Finite Group Databasebeta

← #20295835

Subgroups — #20295835

Order 6976.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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#16812747C211yesmaximal, characteristic, complemented
2#16812745C211yesmaximal, characteristic, complemented
3#16812731C212yesmaximal, complemented
4#16812727C211yesmaximal, characteristic, complemented
5#16812726C212yesmaximal, complemented
6#16812722C212yesmaximal, complemented
7#16812719C211yesmaximal, characteristic, complemented
8C2×#13182016C211yesmaximal, characteristic, complemented
9SD16×D109C212yesmaximal, complemented
10#16812707C211yesmaximal, characteristic, complemented
11#137×C109C211yesmaximal, characteristic, complemented
12#13182030—21no—
13#13182030C2212yes—
14#13182030C2212yescomplemented
15#13182029—22no—
16#13182029—21no—
17#13182029C2211yescharacteristic
18#13182029C2211yescharacteristic, complemented
19Q8×D109—21no—
20Q8×D109C2212yescomplemented
21C2×D436—21no—
22C2×D436C2211yescharacteristic
23D4×D109—22no—
24D4×D109C2212yes—
25#13182019C2211yescharacteristic, complemented
26#13182018C2212yescomplemented
27#13182017C2211yescharacteristic, complemented
28#13182016C2212yescomplemented
29#13182016C2212yescomplemented
30#13182016C2211yescharacteristic, complemented
31D872C2212yescomplemented
32#13182014C2211yescharacteristic, complemented
33#13182012C2212yescomplemented
34C2×#11724784C2211yescharacteristic, complemented
35C8×D109C2212yescomplemented
36#41×C109C2211yescharacteristic
37C2×Q8×C109C2211yescharacteristic, complemented
38Q16×C109C2212yescomplemented
39SD16×C109C2212yescomplemented
40M16×C109C2211yescharacteristic, complemented
41C22×D109—21no—
42C22×D109—22no—
43Q8×C109—21no—
44Q8×C109C2312yes—
45Q8×C109C2311yescharacteristic
46D4×C109—21no—
47D4×C109C2311yescharacteristic
48C2×C4×C109—21no—
49C2×C4×C109—21no—
50C2×C4×C109C2311yescharacteristic
51C22⋊D109—21no—
52C22⋊D109—22no—
53C22⋊D109D412yes—
54D436—22no—
55D436—22no—
56D436—21no—
57D436D411yescharacteristic
58D436C2311yescharacteristic
59D436C2312yes—
60C4×D109—22no—
61C4×D109—21no—
62C4×D109—22no—
63C4×D109—22no—
64C4×D109C2312yes—
65Dic218—22no—
66Dic218D411yescharacteristic
67Dic218C2311yescharacteristic
68C8×C109C2312yes—
69#11724784C2311yescharacteristic
70#11724784C2311yescharacteristic
71C22×C109—21no—
72C22×C109C2×D411yescharacteristic
73C2×D109—21no—
74C2×D109—22no—
75C2×D109—82no—
76C2×D109—81no—
77C2×D109C2×D412yes—
78C4×C109—22no—
79C4×C109—21no—
80C4×C109C2×D411yescharacteristic
81C4×C109C2411yescharacteristic
82Dic109—21no—
83Dic109C2×D412yes—
84C2×C109—21no—
85C2×C109—41no—
86C2×C109C22×D411yescharacteristic
87D109—22no—
88D109—41no—
89D109—42no—
90C109#57711yescharacteristic, complemented, Sylow
91#577—1091nomaximal, Sylow
922−1+4—1091no—
93H(5,𝔽2)—1091no—
94#137—1092no—
95#137D10911yescharacteristic, complemented
96AGL(1,ℤ/8)—1092no—
97AGL(1,ℤ/8)—1091no—
98#135—1091no—
99#135—1092no—
100C2×SD16—1091no—
101C2×SD16—1092no—
102#131—1091no—
103#41—1091no—
104#41—1092no—
105#41—1092no—
106#41—1091no—
107#41—2182no—
108#41—2181no—
109#41—2181no—
110#41C2×D10911yescharacteristic
111C2×Q8—1092no—
112C2×Q8—2181no—
113C2×Q8C2×D10911yescharacteristic, complemented
114C2×D4—1092no—
115C2×D4—1091no—
116C2×D4—2181no—
117C2×D4—2182no—
118Q16—1091no—
119Q16C2×D10912yescomplemented
120SD16—1092no—
121SD16—1092no—
122SD16—1091no—
123SD16—1091no—
124SD16—1092no—
125SD16C2×D10912yescomplemented
126D8—1091no—
127D8—1092no—
128M16—1092no—
129M16C2×D10911yescharacteristic, complemented
130C2×C8—1091no—
131C2×C8—1092no—
132C23—2182no—
133C23—2181no—
134Q8—21no—
135Q8—1091no—
136Q8—1091no—
137Q8—2182no—
138Q8C22×D10912yes—
139Q8C22×D10911yescharacteristic
140D4—21no—
141D4—1092no—
142D4—1091no—
143D4—1091no—
144D4—1092no—
145D4—2182no—
146D4—2182no—
147D4—2182no—
148D4—2181no—
149D4—2181no—
150D4C22×D10911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—1092no—
154C2×C4—2182no—
155C2×C4—2182no—
156C2×C4—2182no—
157C2×C4—2181no—
158C2×C4C22×D10911yescharacteristic
159C8—1091no—
160C8—1091no—
161C8C22×D10912yes—
162C22—21no—
163C22—8721no—
164C22—8722no—
165C22—1092no—
166C22—2181no—
167C22—2182no—
168C22D4×D10911yescharacteristic
169C4—21no—
170C4—22no—
171C4—1092no—
172C4—2181no—
173C4D4×D10911yescharacteristic
174C4C23×D10911yescharacteristic
175C2—21no—
176C2—41no—
177C2—4362no—
178C2—4361no—
179C2—2182no—
180C2C2×D4×D10911yescharacteristic
subgroupG/H
centerC2C2×D4×D109
commutatorC4×C109C24
FrattiniC4C23×D109
Fitting#137×C109C2
socleC2×C109C22×D4
hypercenter#137D109
hypocenterC109#577
LCSG ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #16812747C2semidirect
2. #16812745C2semidirect
3. #16812731C2semidirect
4. #16812727C2semidirect
5. #16812726C2semidirect
6. #16812722C2semidirect
7. #16812719C2semidirect
8. C2×#13182016C2semidirect
9. SD16×D109C2semidirect
10. #16812707C2semidirect
11. #137×C109C2semidirect
14. #13182030C22semidirect
18. #13182029C22semidirect
20. Q8×D109C22semidirect
25. #13182019C22semidirect
26. #13182018C22semidirect
27. #13182017C22semidirect
28. #13182016C22semidirect
29. #13182016C22semidirect
30. #13182016C22semidirect
31. D872C22semidirect
32. #13182014C22semidirect
33. #13182012C22semidirect
34. C2×#11724784C22semidirect
35. C8×D109C22semidirect
37. C2×Q8×C109C22semidirect
38. Q16×C109C22semidirect
39. SD16×C109C22semidirect
40. M16×C109C22semidirect
90. C109#577semidirect
95. #137D109semidirect
113. C2×Q8C2×D109semidirect
119. Q16C2×D109semidirect
125. SD16C2×D109semidirect
129. M16C2×D109semidirect