Finite Group Databasebeta

← #20295669

Subgroups — #20295669

Order 6976.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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
2C2×D4×D109C211yesmaximal, characteristic, complemented
3#16812681C212yesmaximal, complemented
4#16812675C211yesmaximal, characteristic, complemented
5#16812672C212yesmaximal, complemented
6#16812670C211yesmaximal, characteristic, complemented
7#16812669C212yesmaximal, complemented
8C4×D436C211yesmaximal, characteristic, complemented
9C4×D436C211yesmaximal, characteristic, complemented
10C22⋊C4×D109C212yesmaximal, complemented
11#124×C109C211yesmaximal, characteristic, complemented
12C23×D109C2212yes—
13#13182030—24no—
14C2×D436C2211yescharacteristic
15C2×D436C2211yescharacteristic
16C2×D436C2212yescomplemented
17C2×C22⋊D109C2212yes—
18C2×C22⋊D109C2212yescomplemented
19D4×D109—24no—
20C2×C4×D109C2211yescharacteristic
21C2×C4×D109C2212yescomplemented
22C2×C4×D109C2211yescharacteristic, complemented
23C2×C4×D109C2212yescomplemented
24C4⋊Dic109C2211yescharacteristic, complemented
25C4⋊Dic109C2211yescharacteristic, complemented
26#13182008C2212yescomplemented
27C22⋊Dic109C2212yescomplemented
28#13182006C2211yescharacteristic
29#13182006C2212yes—
30#13182006C2212yescomplemented
31#13182006C2211yescharacteristic, complemented
32C2×Q8×C109C2211yescharacteristic, complemented
33C2×D4×C109C2211yescharacteristic
34C22⋊C4×C109C2212yescomplemented
35C22⋊C4×C109C2212yescomplemented
36C42×C109C2211yescharacteristic, complemented
37C23×C109C2312yes—
38C22×D109—24no—
39C22×D109—24no—
40C22×D109—42no—
41C22×D109C2311yescharacteristic
42C22×D109C2312yes—
43C22×D109C2311yescharacteristic
44Q8×C109—22no—
45D4×C109—22no—
46C2×C4×C109C2311yescharacteristic
47C2×C4×C109C2311yescharacteristic
48C2×C4×C109C2312yes—
49C2×C4×C109C2311yescharacteristic
50C22⋊D109—24no—
51C22⋊D109—42no—
52C2×Dic109C2312yes—
53C2×Dic109C2311yescharacteristic
54C2×Dic109C2311yescharacteristic
55D436—24no—
56D436—22no—
57D436D414yescomplemented
58C4×D109—24no—
59C4×D109—22no—
60C4×D109—22no—
61C4×D109—42no—
62C22×C109—22no—
63C22×C109—22no—
64C22×C109—22no—
65C22×C109C2411yescharacteristic
66C2×D109—21no—
67C2×D109—22no—
68C2×D109—22no—
69C2×D109—21no—
70C2×D109—21no—
71C2×D109—21no—
72C2×D109—21no—
73C2×D109—48no—
74C2×D109C2×D414yes—
75C2×D109#4112yescomplemented
76C4×C109—22no—
77C4×C109—21no—
78C4×C109—21no—
79C4×C109C2×D412yes—
80Dic109—21no—
81Dic109—21no—
82Dic109—22no—
83C2×C109—42no—
84C2×C109C22×D411yescharacteristic
85C2×C109C2×#4111yescharacteristic
86C2×C109H(5,𝔽2)11yescharacteristic
87D109—22no—
88D109—24no—
89D109—41no—
90C109#54611yescharacteristic, complemented, Sylow
91#546—1091nomaximal, Sylow
92C2×#41—1091no—
93C22×D4—1091no—
94#124D10911yescharacteristic, complemented
95#123—1092no—
96C22⋊Q8—1091no—
97#121—1091no—
98#121—1092no—
99#120—1092no—
100C4×D4—1091no—
101C4×D4—1091no—
102C2×C22⋊C4—1092no—
103C24—1092no—
104#41—2184no—
105C2×Q8C2×D10911yescharacteristic, complemented
106C2×D4—1092no—
107C2×D4—1092no—
108C2×D4—1091no—
109C2×D4—1091no—
110C2×D4—1092no—
111C2×D4—2184no—
112C2×D4C2×D10911yescharacteristic
113C22×C4—1092no—
114C22×C4—1091no—
115C22×C4—1092no—
116C22×C4—1091no—
117C4⋊C4—1092no—
118C4⋊C4—1091no—
119C4⋊C4—1091no—
120C22⋊C4—1092no—
121C22⋊C4—1091no—
122C22⋊C4—1092no—
123C22⋊C4—1092no—
124C22⋊C4—1091no—
125C22⋊C4C2×D10912yescomplemented
126C22⋊C4C2×D10912yescomplemented
127C42C2×D10911yescharacteristic, complemented
128C23—1091no—
129C23—1092no—
130C23—1091no—
131C23—4362no—
132C23—2184no—
133C23—2184no—
134C23C22×D10912yes—
135Q8—22no—
136D4—22no—
137D4—1094no—
138D4—4362no—
139D4—2184no—
140D4—2182no—
141D4—2184no—
142C2×C4—1092no—
143C2×C4—1091no—
144C2×C4—1091no—
145C2×C4—4362no—
146C2×C4—2184no—
147C2×C4—2182no—
148C2×C4—2182no—
149C2×C4C22×D10911yescharacteristic
150C2×C4C22×D10911yescharacteristic
151C2×C4C22×D10912yes—
152C2×C4C22×D10911yescharacteristic
153C22—22no—
154C22—22no—
155C22—22no—
156C22—1094no—
157C22—1092no—
158C22—4368no—
159C22—2181no—
160C22—2181no—
161C22—2181no—
162C22—2181no—
163C22—2182no—
164C22—2182no—
165C22—2181no—
166C22C23×D10911yescharacteristic
167C4—22no—
168C4—21no—
169C4—21no—
170C4—2181no—
171C4—2181no—
172C4—2182no—
173C4D4×D10912yes—
174C2—42no—
175C2—4361no—
176C2—2182no—
177C2—2184no—
178C2C2×D4×D10911yescharacteristic
179C2#41×D10911yescharacteristic
180C2#1681274511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D109
commutator, socleC22×C109C24
Fitting#124×C109C2
hypercenter#124D109
hypocenterC109#546
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#13182030C2semidirect
2. C2×D4×D109C2semidirect
3. #16812681C2semidirect
4. #16812675C2semidirect
5. #16812672C2semidirect
6. #16812670C2semidirect
7. #16812669C2semidirect
8. C4×D436C2semidirect
9. C4×D436C2semidirect
10. C22⋊C4×D109C2semidirect
11. #124×C109C2semidirect
16. C2×D436C22semidirect
18. C2×C22⋊D109C22semidirect
21. C2×C4×D109C22semidirect
22. C2×C4×D109C22semidirect
23. C2×C4×D109C22semidirect
24. C4⋊Dic109C22semidirect
25. C4⋊Dic109C22semidirect
26. #13182008C22semidirect
27. C22⋊Dic109C22semidirect
30. #13182006C22semidirect
31. #13182006C22semidirect
32. C2×Q8×C109C22semidirect
34. C22⋊C4×C109C22semidirect
35. C22⋊C4×C109C22semidirect
36. C42×C109C22semidirect
57. D436D4semidirect
75. C2×D109#41semidirect
90. C109#546semidirect
94. #124D109semidirect
105. C2×Q8C2×D109semidirect
125. C22⋊C4C2×D109semidirect
126. C22⋊C4C2×D109semidirect
127. C42C2×D109semidirect