Finite Group Databasebeta

← #31824048

Subgroups — #31824048

Order 17984.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#22582872C211yesmaximal, characteristic, complemented
2#22582872C211yesmaximal, characteristic, complemented
3#22582866C211yesmaximal, characteristic, complemented
4#22582865C211yesmaximal, characteristic, complemented
5#22582861C211yesmaximal, characteristic, complemented
6#22582860C211yesmaximal, characteristic, complemented
7#22582856C211yesmaximal, characteristic, complemented
8#22582851C211yesmaximal, characteristic, complemented
9#22582851C211yesmaximal, characteristic, complemented
10#22582850C211yesmaximal, characteristic, complemented
11#22582848C211yesmaximal, characteristic, complemented
12D4×Dic281C211yesmaximal, characteristic, complemented
13#22582839C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic281C211yesmaximal, characteristic, complemented
15#120×C281C211yesmaximal, characteristic, complemented
16C2×Dic562C2211yescharacteristic, complemented
17C2×C22⋊D281C2211yescharacteristic
18C2×C22⋊D281C2211yescharacteristic
19C2×C22⋊D281C2211yescharacteristic
20C2×C22⋊D281C2211yescharacteristic, complemented
21C22×Dic281C2211yescharacteristic
22C22×Dic281C2211yescharacteristic
23C22×Dic281C2211yescharacteristic, complemented
24C2×C4×D281C2211yescharacteristic, complemented
25C4⋊Dic281C2211yescharacteristic, complemented
26C4⋊Dic281C2211yescharacteristic, complemented
27#17985117C2211yescharacteristic, complemented
28#17985117C2211yescharacteristic, complemented
29#17985117C2211yescharacteristic, complemented
30#17985117C2211yescharacteristic, complemented
31C22⋊Dic281C2211yescharacteristic
32C22⋊Dic281C2211yescharacteristic
33C22⋊Dic281C2211yescharacteristic
34C22⋊Dic281C2211yescharacteristic
35C22⋊Dic281C2211yescharacteristic
36C22⋊Dic281C2211yescharacteristic
37C22⋊Dic281C2211yescharacteristic
38C22⋊Dic281C2211yescharacteristic, complemented
39C22⋊Dic281C2211yescharacteristic, complemented
40#17985115C2211yescharacteristic, complemented
41#17985115C2211yescharacteristic, complemented
42C4×Dic281C2211yescharacteristic, complemented
43C4×Dic281C2211yescharacteristic, complemented
44C24×C281C2211yescharacteristic
45C2×D4×C281C2211yescharacteristic
46C2×D4×C281C2211yescharacteristic
47C2×D4×C281C2211yescharacteristic
48C22⋊C4×C281C2211yescharacteristic
49C22⋊C4×C281C2211yescharacteristic
50C22⋊C4×C281C2211yescharacteristic
51C22×D281C2311yescharacteristic
52Dic562—41no—
53C22⋊D281—22no—
54C22⋊D281—41no—
55C22⋊D281—41no—
56C22⋊D281—41no—
57C2×Dic281—41no—
58C2×Dic281—41no—
59C2×Dic281—41no—
60C2×Dic281C2311yescharacteristic
61C2×Dic281C2311yescharacteristic
62C2×Dic281C2311yescharacteristic
63C2×Dic281C2311yescharacteristic
64C2×Dic281C2311yescharacteristic
65C2×Dic281C2311yescharacteristic
66C2×Dic281C2311yescharacteristic
67C4×D281—41no—
68C23×C281—22no—
69C23×C281—41no—
70C23×C281—41no—
71C23×C281C2311yescharacteristic
72C23×C281C2311yescharacteristic
73C23×C281C2311yescharacteristic
74C23×C281C2311yescharacteristic
75D4×C281—22no—
76D4×C281—41no—
77D4×C281—41no—
78C2×C4×C281C2311yescharacteristic
79C2×C4×C281C2311yescharacteristic
80C2×C4×C281C2311yescharacteristic
81C22×C281—21no—
82C22×C281—21no—
83C22×C281—21no—
84C22×C281—21no—
85C22×C281—21no—
86C22×C281—21no—
87C22×C281—21no—
88C22×C281—21no—
89C22×C281—21no—
90C22×C281—21no—
91C22×C281—21no—
92C22×C281—44no—
93C22×C281#4112yes—
94C22×C281C2411yescharacteristic
95C2×D281—21no—
96C2×D281—21no—
97C2×D281—21no—
98C4×C281—21no—
99C4×C281—21no—
100C4×C281—21no—
101Dic281—21no—
102Dic281—21no—
103Dic281—21no—
104Dic281—21no—
105Dic281—21no—
106Dic281—21no—
107Dic281—21no—
108C2×C281—22no—
109C2×C281—41no—
110C2×C281—41no—
111C2×C281—41no—
112C2×C281C2×#4111yescharacteristic
113C2×C281H(5,𝔽2)11yescharacteristic
114C2×C281H(5,𝔽2)11yescharacteristic
115D281—41no—
116C281#53811yescharacteristic, complemented, Sylow
117#538—2811nomaximal, Sylow
118#126—2811no—
119#126—2811no—
120#124—2811no—
121#124—2811no—
122#123—2811no—
123#123—2811no—
124C22⋊Q8—2811no—
125#121—2811no—
126#121—2811no—
127#121—2811no—
128#120—2811no—
129#120D28111yescharacteristic, complemented
130C4×D4—2811no—
131C4×D4—2811no—
132C2×C22⋊C4—2811no—
133C24C2×D28111yescharacteristic
134C2×Q8—2811no—
135C2×D4—2811no—
136C2×D4—2811no—
137C2×D4—2811no—
138C2×D4—2811no—
139C2×D4C2×D28111yescharacteristic
140C2×D4C2×D28111yescharacteristic
141C2×D4C2×D28111yescharacteristic
142C22×C4—2811no—
143C22×C4—2811no—
144C22×C4—2811no—
145C22×C4—2811no—
146C4⋊C4—2811no—
147C4⋊C4—2811no—
148C4⋊C4—2811no—
149C4⋊C4—2811no—
150C4⋊C4—2811no—
151C4⋊C4—2811no—
152C22⋊C4—2811no—
153C22⋊C4—2811no—
154C22⋊C4—2811no—
155C22⋊C4—2811no—
156C22⋊C4—2811no—
157C22⋊C4—2811no—
158C22⋊C4—2811no—
159C22⋊C4—2811no—
160C22⋊C4—2811no—
161C22⋊C4—2811no—
162C22⋊C4—2811no—
163C22⋊C4C2×D28111yescharacteristic
164C22⋊C4C2×D28111yescharacteristic
165C22⋊C4C2×D28111yescharacteristic
166C42—2811no—
167C42—2811no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—2811no—
172C23C22×D28111yescharacteristic
173C23C22×D28111yescharacteristic
174C23C22×D28111yescharacteristic
175C23C22×D28111yescharacteristic
176Q8—11241no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—5622no—
181D4—11241no—
182D4—11241no—
183D4—11241no—
184C2×C4—2811no—
185C2×C4—2811no—
186C2×C4—2811no—
187C2×C4—2811no—
188C2×C4—2811no—
189C2×C4—2811no—
190C2×C4—2811no—
191C2×C4—11241no—
192C2×C4—11241no—
193C2×C4—11241no—
194C2×C4—11241no—
195C2×C4C22×D28111yescharacteristic
196C2×C4C22×D28111yescharacteristic
197C2×C4C22×D28111yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—5621no—
211C22—5621no—
212C22—5621no—
213C22#1798513712yes—
214C22C23×D28111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—5621no—
219C4—5621no—
220C4—5621no—
221C4—5621no—
222C4—5621no—
223C4—5621no—
224C4—5621no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—11241no—
230C2C2×#1798513711yescharacteristic
231C2#2258292411yescharacteristic
232C2#2258292411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D281
commutator, socle, cosocleC22×C281C24
Fitting#120×C281C2
hypercenter#120D281
hypocenterC281#538
LCSG ▷ C22×C281 ▷ C281
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #22582872C2semidirect
2. #22582872C2semidirect
3. #22582866C2semidirect
4. #22582865C2semidirect
5. #22582861C2semidirect
6. #22582860C2semidirect
7. #22582856C2semidirect
8. #22582851C2semidirect
9. #22582851C2semidirect
10. #22582850C2semidirect
11. #22582848C2semidirect
12. D4×Dic281C2semidirect
13. #22582839C2semidirect
14. C2×C22⋊Dic281C2semidirect
15. #120×C281C2semidirect
16. C2×Dic562C22semidirect
20. C2×C22⋊D281C22semidirect
23. C22×Dic281C22semidirect
24. C2×C4×D281C22semidirect
25. C4⋊Dic281C22semidirect
26. C4⋊Dic281C22semidirect
27. #17985117C22semidirect
28. #17985117C22semidirect
29. #17985117C22semidirect
30. #17985117C22semidirect
38. C22⋊Dic281C22semidirect
39. C22⋊Dic281C22semidirect
40. #17985115C22semidirect
41. #17985115C22semidirect
42. C4×Dic281C22semidirect
43. C4×Dic281C22semidirect
116. C281#538semidirect
129. #120D281semidirect

Maximal subgroups

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

subgroupindex
#225828722
#225828662
#225828652
#225828612
#225828602
#225828562
#225828512
#225828502
#225828482
D4×Dic2812
#225828392
C2×C22⋊Dic2812
#120×C2812
#538281