Finite Group Databasebeta

← #20293423

Subgroups — #20293423

Order 6848.

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×#13133817C211yesmaximal, characteristic, complemented
2#41×D107C211yesmaximal, characteristic, complemented
3#16812142C211yesmaximal, characteristic, complemented
4#16812139C211yesmaximal, characteristic, complemented
5#16812138C212yesmaximal, complemented
6#16812135C211yesmaximal, characteristic, complemented
7#16812133C212yesmaximal, complemented
8#16812130C212yesmaximal, complemented
9#16812128C212yesmaximal, complemented
10M16×D107C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12#13133819C2211yescharacteristic, complemented
13#13133818—21no—
14#13133818C2211yescharacteristic
15#13133817—21no—
16#13133817—21no—
17#13133817—21no—
18#13133817C2212yes—
19#13133817C2211yescharacteristic, complemented
20#13133817C2212yescomplemented
21C2×Dic214C2211yescharacteristic, complemented
22Q8×D107C2211yescharacteristic, complemented
23C2×C22⋊D107—21no—
24D4×D107—21no—
25D4×D107C2211yescharacteristic, complemented
26C22×Dic107—21no—
27C2×C4×D107—21no—
28C2×C4×D107C2211yescharacteristic
29Dic428C2212yescomplemented
30#13133808C2212yescomplemented
31#13133807C2212yescomplemented
32#13133806C2212yescomplemented
33#13133806C2212yescomplemented
34#13133803C2212yescomplemented
35#13133802C2211yescharacteristic, complemented
36#13133801C2212yescomplemented
37C8×D107C2212yescomplemented
38#41×C107C2211yescharacteristic
39C2×D4×C107C2211yescharacteristic, complemented
40SD16×C107C2212yescomplemented
41D8×C107C2212yescomplemented
42M16×C107C2211yescharacteristic, complemented
43C23×C107—21no—
44C22×D107—21no—
45C22×D107D411yescharacteristic
46Q8×C107C2311yescharacteristic
47D4×C107—21no—
48D4×C107—21no—
49D4×C107C2311yescharacteristic
50D4×C107C2312yes—
51C2×C4×C107—21no—
52C2×C4×C107C2311yescharacteristic
53C22⋊D107—21no—
54C22⋊D107—21no—
55C22⋊D107—22no—
56C22⋊D107—22no—
57C22⋊D107—21no—
58C2×Dic107—21no—
59C2×Dic107—21no—
60C2×Dic107—22no—
61C2×Dic107—22no—
62C2×Dic107D411yescharacteristic
63D428—21no—
64D428C2311yescharacteristic
65C4×D107—21no—
66C4×D107—21no—
67C4×D107—21no—
68C4×D107D411yescharacteristic
69C4×D107D411yescharacteristic
70C4×D107C2311yescharacteristic
71C4×D107C2311yescharacteristic
72Dic214—21no—
73Dic214—21no—
74Dic214C2311yescharacteristic
75Dic214C2312yes—
76C8×C107C2312yes—
77C107⋊C8C2312yes—
78C22×C107—22no—
79C22×C107—21no—
80C22×C107—42no—
81C22×C107C2×D411yescharacteristic
82C2×D107—21no—
83C2×D107—41no—
84C2×D107—42no—
85C2×D107C2×D411yescharacteristic
86C2×D107C2×D411yescharacteristic
87C4×C107—21no—
88C4×C107C2×D411yescharacteristic
89C4×C107C2411yescharacteristic
90Dic107—21no—
91Dic107—22no—
92Dic107C2×D411yescharacteristic
93Dic107C2×D411yescharacteristic
94C2×C107—21no—
95C2×C107—41no—
96C2×C107—42no—
97C2×C107C22×D411yescharacteristic
98D107—21no—
99D107—21no—
100D107—41no—
101C107#57511yescharacteristic, complemented, Sylow
102#575—1071nomaximal, Sylow
103C2×#41—1071no—
104C2×#41—1071no—
105#137—1071no—
106#137—1072no—
107#137—1071no—
108AGL(1,ℤ/8)—1071no—
109AGL(1,ℤ/8)—1072no—
110AGL(1,ℤ/8)D10711yescharacteristic, complemented
111#135—1072no—
112#135—1072no—
113C2×M16—1071no—
114#41—1071no—
115#41—1072no—
116#41—1071no—
117#41—1072no—
118#41—1071no—
119#41—2141no—
120#41—2141no—
121#41—2141no—
122#41—2141no—
123#41C2×D10711yescharacteristic
124C2×Q8—1071no—
125C2×Q8—1071no—
126C2×D4—1071no—
127C2×D4—2141no—
128C2×D4—2141no—
129C2×D4C2×D10711yescharacteristic, complemented
130C22×C4—1071no—
131C22×C4—2141no—
132C22×C4—2141no—
133Q16—1072no—
134Q16—1072no—
135SD16—1072no—
136SD16—1072no—
137SD16—1072no—
138SD16C2×D10712yescomplemented
139D8—1072no—
140D8C2×D10712yescomplemented
141M16—1071no—
142M16—1072no—
143M16C2×D10711yescharacteristic, complemented
144C2×C8—1072no—
145C23—21no—
146C23—1071no—
147C23—2141no—
148Q8—1071no—
149Q8—1072no—
150Q8—2141no—
151Q8—2141no—
152Q8C22×D10711yescharacteristic
153D4—21no—
154D4—21no—
155D4—1071no—
156D4—2141no—
157D4—2141no—
158D4—2141no—
159D4—2142no—
160D4—2142no—
161D4—2141no—
162D4C22×D10712yes—
163D4C22×D10711yescharacteristic
164C2×C4—21no—
165C2×C4—1071no—
166C2×C4—1071no—
167C2×C4—1071no—
168C2×C4—1071no—
169C2×C4—1071no—
170C2×C4—2141no—
171C2×C4—2141no—
172C2×C4—2141no—
173C2×C4—2142no—
174C2×C4—2142no—
175C2×C4—2141no—
176C2×C4—2141no—
177C2×C4C22×D10711yescharacteristic
178C8—1072no—
179C8C22×D10712yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—1071no—
184C22—1071no—
185C22—4281no—
186C22—4282no—
187C22—2141no—
188C22D4×D10711yescharacteristic
189C4—21no—
190C4—1071no—
191C4—1071no—
192C4—2141no—
193C4—2142no—
194C4D4×D10711yescharacteristic
195C4C23×D10711yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—4281no—
200C2—2141no—
201C2—2141no—
202C2C2×D4×D10711yescharacteristic
subgroupG/H
centerC2C2×D4×D107
commutatorC4×C107C24
FrattiniC4C23×D107
FittingAGL(1,ℤ/8)×C107C2
socleC2×C107C22×D4
hypercenterAGL(1,ℤ/8)D107
hypocenterC107#575
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#13133817C2semidirect
2. #41×D107C2semidirect
3. #16812142C2semidirect
4. #16812139C2semidirect
5. #16812138C2semidirect
6. #16812135C2semidirect
7. #16812133C2semidirect
8. #16812130C2semidirect
9. #16812128C2semidirect
10. M16×D107C2semidirect
11. AGL(1,ℤ/8)×C107C2semidirect
12. #13133819C22semidirect
19. #13133817C22semidirect
20. #13133817C22semidirect
21. C2×Dic214C22semidirect
22. Q8×D107C22semidirect
25. D4×D107C22semidirect
29. Dic428C22semidirect
30. #13133808C22semidirect
31. #13133807C22semidirect
32. #13133806C22semidirect
33. #13133806C22semidirect
34. #13133803C22semidirect
35. #13133802C22semidirect
36. #13133801C22semidirect
37. C8×D107C22semidirect
39. C2×D4×C107C22semidirect
40. SD16×C107C22semidirect
41. D8×C107C22semidirect
42. M16×C107C22semidirect
101. C107#575semidirect
110. AGL(1,ℤ/8)D107semidirect
129. C2×D4C2×D107semidirect
138. SD16C2×D107semidirect
140. D8C2×D107semidirect
143. M16C2×D107semidirect