Finite Group Databasebeta

← #28374825

Subgroups — #28374825

Order 13952.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#20295832C214yesmaximal, complemented
2#20295286C214yesmaximal, complemented
3#20295208C212yesmaximal, complemented
4#20295193C212yesmaximal, complemented
5#20295144C212yesmaximal, complemented
6#492×C109C211yesmaximal, characteristic, complemented
7#16812747C2212yes—
8#16812744C2212yescomplemented
9#16812730C2214yescomplemented
10#16812728—24no—
11#16812726—24no—
12#16812725C2214yes—
13#16812721C2212yes—
14#16812720—24no—
15SD16×D109—24no—
16#16812705C2214yescomplemented
17#16812635C2212yescomplemented
18#16812627C2214yescomplemented
19#16812618C2212yescomplemented
20#16812616C2212yescomplemented
21C2×SD16×C109C2214yescomplemented
22C4⋊Q8×C109C2211yescharacteristic, complemented
23#127×C109C2211yescharacteristic
24C4×C8×C109C2211yescharacteristic, complemented
25#13182030—24no—
26#13182029—42no—
27#13182029C2314yes—
28#13182028—24no—
29Q8×D109—24no—
30C2×C22⋊D109—42no—
31D4×D109—24no—
32Dic436D412yescomplemented
33#13182019—24no—
34#13182018D414yescomplemented
35#13182017—24no—
36#13182016—24no—
37D872D412yescomplemented
38#13182014—24no—
39#13182013C2314yes—
40#13182012—24no—
41C8×D109—24no—
42C2×Q8×C109C2312yes—
43C2×D4×C109—41no—
44C2×D4×C109C2312yes—
45SD16×C109—24no—
46SD16×C109—24no—
47C2×C8×C109C2312yes—
48C4⋊C4×C109—41no—
49C42×C109C2311yescharacteristic
50C23×C109—22no—
51C22×D109—42no—
52Q8×C109—24no—
53D4×C109—24no—
54D4×C109—82no—
55C2×C4×C109—21no—
56C2×C4×C109—22no—
57C2×C4×C109C2411yescharacteristic
58C22⋊D109—24no—
59C22⋊D109—44no—
60C2×Dic109—42no—
61D436—42no—
62D436C2×D414yes—
63C4×D109—24no—
64C4×D109—44no—
65Dic218—42no—
66Dic218C2×D414yes—
67C8×C109C2×D412yes—
68C8×C109C2×D412yes—
69#11724784—24no—
70C22×C109—42no—
71C22×C109—82no—
72C22×C109H(5,𝔽2)11yescharacteristic
73C2×D109—24no—
74C2×D109—162no—
75C4×C109—22no—
76C4×C109—42no—
77C4×C109C22×D412yes—
78Dic109—24no—
79C2×C109—21no—
80C2×C109—82no—
81C2×C109D4211yescharacteristic
82D109—44no—
83#3291—1091nomaximal, Sylow
84C109#329111yescharacteristic, complemented, Sylow
85#577—1094no—
86#492D10911yescharacteristic, complemented
87#471—1094no—
88#456—1092no—
89#453—1092no—
90#443—1092no—
912−1+4—1092no—
92H(5,𝔽2)—1092no—
93#137—1094no—
94#137—2184no—
95AGL(1,ℤ/8)—1094no—
96AGL(1,ℤ/8)—2184no—
97#135—1092no—
98#135—2184no—
99C2×SD16—2184no—
100C2×SD16C2×D10914yescomplemented
101#131—1094no—
102C4⋊Q8C2×D10911yescharacteristic, complemented
103#127C2×D10911yescharacteristic
104#108—1092no—
105#104—1094no—
106#101—1092no—
107#100—1092no—
108C4×C8C2×D10911yescharacteristic, complemented
109#41—1094no—
110#41—4362no—
111#41—2184no—
112#41—2184no—
113C2×Q8—2184no—
114C2×Q8C22×D10912yes—
115C2×D4—41no—
116C2×D4—4362no—
117C2×D4—2184no—
118C2×D4C22×D10912yes—
119Q16—1092no—
120Q16—2184no—
121SD16—24no—
122SD16—24no—
123SD16—1094no—
124SD16—2184no—
125SD16—2184no—
126D8—1092no—
127D8—2184no—
128M16—1094no—
129M16—2184no—
130C2×C8—2184no—
131C2×C8C22×D10912yes—
132C4⋊C4—41no—
133C42C22×D10911yescharacteristic
134C23—22no—
135C23—4362no—
136Q8—24no—
137Q8—1094no—
138Q8—4362no—
139D4—24no—
140D4—82no—
141D4—1094no—
142D4—4362no—
143D4—4364no—
144D4—2184no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—4364no—
148C2×C4—4362no—
149C2×C4—2184no—
150C2×C4C23×D10911yescharacteristic
151C8—2184no—
152C8D4×D10912yes—
153C8D4×D10912yes—
154C22—42no—
155C22—82no—
156C22—17442no—
157C22—2184no—
158C22#1681274411yescharacteristic
159C4—22no—
160C4—42no—
161C4—2184no—
162C4C2×D4×D10912yes—
163C2—21no—
164C2—82no—
165C2—4364no—
166C2#2029566111yescharacteristic
subgroupG/H
centerC2#20295661
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#492×C109C2
socleC2×C109D42
hypercenter#492D109
hypocenterC109#3291
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #20295832C2semidirect
2. #20295286C2semidirect
3. #20295208C2semidirect
4. #20295193C2semidirect
5. #20295144C2semidirect
6. #492×C109C2semidirect
8. #16812744C22semidirect
9. #16812730C22semidirect
16. #16812705C22semidirect
17. #16812635C22semidirect
18. #16812627C22semidirect
19. #16812618C22semidirect
20. #16812616C22semidirect
21. C2×SD16×C109C22semidirect
22. C4⋊Q8×C109C22semidirect
24. C4×C8×C109C22semidirect
32. Dic436D4semidirect
34. #13182018D4semidirect
37. D872D4semidirect
84. C109#3291semidirect
86. #492D109semidirect
100. C2×SD16C2×D109semidirect
102. C4⋊Q8C2×D109semidirect
108. C4×C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958322
#202952862
#202952082
#202951932
#202951442
#492×C1092
#3291109