Finite Group Databasebeta

← #44408014

Subgroups — #44408014

Order 29056.

1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C24×Dic227C211yesmaximal, characteristic, complemented
2C2×#20300260C213yesmaximal, complemented
3C2×#20300257C211yesmaximal, characteristic, complemented
4#28408912C213yesmaximal, complemented
5#28408909C213yesmaximal, complemented
6#28408890C213yesmaximal, complemented
7C2×#120×C227C211yesmaximal, characteristic, complemented
8C22×C22⋊D227C2213yes—
9C22×C22⋊D227C2211yescharacteristic, complemented
10C23×Dic227—212no—
11C23×Dic227C2213yes—
12C23×Dic227C2213yes—
13#20300260—212no—
14#20300257—24no—
15C2×Dic227⋊C4C2213yescomplemented
16C2×C22⋊Dic227C2213yes—
17C2×C22⋊Dic227C2213yescomplemented
18C2×(C2×D227)⋊C4C2213yescomplemented
19C227⋊#95C2216yescomplemented
20C25×C227C2211yescharacteristic
21C22×D4×C227C2213yes—
22#120×C227—24no—
23C2×C22⋊C4×C227C2213yes—
24C23×D227C2311yescharacteristic
25C2×C22⋊D227—26no—
26C2×C22⋊D227—26no—
27C2×C22⋊D227—26no—
28C2×C22⋊D227—26no—
29C22×Dic227—232no—
30C22×Dic227—212no—
31C22×Dic227—212no—
32C22×Dic227—26no—
33C22×Dic227D4112yescomplemented
34C22×Dic227C2313yes—
35C22×Dic227C2313yes—
36C22×Dic227C2311yescharacteristic
37Dic227⋊C4—26no—
38C22⋊Dic227—26no—
39C22⋊Dic227—26no—
40(C2×D227)⋊C4—26no—
41C24×C227—26no—
42C24×C227—26no—
43C24×C227D414yescomplemented
44C24×C227C2311yescharacteristic
45C24×C227C2313yes—
46C2×D4×C227—26no—
47C2×D4×C227—26no—
48C2×D4×C227—26no—
49C22×C4×C227C2313yes—
50C22⋊C4×C227—26no—
51C22×D227—23no—
52C22×D227—21no—
53C22×D227—23no—
54C22⋊D227—412no—
55C22⋊D227—44no—
56C2×Dic227—224no—
57C2×Dic227—26no—
58C2×Dic227—224no—
59C2×Dic227—23no—
60C2×Dic227—23no—
61C2×Dic227—23no—
62C2×Dic227C2×D416yes—
63C2×Dic227C2×D416yes—
64C23×C227—224no—
65C23×C227—28no—
66C23×C227—224no—
67C23×C227—26no—
68C23×C227—26no—
69C23×C227—21no—
70C23×C227—23no—
71C23×C227—23no—
72C23×C227C2×D416yes—
73C23×C227C2×D416yes—
74C23×C227#4116yes—
75C23×C227C2411yescharacteristic
76D4×C227—412no—
77C2×C4×C227—23no—
78C2×C4×C227—23no—
79C2×C4×C227—23no—
80C22×C227—232no—
81C22×C227—26no—
82C22×C227—212no—
83C22×C227—26no—
84C22×C227—212no—
85C22×C227—41no—
86C22×C227—43no—
87C22×C227—43no—
88C22×C227#121112yescomplemented
89C22×C227C22×D413yes—
90C22×C227C22×D411yescharacteristic
91C22×C227C2×#4113yes—
92C2×D227—41no—
93C2×D227—43no—
94C2×D227—43no—
95C4×C227—43no—
96Dic227—26no—
97Dic227—43no—
98Dic227#12014yescomplemented
99C2×C227—212no—
100C2×C227—81no—
101C2×C227C2×#12011yescharacteristic
102C2×C227C2×#12113yes—
103C2×C227#52513yes—
104D227—81no—
105C227#284711yescharacteristic, complemented, Sylow
106#2847—2271nomaximal, Sylow
107C24×C4—2271no—
108C2×#121—2273no—
109C2×#120—2271no—
110C2×#120D22711yescharacteristic, complemented
111#386—2273no—
112#386—2273no—
113#380—2273no—
114C25C2×D22711yescharacteristic
115C22×D4—2273no—
116C22×D4—2271no—
117C22×D4C2×D22713yes—
118C23×C4—45412no—
119C23×C4—2273no—
120C23×C4—2273no—
121#121—45412no—
122#120—24no—
123#120—4544no—
124C2×C4⋊C4—2273no—
125C2×C22⋊C4—2273no—
126C2×C22⋊C4—2273no—
127C2×C22⋊C4—2273no—
128C2×C22⋊C4C2×D22713yes—
129#95—2276no—
130C24—26no—
131C24—26no—
132C24—2271no—
133C24C22⋊D22714yescomplemented
134C24C22×D22711yescharacteristic
135C24C22×D22713yes—
136C2×D4—26no—
137C2×D4—26no—
138C2×D4—26no—
139C2×D4—4546no—
140C2×D4—4546no—
141C2×D4—4546no—
142C2×D4—4546no—
143C22×C4—45432no—
144C22×C4—4546no—
145C22×C4—45412no—
146C22×C4—45412no—
147C22×C4—2273no—
148C22×C4—22712no—
149C22×C4—2273no—
150C22×C4—2271no—
151C22×C4C22×D22713yes—
152C4⋊C4—4546no—
153C22⋊C4—26no—
154C22⋊C4—4546no—
155C22⋊C4—4546no—
156C22⋊C4—4546no—
157C23—28no—
158C23—224no—
159C23—26no—
160C23—26no—
161C23—224no—
162C23—23no—
163C23—21no—
164C23—23no—
165C23—4543no—
166C23—4541no—
167C23—4543no—
168C23D4×D22716yes—
169C23C2×C22⋊D22716yes—
170C23#1681388316yes—
171C23C23×D22711yescharacteristic
172D4—412no—
173D4—90812no—
174D4—9084no—
175C2×C4—23no—
176C2×C4—23no—
177C2×C4—23no—
178C2×C4—4543no—
179C2×C4—4543no—
180C2×C4—4543no—
181C2×C4—45424no—
182C2×C4—4546no—
183C2×C4—45424no—
184C2×C4—2276no—
185C2×C4—2276no—
186C22—232no—
187C22—26no—
188C22—212no—
189C22—26no—
190C22—212no—
191C22—41no—
192C22—43no—
193C22—43no—
194C22—9081no—
195C22—9083no—
196C22—9083no—
197C22#20300260112yescomplemented
198C22C2×D4×D22713yes—
199C22C22×C22⋊D22711yescharacteristic
200C22C2×#1681388313yes—
201C4—43no—
202C4—9083no—
203C4—4546no—
204C4—2274no—
205C2—212no—
206C2—81no—
207C2—18161no—
208C2#120×D22711yescharacteristic
209C2C2×#2030026013yes—
210C2#2840953813yes—
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
FittingC2×#120×C227C2
hypercenterC2×#120D227
hypocenterC227#2847
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic227C2semidirect
2. C2×#20300260C2semidirect
3. C2×#20300257C2semidirect
4. #28408912C2semidirect
5. #28408909C2semidirect
6. #28408890C2semidirect
7. C2×#120×C227C2semidirect
9. C22×C22⋊D227C22semidirect
15. C2×Dic227⋊C4C22semidirect
17. C2×C22⋊Dic227C22semidirect
18. C2×(C2×D227)⋊C4C22semidirect
19. C227⋊#95C22semidirect
33. C22×Dic227D4semidirect
43. C24×C227D4semidirect
88. C22×C227#121semidirect
98. Dic227#120semidirect
105. C227#2847semidirect
110. C2×#120D227semidirect
133. C24C22⋊D227semidirect
197. C22#20300260semidirect

Maximal subgroups

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

subgroupindex
C24×Dic2272
C2×#203002602
C2×#203002572
#284089122
#284089092
#284088902
C2×#120×C2272
#2847227