Finite Group Databasebeta

← #20295833

Subgroups — #20295833

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#16812745C211yesmaximal, characteristic, complemented
3#16812731C212yesmaximal, complemented
4#16812729C211yesmaximal, characteristic, complemented
5#16812724C211yesmaximal, characteristic, complemented
6#16812723C212yesmaximal, complemented
7#16812720C212yesmaximal, complemented
8C2×#13182018C211yesmaximal, characteristic, complemented
9SD16×D109C212yesmaximal, complemented
10#16812705C211yesmaximal, characteristic, complemented
11#135×C109C211yesmaximal, characteristic, complemented
12#13182030C2212yes—
13#13182029—21no—
14#13182029—21no—
15#13182029C2211yescharacteristic
16#13182028—22no—
17#13182028C2212yescomplemented
18C2×Dic218—21no—
19C2×Dic218C2211yescharacteristic, complemented
20Q8×D109C2212yescomplemented
21C2×D436—21no—
22C2×D436C2211yescharacteristic
23D4×D109—22no—
24D4×D109C2212yes—
25#13182019C2212yescomplemented
26#13182018C2212yescomplemented
27#13182018C2212yescomplemented
28#13182017C2212yescomplemented
29#13182016C2212yescomplemented
30#13182014C2212yescomplemented
31#13182013C2211yescharacteristic, complemented
32#13182012C2212yescomplemented
33C8×D109C2212yescomplemented
34#41×C109C2211yescharacteristic
35#41×C109C2211yescharacteristic, complemented
36Q16×C109C2211yescharacteristic, complemented
37SD16×C109C2211yescharacteristic, complemented
38SD16×C109C2211yescharacteristic, complemented
39D8×C109C2211yescharacteristic, complemented
40C2×C8×C109C2211yescharacteristic, complemented
41C22×D109—21no—
42C22×D109—22no—
43Q8×C109C2311yescharacteristic
44Q8×C109C2311yescharacteristic
45D4×C109—21no—
46D4×C109—21no—
47D4×C109C2311yescharacteristic
48D4×C109C2311yescharacteristic
49C2×C4×C109—21no—
50C2×C4×C109—21no—
51C2×C4×C109C2311yescharacteristic
52C22⋊D109—22no—
53C22⋊D109—22no—
54C22⋊D109D412yes—
55C2×Dic109—21no—
56C2×Dic109—22no—
57D436—22no—
58D436—21no—
59D436D411yescharacteristic
60D436C2312yes—
61C4×D109—22no—
62C4×D109—22no—
63C4×D109C2312yes—
64Dic218—21no—
65Dic218—22no—
66Dic218D411yescharacteristic
67Dic218C2312yes—
68C8×C109C2311yescharacteristic
69C8×C109C2311yescharacteristic
70#11724784C2312yes—
71C22×C109—21no—
72C22×C109—21no—
73C22×C109C2×D411yescharacteristic
74C2×D109—22no—
75C2×D109—82no—
76C2×D109—81no—
77C2×D109C2×D412yes—
78C4×C109—21no—
79C4×C109—21no—
80C4×C109C2×D411yescharacteristic
81C4×C109C2411yescharacteristic
82Dic109—22no—
83Dic109C2×D412yes—
84C2×C109—21no—
85C2×C109—41no—
86C2×C109—41no—
87C2×C109C22×D411yescharacteristic
88D109—22no—
89D109—42no—
90C109#57711yescharacteristic, complemented, Sylow
91#577—1091nomaximal, Sylow
922−1+4—1091no—
93H(5,𝔽2)—1091no—
94#137—1091no—
95#137—1092no—
96AGL(1,ℤ/8)—1091no—
97AGL(1,ℤ/8)—1092no—
98#135—1092no—
99#135D10911yescharacteristic, complemented
100C2×SD16—1092no—
101C2×SD16—1091no—
102#131—1091no—
103#41—1091no—
104#41—1092no—
105#41—1092no—
106#41—2181no—
107#41—2181no—
108#41—2182no—
109#41C2×D10911yescharacteristic
110#41C2×D10911yescharacteristic, complemented
111C2×Q8—1092no—
112C2×Q8—1091no—
113C2×Q8—2181no—
114C2×D4—1092no—
115C2×D4—1091no—
116C2×D4—2181no—
117C2×D4—2182no—
118Q16—1092no—
119Q16C2×D10911yescharacteristic, complemented
120SD16—1092no—
121SD16—1092no—
122SD16—1092no—
123SD16—1092no—
124SD16C2×D10911yescharacteristic, complemented
125SD16C2×D10911yescharacteristic, complemented
126D8—1092no—
127D8C2×D10911yescharacteristic, complemented
128M16—1091no—
129M16—1092no—
130C2×C8—1092no—
131C2×C8C2×D10911yescharacteristic, complemented
132C23—2182no—
133C23—2181no—
134Q8—1091no—
135Q8—1092no—
136Q8—2182no—
137Q8—2181no—
138Q8C22×D10911yescharacteristic
139Q8C22×D10911yescharacteristic
140D4—21no—
141D4—21no—
142D4—1092no—
143D4—1091no—
144D4—1092no—
145D4—2182no—
146D4—2182no—
147D4—2182no—
148D4—2181no—
149D4C22×D10911yescharacteristic
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—1092no—
160C8C22×D10911yescharacteristic
161C8C22×D10911yescharacteristic
162C22—21no—
163C22—21no—
164C22—8721no—
165C22—8722no—
166C22—1092no—
167C22—2182no—
168C22D4×D10911yescharacteristic
169C4—21no—
170C4—21no—
171C4—1092no—
172C4—2182no—
173C4D4×D10911yescharacteristic
174C4C23×D10911yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—4362no—
179C2—2182no—
180C2C2×D4×D10911yescharacteristic
subgroupG/H
centerC2C2×D4×D109
commutatorC4×C109C24
FrattiniC4C23×D109
Fitting#135×C109C2
socleC2×C109C22×D4
hypercenter#135D109
hypocenterC109#577
LCSG ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #16812746C2semidirect
2. #16812745C2semidirect
3. #16812731C2semidirect
4. #16812729C2semidirect
5. #16812724C2semidirect
6. #16812723C2semidirect
7. #16812720C2semidirect
8. C2×#13182018C2semidirect
9. SD16×D109C2semidirect
10. #16812705C2semidirect
11. #135×C109C2semidirect
17. #13182028C22semidirect
19. C2×Dic218C22semidirect
20. Q8×D109C22semidirect
25. #13182019C22semidirect
26. #13182018C22semidirect
27. #13182018C22semidirect
28. #13182017C22semidirect
29. #13182016C22semidirect
30. #13182014C22semidirect
31. #13182013C22semidirect
32. #13182012C22semidirect
33. C8×D109C22semidirect
35. #41×C109C22semidirect
36. Q16×C109C22semidirect
37. SD16×C109C22semidirect
38. SD16×C109C22semidirect
39. D8×C109C22semidirect
40. C2×C8×C109C22semidirect
90. C109#577semidirect
99. #135D109semidirect
110. #41C2×D109semidirect
119. Q16C2×D109semidirect
124. SD16C2×D109semidirect
125. SD16C2×D109semidirect
127. D8C2×D109semidirect
131. C2×C8C2×D109semidirect