41×C271C2212yes—
20#137×C271C414yescomplemented
21AGL(1,ℤ/8)×C271C414yescomplemented
22#135×C271C2214yes—
23C2×M16×C271C2211yescharacteristic
24C22×Dic271—21no—
25#16864638—28no—
26#16864638—22no—
27#16864638D414yes—
28#16864638C2314yes—
29C2×C271⋊C8—28no—
30C2×C271⋊C8—22no—
31C2×C271⋊C8C2312yes—
32C4×Dic271—21no—
33C4×Dic271C2311yescharacteristic
34C4×Dic271C2311yescharacteristic
35#41×C271—22no—
36#41×C271—22no—
37#41×C271C2×C414yes—
38#41×C271C2314yes—
39C2×Q8×C271C2×C412yes—
40C2×D4×C271—22no—
41C2×D4×C271C2×C412yes—
42C22×C4×C271—22no—
43C22×C4×C271C2311yescharacteristic
44Q16×C271C2×C414yescomplemented
45SD16×C271C2×C418yescomplemented
46D8×C271C2×C414yescomplemented
47M16×C271C2×C414yes—
48C2×C8×C271C2312yes—
49C2×Dic271—21no—
50C2×Dic271—21no—
51C2×Dic271—42no—
52C271⋊C8—24no—
53C271⋊C8C2×D414yes—
54C23×C271—22no—
55C23×C271#4111yescharacteristic
56Q8×C271—22no—
57Q8×C271C22×C414yes—
58D4×C271—24no—
59D4×C271—24no—
60D4×C271—22no—
61D4×C271C22×C414yes—
62C2×C4×C271—22no—
63C2×C4×C271—24no—
64C2×C4×C271—24no—
65C2×C4×C271C22×C412yes—
66C2×C4×C271C2×D412yes—
67C2×C4×C271#4111yescharacteristic
68C2×C4×C271C2411yescharacteristic
69C8×C271C22×C414yes—
70C22×C271—24no—
71C22×C271—41no—
72C22×C271—82no—
73C22×C271C4×D412yes—
74C22×C271C2×#4111yescharacteristic
75C4×C271—24no—
76C4×C271C4×D412yes—
77C4×C271C23×C411yescharacteristic
78C4×C271C22×D411yescharacteristic
79Dic271—22no—
80Dic271—41no—
81C2×C271—21no—
82C2×C271—22no—
83C2×C271—44no—
84C2×C271C2×C4×D411yescharacteristic
85C271#295611yescharacteristic, complemented, Sylow
86#2956—2711nomaximal, Sylow
87#575D27111yescharacteristic
88#568—2712no—
89#444—2714no—
90#443—2714no—
91#430—2711no—
92C2×#104—2712no—
93C4×M16—2711no—
94C2×#41C2×D27112yes—
95#137Dic27114yescomplemented
96AGL(1,ℤ/8)Dic27114yescomplemented
97#135C2×D27114yes—
98#131—2718no—
99#131—5424no—
100C2×M16—2711no—
101C2×M16—2712no—
102C2×M16—5424no—
103C2×M16C2×D27111yescharacteristic
104C2×C42—2711no—
105#108—2714no—
106#104—2714no—
107#104—2714no—
108#97—2712no—
109C4×C8—2712no—
110#41—22no—
111#41—22no—
112#41C2×Dic27114yes—
113#41C22×D27114yes—
114C2×Q8C2×Dic27112yes—
115C2×D4—22no—
116C2×D4C2×Dic27112yes—
117C22×C4—22no—
118C22×C4—5421no—
119C22×C4C22×D27111yescharacteristic
120Q16C2×Dic27114yescomplemented
121SD16C2×Dic27118yescomplemented
122D8C2×Dic27114yescomplemented
123M16—2714no—
124M16—2714no—
125M16—5428no—
126M16—5422no—
127M16C2×Dic27114yes—
128C2×C8—2712no—
129C2×C8—5428no—
130C2×C8—5422no—
131C2×C8C22×D27112yes—
132C42—2711no—
133C42—2711no—
134C42—5421no—
135C23—22no—
136C23#1686465311yescharacteristic
137Q8—22no—
138Q8C22×Dic27114yes—
139D4—24no—
140D4—24no—
141D4—22no—
142D4C22×Dic27114yes—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—5421no—
147C2×C4—5421no—
148C2×C4—10842no—
149C2×C4C22×Dic27112yes—
150C2×C4D4×D27112yes—
151C2×C4#1686465311yescharacteristic
152C2×C4C23×D27111yescharacteristic
153C8—2714no—
154C8—5424no—
155C8C22×Dic27114yes—
156C22—24no—
157C22—41no—
158C22—82no—
159C22D4×Dic27112yes—
160C22C2×#1686465311yescharacteristic
161C4—24no—
162C4—5422no—
163C4—10841no—
164C4D4×Dic27112yes—
165C4C23×Dic27111yescharacteristic
166C4C2×D4×D27111yescharacteristic
167C2—21no—
168C2—22no—
169C2—44no—
170C2C2×D4×Dic27111yescharacteristic
subgroupG/H
centerC4C2×D4×D271
commutatorC4×C271C23×C4
FrattiniC2×C4C23×D271
Fitting, prime radical#575×C271C2
socleC2×C271C2×C4×D4
cosocleC2×C4×C271C24
hypercenter#575D271
hypocenterC271#2956
LCSG ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #31774664C2semidirect
2. #31774019C2semidirect
3. #31774014C2semidirect
4. #31773946C2semidirect
5. C2×#22576087C2semidirect
6. M16×Dic271C2semidirect
9. #22576167C22semidirect
14. #22576096C22semidirect
15. #22576087C22semidirect
16. #22576087C22semidirect
17. #22576070C22semidirect
18. C8×Dic271C22semidirect
20. #137×C271C4semidirect
21. AGL(1,ℤ/8)×C271C4semidirect
44. Q16×C271C2×C4semidirect
45. SD16×C271C2×C4semidirect
46. D8×C271C2×C4semidirect
85. C271#2956semidirect
95. #137Dic271semidirect
96. AGL(1,ℤ/8)Dic271semidirect
120. Q16C2×Dic271semidirect
121. SD16C2×Dic271semidirect
122. D8C2×Dic271semidirect

Maximal subgroups

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

subgroupindex
#317746642
#317740192
#317740142
#317739462
C2×#225760872
M16×Dic2712
#575×C2712
#2956271