Finite Group Databasebeta

← #27150019

Subgroups — #27150019

Order 12416.

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×D97C211yesmaximal, characteristic, complemented
2C2×#15690785C212yesmaximal, complemented
3C2×#15690784C211yesmaximal, characteristic, complemented
4C2×#15690780C212yesmaximal, complemented
5#20260613C214yesmaximal, complemented
6#20260612C212yesmaximal, complemented
7#20260592C212yesmaximal, complemented
8#380×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C22×D388C2214yes—
11C22×D388C2211yescharacteristic, complemented
12C22×C22⋊D97C2212yes—
13C23×Dic97C2211yescharacteristic
14C22×C4×D97—28no—
15C22×C4×D97—24no—
16C22×C4×D97C2212yes—
17C22×C4×D97C2212yes—
18#15690785—28no—
19#15690784—24no—
20#15690780—28no—
21C2×C4⋊Dic97C2211yescharacteristic, complemented
22C2×#13095488C2214yes—
23C2×#13095488C2214yescomplemented
24C2×#13095488C2212yescomplemented
25#15690717C2214yescomplemented
26C23×C4×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2212yescomplemented
28C2×C22⋊C4×C97C2212yes—
29#95×C97C2212yescomplemented
30C23×D97—22no—
31C23×D97—22no—
32C23×D97—24no—
33C23×D97—24no—
34C23×D97D414yescomplemented
35C23×D97C2312yes—
36C23×D97C2312yes—
37C2×D388—28no—
38C2×D388—28no—
39C2×D388—28no—
40C2×D388—22no—
41C2×D388—24no—
42C2×C22⋊D97—24no—
43C2×C22⋊D97—24no—
44C2×C22⋊D97—24no—
45C22×Dic97—22no—
46C22×Dic97—24no—
47C22×Dic97C2312yes—
48C22×Dic97C2312yes—
49C2×C4×D97—232no—
50C2×C4×D97—24no—
51C2×C4×D97—24no—
52C2×C4×D97—24no—
53C2×C4×D97—24no—
54C2×C4×D97—24no—
55C2×C4×D97D418yescomplemented
56C4⋊Dic97—22no—
57#13095488—28no—
58#13095488—24no—
59#13095488—28no—
60C24×C97C2311yescharacteristic
61C22×C4×C97—24no—
62C22×C4×C97D414yescomplemented
63C22×C4×C97C2314yes—
64C22×C4×C97C2311yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—24no—
67C22⋊C4×C97—24no—
68C23×C97—22no—
69C23×C97—22no—
70C23×C97C2×D412yes—
71C23×C97C2×D412yes—
72C23×C97#4112yes—
73C23×C97C2411yescharacteristic
74C22×D97—28no—
75C22×D97—216no—
76C22×D97—28no—
77C22×D97—28no—
78C22×D97—22no—
79C22×D97—22no—
80C22×D97—22no—
81C22×D97—22no—
82C22×D97—216no—
83C22×D97—22no—
84C22×D97—24no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—24no—
88C22×D97C2×D414yes—
89C22×D97C2×D414yes—
90C22×D97#4114yes—
91C2×C4×C97—22no—
92C2×C4×C97—28no—
93C2×C4×C97—24no—
94C2×C4×C97—24no—
95C2×C4×C97—24no—
96C2×C4×C97C2×D414yes—
97C2×C4×C97C2×D412yes—
98C2×C4×C97C2×D412yes—
99C22⋊D97—48no—
100C2×Dic97—28no—
101C2×Dic97—22no—
102C2×Dic97—22no—
103C2×Dic97—22no—
104C2×Dic97—22no—
105C2×Dic97—22no—
106C2×Dic97C2×D414yes—
107D388—416no—
108D388—44no—
109C4×D97—216no—
110C4×D97—216no—
111C22×C97—22no—
112C22×C97—24no—
113C22×C97—22no—
114C22×C97—24no—
115C22×C97#12114yescomplemented
116C22×C97C22×D412yes—
117C22×C97C22×D411yescharacteristic
118C22×C97C22×D411yescharacteristic
119C22×C97C2×#4112yes—
120C22×C97C2×#4111yescharacteristic
121C2×D97—232no—
122C2×D97—24no—
123C2×D97—24no—
124C2×D97—24no—
125C2×D97—24no—
126C2×D97—24no—
127C2×D97—24no—
128C2×D97—42no—
129C2×D97—44no—
130C2×D97—42no—
131C2×D97—42no—
132C2×D97—44no—
133C2×D97#12118yescomplemented
134C4×C97—22no—
135C4×C97—44no—
136C4×C97#12014yescomplemented
137Dic97—24no—
138Dic97—42no—
139C2×C97—24no—
140C2×C97C2×#12011yescharacteristic
141C2×C97C2×#12112yes—
142C2×C97C2×#12111yescharacteristic
143C2×C97#52511yescharacteristic
144C2×C97#52512yes—
145D97—28no—
146D97—82no—
147#2847—971nomaximal, Sylow
148C97#284711yescharacteristic, complemented, Sylow
149C24×C4—971no—
150C2×#121—971no—
151C2×#121—972no—
152C2×#120—972no—
153#386—972no—
154#386—974no—
155#380—972no—
156#380D9711yescharacteristic, complemented
157C25—971no—
158C22×D4—972no—
159C22×D4—974no—
160C22×D4—971no—
161C23×C4—972no—
162C23×C4—971no—
163C23×C4—972no—
164C23×C4—1948no—
165C23×C4—1944no—
166C23×C4C2×D9711yescharacteristic
167#121—1948no—
168#121—1944no—
169#120—1948no—
170C2×C4⋊C4—971no—
171C2×C4⋊C4C2×D9712yescomplemented
172C2×C22⋊C4—974no—
173C2×C22⋊C4—974no—
174C2×C22⋊C4—972no—
175C2×C22⋊C4C2×D9712yes—
176#95—974no—
177#95C2×D9712yescomplemented
178C24—972no—
179C24—974no—
180C24—972no—
181C24—1944no—
182C24—1944no—
183C24—1942no—
184C24—1942no—
185C24C22×D9711yescharacteristic
186C2×D4—1948no—
187C2×D4—1948no—
188C2×D4—1948no—
189C2×D4—1944no—
190C2×D4—1944no—
191C2×D4—1944no—
192C2×D4—1944no—
193C2×D4—1942no—
194C22×C4—24no—
195C22×C4—972no—
196C22×C4—978no—
197C22×C4—972no—
198C22×C4—1944no—
199C22×C4—1944no—
200C22×C4—19432no—
201C22×C4—1944no—
202C22×C4—1942no—
203C22×C4—1944no—
204C22×C4—1944no—
205C22×C4—1944no—
206C22×C4D38814yescomplemented
207C22×C4C22×D9714yes—
208C22×C4C22×D9711yescharacteristic
209C22×C4C22×D9711yescharacteristic
210C4⋊C4—24no—
211C4⋊C4—1942no—
212C22⋊C4—24no—
213C22⋊C4—1948no—
214C22⋊C4—1948no—
215C22⋊C4—1944no—
216C23—22no—
217C23—22no—
218C23—974no—
219C23—974no—
220C23—974no—
221C23—19416no—
222C23—1948no—
223C23—1948no—
224C23—19416no—
225C23—1944no—
226C23—1942no—
227C23—1942no—
228C23—1944no—
229C23—1942no—
230C23—1948no—
231C23—1942no—
232C23—1942no—
233C23—1942no—
234C23—1942no—
235C23D4×D9712yes—
236C23C2×D38812yes—
237C23#1309551212yes—
238C23C23×D9711yescharacteristic
239D4—38816no—
240D4—3888no—
241D4—3884no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—974no—
248C2×C4—19416no—
249C2×C4—1942no—
250C2×C4—1942no—
251C2×C4—1942no—
252C2×C4—1948no—
253C2×C4—1942no—
254C2×C4—1942no—
255C2×C4—19416no—
256C2×C4D4×D9714yes—
257C2×C4C2×D38812yes—
258C2×C4C2×D38812yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—978no—
264C22—3882no—
265C22—3884no—
266C22—3882no—
267C22—3882no—
268C22—3884no—
269C22—19432no—
270C22—1944no—
271C22—1944no—
272C22—1944no—
273C22—1944no—
274C22—1944no—
275C22—1944no—
276C22#1569078514yescomplemented
277C22C2×D4×D9712yes—
278C22C2×D4×D9711yescharacteristic
279C22C22×D38811yescharacteristic
280C22#41×D9712yes—
281C22C2×#1309551211yescharacteristic
282C4—22no—
283C4—44no—
284C4—3882no—
285C4—1944no—
286C4#1569078014yescomplemented
287C2—24no—
288C2—7762no—
289C2—1948no—
290C2C2×#1569078011yescharacteristic
291C2#121×D9712yes—
292C2C2×#1569078511yescharacteristic
293C2#2026124711yescharacteristic
294C2#2026124812yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#380×C97C2
hypercenter#380D97
hypocenterC97#2847
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×C4×D97C2semidirect
2. C2×#15690785C2semidirect
3. C2×#15690784C2semidirect
4. C2×#15690780C2semidirect
5. #20260613C2semidirect
6. #20260612C2semidirect
7. #20260592C2semidirect
8. #380×C97C2semidirect
11. C22×D388C22semidirect
21. C2×C4⋊Dic97C22semidirect
23. C2×#13095488C22semidirect
24. C2×#13095488C22semidirect
25. #15690717C22semidirect
27. C2×C4⋊C4×C97C22semidirect
29. #95×C97C22semidirect
34. C23×D97D4semidirect
55. C2×C4×D97D4semidirect
62. C22×C4×C97D4semidirect
115. C22×C97#121semidirect
133. C2×D97#121semidirect
136. C4×C97#120semidirect
148. C97#2847semidirect
156. #380D97semidirect
171. C2×C4⋊C4C2×D97semidirect
177. #95C2×D97semidirect
206. C22×C4D388semidirect
276. C22#15690785semidirect
286. C4#15690780semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D972
C2×#156907852
C2×#156907842
C2×#156907802
#202606132
#202606122
#202605922
#380×C972
#284797