Finite Group Databasebeta

← #44408023

Subgroups — #44408023

Order 29056.

1102 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C4×D227C211yesmaximal, characteristic, complemented
2C2×#20300269C211yesmaximal, characteristic, complemented
3C2×#20300256C211yesmaximal, characteristic
4#28408908C214yesmaximal, complemented
5#28408903C214yesmaximal, complemented
6#28408895C211yesmaximal, characteristic, complemented
7#28408893C211yesmaximal, characteristic, complemented
8#28408887C211yesmaximal, characteristic, complemented
9C2×#123×C227C211yesmaximal, characteristic, complemented
10C24×D227C2211yescharacteristic
11C22×C22⋊D227C2212yes—
12C23×Dic227C2211yescharacteristic
13C22×C4×D227—24no—
14C22×C4×D227—24no—
15C22×C4×D227—24no—
16C22×C4×D227C2214yes—
17#20300269—24no—
18#20300256—24no—
19C2×Dic227⋊C4C2212yescomplemented
20C2×C22⋊Dic227C2211yescharacteristic
21C2×C22⋊Dic227C2211yescharacteristic
22C2×C22⋊Dic227C2212yescomplemented
23C2×(C2×D227)⋊C4C2214yes—
24C2×(C2×D227)⋊C4C2212yes—
25C227⋊#95C2214yescomplemented
26C227⋊#95C2214yescomplemented
27C22×D4×C227C2211yescharacteristic
28C23×C4×C227C2211yescharacteristic
29#123×C227—24no—
30C2×C4⋊C4×C227C2212yescomplemented
31C2×C22⋊C4×C227C2211yescharacteristic
32C2×C22⋊C4×C227C2212yescomplemented
33C23×D227—28no—
34C23×D227—22no—
35C23×D227—22no—
36C23×D227D414yes—
37C23×D227C2312yes—
38C2×C22⋊D227—24no—
39C2×C22⋊D227—28no—
40C22×Dic227—22no—
41C22×Dic227—22no—
42C22×Dic227D414yescomplemented
43C22×Dic227C2312yes—
44C22×Dic227C2312yes—
45C22×Dic227C2312yes—
46C2×C4×D227—232no—
47C2×C4×D227—28no—
48C2×C4×D227—28no—
49C2×C4×D227—28no—
50Dic227⋊C4—24no—
51C22⋊Dic227—24no—
52C22⋊Dic227—22no—
53C22⋊Dic227—22no—
54(C2×D227)⋊C4—24no—
55(C2×D227)⋊C4—28no—
56C24×C227C2311yescharacteristic
57C24×C227C2311yescharacteristic
58C2×D4×C227—22no—
59C2×D4×C227—22no—
60C2×D4×C227—22no—
61C22×C4×C227—22no—
62C22×C4×C227—22no—
63C22×C4×C227D414yescomplemented
64C22×C4×C227C2312yes—
65C22×C4×C227C2311yescharacteristic
66C22×C4×C227C2312yes—
67C4⋊C4×C227—24no—
68C22⋊C4×C227—22no—
69C22⋊C4×C227—24no—
70C22×D227—216no—
71C22×D227—216no—
72C22×D227—28no—
73C22×D227—22no—
74C22×D227—22no—
75C22×D227—24no—
76C22×D227—216no—
77C22×D227C2×D414yes—
78C22×D227#4118yes—
79C22⋊D227—48no—
80C2×Dic227—28no—
81C2×Dic227—22no—
82C2×Dic227—22no—
83C2×Dic227—22no—
84C2×Dic227—22no—
85C2×Dic227—24no—
86C2×Dic227—24no—
87C2×Dic227C2×D414yes—
88C4×D227—232no—
89C23×C227—24no—
90C23×C227—21no—
91C23×C227—21no—
92C23×C227—21no—
93C23×C227—24no—
94C23×C227C2×D412yes—
95C23×C227C2×D412yes—
96C23×C227C2×D412yes—
97C23×C227C2411yescharacteristic
98D4×C227—44no—
99C2×C4×C227—28no—
100C2×C4×C227—22no—
101C2×C4×C227—22no—
102C2×C4×C227—22no—
103C2×C4×C227—21no—
104C2×C4×C227—21no—
105C2×C4×C227—21no—
106C2×C4×C227—24no—
107C2×C4×C227C2×D414yes—
108C22×C227—24no—
109C22×C227—24no—
110C22×C227—24no—
111C22×C227—41no—
112C22×C227—42no—
113C22×C227—42no—
114C22×C227—42no—
115C22×C227#12014yes—
116C22×C227C22×D411yescharacteristic
117C22×C227C22×D411yescharacteristic
118C22×C227C22×D411yescharacteristic
119C22×C227C2×#4114yes—
120C2×D227—232no—
121C2×D227—24no—
122C2×D227—24no—
123C2×D227—28no—
124C2×D227—28no—
125C2×D227#12318yescomplemented
126C4×C227—24no—
127C4×C227—41no—
128C4×C227—42no—
129Dic227—24no—
130Dic227—42no—
131Dic227—42no—
132C2×C227—24no—
133C2×C227—81no—
134C2×C227C2×#12011yescharacteristic
135C2×C227C2×#12312yes—
136C2×C227#52512yes—
137C2×C227#52512yes—
138D227—28no—
139C227#284911yescharacteristic, complemented, Sylow
140#2849—2271nomaximal, Sylow
141C24×C4—2271no—
142C2×#123—2271no—
143C2×#123D22711yescharacteristic, complemented
144C2×#120—2271no—
145#386—2274no—
146#385—2274no—
147#381—2271no—
148#381—2271no—
149#379—2271no—
150C25—2271no—
151C22×D4—2272no—
152C22×D4C2×D22711yescharacteristic
153C23×C4—4544no—
154C23×C4—4544no—
155C23×C4—4544no—
156C23×C4—2274no—
157C23×C4—2271no—
158C23×C4C2×D22711yescharacteristic
159#123—24no—
160#123—4544no—
161#120—4544no—
162C2×C4⋊C4—2272no—
163C2×C4⋊C4C2×D22712yescomplemented
164C2×C22⋊C4—2271no—
165C2×C22⋊C4—2271no—
166C2×C22⋊C4—2272no—
167C2×C22⋊C4—2272no—
168C2×C22⋊C4—2274no—
169C2×C22⋊C4C2×D22711yescharacteristic
170C2×C22⋊C4C2×D22712yescomplemented
171#95—2274no—
172#95—2274no—
173C24—4548no—
174C24—4542no—
175C24—4542no—
176C24—2274no—
177C24—2272no—
178C24C22×D22711yescharacteristic
179C24C22×D22711yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—4544no—
184C2×D4—4548no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—45432no—
188C22×C4—4548no—
189C22×C4—4548no—
190C22×C4—4542no—
191C22×C4—4542no—
192C22×C4—4548no—
193C22×C4—2272no—
194C22×C4—2272no—
195C22×C4—2274no—
196C22×C4—2272no—
197C22×C4C22⋊D22714yescomplemented
198C22×C4C22×D22712yes—
199C22×C4C22×D22711yescharacteristic
200C22×C4C22×D22712yes—
201C4⋊C4—24no—
202C4⋊C4—4544no—
203C22⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—4548no—
206C22⋊C4—4544no—
207C22⋊C4—4542no—
208C22⋊C4—4542no—
209C22⋊C4—4544no—
210C23—24no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—24no—
215C23—45416no—
216C23—45416no—
217C23—45416no—
218C23—4542no—
219C23—4542no—
220C23—4544no—
221C23—4548no—
222C23—2274no—
223C23—2278no—
224C23D4×D22712yes—
225C23D4×D22712yes—
226C23C2×C22⋊D22712yes—
227C23C23×D22711yescharacteristic
228D4—44no—
229D4—9088no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—28no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—45432no—
239C2×C4—4542no—
240C2×C4—4542no—
241C2×C4—4544no—
242C2×C4—4544no—
243C2×C4—4542no—
244C2×C4—4542no—
245C2×C4—4548no—
246C2×C4—2274no—
247C2×C4C2×C22⋊D22714yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—45432no—
256C22—4544no—
257C22—4544no—
258C22—4548no—
259C22—4548no—
260C22—2278no—
261C22#2030025614yes—
262C22C2×D4×D22711yescharacteristic
263C22C2×D4×D22711yescharacteristic
264C22C22×C22⋊D22711yescharacteristic
265C22#41×D22714yes—
266C4—24no—
267C4—41no—
268C4—42no—
269C4—9082no—
270C4—9082no—
271C4—4544no—
272C2—24no—
273C2—81no—
274C2—4548no—
275C2C2×#2030025611yescharacteristic
276C2#123×D22712yes—
277C2#2840953912yes—
278C2#2840954012yes—
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
FittingC2×#123×C227C2
prime radicalC2×#20300256C2
hypercenterC2×#123D227
hypocenterC227#2849
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×C4×D227C2semidirect
2. C2×#20300269C2semidirect
4. #28408908C2semidirect
5. #28408903C2semidirect
6. #28408895C2semidirect
7. #28408893C2semidirect
8. #28408887C2semidirect
9. C2×#123×C227C2semidirect
19. C2×Dic227⋊C4C22semidirect
22. C2×C22⋊Dic227C22semidirect
25. C227⋊#95C22semidirect
26. C227⋊#95C22semidirect
30. C2×C4⋊C4×C227C22semidirect
32. C2×C22⋊C4×C227C22semidirect
42. C22×Dic227D4semidirect
63. C22×C4×C227D4semidirect
125. C2×D227#123semidirect
139. C227#2849semidirect
143. C2×#123D227semidirect
163. C2×C4⋊C4C2×D227semidirect
170. C2×C22⋊C4C2×D227semidirect
197. C22×C4C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D2272
C2×#203002692
C2×#203002562
#284089082
#284089032
#284088952
#284088932
#284088872
C2×#123×C2272
#2849227