Finite Group Databasebeta

← #28376170

Subgroups — #28376170

Order 13952.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#20295828C212yesmaximal, complemented
2#20295463C214yesmaximal, complemented
3#20295461C212yesmaximal, complemented
4#20295459C212yesmaximal, complemented
5C2×D1744C211yesmaximal, characteristic, complemented
6D16×D109C212yesmaximal, complemented
7#20295442C211yesmaximal, characteristic, complemented
8#508×C109C211yesmaximal, characteristic, complemented
9#16812745—22no—
10#16812726—24no—
11#16812724—22no—
12#16812722C2214yescomplemented
13C2×D872C2212yescomplemented
14D8×D109C2214yescomplemented
15#16812705C2211yescharacteristic
16#16812642C2214yescomplemented
17D1744C2212yescomplemented
18D1744C2212yescomplemented
19#16812640C2214yescomplemented
20#16812639C2211yescharacteristic, complemented
21#16812638C2212yescomplemented
22C16×D109C2212yescomplemented
23#135×C109C2212yescomplemented
24Q32×C109C2211yescharacteristic, complemented
25SD32×C109C2212yescomplemented
26D16×C109C2211yescharacteristic, complemented
27C2×C16×C109C2211yescharacteristic, complemented
28#13182030—24no—
29#13182029—42no—
30#13182029D411yescharacteristic
31C2×D436—22no—
32C2×D436—42no—
33D4×D109—24no—
34D4×D109—44no—
35#13182016—24no—
36D872—22no—
37D872C2314yes—
38#13182014—24no—
39#13182013D411yescharacteristic
40#13182012D412yes—
41C8×D109C2312yes—
42#13182004C2312yes—
43#41×C109—22no—
44Q16×C109C2312yes—
45SD16×C109—22no—
46D8×C109C2312yes—
47C2×C8×C109C2311yescharacteristic
48C16×C109C2311yescharacteristic
49C16×C109C2311yescharacteristic
50C22×D109—42no—
51C22×D109—44no—
52Q8×C109—22no—
53D4×C109—22no—
54D4×C109—42no—
55C2×C4×C109—42no—
56C2×C4×C109C2×D411yescharacteristic
57C22⋊D109—44no—
58C22⋊D109D812yes—
59D436—24no—
60D436—44no—
61D436—42no—
62D436D811yescharacteristic
63C4×D109—44no—
64C4×D109C2×D412yes—
65Dic218D811yescharacteristic
66C8×C109C2×D411yescharacteristic
67C8×C109C2411yescharacteristic
68#11724784C2×D412yes—
69C22×C109—42no—
70C22×C109C2×D811yescharacteristic
71C2×D109—44no—
72C2×D109—164no—
73C2×D109—162no—
74C2×D109C2×D812yes—
75C4×C109—42no—
76C4×C109C2×D811yescharacteristic
77C4×C109C22×D411yescharacteristic
78Dic109C2×D812yes—
79C2×C109—21no—
80C2×C109—82no—
81C2×C109C22×D811yescharacteristic
82D109—22no—
83D109—84no—
84#3415—1091nomaximal, Sylow
85C109#341511yescharacteristic, complemented, Sylow
86#576—1092no—
87#509—1092no—
88#509—1094no—
89#508—1092no—
90#508D10911yescharacteristic, complemented
91C2×D16—1092no—
92C2×D16—1091no—
93#504—1091no—
94H(5,𝔽2)—2182no—
95AGL(1,ℤ/8)—2184no—
96AGL(1,ℤ/8)—2182no—
97#135—1094no—
98#135C2×D10912yescomplemented
99C2×D8—1094no—
100C2×D8—1092no—
101#131—1091no—
102Q32C2×D10911yescharacteristic, complemented
103SD32—1094no—
104SD32C2×D10912yescomplemented
105D16—1094no—
106D16—1092no—
107D16—1092no—
108D16C2×D10911yescharacteristic, complemented
109M32—1091no—
110M32—1092no—
111C2×C16—1092no—
112C2×C16C2×D10911yescharacteristic, complemented
113#41—22no—
114#41—1091no—
115#41—4362no—
116#41—2184no—
117C2×D4—4362no—
118C2×D4—4364no—
119C2×D4—2184no—
120C2×D4—2182no—
121Q16C22×D10912yes—
122SD16—22no—
123SD16—2184no—
124D8—1094no—
125D8—2184no—
126D8—2182no—
127D8C22×D10912yes—
128M16—1091no—
129M16—1092no—
130C2×C8—1092no—
131C2×C8C22×D10911yescharacteristic
132C16—1092no—
133C16C22×D10911yescharacteristic
134C16C22×D10911yescharacteristic
135C23—4364no—
136C23—4362no—
137Q8—22no—
138Q8—1091no—
139D4—22no—
140D4—42no—
141D4—1092no—
142D4—1091no—
143D4—4364no—
144D4—4364no—
145D4—4362no—
146D4—2184no—
147C2×C4—42no—
148C2×C4—1092no—
149C2×C4—4364no—
150C2×C4D4×D10911yescharacteristic
151C8—1092no—
152C8D4×D10911yescharacteristic
153C8C23×D10911yescharacteristic
154C22—42no—
155C22—1092no—
156C22—4364no—
157C22—17442no—
158C22—17444no—
159C22D8×D10911yescharacteristic
160C4—42no—
161C4—1092no—
162C4D8×D10911yescharacteristic
163C4C2×D4×D10911yescharacteristic
164C2—21no—
165C2—82no—
166C2—8724no—
167C2—2182no—
168C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC2C2×D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#508×C109C2
socleC2×C109C22×D8
hypercenter#508D109
hypocenterC109#3415
LCSG ▷ C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #20295828C2semidirect
2. #20295463C2semidirect
3. #20295461C2semidirect
4. #20295459C2semidirect
5. C2×D1744C2semidirect
6. D16×D109C2semidirect
7. #20295442C2semidirect
8. #508×C109C2semidirect
12. #16812722C22semidirect
13. C2×D872C22semidirect
14. D8×D109C22semidirect
16. #16812642C22semidirect
17. D1744C22semidirect
18. D1744C22semidirect
19. #16812640C22semidirect
20. #16812639C22semidirect
21. #16812638C22semidirect
22. C16×D109C22semidirect
23. #135×C109C22semidirect
24. Q32×C109C22semidirect
25. SD32×C109C22semidirect
26. D16×C109C22semidirect
27. C2×C16×C109C22semidirect
85. C109#3415semidirect
90. #508D109semidirect
98. #135C2×D109semidirect
102. Q32C2×D109semidirect
104. SD32C2×D109semidirect
108. D16C2×D109semidirect
112. C2×C16C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958282
#202954632
#202954612
#202954592
C2×D17442
D16×D1092
#202954422
#508×C1092
#3415109