Finite Group Databasebeta

← #20293216

Subgroups — #20293216

Order 6848.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#16812104C211yesmaximal, characteristic, complemented
2#16812097C211yesmaximal, characteristic, complemented
3#16812095C211yesmaximal, characteristic, complemented
4#16812094C211yesmaximal, characteristic, complemented
5#16812091C211yesmaximal, characteristic, complemented
6#16812087C211yesmaximal, characteristic, complemented
7#16812084C211yesmaximal, characteristic, complemented
8#16812084C211yesmaximal, characteristic, complemented
9#16812082C211yesmaximal, characteristic, complemented
10#16812079C211yesmaximal, characteristic, complemented
11#16812079C211yesmaximal, characteristic, complemented
12C4×D428C211yesmaximal, characteristic, complemented
13#16812071C211yesmaximal, characteristic, complemented
14C22⋊C4×D107C211yesmaximal, characteristic, complemented
15#126×C107C211yesmaximal, characteristic, complemented
16C23×D107C2211yescharacteristic
17C2×Dic214C2211yescharacteristic, complemented
18C2×D428C2211yescharacteristic
19C2×D428C2211yescharacteristic
20C2×D428C2211yescharacteristic
21C2×D428C2211yescharacteristic
22C2×D428C2211yescharacteristic, complemented
23C2×C22⋊D107C2211yescharacteristic
24C2×C22⋊D107C2211yescharacteristic
25C2×C4×D107C2211yescharacteristic
26C2×C4×D107C2211yescharacteristic
27C2×C4×D107C2211yescharacteristic, complemented
28C2×C4×D107C2211yescharacteristic, complemented
29C4⋊Dic107C2211yescharacteristic, complemented
30Dic107⋊C4C2211yescharacteristic, complemented
31Dic107⋊C4C2211yescharacteristic, complemented
32C22⋊Dic107C2211yescharacteristic
33(C2×D107)⋊C4C2211yescharacteristic
34(C2×D107)⋊C4C2211yescharacteristic
35(C2×D107)⋊C4C2211yescharacteristic
36(C2×D107)⋊C4C2211yescharacteristic
37(C2×D107)⋊C4C2211yescharacteristic
38(C2×D107)⋊C4C2211yescharacteristic
39(C2×D107)⋊C4C2211yescharacteristic, complemented
40(C2×D107)⋊C4C2211yescharacteristic, complemented
41(C2×D107)⋊C4C2211yescharacteristic, complemented
42(C2×D107)⋊C4C2211yescharacteristic, complemented
43C4×Dic107C2211yescharacteristic, complemented
44C4⋊C4×C107C2211yescharacteristic, complemented
45C4⋊C4×C107C2211yescharacteristic, complemented
46C4⋊C4×C107C2211yescharacteristic, complemented
47C22⋊C4×C107C2211yescharacteristic
48C22⋊C4×C107C2211yescharacteristic
49C22⋊C4×C107C2211yescharacteristic
50C42×C107C2211yescharacteristic, complemented
51C23×C107C2311yescharacteristic
52C22×D107—22no—
53C22×D107—41no—
54C22×D107—41no—
55C22×D107C2311yescharacteristic
56C22×D107C2311yescharacteristic
57C22×D107C2311yescharacteristic
58C22×D107C2311yescharacteristic
59C2×C4×C107C2311yescharacteristic
60C2×C4×C107C2311yescharacteristic
61C2×C4×C107C2311yescharacteristic
62C2×C4×C107C2311yescharacteristic
63C2×C4×C107C2311yescharacteristic
64C2×C4×C107C2311yescharacteristic
65C22⋊D107—41no—
66C22⋊D107—41no—
67C2×Dic107C2311yescharacteristic
68C2×Dic107C2311yescharacteristic
69C2×Dic107C2311yescharacteristic
70C2×Dic107C2311yescharacteristic
71D428—22no—
72D428—22no—
73D428—41no—
74D428—41no—
75D428—41no—
76C4×D107—41no—
77C4×D107—41no—
78C4×D107—41no—
79C4×D107—41no—
80Dic214—41no—
81C22×C107—21no—
82C22×C107—21no—
83C22×C107—21no—
84C22×C107C2411yescharacteristic
85C2×D107—21no—
86C2×D107—21no—
87C2×D107—21no—
88C2×D107—21no—
89C2×D107—21no—
90C2×D107—21no—
91C2×D107—21no—
92C2×D107—21no—
93C2×D107—21no—
94C2×D107—21no—
95C2×D107—21no—
96C2×D107—44no—
97C2×D107#4112yes—
98C4×C107—21no—
99C4×C107—21no—
100C4×C107—21no—
101C4×C107—21no—
102C4×C107—21no—
103C4×C107—21no—
104Dic107—21no—
105Dic107—21no—
106Dic107—21no—
107Dic107—21no—
108C2×C107—41no—
109C2×C107C2×#4111yescharacteristic
110C2×C107H(5,𝔽2)11yescharacteristic
111C2×C107H(5,𝔽2)11yescharacteristic
112D107—22no—
113D107—41no—
114D107—41no—
115D107—41no—
116C107#53811yescharacteristic, complemented, Sylow
117#538—1071nomaximal, Sylow
118#126—1071no—
119#126D10711yescharacteristic, complemented
120#124—1071no—
121#124—1071no—
122#123—1071no—
123#123—1071no—
124C22⋊Q8—1071no—
125#121—1071no—
126#121—1071no—
127#121—1071no—
128#120—1071no—
129#120—1071no—
130C4×D4—1071no—
131C4×D4—1071no—
132C2×C22⋊C4—1071no—
133C24—1071no—
134C2×Q8—1071no—
135C2×D4—1071no—
136C2×D4—1071no—
137C2×D4—1071no—
138C2×D4—1071no—
139C2×D4—1071no—
140C2×D4—1071no—
141C2×D4—1071no—
142C22×C4—1071no—
143C22×C4—1071no—
144C22×C4—1071no—
145C22×C4—1071no—
146C4⋊C4—1071no—
147C4⋊C4—1071no—
148C4⋊C4—1071no—
149C4⋊C4C2×D10711yescharacteristic, complemented
150C4⋊C4C2×D10711yescharacteristic, complemented
151C4⋊C4C2×D10711yescharacteristic, complemented
152C22⋊C4—1071no—
153C22⋊C4—1071no—
154C22⋊C4—1071no—
155C22⋊C4—1071no—
156C22⋊C4—1071no—
157C22⋊C4—1071no—
158C22⋊C4—1071no—
159C22⋊C4—1071no—
160C22⋊C4—1071no—
161C22⋊C4—1071no—
162C22⋊C4—1071no—
163C22⋊C4C2×D10711yescharacteristic
164C22⋊C4C2×D10711yescharacteristic
165C22⋊C4C2×D10711yescharacteristic
166C42—1071no—
167C42C2×D10711yescharacteristic, complemented
168C23—1071no—
169C23—1071no—
170C23—1071no—
171C23—1071no—
172C23—4281no—
173C23—4281no—
174C23—2142no—
175C23C22×D10711yescharacteristic
176Q8—4281no—
177D4—4281no—
178D4—4281no—
179D4—4281no—
180D4—4281no—
181D4—4281no—
182D4—2142no—
183D4—2142no—
184C2×C4—1071no—
185C2×C4—1071no—
186C2×C4—1071no—
187C2×C4—1071no—
188C2×C4—4281no—
189C2×C4—4281no—
190C2×C4—4281no—
191C2×C4—4281no—
192C2×C4C22×D10711yescharacteristic
193C2×C4C22×D10711yescharacteristic
194C2×C4C22×D10711yescharacteristic
195C2×C4C22×D10711yescharacteristic
196C2×C4C22×D10711yescharacteristic
197C2×C4C22×D10711yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—1072no—
202C22—4284no—
203C22—2141no—
204C22—2141no—
205C22—2141no—
206C22—2141no—
207C22—2141no—
208C22—2141no—
209C22—2141no—
210C22—2141no—
211C22—2141no—
212C22—2141no—
213C22—2141no—
214C22C23×D10711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—2141no—
222C4—2141no—
223C4—2141no—
224C4—2141no—
225C2—41no—
226C2—4281no—
227C2—4281no—
228C2—4281no—
229C2—2142no—
230C2#41×D10711yescharacteristic
231C2#1681215511yescharacteristic
232C2#1681215511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
Fitting#126×C107C2
hypercenter#126D107
hypocenterC107#538
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #16812104C2semidirect
2. #16812097C2semidirect
3. #16812095C2semidirect
4. #16812094C2semidirect
5. #16812091C2semidirect
6. #16812087C2semidirect
7. #16812084C2semidirect
8. #16812084C2semidirect
9. #16812082C2semidirect
10. #16812079C2semidirect
11. #16812079C2semidirect
12. C4×D428C2semidirect
13. #16812071C2semidirect
14. C22⋊C4×D107C2semidirect
15. #126×C107C2semidirect
17. C2×Dic214C22semidirect
22. C2×D428C22semidirect
27. C2×C4×D107C22semidirect
28. C2×C4×D107C22semidirect
29. C4⋊Dic107C22semidirect
30. Dic107⋊C4C22semidirect
31. Dic107⋊C4C22semidirect
39. (C2×D107)⋊C4C22semidirect
40. (C2×D107)⋊C4C22semidirect
41. (C2×D107)⋊C4C22semidirect
42. (C2×D107)⋊C4C22semidirect
43. C4×Dic107C22semidirect
44. C4⋊C4×C107C22semidirect
45. C4⋊C4×C107C22semidirect
46. C4⋊C4×C107C22semidirect
50. C42×C107C22semidirect
116. C107#538semidirect
119. #126D107semidirect
149. C4⋊C4C2×D107semidirect
150. C4⋊C4C2×D107semidirect
151. C4⋊C4C2×D107semidirect
167. C42C2×D107semidirect