Finite Group Databasebeta

← #44408017

Subgroups — #44408017

Order 29056.

1224 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 201 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×#20300263C212yesmaximal, complemented
3C2×#20300262C211yesmaximal, characteristic, complemented
4C2×#20300258C212yesmaximal, complemented
5#28408910C214yesmaximal, complemented
6#28408909C212yesmaximal, complemented
7#28408889C212yesmaximal, complemented
8#380×C227C211yesmaximal, characteristic, complemented
9C24×D227C2211yescharacteristic
10C22×D908C2214yes—
11C22×D908C2211yescharacteristic, complemented
12C22×C22⋊D227C2212yes—
13C23×Dic227C2211yescharacteristic
14C22×C4×D227—28no—
15C22×C4×D227—24no—
16C22×C4×D227C2212yes—
17C22×C4×D227C2212yes—
18#20300263—28no—
19#20300262—24no—
20#20300258—28no—
21C2×C4⋊Dic227C2211yescharacteristic, complemented
22C2×(C2×D227)⋊C4C2214yes—
23C2×(C2×D227)⋊C4C2214yescomplemented
24C2×(C2×D227)⋊C4C2212yescomplemented
25C227⋊#95C2214yescomplemented
26C23×C4×C227C2211yescharacteristic
27C2×C4⋊C4×C227C2212yescomplemented
28C2×C22⋊C4×C227C2212yes—
29#95×C227C2212yescomplemented
30C23×D227—24no—
31C23×D227—24no—
32C23×D227—22no—
33C23×D227—22no—
34C23×D227D414yescomplemented
35C23×D227C2312yes—
36C23×D227C2312yes—
37C2×D908—28no—
38C2×D908—28no—
39C2×D908—28no—
40C2×D908—22no—
41C2×D908—24no—
42C2×C22⋊D227—24no—
43C2×C22⋊D227—24no—
44C2×C22⋊D227—24no—
45C22×Dic227—24no—
46C22×Dic227—22no—
47C22×Dic227C2312yes—
48C22×Dic227C2312yes—
49C2×C4×D227—232no—
50C2×C4×D227—24no—
51C2×C4×D227—24no—
52C2×C4×D227—24no—
53C2×C4×D227—24no—
54C2×C4×D227—24no—
55C2×C4×D227D418yescomplemented
56C4⋊Dic227—22no—
57(C2×D227)⋊C4—28no—
58(C2×D227)⋊C4—28no—
59(C2×D227)⋊C4—24no—
60C24×C227C2311yescharacteristic
61C22×C4×C227—24no—
62C22×C4×C227D414yescomplemented
63C22×C4×C227C2314yes—
64C22×C4×C227C2311yescharacteristic
65C22×C4×C227C2311yescharacteristic
66C4⋊C4×C227—24no—
67C22⋊C4×C227—24no—
68C22×D227—28no—
69C22×D227—216no—
70C22×D227—28no—
71C22×D227—28no—
72C22×D227—22no—
73C22×D227—22no—
74C22×D227—22no—
75C22×D227—22no—
76C22×D227—216no—
77C22×D227—22no—
78C22×D227—24no—
79C22×D227—22no—
80C22×D227—22no—
81C22×D227—24no—
82C22×D227C2×D414yes—
83C22×D227C2×D414yes—
84C22×D227#4114yes—
85D908—416no—
86D908—44no—
87C22⋊D227—48no—
88C2×Dic227—28no—
89C2×Dic227—22no—
90C2×Dic227—22no—
91C2×Dic227—22no—
92C2×Dic227—22no—
93C2×Dic227—22no—
94C2×Dic227C2×D414yes—
95C4×D227—216no—
96C4×D227—216no—
97C23×C227—22no—
98C23×C227—22no—
99C23×C227C2×D412yes—
100C23×C227C2×D412yes—
101C23×C227#4112yes—
102C23×C227C2411yescharacteristic
103C2×C4×C227—22no—
104C2×C4×C227—28no—
105C2×C4×C227—24no—
106C2×C4×C227—24no—
107C2×C4×C227—24no—
108C2×C4×C227C2×D412yes—
109C2×C4×C227C2×D414yes—
110C2×C4×C227C2×D412yes—
111C22×C227—22no—
112C22×C227—24no—
113C22×C227—22no—
114C22×C227—24no—
115C22×C227#12114yescomplemented
116C22×C227C22×D412yes—
117C22×C227C22×D411yescharacteristic
118C22×C227C22×D411yescharacteristic
119C22×C227C2×#4112yes—
120C22×C227C2×#4111yescharacteristic
121C2×D227—232no—
122C2×D227—24no—
123C2×D227—24no—
124C2×D227—24no—
125C2×D227—24no—
126C2×D227—24no—
127C2×D227—24no—
128C2×D227—42no—
129C2×D227—42no—
130C2×D227—44no—
131C2×D227—42no—
132C2×D227—44no—
133C2×D227#12118yescomplemented
134C4×C227—22no—
135C4×C227—44no—
136C4×C227#12014yescomplemented
137Dic227—24no—
138Dic227—42no—
139C2×C227—24no—
140C2×C227C2×#12011yescharacteristic
141C2×C227C2×#12112yes—
142C2×C227C2×#12111yescharacteristic
143C2×C227#52511yescharacteristic
144C2×C227#52512yes—
145D227—28no—
146D227—82no—
147C227#284711yescharacteristic, complemented, Sylow
148#2847—2271nomaximal, Sylow
149C24×C4—2271no—
150C2×#121—2271no—
151C2×#121—2272no—
152C2×#120—2272no—
153#386—2272no—
154#386—2274no—
155#380—2272no—
156#380D22711yescharacteristic, complemented
157C25—2271no—
158C22×D4—2272no—
159C22×D4—2274no—
160C22×D4—2271no—
161C23×C4—4548no—
162C23×C4—4544no—
163C23×C4—2271no—
164C23×C4—2272no—
165C23×C4—2272no—
166C23×C4C2×D22711yescharacteristic
167#121—4548no—
168#121—4544no—
169#120—4548no—
170C2×C4⋊C4—2271no—
171C2×C4⋊C4C2×D22712yescomplemented
172C2×C22⋊C4—2274no—
173C2×C22⋊C4—2274no—
174C2×C22⋊C4—2272no—
175C2×C22⋊C4C2×D22712yes—
176#95—2274no—
177#95C2×D22712yescomplemented
178C24—4544no—
179C24—4544no—
180C24—4542no—
181C24—4542no—
182C24—2272no—
183C24—2274no—
184C24—2272no—
185C24C22×D22711yescharacteristic
186C2×D4—4548no—
187C2×D4—4548no—
188C2×D4—4548no—
189C2×D4—4544no—
190C2×D4—4544no—
191C2×D4—4544no—
192C2×D4—4544no—
193C2×D4—4542no—
194C22×C4—24no—
195C22×C4—45432no—
196C22×C4—4544no—
197C22×C4—4544no—
198C22×C4—4542no—
199C22×C4—4544no—
200C22×C4—4544no—
201C22×C4—4544no—
202C22×C4—4544no—
203C22×C4—2272no—
204C22×C4—2272no—
205C22×C4—2278no—
206C22×C4D90814yescomplemented
207C22×C4C22×D22714yes—
208C22×C4C22×D22711yescharacteristic
209C22×C4C22×D22711yescharacteristic
210C4⋊C4—24no—
211C4⋊C4—4542no—
212C22⋊C4—24no—
213C22⋊C4—4548no—
214C22⋊C4—4548no—
215C22⋊C4—4544no—
216C23—22no—
217C23—22no—
218C23—4548no—
219C23—4548no—
220C23—45416no—
221C23—45416no—
222C23—4544no—
223C23—4542no—
224C23—4542no—
225C23—4544no—
226C23—4542no—
227C23—4542no—
228C23—4542no—
229C23—4542no—
230C23—4542no—
231C23—4548no—
232C23—2274no—
233C23—2274no—
234C23—2274no—
235C23D4×D22712yes—
236C23C2×D90812yes—
237C23#1681388512yes—
238C23C23×D22711yescharacteristic
239D4—90816no—
240D4—9088no—
241D4—9084no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—45416no—
248C2×C4—4542no—
249C2×C4—4542no—
250C2×C4—4542no—
251C2×C4—4548no—
252C2×C4—4542no—
253C2×C4—4542no—
254C2×C4—45416no—
255C2×C4—2274no—
256C2×C4D4×D22714yes—
257C2×C4C2×D90812yes—
258C2×C4C2×D90812yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—9082no—
264C22—9084no—
265C22—9082no—
266C22—9082no—
267C22—9084no—
268C22—45432no—
269C22—4544no—
270C22—4544no—
271C22—4544no—
272C22—4544no—
273C22—4544no—
274C22—4544no—
275C22—2278no—
276C22#2030026314yescomplemented
277C22C2×D4×D22712yes—
278C22C2×D4×D22711yescharacteristic
279C22C22×D90811yescharacteristic
280C22#41×D22712yes—
281C22C2×#1681388511yescharacteristic
282C4—22no—
283C4—44no—
284C4—9082no—
285C4—4544no—
286C4#2030025814yescomplemented
287C2—24no—
288C2—18162no—
289C2—4548no—
290C2C2×#2030025811yescharacteristic
291C2#121×D22712yes—
292C2C2×#2030026311yescharacteristic
293C2#2840954411yescharacteristic
294C2#2840954512yes—
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#380×C227C2
hypercenter#380D227
hypocenterC227#2847
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×C4×D227C2semidirect
2. C2×#20300263C2semidirect
3. C2×#20300262C2semidirect
4. C2×#20300258C2semidirect
5. #28408910C2semidirect
6. #28408909C2semidirect
7. #28408889C2semidirect
8. #380×C227C2semidirect
11. C22×D908C22semidirect
21. C2×C4⋊Dic227C22semidirect
23. C2×(C2×D227)⋊C4C22semidirect
24. C2×(C2×D227)⋊C4C22semidirect
25. C227⋊#95C22semidirect
27. C2×C4⋊C4×C227C22semidirect
29. #95×C227C22semidirect
34. C23×D227D4semidirect
55. C2×C4×D227D4semidirect
62. C22×C4×C227D4semidirect
115. C22×C227#121semidirect
133. C2×D227#121semidirect
136. C4×C227#120semidirect
147. C227#2847semidirect
156. #380D227semidirect
171. C2×C4⋊C4C2×D227semidirect
177. #95C2×D227semidirect
206. C22×C4D908semidirect
276. C22#20300263semidirect
286. C4#20300258semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D2272
C2×#203002632
C2×#203002622
C2×#203002582
#284089102
#284089092
#284088892
#380×C2272
#2847227