Finite Group Databasebeta

← #27164694

Subgroups — #27164694

Order 12736.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 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
1#20263948C211yesmaximal, characteristic, complemented
2#20263948C211yesmaximal, characteristic, complemented
3#20263941C211yesmaximal, characteristic, complemented
4#20263939C211yesmaximal, characteristic, complemented
5#20263938C211yesmaximal, characteristic, complemented
6#20263937C211yesmaximal, characteristic, complemented
7#20263934C211yesmaximal, characteristic, complemented
8#20263932C211yesmaximal, characteristic, complemented
9#20263924C211yesmaximal, characteristic
10C4×C22⋊D199C211yesmaximal, characteristic, complemented
11(C22⋊D199)⋊C4C211yesmaximal, characteristic, complemented
12#20263914C211yesmaximal, characteristic, complemented
13#20263911C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic199C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C199C211yesmaximal, characteristic, complemented
16C2×Dic398C2211yescharacteristic, complemented
17C2×C22⋊D199C2211yescharacteristic
18C2×C22⋊D199C2211yescharacteristic
19C2×C22⋊D199C2211yescharacteristic
20C22×Dic199C2211yescharacteristic
21C22×Dic199C2211yescharacteristic
22C2×C4×D199C2211yescharacteristic, complemented
23C4⋊Dic199C2211yescharacteristic, complemented
24C4⋊Dic199C2211yescharacteristic, complemented
25C4⋊Dic199C2211yescharacteristic, complemented
26Dic199⋊C4C2211yescharacteristic, complemented
27Dic199⋊C4C2211yescharacteristic, complemented
28Dic199⋊C4C2211yescharacteristic, complemented
29Dic199⋊C4C2211yescharacteristic, complemented
30Dic199⋊C4C2211yescharacteristic, complemented
31C22⋊Dic199C2211yescharacteristic
32C22⋊Dic199C2211yescharacteristic
33C22⋊Dic199C2211yescharacteristic
34C22⋊Dic199C2211yescharacteristic
35C22⋊Dic199C2211yescharacteristic
36C22⋊Dic199C2211yescharacteristic
37C22⋊Dic199C2211yescharacteristic
38(C2×D199)⋊C4C2211yescharacteristic, complemented
39(C2×D199)⋊C4C2211yescharacteristic, complemented
40(C2×D199)⋊C4C2211yescharacteristic, complemented
41C4×Dic199C2211yescharacteristic, complemented
42C4×Dic199C2211yescharacteristic, complemented
43C4×Dic199C2211yescharacteristic, complemented
44C24×C199C2211yescharacteristic
45C22×C4×C199C2211yescharacteristic
46C22×C4×C199C2211yescharacteristic
47C22⋊C4×C199C2211yescharacteristic
48C22⋊C4×C199C2211yescharacteristic
49C22⋊C4×C199C2211yescharacteristic
50C22⋊C4×C199C2211yescharacteristic
51C22×D199C2311yescharacteristic
52Dic398—41no—
53C22⋊D199—22no—
54C22⋊D199—22no—
55C22⋊D199—41no—
56C2×Dic199—22no—
57C2×Dic199—41no—
58C2×Dic199C2311yescharacteristic
59C2×Dic199C2311yescharacteristic
60C2×Dic199C2311yescharacteristic
61C2×Dic199C2311yescharacteristic
62C2×Dic199C2311yescharacteristic
63C2×Dic199C2311yescharacteristic
64C2×Dic199C2311yescharacteristic
65C4×D199—41no—
66C23×C199—22no—
67C23×C199—22no—
68C23×C199—41no—
69C23×C199C2311yescharacteristic
70C23×C199C2311yescharacteristic
71C23×C199C2311yescharacteristic
72C2×C4×C199—22no—
73C2×C4×C199—41no—
74C2×C4×C199C2311yescharacteristic
75C2×C4×C199C2311yescharacteristic
76C2×C4×C199C2311yescharacteristic
77C2×C4×C199C2311yescharacteristic
78C22×C199—21no—
79C22×C199—21no—
80C22×C199—21no—
81C22×C199—21no—
82C22×C199—21no—
83C22×C199—21no—
84C22×C199—21no—
85C22×C199—44no—
86C22×C199#4112yes—
87C22×C199#4112yes—
88C22×C199C2411yescharacteristic
89C2×D199—21no—
90C2×D199—21no—
91C2×D199—21no—
92C4×C199—21no—
93C4×C199—21no—
94C4×C199—21no—
95C4×C199—21no—
96Dic199—21no—
97Dic199—21no—
98Dic199—21no—
99Dic199—21no—
100Dic199—21no—
101Dic199—21no—
102Dic199—21no—
103C2×C199—22no—
104C2×C199—22no—
105C2×C199—41no—
106C2×C199C2×#4111yescharacteristic
107C2×C199C2×#4111yescharacteristic
108C2×C199H(5,𝔽2)11yescharacteristic
109D199—41no—
110C199#55111yescharacteristic, complemented, Sylow
111#551—1991nomaximal, Sylow
112#126—1991no—
113#126—1991no—
114#124—1991no—
115#123—1991no—
116#123—1991no—
117#123—1991no—
118C22⋊Q8—1991no—
119C22⋊Q8—1991no—
120#120—1991no—
121C4×D4—1991no—
122C4×D4—1991no—
123#117—1991no—
124#117—1991no—
125C2×C22⋊C4—1991no—
126C2×C22⋊C4D19911yescharacteristic, complemented
127C24C2×D19911yescharacteristic
128C2×Q8—1991no—
129C2×D4—1991no—
130C2×D4—1991no—
131C2×D4—1991no—
132C22×C4—1991no—
133C22×C4—1991no—
134C22×C4—1991no—
135C22×C4C2×D19911yescharacteristic
136C22×C4C2×D19911yescharacteristic
137C4⋊C4—1991no—
138C4⋊C4—1991no—
139C4⋊C4—1991no—
140C4⋊C4—1991no—
141C4⋊C4—1991no—
142C4⋊C4—1991no—
143C4⋊C4—1991no—
144C4⋊C4—1991no—
145C22⋊C4—1991no—
146C22⋊C4—1991no—
147C22⋊C4—1991no—
148C22⋊C4—1991no—
149C22⋊C4—1991no—
150C22⋊C4—1991no—
151C22⋊C4—1991no—
152C22⋊C4—1991no—
153C22⋊C4—1991no—
154C22⋊C4—1991no—
155C22⋊C4C2×D19911yescharacteristic
156C22⋊C4C2×D19911yescharacteristic
157C22⋊C4C2×D19911yescharacteristic
158C22⋊C4C2×D19911yescharacteristic
159C42—1991no—
160C42—1991no—
161C42—1991no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—1991no—
166C23C22×D19911yescharacteristic
167C23C22×D19911yescharacteristic
168C23C22×D19911yescharacteristic
169Q8—7961no—
170D4—7961no—
171D4—3982no—
172D4—3982no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—7961no—
176C2×C4—7961no—
177C2×C4—3982no—
178C2×C4—1991no—
179C2×C4—1991no—
180C2×C4—1991no—
181C2×C4—1991no—
182C2×C4—1991no—
183C2×C4—1991no—
184C2×C4—1991no—
185C2×C4C22×D19911yescharacteristic
186C2×C4C22×D19911yescharacteristic
187C2×C4C22×D19911yescharacteristic
188C2×C4C22×D19911yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—44no—
197C22—3981no—
198C22—3981no—
199C22—3981no—
200C22#1569188512yes—
201C22#1569188612yes—
202C22C23×D19911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—3981no—
208C4—3981no—
209C4—3981no—
210C4—3981no—
211C4—3981no—
212C4—3981no—
213C4—3981no—
214C2—22no—
215C2—22no—
216C2—41no—
217C2—7961no—
218C2C2×#1569188511yescharacteristic
219C2C2×#1569188611yescharacteristic
220C2#2026400011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
FittingC2×C22⋊C4×C199C2
prime radical#20263924C2
hypercenterC2×C22⋊C4D199
hypocenterC199#551
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20263948C2semidirect
2. #20263948C2semidirect
3. #20263941C2semidirect
4. #20263939C2semidirect
5. #20263938C2semidirect
6. #20263937C2semidirect
7. #20263934C2semidirect
8. #20263932C2semidirect
10. C4×C22⋊D199C2semidirect
11. (C22⋊D199)⋊C4C2semidirect
12. #20263914C2semidirect
13. #20263911C2semidirect
14. C2×C22⋊Dic199C2semidirect
15. C2×C22⋊C4×C199C2semidirect
16. C2×Dic398C22semidirect
22. C2×C4×D199C22semidirect
23. C4⋊Dic199C22semidirect
24. C4⋊Dic199C22semidirect
25. C4⋊Dic199C22semidirect
26. Dic199⋊C4C22semidirect
27. Dic199⋊C4C22semidirect
28. Dic199⋊C4C22semidirect
29. Dic199⋊C4C22semidirect
30. Dic199⋊C4C22semidirect
38. (C2×D199)⋊C4C22semidirect
39. (C2×D199)⋊C4C22semidirect
40. (C2×D199)⋊C4C22semidirect
41. C4×Dic199C22semidirect
42. C4×Dic199C22semidirect
43. C4×Dic199C22semidirect
110. C199#551semidirect
126. C2×C22⋊C4D199semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202639482
#202639412
#202639392
#202639382
#202639372
#202639342
#202639322
#202639242
C4×C22⋊D1992
(C22⋊D199)⋊C42
#202639142
#202639112
C2×C22⋊Dic1992
C2×C22⋊C4×C1992
#551199