Finite Group Databasebeta

← #20293133

Subgroups — #20293133

Order 6848.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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
2C23×Dic107C211yesmaximal, characteristic, complemented
3#16812093C211yesmaximal, characteristic, complemented
4#16812090C211yesmaximal, characteristic, complemented
5#16812086C212yesmaximal, complemented
6#16812081C212yesmaximal, complemented
7#16812078C211yesmaximal, characteristic, complemented
8D4×Dic107C212yesmaximal, complemented
9(C22⋊D107)⋊C4C212yesmaximal, complemented
10#16812065C211yesmaximal, characteristic, complemented
11#120×C107C211yesmaximal, characteristic, complemented
12#13133817—24no—
13C2×Dic214C2211yescharacteristic, complemented
14C2×C22⋊D107C2212yes—
15C2×C22⋊D107C2211yescharacteristic
16C2×C22⋊D107C2211yescharacteristic, complemented
17C22×Dic107—24no—
18C22×Dic107C2212yes—
19C22×Dic107C2212yes—
20C22×Dic107C2211yescharacteristic
21C22×Dic107C2211yescharacteristic
22C22×Dic107C2211yescharacteristic, complemented
23C2×C4×D107C2211yescharacteristic, complemented
24C4⋊Dic107C2212yescomplemented
25Dic107⋊C4C2212yescomplemented
26Dic107⋊C4C2212yescomplemented
27C22⋊Dic107C2211yescharacteristic
28C22⋊Dic107C2212yes—
29C22⋊Dic107C2212yescomplemented
30(C2×D107)⋊C4C2212yescomplemented
31C4×Dic107C2212yescomplemented
32C24×C107C2211yescharacteristic
33C2×D4×C107C2212yes—
34C2×D4×C107C2211yescharacteristic
35C22⋊C4×C107C2211yescharacteristic
36C22⋊C4×C107C2212yes—
37C23×C107—22no—
38C23×C107—24no—
39C23×C107C2311yescharacteristic
40C23×C107C2311yescharacteristic
41C23×C107C2312yes—
42C22×D107C2311yescharacteristic
43D4×C107—22no—
44D4×C107—24no—
45C2×C4×C107C2312yes—
46C2×C4×C107C2311yescharacteristic
47C22⋊D107—22no—
48C22⋊D107—24no—
49C22⋊D107—22no—
50C2×Dic107—22no—
51C2×Dic107—24no—
52C2×Dic107—24no—
53C2×Dic107—22no—
54C2×Dic107D414yescomplemented
55C2×Dic107C2312yes—
56C2×Dic107C2312yes—
57C2×Dic107C2311yescharacteristic
58C2×Dic107C2311yescharacteristic
59C2×Dic107C2311yescharacteristic
60C4×D107—22no—
61Dic214—22no—
62C22×C107—28no—
63C22×C107—22no—
64C22×C107—22no—
65C22×C107—22no—
66C22×C107—21no—
67C22×C107—22no—
68C22×C107C2×D412yes—
69C22×C107#4114yescomplemented
70C22×C107C2411yescharacteristic
71C2×D107—21no—
72C2×D107—22no—
73C4×C107—21no—
74C4×C107—22no—
75Dic107—21no—
76Dic107—22no—
77Dic107—22no—
78Dic107C2×D412yes—
79Dic107C2×D412yes—
80C2×C107—22no—
81C2×C107—24no—
82C2×C107—41no—
83C2×C107C22×D411yescharacteristic
84C2×C107C2×#4112yes—
85D107—41no—
86C107#52511yescharacteristic, complemented, Sylow
87#525—1071nomaximal, Sylow
88C2×#41—1071no—
89C23×C4—1071no—
90#123—1071no—
91#123—1071no—
92C22⋊Q8—1072no—
93#121—1072no—
94#120—1071no—
95#120D10711yescharacteristic, complemented
96C4×D4—1072no—
97C4×D4—1072no—
98#117—1071no—
99C24C2×D10711yescharacteristic
100#41—2144no—
101C2×Q8—1071no—
102C2×D4—1072no—
103C2×D4—1071no—
104C2×D4—1071no—
105C2×D4C2×D10712yes—
106C2×D4C2×D10711yescharacteristic
107C22×C4—1071no—
108C22×C4—1072no—
109C22×C4—1071no—
110C22×C4—1072no—
111C22×C4—1071no—
112C22×C4—1071no—
113C22×C4—2144no—
114C4⋊C4—1072no—
115C4⋊C4—1072no—
116C4⋊C4—1072no—
117C22⋊C4—1071no—
118C22⋊C4—1072no—
119C22⋊C4—1072no—
120C22⋊C4—1072no—
121C22⋊C4C2×D10711yescharacteristic
122C22⋊C4C2×D10712yes—
123C42—1072no—
124C23—22no—
125C23—24no—
126C23—1071no—
127C23C22×D10711yescharacteristic
128C23C22×D10711yescharacteristic
129C23C22×D10712yes—
130Q8—2142no—
131D4—22no—
132D4—24no—
133D4—2142no—
134D4—2142no—
135D4—2144no—
136C2×C4—1074no—
137C2×C4—1071no—
138C2×C4—1071no—
139C2×C4—1072no—
140C2×C4—1072no—
141C2×C4—1071no—
142C2×C4—2142no—
143C2×C4—2144no—
144C2×C4—2144no—
145C2×C4—2142no—
146C2×C4—2142no—
147C2×C4C22×D10712yes—
148C2×C4C22×D10711yescharacteristic
149C22—28no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—22no—
155C22—2141no—
156C22—2142no—
157C22D4×D10712yes—
158C22#1313381714yescomplemented
159C22C23×D10711yescharacteristic
160C4—21no—
161C4—22no—
162C4—1072no—
163C4—1072no—
164C4—2142no—
165C4—2141no—
166C4—2142no—
167C2—22no—
168C2—24no—
169C2—41no—
170C2—4281no—
171C2C2×D4×D10711yescharacteristic
172C2C2×#1313381712yes—
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
Fitting#120×C107C2
hypercenter#120D107
hypocenterC107#525
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#13133817C2semidirect
2. C23×Dic107C2semidirect
3. #16812093C2semidirect
4. #16812090C2semidirect
5. #16812086C2semidirect
6. #16812081C2semidirect
7. #16812078C2semidirect
8. D4×Dic107C2semidirect
9. (C22⋊D107)⋊C4C2semidirect
10. #16812065C2semidirect
11. #120×C107C2semidirect
13. C2×Dic214C22semidirect
16. C2×C22⋊D107C22semidirect
22. C22×Dic107C22semidirect
23. C2×C4×D107C22semidirect
24. C4⋊Dic107C22semidirect
25. Dic107⋊C4C22semidirect
26. Dic107⋊C4C22semidirect
29. C22⋊Dic107C22semidirect
30. (C2×D107)⋊C4C22semidirect
31. C4×Dic107C22semidirect
54. C2×Dic107D4semidirect
69. C22×C107#41semidirect
86. C107#525semidirect
95. #120D107semidirect
158. C22#13133817semidirect