Finite Group Databasebeta

← #42119736

Subgroups — #42119736

Order 27584.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#28352184C211yesmaximal, characteristic, complemented
2#28352184C211yesmaximal, characteristic, complemented
3#28352178C211yesmaximal, characteristic, complemented
4#28352174C211yesmaximal, characteristic, complemented
5#28352174C211yesmaximal, characteristic, complemented
6#28352173C211yesmaximal, characteristic, complemented
7#28352165C211yesmaximal, characteristic, complemented
8#28352159C211yesmaximal, characteristic
9#28352151C211yesmaximal, characteristic, complemented
10C4×C22⋊D431C211yesmaximal, characteristic, complemented
11#28352147C211yesmaximal, characteristic, complemented
12#28352145C211yesmaximal, characteristic, complemented
13C2×(C2×D431)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D431C211yesmaximal, characteristic, complemented
15C22⋊Q8×C431C211yesmaximal, characteristic, complemented
16C23×D431C2211yescharacteristic
17C2×D1724C2211yescharacteristic
18C2×D1724C2211yescharacteristic
19C2×C22⋊D431C2211yescharacteristic
20C22×Dic431C2211yescharacteristic
21C2×C4×D431C2211yescharacteristic
22C2×C4×D431C2211yescharacteristic
23C2×C4×D431C2211yescharacteristic, complemented
24C4⋊Dic431C2211yescharacteristic, complemented
25C4⋊Dic431C2211yescharacteristic, complemented
26Dic431⋊C4C2211yescharacteristic, complemented
27Dic431⋊C4C2211yescharacteristic, complemented
28Dic431⋊C4C2211yescharacteristic, complemented
29C22⋊Dic431C2211yescharacteristic
30(C2×D431)⋊C4C2211yescharacteristic
31(C2×D431)⋊C4C2211yescharacteristic
32(C2×D431)⋊C4C2211yescharacteristic
33(C2×D431)⋊C4C2211yescharacteristic
34(C2×D431)⋊C4C2211yescharacteristic
35(C2×D431)⋊C4C2211yescharacteristic
36(C2×D431)⋊C4C2211yescharacteristic
37(C2×D431)⋊C4C2211yescharacteristic
38(C2×D431)⋊C4C2211yescharacteristic, complemented
39(C2×D431)⋊C4C2211yescharacteristic, complemented
40(C2×D431)⋊C4C2211yescharacteristic, complemented
41C4×Dic431C2211yescharacteristic, complemented
42C4×Dic431C2211yescharacteristic, complemented
43C4×Dic431C2211yescharacteristic, complemented
44C2×Q8×C431C2211yescharacteristic, complemented
45C22×C4×C431C2211yescharacteristic
46C4⋊C4×C431C2211yescharacteristic, complemented
47C4⋊C4×C431C2211yescharacteristic, complemented
48C4⋊C4×C431C2211yescharacteristic, complemented
49C22⋊C4×C431C2211yescharacteristic
50C22⋊C4×C431C2211yescharacteristic
51C22×D431—22no—
52C22×D431—22no—
53C22×D431—41no—
54C22×D431C2311yescharacteristic
55C22×D431C2311yescharacteristic
56C22×D431C2311yescharacteristic
57D1724—22no—
58D1724—41no—
59C22⋊D431—22no—
60C2×Dic431—22no—
61C2×Dic431C2311yescharacteristic
62C2×Dic431C2311yescharacteristic
63C2×Dic431C2311yescharacteristic
64C2×Dic431C2311yescharacteristic
65C2×Dic431C2311yescharacteristic
66C4×D431—22no—
67C4×D431—41no—
68C4×D431—41no—
69C23×C431C2311yescharacteristic
70Q8×C431—41no—
71C2×C4×C431—41no—
72C2×C4×C431C2311yescharacteristic
73C2×C4×C431C2311yescharacteristic
74C2×C4×C431C2311yescharacteristic
75C2×C4×C431C2311yescharacteristic
76C2×C4×C431C2311yescharacteristic
77C2×C4×C431C2311yescharacteristic
78C22×C431—21no—
79C22×C431—21no—
80C22×C431#4112yes—
81C22×C431C2411yescharacteristic
82C2×D431—21no—
83C2×D431—21no—
84C2×D431—21no—
85C2×D431—21no—
86C2×D431—21no—
87C2×D431—21no—
88C2×D431—21no—
89C2×D431—21no—
90C2×D431—44no—
91C2×D431#4112yes—
92C4×C431—21no—
93C4×C431—21no—
94C4×C431—21no—
95C4×C431—21no—
96C4×C431—21no—
97C4×C431—21no—
98Dic431—21no—
99Dic431—21no—
100Dic431—21no—
101Dic431—21no—
102Dic431—21no—
103C2×C431—22no—
104C2×C431C2×#4111yescharacteristic
105C2×C431C2×#4111yescharacteristic
106C2×C431H(5,𝔽2)11yescharacteristic
107D431—22no—
108D431—41no—
109D431—41no—
110C431#55111yescharacteristic, complemented, Sylow
111#551—4311nomaximal, Sylow
112#126—4311no—
113#126—4311no—
114#124—4311no—
115#123—4311no—
116#123—4311no—
117#123—4311no—
118C22⋊Q8—4311no—
119C22⋊Q8D43111yescharacteristic, complemented
120#120—4311no—
121C4×D4—4311no—
122C4×D4—4311no—
123#117—4311no—
124#117—4311no—
125C2×C22⋊C4—4311no—
126C2×C22⋊C4—4311no—
127C24—4311no—
128C2×Q8C2×D43111yescharacteristic, complemented
129C2×D4—4311no—
130C2×D4—4311no—
131C2×D4—4311no—
132C22×C4—4311no—
133C22×C4—4311no—
134C22×C4—4311no—
135C22×C4—4311no—
136C22×C4C2×D43111yescharacteristic
137C4⋊C4—4311no—
138C4⋊C4—4311no—
139C4⋊C4—4311no—
140C4⋊C4—4311no—
141C4⋊C4—4311no—
142C4⋊C4C2×D43111yescharacteristic, complemented
143C4⋊C4C2×D43111yescharacteristic, complemented
144C4⋊C4C2×D43111yescharacteristic, complemented
145C22⋊C4—4311no—
146C22⋊C4—4311no—
147C22⋊C4—4311no—
148C22⋊C4—4311no—
149C22⋊C4—4311no—
150C22⋊C4—4311no—
151C22⋊C4—4311no—
152C22⋊C4—4311no—
153C22⋊C4—4311no—
154C22⋊C4—4311no—
155C22⋊C4—4311no—
156C22⋊C4—4311no—
157C22⋊C4C2×D43111yescharacteristic
158C22⋊C4C2×D43111yescharacteristic
159C42—4311no—
160C42—4311no—
161C42—4311no—
162C23—8622no—
163C23—8622no—
164C23—4311no—
165C23—4311no—
166C23—4311no—
167C23—17241no—
168C23C22×D43111yescharacteristic
169Q8—41no—
170D4—8622no—
171D4—8622no—
172D4—17241no—
173C2×C4—41no—
174C2×C4—8622no—
175C2×C4—8622no—
176C2×C4—4311no—
177C2×C4—4311no—
178C2×C4—4311no—
179C2×C4—4311no—
180C2×C4—4311no—
181C2×C4—17241no—
182C2×C4—17241no—
183C2×C4C22×D43111yescharacteristic
184C2×C4C22×D43111yescharacteristic
185C2×C4C22×D43111yescharacteristic
186C2×C4C22×D43111yescharacteristic
187C2×C4C22×D43111yescharacteristic
188C2×C4C22×D43111yescharacteristic
189C22—21no—
190C22—21no—
191C22—8621no—
192C22—8621no—
193C22—8621no—
194C22—8621no—
195C22—8621no—
196C22—8621no—
197C22—8621no—
198C22—8621no—
199C22—4312no—
200C22—17244no—
201C22#2029380112yes—
202C22C23×D43111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—8621no—
210C4—8621no—
211C4—8621no—
212C4—8621no—
213C4—8621no—
214C2—22no—
215C2—8622no—
216C2—17241no—
217C2—17241no—
218C2#41×D43111yescharacteristic
219C2C2×#2029380111yescharacteristic
220C2#2835223411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
FittingC22⋊Q8×C431C2
prime radical#28352159C2
hypercenterC22⋊Q8D431
hypocenterC431#551
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #28352184C2semidirect
2. #28352184C2semidirect
3. #28352178C2semidirect
4. #28352174C2semidirect
5. #28352174C2semidirect
6. #28352173C2semidirect
7. #28352165C2semidirect
9. #28352151C2semidirect
10. C4×C22⋊D431C2semidirect
11. #28352147C2semidirect
12. #28352145C2semidirect
13. C2×(C2×D431)⋊C4C2semidirect
14. C22⋊C4×D431C2semidirect
15. C22⋊Q8×C431C2semidirect
23. C2×C4×D431C22semidirect
24. C4⋊Dic431C22semidirect
25. C4⋊Dic431C22semidirect
26. Dic431⋊C4C22semidirect
27. Dic431⋊C4C22semidirect
28. Dic431⋊C4C22semidirect
38. (C2×D431)⋊C4C22semidirect
39. (C2×D431)⋊C4C22semidirect
40. (C2×D431)⋊C4C22semidirect
41. C4×Dic431C22semidirect
42. C4×Dic431C22semidirect
43. C4×Dic431C22semidirect
44. C2×Q8×C431C22semidirect
46. C4⋊C4×C431C22semidirect
47. C4⋊C4×C431C22semidirect
48. C4⋊C4×C431C22semidirect
110. C431#551semidirect
119. C22⋊Q8D431semidirect
128. C2×Q8C2×D431semidirect
142. C4⋊C4C2×D431semidirect
143. C4⋊C4C2×D431semidirect
144. C4⋊C4C2×D431semidirect

Maximal subgroups

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

subgroupindex
#283521842
#283521782
#283521742
#283521732
#283521652
#283521592
#283521512
C4×C22⋊D4312
#283521472
#283521452
C2×(C2×D431)⋊C42
C22⋊C4×D4312
C22⋊Q8×C4312
#551431