Finite Group Databasebeta

← #20295830

Subgroups — #20295830

Order 6976.

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

normal subgroups only

This group has 268 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#16812745C211yesmaximal, characteristic, complemented
2#16812744C211yesmaximal, characteristic, complemented
3#16812727C211yesmaximal, characteristic, complemented
4#16812726C212yesmaximal, complemented
5#16812723C212yesmaximal, complemented
6#16812720C212yesmaximal, complemented
7#16812719C211yesmaximal, characteristic, complemented
8C2×#13182014C211yesmaximal, characteristic, complemented
9D8×D109C212yesmaximal, complemented
10#16812707C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12#13182030C2212yescomplemented
13#13182029—21no—
14#13182029C2211yescharacteristic
15#13182029C2211yescharacteristic, complemented
16#13182028—21no—
17#13182028C2212yescomplemented
18C2×D436—21no—
19C2×D436C2211yescharacteristic, complemented
20C2×C22⋊D109—22no—
21D4×D109—22no—
22D4×D109—21no—
23D4×D109C2212yescomplemented
24D4×D109C2212yescomplemented
25#13182019C2211yescharacteristic, complemented
26#13182018C2212yescomplemented
27#13182017C2211yescharacteristic, complemented
28#13182016C2211yescharacteristic, complemented
29D872C2212yescomplemented
30#13182014C2211yescharacteristic, complemented
31#13182014C2212yescomplemented
32#13182014C2212yescomplemented
33#13182012C2212yescomplemented
34C2×#11724784C2211yescharacteristic, complemented
35C8×D109C2212yescomplemented
36#41×C109C2211yescharacteristic, complemented
37C2×D4×C109C2211yescharacteristic, complemented
38SD16×C109C2212yescomplemented
39D8×C109C2212yescomplemented
40M16×C109C2211yescharacteristic, complemented
41C23×C109—21no—
42C22×D109—21no—
43C22×D109—22no—
44C22×D109—22no—
45Q8×C109C2311yescharacteristic
46D4×C109—21no—
47D4×C109—21no—
48D4×C109C2311yescharacteristic
49D4×C109C2312yes—
50C2×C4×C109—21no—
51C2×C4×C109C2311yescharacteristic
52C22⋊D109—21no—
53C22⋊D109—22no—
54C22⋊D109—22no—
55C22⋊D109—22no—
56C22⋊D109D412yes—
57C2×Dic109—22no—
58D436—22no—
59D436—21no—
60D436D411yescharacteristic
61D436C2312yes—
62D436C2311yescharacteristic
63C4×D109—21no—
64C4×D109—22no—
65C4×D109C2312yes—
66Dic218D411yescharacteristic
67Dic218C2311yescharacteristic
68C8×C109C2312yes—
69#11724784C2311yescharacteristic
70#11724784C2311yescharacteristic
71C22×C109—22no—
72C22×C109—21no—
73C22×C109—81no—
74C22×C109C2×D411yescharacteristic
75C2×D109—22no—
76C2×D109—21no—
77C2×D109—82no—
78C2×D109—82no—
79C2×D109—81no—
80C2×D109C2×D412yes—
81C4×C109—21no—
82C4×C109C2×D411yescharacteristic
83C4×C109C2411yescharacteristic
84Dic109—21no—
85Dic109C2×D412yes—
86C2×C109—21no—
87C2×C109—41no—
88C2×C109—42no—
89C2×C109C22×D411yescharacteristic
90D109—22no—
91D109—42no—
92D109—41no—
93C109#57611yescharacteristic, complemented, Sylow
94#576—1091nomaximal, Sylow
95H(5,𝔽2)—1091no—
96H(5,𝔽2)—1091no—
97AGL(1,ℤ/8)—1092no—
98AGL(1,ℤ/8)—1092no—
99AGL(1,ℤ/8)—1091no—
100AGL(1,ℤ/8)D10911yescharacteristic, complemented
101#135—1091no—
102#135—1092no—
103C2×D8—1091no—
104C2×D8—1092no—
105#131—1091no—
106#41—1091no—
107#41—1092no—
108#41—1092no—
109#41—1091no—
110#41—2181no—
111#41—2181no—
112#41C2×D10911yescharacteristic, complemented
113C2×D4—1092no—
114C2×D4—1092no—
115C2×D4—1091no—
116C2×D4—2181no—
117C2×D4—2182no—
118C2×D4—2181no—
119C2×D4—2182no—
120C2×D4C2×D10911yescharacteristic, complemented
121Q16—1091no—
122SD16—1091no—
123SD16—1091no—
124SD16—1092no—
125SD16C2×D10912yescomplemented
126D8—1092no—
127D8—1091no—
128D8—1092no—
129D8—1092no—
130D8C2×D10912yescomplemented
131M16—1092no—
132M16C2×D10911yescharacteristic, complemented
133C2×C8—1091no—
134C2×C8—1092no—
135C23—21no—
136C23—2182no—
137C23—2182no—
138C23—2181no—
139Q8—1091no—
140Q8—1091no—
141Q8C22×D10911yescharacteristic
142D4—21no—
143D4—21no—
144D4—1092no—
145D4—1091no—
146D4—1092no—
147D4—1091no—
148D4—2182no—
149D4—2182no—
150D4—2182no—
151D4—2182no—
152D4—2181no—
153D4—2181no—
154D4C22×D10912yes—
155D4C22×D10911yescharacteristic
156C2×C4—21no—
157C2×C4—1092no—
158C2×C4—2182no—
159C2×C4—2182no—
160C2×C4—2181no—
161C2×C4C22×D10911yescharacteristic
162C8—1091no—
163C8—1091no—
164C8C22×D10912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—8721no—
169C22—8722no—
170C22—8722no—
171C22—1092no—
172C22—2182no—
173C22—2181no—
174C22D4×D10911yescharacteristic
175C4—21no—
176C4—1092no—
177C4—2181no—
178C4D4×D10911yescharacteristic
179C4C23×D10911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—4362no—
184C2—4361no—
185C2—2182no—
186C2C2×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#576
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. #16812745C2semidirect
2. #16812744C2semidirect
3. #16812727C2semidirect
4. #16812726C2semidirect
5. #16812723C2semidirect
6. #16812720C2semidirect
7. #16812719C2semidirect
8. C2×#13182014C2semidirect
9. D8×D109C2semidirect
10. #16812707C2semidirect
11. AGL(1,ℤ/8)×C109C2semidirect
12. #13182030C22semidirect
15. #13182029C22semidirect
17. #13182028C22semidirect
19. C2×D436C22semidirect
23. D4×D109C22semidirect
24. D4×D109C22semidirect
25. #13182019C22semidirect
26. #13182018C22semidirect
27. #13182017C22semidirect
28. #13182016C22semidirect
29. D872C22semidirect
30. #13182014C22semidirect
31. #13182014C22semidirect
32. #13182014C22semidirect
33. #13182012C22semidirect
34. C2×#11724784C22semidirect
35. C8×D109C22semidirect
36. #41×C109C22semidirect
37. C2×D4×C109C22semidirect
38. SD16×C109C22semidirect
39. D8×C109C22semidirect
40. M16×C109C22semidirect
93. C109#576semidirect
100. AGL(1,ℤ/8)D109semidirect
112. #41C2×D109semidirect
120. C2×D4C2×D109semidirect
125. SD16C2×D109semidirect
130. D8C2×D109semidirect
132. M16C2×D109semidirect