Finite Group Databasebeta

← #20295825

Subgroups — #20295825

Order 6976.

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

normal subgroups only

This group has 262 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
1C2×#13182030C211yesmaximal, characteristic, complemented
2#41×D109C211yesmaximal, characteristic, complemented
3#16812731C212yesmaximal, complemented
4#16812730C211yesmaximal, characteristic, complemented
5#16812727C211yesmaximal, characteristic, complemented
6#16812726C212yesmaximal, complemented
7#16812724C211yesmaximal, characteristic, complemented
8#16812722C212yesmaximal, complemented
9#16812720C212yesmaximal, complemented
10M16×D109C211yesmaximal, characteristic, complemented
11#137×C109C211yesmaximal, characteristic, complemented
12#13182030—21no—
13#13182030—21no—
14#13182030C2212yes—
15#13182030C2212yescomplemented
16#13182030C2211yescharacteristic, complemented
17#13182029—21no—
18#13182029C2211yescharacteristic
19#13182028—21no—
20#13182028C2211yescharacteristic, complemented
21Q8×D109C2211yescharacteristic, complemented
22C2×D436—21no—
23C2×D436C2211yescharacteristic, complemented
24D4×D109—21no—
25D4×D109C2211yescharacteristic, complemented
26C2×C4×D109—21no—
27C2×C4×D109—21no—
28C2×C4×D109C2211yescharacteristic
29#13182019C2212yescomplemented
30#13182018C2212yescomplemented
31#13182016C2212yescomplemented
32#13182016C2212yescomplemented
33D872C2212yescomplemented
34#13182014C2212yescomplemented
35#13182013C2211yescharacteristic, complemented
36#13182012C2212yescomplemented
37C8×D109C2212yescomplemented
38#41×C109C2211yescharacteristic
39C2×Q8×C109C2211yescharacteristic, complemented
40Q16×C109C2212yescomplemented
41SD16×C109C2212yescomplemented
42M16×C109C2211yescharacteristic, complemented
43C22×D109—21no—
44C22×D109—21no—
45C22×D109D411yescharacteristic
46Q8×C109—21no—
47Q8×C109C2312yes—
48Q8×C109C2311yescharacteristic
49D4×C109—21no—
50D4×C109C2311yescharacteristic
51C2×C4×C109—21no—
52C2×C4×C109—21no—
53C2×C4×C109C2311yescharacteristic
54C22⋊D109—21no—
55C22⋊D109—21no—
56C22⋊D109—21no—
57C2×Dic109—21no—
58C2×Dic109D411yescharacteristic
59D436—21no—
60D436—22no—
61D436—22no—
62D436—21no—
63D436C2312yes—
64D436C2311yescharacteristic
65C4×D109—21no—
66C4×D109—21no—
67C4×D109—21no—
68C4×D109—22no—
69C4×D109—22no—
70C4×D109D411yescharacteristic
71C4×D109D411yescharacteristic
72C4×D109C2311yescharacteristic
73C4×D109C2311yescharacteristic
74Dic218—21no—
75Dic218C2311yescharacteristic
76C8×C109C2312yes—
77#11724784C2312yes—
78C22×C109—21no—
79C22×C109C2×D411yescharacteristic
80C2×D109—22no—
81C2×D109—21no—
82C2×D109—41no—
83C2×D109—42no—
84C2×D109—42no—
85C2×D109C2×D411yescharacteristic
86C2×D109C2×D411yescharacteristic
87C4×C109—21no—
88C4×C109—22no—
89C4×C109C2×D411yescharacteristic
90C4×C109C2411yescharacteristic
91Dic109—21no—
92Dic109C2×D411yescharacteristic
93Dic109C2×D411yescharacteristic
94C2×C109—21no—
95C2×C109—41no—
96C2×C109C22×D411yescharacteristic
97D109—21no—
98D109—21no—
99D109—42no—
100D109—41no—
101C109#57511yescharacteristic, complemented, Sylow
102#575—1091nomaximal, Sylow
103C2×#41—1091no—
104C2×#41—1091no—
105#137—1091no—
106#137—1092no—
107#137D10911yescharacteristic, complemented
108AGL(1,ℤ/8)—1091no—
109AGL(1,ℤ/8)—1092no—
110AGL(1,ℤ/8)—1091no—
111#135—1092no—
112#135—1092no—
113C2×M16—1091no—
114#41—1092no—
115#41—1091no—
116#41—1092no—
117#41—1091no—
118#41—1091no—
119#41—2181no—
120#41—2181no—
121#41—2181no—
122#41—2181no—
123#41C2×D10911yescharacteristic
124C2×Q8—1091no—
125C2×Q8C2×D10911yescharacteristic, complemented
126C2×D4—1091no—
127C2×D4—1091no—
128C2×D4—2181no—
129C2×D4—2181no—
130C22×C4—1091no—
131C22×C4—2181no—
132C22×C4—2181no—
133Q16—1092no—
134Q16C2×D10912yescomplemented
135SD16—1092no—
136SD16—1092no—
137SD16—1092no—
138SD16C2×D10912yescomplemented
139D8—1092no—
140D8—1092no—
141M16—1091no—
142M16—1092no—
143M16C2×D10911yescharacteristic, complemented
144C2×C8—1092no—
145C23—1091no—
146C23—2181no—
147C23—2181no—
148Q8—21no—
149Q8—1091no—
150Q8—2181no—
151Q8C22×D10911yescharacteristic
152Q8C22×D10912yes—
153D4—21no—
154D4—1092no—
155D4—1091no—
156D4—2181no—
157D4—2182no—
158D4—2182no—
159D4—2181no—
160D4—2181no—
161D4—2181no—
162D4—2181no—
163D4C22×D10911yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—1091no—
167C2×C4—1091no—
168C2×C4—1091no—
169C2×C4—1091no—
170C2×C4—1091no—
171C2×C4—2181no—
172C2×C4—2182no—
173C2×C4—2182no—
174C2×C4—2181no—
175C2×C4—2181no—
176C2×C4—2181no—
177C2×C4C22×D10911yescharacteristic
178C8—1092no—
179C8C22×D10912yes—
180C22—21no—
181C22—1091no—
182C22—1091no—
183C22—4362no—
184C22—4361no—
185C22—4362no—
186C22—2182no—
187C22—2181no—
188C22D4×D10911yescharacteristic
189C4—21no—
190C4—22no—
191C4—1091no—
192C4—1091no—
193C4—2181no—
194C4D4×D10911yescharacteristic
195C4C23×D10911yescharacteristic
196C2—21no—
197C2—41no—
198C2—4362no—
199C2—4361no—
200C2—2181no—
201C2—2181no—
202C2C2×D4×D10911yescharacteristic
subgroupG/H
centerC2C2×D4×D109
commutatorC4×C109C24
FrattiniC4C23×D109
Fitting#137×C109C2
socleC2×C109C22×D4
hypercenter#137D109
hypocenterC109#575
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. C2×#13182030C2semidirect
2. #41×D109C2semidirect
3. #16812731C2semidirect
4. #16812730C2semidirect
5. #16812727C2semidirect
6. #16812726C2semidirect
7. #16812724C2semidirect
8. #16812722C2semidirect
9. #16812720C2semidirect
10. M16×D109C2semidirect
11. #137×C109C2semidirect
15. #13182030C22semidirect
16. #13182030C22semidirect
20. #13182028C22semidirect
21. Q8×D109C22semidirect
23. C2×D436C22semidirect
25. D4×D109C22semidirect
29. #13182019C22semidirect
30. #13182018C22semidirect
31. #13182016C22semidirect
32. #13182016C22semidirect
33. D872C22semidirect
34. #13182014C22semidirect
35. #13182013C22semidirect
36. #13182012C22semidirect
37. C8×D109C22semidirect
39. C2×Q8×C109C22semidirect
40. Q16×C109C22semidirect
41. SD16×C109C22semidirect
42. M16×C109C22semidirect
101. C109#575semidirect
107. #137D109semidirect
125. C2×Q8C2×D109semidirect
134. Q16C2×D109semidirect
138. SD16C2×D109semidirect
143. M16C2×D109semidirect