Finite Group Databasebeta

← #20293137

Subgroups — #20293137

Order 6848.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#13133817C211yesmaximal, characteristic, complemented
2C22×C4×D107C211yesmaximal, characteristic, complemented
3#16812091C212yesmaximal, complemented
4#16812089C211yesmaximal, characteristic, complemented
5#16812088C211yesmaximal, characteristic, complemented
6#16812080C211yesmaximal, characteristic, complemented
7#16812077C212yesmaximal, complemented
8D4×Dic107C212yesmaximal, complemented
9(C22⋊D107)⋊C4C212yesmaximal, complemented
10#16812067C211yesmaximal, characteristic, complemented
11#121×C107C211yesmaximal, characteristic, complemented
12C23×D107C2211yescharacteristic
13#13133817—24no—
14C2×Dic214C2211yescharacteristic, complemented
15C2×C22⋊D107C2212yes—
16C2×C22⋊D107C2212yes—
17C22×Dic107C2211yescharacteristic
18C22×Dic107C2212yescomplemented
19C2×C4×D107—24no—
20C2×C4×D107C2211yescharacteristic
21C2×C4×D107C2211yescharacteristic
22C2×C4×D107C2211yescharacteristic
23C2×C4×D107C2211yescharacteristic
24C4⋊Dic107C2211yescharacteristic, complemented
25C4⋊Dic107C2212yescomplemented
26Dic107⋊C4C2212yescomplemented
27C22⋊Dic107C2212yes—
28C22⋊Dic107C2212yescomplemented
29(C2×D107)⋊C4C2212yes—
30(C2×D107)⋊C4C2212yes—
31C4×Dic107C2212yescomplemented
32C2×D4×C107C2212yes—
33C2×D4×C107C2211yescharacteristic, complemented
34C22×C4×C107C2211yescharacteristic
35C4⋊C4×C107C2211yescharacteristic, complemented
36C22⋊C4×C107C2212yescomplemented
37C23×C107C2312yes—
38C23×C107C2311yescharacteristic
39C22×D107—22no—
40C22×D107—22no—
41C22×D107—22no—
42C22×D107C2311yescharacteristic
43C22×D107C2311yescharacteristic
44D4×C107—24no—
45D4×C107—22no—
46C2×C4×C107—22no—
47C2×C4×C107C2311yescharacteristic
48C2×C4×C107C2312yes—
49C2×C4×C107C2311yescharacteristic
50C22⋊D107—24no—
51C22⋊D107—24no—
52C2×Dic107—22no—
53C2×Dic107—24no—
54C2×Dic107C2312yes—
55C2×Dic107C2312yes—
56C2×Dic107C2311yescharacteristic
57C2×Dic107C2311yescharacteristic
58C4×D107—22no—
59C4×D107—22no—
60C4×D107—22no—
61C4×D107D414yescomplemented
62Dic214—22no—
63C22×C107—21no—
64C22×C107—21no—
65C22×C107—22no—
66C22×C107—22no—
67C22×C107—22no—
68C22×C107#4112yescomplemented
69C22×C107C2411yescharacteristic
70C2×D107—28no—
71C2×D107—21no—
72C2×D107—21no—
73C2×D107—21no—
74C2×D107—21no—
75C2×D107C2×D412yes—
76C2×D107#4112yescomplemented
77C4×C107—21no—
78C4×C107—22no—
79C4×C107C2×D412yes—
80Dic107—22no—
81Dic107—21no—
82Dic107—22no—
83Dic107C2×D412yes—
84C2×C107—22no—
85C2×C107—42no—
86C2×C107C22×D411yescharacteristic
87C2×C107C2×#4111yescharacteristic
88C2×C107C2×#4111yescharacteristic
89D107—22no—
90D107—22no—
91C107#52511yescharacteristic, complemented, Sylow
92#525—1071nomaximal, Sylow
93C2×#41—1071no—
94C23×C4—1071no—
95#123—1072no—
96C22⋊Q8—1071no—
97C22⋊Q8—1071no—
98#121—1071no—
99#121D10711yescharacteristic, complemented
100#120—1072no—
101C4×D4—1072no—
102C4×D4—1072no—
103#117—1071no—
104C24—1071no—
105#41—2144no—
106C2×Q8—1071no—
107C2×D4—1072no—
108C2×D4—1072no—
109C2×D4C2×D10712yes—
110C2×D4C2×D10711yescharacteristic, complemented
111C22×C4—1071no—
112C22×C4—1071no—
113C22×C4—1072no—
114C22×C4—1071no—
115C22×C4—1071no—
116C22×C4—1071no—
117C22×C4—2144no—
118C22×C4C2×D10711yescharacteristic
119C4⋊C4—1071no—
120C4⋊C4—1072no—
121C4⋊C4—1072no—
122C4⋊C4C2×D10711yescharacteristic, complemented
123C22⋊C4—1072no—
124C22⋊C4—1072no—
125C22⋊C4—1072no—
126C22⋊C4—1072no—
127C22⋊C4C2×D10712yescomplemented
128C42—1072no—
129C23—1071no—
130C23—1071no—
131C23—2142no—
132C23—2142no—
133C23—2142no—
134C23C22×D10712yes—
135C23C22×D10711yescharacteristic
136Q8—2142no—
137D4—24no—
138D4—22no—
139D4—2144no—
140D4—2144no—
141C2×C4—22no—
142C2×C4—1071no—
143C2×C4—1074no—
144C2×C4—1072no—
145C2×C4—1071no—
146C2×C4—1072no—
147C2×C4—2142no—
148C2×C4—2142no—
149C2×C4—2142no—
150C2×C4—2144no—
151C2×C4—2142no—
152C2×C4C22×D10712yes—
153C2×C4C22×D10711yescharacteristic
154C2×C4C22×D10711yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—1072no—
161C22—1072no—
162C22—2148no—
163C22—2141no—
164C22—2141no—
165C22—2141no—
166C22—2141no—
167C22#1313381712yescomplemented
168C22C23×D10711yescharacteristic
169C4—21no—
170C4—22no—
171C4—1072no—
172C4—2141no—
173C4—2142no—
174C4—2142no—
175C4D4×D10712yes—
176C2—22no—
177C2—42no—
178C2—2142no—
179C2—2142no—
180C2C2×D4×D10711yescharacteristic
181C2#41×D10711yescharacteristic
182C2C2×#1313381711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
Fitting#121×C107C2
hypercenter#121D107
hypocenterC107#525
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#13133817C2semidirect
2. C22×C4×D107C2semidirect
3. #16812091C2semidirect
4. #16812089C2semidirect
5. #16812088C2semidirect
6. #16812080C2semidirect
7. #16812077C2semidirect
8. D4×Dic107C2semidirect
9. (C22⋊D107)⋊C4C2semidirect
10. #16812067C2semidirect
11. #121×C107C2semidirect
14. C2×Dic214C22semidirect
18. C22×Dic107C22semidirect
24. C4⋊Dic107C22semidirect
25. C4⋊Dic107C22semidirect
26. Dic107⋊C4C22semidirect
28. C22⋊Dic107C22semidirect
31. C4×Dic107C22semidirect
33. C2×D4×C107C22semidirect
35. C4⋊C4×C107C22semidirect
36. C22⋊C4×C107C22semidirect
61. C4×D107D4semidirect
68. C22×C107#41semidirect
76. C2×D107#41semidirect
91. C107#525semidirect
99. #121D107semidirect
110. C2×D4C2×D107semidirect
122. C4⋊C4C2×D107semidirect
127. C22⋊C4C2×D107semidirect
167. C22#13133817semidirect