Finite Group Databasebeta

← #20295836

Subgroups — #20295836

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#16812746C211yesmaximal, characteristic, complemented
2#16812744C211yesmaximal, characteristic, complemented
3#16812732C211yesmaximal, characteristic, complemented
4#16812728C212yesmaximal, complemented
5#16812723C212yesmaximal, complemented
6#16812719C211yesmaximal, characteristic, complemented
7#16812718C212yesmaximal, complemented
8C2×#13182017C211yesmaximal, characteristic, complemented
9SD16×D109C212yesmaximal, complemented
10#16812707C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12#13182029—21no—
13#13182029C2211yescharacteristic
14#13182029C2211yescharacteristic
15#13182028—22no—
16#13182028—21no—
17#13182028C2212yes—
18#13182028C2212yescomplemented
19C2×Dic218—21no—
20C2×Dic218C2211yescharacteristic, complemented
21Q8×D109C2212yescomplemented
22C2×C22⋊D109—22no—
23D4×D109—21no—
24D4×D109C2212yes—
25Dic436C2212yescomplemented
26#13182019C2211yescharacteristic, complemented
27#13182018C2212yescomplemented
28#13182017C2212yescomplemented
29#13182017C2212yescomplemented
30#13182017C2211yescharacteristic, complemented
31#13182016C2211yescharacteristic, complemented
32#13182014C2211yescharacteristic, complemented
33#13182012C2212yescomplemented
34C2×#11724784C2211yescharacteristic, complemented
35C8×D109C2212yescomplemented
36#41×C109C2211yescharacteristic, complemented
37C2×D4×C109C2211yescharacteristic
38SD16×C109C2212yescomplemented
39D8×C109C2212yescomplemented
40M16×C109C2211yescharacteristic, complemented
41C23×C109—21no—
42C22×D109—22no—
43Q8×C109C2311yescharacteristic
44D4×C109—21no—
45D4×C109—21no—
46D4×C109C2311yescharacteristic
47D4×C109C2312yes—
48C2×C4×C109—21no—
49C2×C4×C109C2311yescharacteristic
50C22⋊D109—21no—
51C22⋊D109—22no—
52C22⋊D109—22no—
53C22⋊D109—22no—
54C22⋊D109D412yes—
55C2×Dic109—21no—
56C2×Dic109—22no—
57C2×Dic109—22no—
58D436D411yescharacteristic
59D436C2311yescharacteristic
60C4×D109—22no—
61C4×D109—21no—
62C4×D109C2312yes—
63Dic218—21no—
64Dic218—22no—
65Dic218D411yescharacteristic
66Dic218C2312yes—
67Dic218C2311yescharacteristic
68C8×C109C2312yes—
69#11724784C2311yescharacteristic
70#11724784C2311yescharacteristic
71C22×C109—21no—
72C22×C109—22no—
73C22×C109—81no—
74C22×C109C2×D411yescharacteristic
75C2×D109—21no—
76C2×D109—82no—
77C2×D109C2×D412yes—
78C4×C109—21no—
79C4×C109C2×D411yescharacteristic
80C4×C109C2411yescharacteristic
81Dic109—22no—
82Dic109—21no—
83Dic109C2×D412yes—
84C2×C109—21no—
85C2×C109—42no—
86C2×C109—41no—
87C2×C109C22×D411yescharacteristic
88D109—22no—
89D109—41no—
90C109#57711yescharacteristic, complemented, Sylow
91#577—1091nomaximal, Sylow
922−1+4—1091no—
93H(5,𝔽2)—1091no—
94#137—1092no—
95#137—1091no—
96AGL(1,ℤ/8)—1092no—
97AGL(1,ℤ/8)D10911yescharacteristic, complemented
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—2181no—
108#41—2182no—
109#41—2181no—
110#41C2×D10911yescharacteristic, complemented
111C2×Q8—1092no—
112C2×Q8—1091no—
113C2×Q8—2181no—
114C2×D4—1092no—
115C2×D4—2181no—
116C2×D4—2182no—
117C2×D4C2×D10911yescharacteristic
118Q16—1091no—
119Q16—1092no—
120SD16—1091no—
121SD16—1092no—
122SD16—1092no—
123SD16—1091no—
124SD16—1092no—
125SD16C2×D10912yescomplemented
126D8—1091no—
127D8C2×D10912yescomplemented
128M16—1092no—
129M16C2×D10911yescharacteristic, complemented
130C2×C8—1091no—
131C2×C8—1092no—
132C23—21no—
133C23—2182no—
134Q8—1091no—
135Q8—1092no—
136Q8—1091no—
137Q8—2182no—
138Q8—2181no—
139Q8C22×D10911yescharacteristic
140D4—21no—
141D4—21no—
142D4—1092no—
143D4—1091no—
144D4—1091no—
145D4—2182no—
146D4—2182no—
147D4—2182no—
148D4—2181no—
149D4C22×D10911yescharacteristic
150D4C22×D10912yes—
151C2×C4—21no—
152C2×C4—1092no—
153C2×C4—2182no—
154C2×C4—2182no—
155C2×C4—2182no—
156C2×C4—2181no—
157C2×C4—2181no—
158C2×C4C22×D10911yescharacteristic
159C8—1091no—
160C8—1091no—
161C8C22×D10912yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—8722no—
166C22—1092no—
167C22—2181no—
168C22D4×D10911yescharacteristic
169C4—21no—
170C4—1092no—
171C4—2181no—
172C4—2182no—
173C4D4×D10911yescharacteristic
174C4C23×D10911yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—4361no—
179C2—2182no—
180C2C2×D4×D10911yescharacteristic
subgroupG/H
centerC2C2×D4×D109
commutatorC4×C109C24
FrattiniC4C23×D109
FittingAGL(1,ℤ/8)×C109C2
socleC2×C109C22×D4
hypercenterAGL(1,ℤ/8)D109
hypocenterC109#577
LCSG ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16812746C2semidirect
2. #16812744C2semidirect
3. #16812732C2semidirect
4. #16812728C2semidirect
5. #16812723C2semidirect
6. #16812719C2semidirect
7. #16812718C2semidirect
8. C2×#13182017C2semidirect
9. SD16×D109C2semidirect
10. #16812707C2semidirect
11. AGL(1,ℤ/8)×C109C2semidirect
18. #13182028C22semidirect
20. C2×Dic218C22semidirect
21. Q8×D109C22semidirect
25. Dic436C22semidirect
26. #13182019C22semidirect
27. #13182018C22semidirect
28. #13182017C22semidirect
29. #13182017C22semidirect
30. #13182017C22semidirect
31. #13182016C22semidirect
32. #13182014C22semidirect
33. #13182012C22semidirect
34. C2×#11724784C22semidirect
35. C8×D109C22semidirect
36. #41×C109C22semidirect
38. SD16×C109C22semidirect
39. D8×C109C22semidirect
40. M16×C109C22semidirect
90. C109#577semidirect
97. AGL(1,ℤ/8)D109semidirect
110. #41C2×D109semidirect
125. SD16C2×D109semidirect
127. D8C2×D109semidirect
129. M16C2×D109semidirect