Finite Group Databasebeta

← #27179415

Subgroups — #27179415

Order 12928.

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×D101C211yesmaximal, characteristic, complemented
2C2×#15692230C212yesmaximal, complemented
3C2×#15692229C211yesmaximal, characteristic, complemented
4C2×#15692225C212yesmaximal, complemented
5#20265055C214yesmaximal, complemented
6#20265054C212yesmaximal, complemented
7#20265034C212yesmaximal, complemented
8#380×C101C211yesmaximal, characteristic, complemented
9C24×D101C2211yescharacteristic
10C22×D404C2214yes—
11C22×D404C2211yescharacteristic, complemented
12C22×C22⋊D101C2212yes—
13C23×Dic101C2211yescharacteristic
14C22×C4×D101—28no—
15C22×C4×D101—24no—
16C22×C4×D101C2212yes—
17C22×C4×D101C2212yes—
18#15692230—28no—
19#15692229—24no—
20#15692225—28no—
21C2×C4⋊Dic101C2211yescharacteristic, complemented
22C2×#13108051C2214yes—
23C2×#13108051C2214yescomplemented
24C2×#13108051C2212yescomplemented
25#15692162C2214yescomplemented
26C23×C4×C101C2211yescharacteristic
27C2×C4⋊C4×C101C2212yescomplemented
28C2×C22⋊C4×C101C2212yes—
29#95×C101C2212yescomplemented
30C23×D101—22no—
31C23×D101—22no—
32C23×D101—24no—
33C23×D101—24no—
34C23×D101D414yescomplemented
35C23×D101C2312yes—
36C23×D101C2312yes—
37C2×D404—28no—
38C2×D404—28no—
39C2×D404—28no—
40C2×D404—22no—
41C2×D404—24no—
42C2×C22⋊D101—24no—
43C2×C22⋊D101—24no—
44C2×C22⋊D101—24no—
45C22×Dic101—22no—
46C22×Dic101—24no—
47C22×Dic101C2312yes—
48C22×Dic101C2312yes—
49C2×C4×D101—232no—
50C2×C4×D101—24no—
51C2×C4×D101—24no—
52C2×C4×D101—24no—
53C2×C4×D101—24no—
54C2×C4×D101—24no—
55C2×C4×D101D418yescomplemented
56C4⋊Dic101—22no—
57#13108051—28no—
58#13108051—24no—
59#13108051—28no—
60C24×C101C2311yescharacteristic
61C22×C4×C101—24no—
62C22×C4×C101D414yescomplemented
63C22×C4×C101C2314yes—
64C22×C4×C101C2311yescharacteristic
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—24no—
67C22⋊C4×C101—24no—
68C23×C101—22no—
69C23×C101—22no—
70C23×C101C2×D412yes—
71C23×C101C2×D412yes—
72C23×C101#4112yes—
73C23×C101C2411yescharacteristic
74C22×D101—28no—
75C22×D101—216no—
76C22×D101—28no—
77C22×D101—28no—
78C22×D101—22no—
79C22×D101—22no—
80C22×D101—22no—
81C22×D101—22no—
82C22×D101—216no—
83C22×D101—22no—
84C22×D101—24no—
85C22×D101—22no—
86C22×D101—22no—
87C22×D101—24no—
88C22×D101C2×D414yes—
89C22×D101C2×D414yes—
90C22×D101#4114yes—
91C2×C4×C101—22no—
92C2×C4×C101—28no—
93C2×C4×C101—24no—
94C2×C4×C101—24no—
95C2×C4×C101—24no—
96C2×C4×C101C2×D414yes—
97C2×C4×C101C2×D412yes—
98C2×C4×C101C2×D412yes—
99C22⋊D101—48no—
100C2×Dic101—28no—
101C2×Dic101—22no—
102C2×Dic101—22no—
103C2×Dic101—22no—
104C2×Dic101—22no—
105C2×Dic101—22no—
106C2×Dic101C2×D414yes—
107D404—416no—
108D404—44no—
109C4×D101—216no—
110C4×D101—216no—
111C22×C101—22no—
112C22×C101—24no—
113C22×C101—22no—
114C22×C101—24no—
115C22×C101#12114yescomplemented
116C22×C101C22×D412yes—
117C22×C101C22×D411yescharacteristic
118C22×C101C22×D411yescharacteristic
119C22×C101C2×#4112yes—
120C22×C101C2×#4111yescharacteristic
121C2×D101—232no—
122C2×D101—24no—
123C2×D101—24no—
124C2×D101—24no—
125C2×D101—24no—
126C2×D101—24no—
127C2×D101—24no—
128C2×D101—42no—
129C2×D101—44no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—44no—
133C2×D101#12118yescomplemented
134C4×C101—22no—
135C4×C101—44no—
136C4×C101#12014yescomplemented
137Dic101—24no—
138Dic101—42no—
139C2×C101—24no—
140C2×C101C2×#12011yescharacteristic
141C2×C101C2×#12112yes—
142C2×C101C2×#12111yescharacteristic
143C2×C101#52511yescharacteristic
144C2×C101#52512yes—
145D101—28no—
146D101—82no—
147#2847—1011nomaximal, Sylow
148C101#284711yescharacteristic, complemented, Sylow
149C24×C4—1011no—
150C2×#121—1011no—
151C2×#121—1012no—
152C2×#120—1012no—
153#386—1012no—
154#386—1014no—
155#380—1012no—
156#380D10111yescharacteristic, complemented
157C25—1011no—
158C22×D4—1012no—
159C22×D4—1014no—
160C22×D4—1011no—
161C23×C4—1012no—
162C23×C4—1011no—
163C23×C4—1012no—
164C23×C4—2028no—
165C23×C4—2024no—
166C23×C4C2×D10111yescharacteristic
167#121—2028no—
168#121—2024no—
169#120—2028no—
170C2×C4⋊C4—1011no—
171C2×C4⋊C4C2×D10112yescomplemented
172C2×C22⋊C4—1014no—
173C2×C22⋊C4—1014no—
174C2×C22⋊C4—1012no—
175C2×C22⋊C4C2×D10112yes—
176#95—1014no—
177#95C2×D10112yescomplemented
178C24—1012no—
179C24—1014no—
180C24—1012no—
181C24—2024no—
182C24—2024no—
183C24—2022no—
184C24—2022no—
185C24C22×D10111yescharacteristic
186C2×D4—2028no—
187C2×D4—2028no—
188C2×D4—2028no—
189C2×D4—2024no—
190C2×D4—2024no—
191C2×D4—2024no—
192C2×D4—2024no—
193C2×D4—2022no—
194C22×C4—24no—
195C22×C4—1012no—
196C22×C4—1018no—
197C22×C4—1012no—
198C22×C4—2024no—
199C22×C4—2024no—
200C22×C4—20232no—
201C22×C4—2024no—
202C22×C4—2022no—
203C22×C4—2024no—
204C22×C4—2024no—
205C22×C4—2024no—
206C22×C4D40414yescomplemented
207C22×C4C22×D10114yes—
208C22×C4C22×D10111yescharacteristic
209C22×C4C22×D10111yescharacteristic
210C4⋊C4—24no—
211C4⋊C4—2022no—
212C22⋊C4—24no—
213C22⋊C4—2028no—
214C22⋊C4—2028no—
215C22⋊C4—2024no—
216C23—22no—
217C23—22no—
218C23—1014no—
219C23—1014no—
220C23—1014no—
221C23—20216no—
222C23—2028no—
223C23—2028no—
224C23—20216no—
225C23—2024no—
226C23—2022no—
227C23—2022no—
228C23—2024no—
229C23—2022no—
230C23—2022no—
231C23—2022no—
232C23—2022no—
233C23—2022no—
234C23—2028no—
235C23D4×D10112yes—
236C23C2×D40412yes—
237C23#1310807512yes—
238C23C23×D10111yescharacteristic
239D4—40416no—
240D4—4048no—
241D4—4044no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—1014no—
248C2×C4—20216no—
249C2×C4—2022no—
250C2×C4—2022no—
251C2×C4—2022no—
252C2×C4—2028no—
253C2×C4—2022no—
254C2×C4—2022no—
255C2×C4—20216no—
256C2×C4D4×D10114yes—
257C2×C4C2×D40412yes—
258C2×C4C2×D40412yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—1018no—
264C22—4042no—
265C22—4044no—
266C22—4042no—
267C22—4042no—
268C22—4044no—
269C22—20232no—
270C22—2024no—
271C22—2024no—
272C22—2024no—
273C22—2024no—
274C22—2024no—
275C22—2024no—
276C22#1569223014yescomplemented
277C22C2×D4×D10112yes—
278C22C2×D4×D10111yescharacteristic
279C22C22×D40411yescharacteristic
280C22#41×D10112yes—
281C22C2×#1310807511yescharacteristic
282C4—22no—
283C4—44no—
284C4—4042no—
285C4—2024no—
286C4#1569222514yescomplemented
287C2—24no—
288C2—8082no—
289C2—2028no—
290C2C2×#1569222511yescharacteristic
291C2#121×D10112yes—
292C2C2×#1569223011yescharacteristic
293C2#2026568911yescharacteristic
294C2#2026569012yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#380×C101C2
hypercenter#380D101
hypocenterC101#2847
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×C4×D101C2semidirect
2. C2×#15692230C2semidirect
3. C2×#15692229C2semidirect
4. C2×#15692225C2semidirect
5. #20265055C2semidirect
6. #20265054C2semidirect
7. #20265034C2semidirect
8. #380×C101C2semidirect
11. C22×D404C22semidirect
21. C2×C4⋊Dic101C22semidirect
23. C2×#13108051C22semidirect
24. C2×#13108051C22semidirect
25. #15692162C22semidirect
27. C2×C4⋊C4×C101C22semidirect
29. #95×C101C22semidirect
34. C23×D101D4semidirect
55. C2×C4×D101D4semidirect
62. C22×C4×C101D4semidirect
115. C22×C101#121semidirect
133. C2×D101#121semidirect
136. C4×C101#120semidirect
148. C101#2847semidirect
156. #380D101semidirect
171. C2×C4⋊C4C2×D101semidirect
177. #95C2×D101semidirect
206. C22×C4D404semidirect
276. C22#15692230semidirect
286. C4#15692225semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1012
C2×#156922302
C2×#156922292
C2×#156922252
#202650552
#202650542
#202650342
#380×C1012
#2847101