Finite Group Databasebeta

← #27179411

Subgroups — #27179411

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×#15692226C212yesmaximal, complemented
3C2×#15692223C212yesmaximal, complemented
4#20265057C212yesmaximal, complemented
5#20265053C214yesmaximal, complemented
6#20265037C211yesmaximal, characteristic, complemented
7#20265034C212yesmaximal, complemented
8C2×#121×C101C211yesmaximal, characteristic, complemented
9C24×D101C2211yescharacteristic
10C22×C22⋊D101C2214yes—
11C23×Dic101C2211yescharacteristic
12C22×C4×D101—28no—
13C22×C4×D101—24no—
14C22×C4×D101C2212yes—
15C22×C4×D101C2212yes—
16#15692226—28no—
17#15692223—28no—
18C2×C4⋊Dic101C2212yescomplemented
19C2×C22⋊Dic101C2212yes—
20C2×C22⋊Dic101C2214yescomplemented
21C2×#13108051C2214yes—
22#15692162C2214yescomplemented
23#15692162C2212yescomplemented
24C22×D4×C101C2212yes—
25C22×D4×C101C2211yescharacteristic, complemented
26C23×C4×C101C2211yescharacteristic
27#121×C101—24no—
28C2×C4⋊C4×C101C2211yescharacteristic, complemented
29C2×C22⋊C4×C101C2212yescomplemented
30C23×D101—22no—
31C23×D101—22no—
32C23×D101—24no—
33C23×D101—24no—
34C23×D101D414yescomplemented
35C23×D101C2312yes—
36C2×C22⋊D101—28no—
37C2×C22⋊D101—28no—
38C2×C22⋊D101—28no—
39C22×Dic101—22no—
40C22×Dic101—24no—
41C22×Dic101C2314yes—
42C22×Dic101C2312yes—
43C2×C4×D101—232no—
44C2×C4×D101—24no—
45C2×C4×D101—24no—
46C2×C4×D101—24no—
47C2×C4×D101—24no—
48C2×C4×D101—24no—
49C2×C4×D101D418yescomplemented
50C4⋊Dic101—24no—
51C22⋊Dic101—24no—
52C22⋊Dic101—28no—
53#13108051—28no—
54C24×C101C2312yes—
55C24×C101C2311yescharacteristic
56C2×D4×C101—24no—
57C2×D4×C101—24no—
58C2×D4×C101—24no—
59C2×D4×C101—22no—
60C2×D4×C101—24no—
61C22×C4×C101—24no—
62C22×C4×C101D414yescomplemented
63C22×C4×C101C2312yes—
64C22×C4×C101C2311yescharacteristic
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—22no—
67C22⋊C4×C101—24no—
68C23×C101—22no—
69C23×C101—22no—
70C23×C101—22no—
71C23×C101—24no—
72C23×C101—22no—
73C23×C101—22no—
74C23×C101—24no—
75C23×C101C2×D412yes—
76C23×C101C2×D412yes—
77C23×C101#4112yes—
78C23×C101C2411yescharacteristic
79C22×D101—28no—
80C22×D101—216no—
81C22×D101—28no—
82C22×D101—28no—
83C22×D101—22no—
84C22×D101—22no—
85C22×D101—22no—
86C22×D101—22no—
87C22×D101—216no—
88C22×D101C2×D414yes—
89C22×D101C2×D414yes—
90C22×D101#4114yes—
91D4×C101—48no—
92D4×C101—44no—
93C2×C4×C101—22no—
94C2×C4×C101—28no—
95C2×C4×C101—22no—
96C2×C4×C101—22no—
97C2×C4×C101—22no—
98C2×C4×C101C2×D414yes—
99C2×C4×C101C2×D412yes—
100C2×C4×C101C2×D412yes—
101C22⋊D101—416no—
102C2×Dic101—28no—
103C2×Dic101—22no—
104C2×Dic101—22no—
105C2×Dic101—24no—
106C2×Dic101—24no—
107C2×Dic101—24no—
108C2×Dic101C2×D414yes—
109C4×D101—216no—
110C4×D101—216no—
111C22×C101—22no—
112C22×C101—24no—
113C22×C101—22no—
114C22×C101—24no—
115C22×C101—42no—
116C22×C101—44no—
117C22×C101—42no—
118C22×C101—42no—
119C22×C101—44no—
120C22×C101#12114yescomplemented
121C22×C101C22×D412yes—
122C22×C101C22×D411yescharacteristic
123C22×C101C22×D411yescharacteristic
124C22×C101C2×#4112yes—
125C22×C101C2×#4111yescharacteristic
126C2×D101—232no—
127C2×D101—24no—
128C2×D101—24no—
129C2×D101—24no—
130C2×D101—24no—
131C2×D101—24no—
132C2×D101—24no—
133C2×D101#12118yescomplemented
134C4×C101—22no—
135C4×C101—42no—
136C4×C101#12014yescomplemented
137Dic101—24no—
138Dic101—44no—
139C2×C101—24no—
140C2×C101—82no—
141C2×C101C2×#12011yescharacteristic
142C2×C101C2×#12112yes—
143C2×C101C2×#12111yescharacteristic
144C2×C101#52511yescharacteristic
145C2×C101#52512yes—
146D101—28no—
147#2847—1011nomaximal, Sylow
148C101#284711yescharacteristic, complemented, Sylow
149C24×C4—1011no—
150C2×#121—1012no—
151C2×#121D10111yescharacteristic, complemented
152C2×#120—1012no—
153#386—1012no—
154#386—1014no—
155#380—1011no—
156#380—1012no—
157C25—1011no—
158C22×D4—1014no—
159C22×D4C2×D10112yes—
160C22×D4C2×D10111yescharacteristic, complemented
161C23×C4—1012no—
162C23×C4—1011no—
163C23×C4—1012no—
164C23×C4—2028no—
165C23×C4—2024no—
166C23×C4C2×D10111yescharacteristic
167#121—24no—
168#121—2028no—
169#120—2028no—
170C2×C4⋊C4—1012no—
171C2×C4⋊C4C2×D10111yescharacteristic, complemented
172C2×C22⋊C4—1012no—
173C2×C22⋊C4—1014no—
174C2×C22⋊C4—1014no—
175C2×C22⋊C4C2×D10112yescomplemented
176#95—1014no—
177#95—1012no—
178C24—1012no—
179C24—1014no—
180C24—2024no—
181C24—2024no—
182C24—2022no—
183C24—2022no—
184C24C22×D10112yes—
185C24C22×D10111yescharacteristic
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—2028no—
192C2×D4—2028no—
193C2×D4—2028no—
194C22×C4—24no—
195C22×C4—1012no—
196C22×C4—1018no—
197C22×C4—1014no—
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×C4C22⋊D10114yescomplemented
207C22×C4C22×D10112yes—
208C22×C4C22×D10111yescharacteristic
209C22×C4C22×D10111yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—2024no—
212C22⋊C4—24no—
213C22⋊C4—2028no—
214C22⋊C4—2028no—
215C22⋊C4—2024no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—1014no—
224C23—1014no—
225C23—1014no—
226C23—20216no—
227C23—2028no—
228C23—2028no—
229C23—20216no—
230C23—2022no—
231C23—2022no—
232C23—2022no—
233C23—2022no—
234C23—2028no—
235C23D4×D10112yes—
236C23C2×C22⋊D10112yes—
237C23#1310807312yes—
238C23C23×D10111yescharacteristic
239D4—48no—
240D4—44no—
241D4—40416no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—1014no—
248C2×C4—20216no—
249C2×C4—2024no—
250C2×C4—2024no—
251C2×C4—2024no—
252C2×C4—2028no—
253C2×C4—2022no—
254C2×C4—2022no—
255C2×C4—20216no—
256C2×C4D4×D10114yes—
257C2×C4C2×C22⋊D10112yes—
258C2×C4C2×C22⋊D10112yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—1018no—
269C22—20232no—
270C22—2024no—
271C22—2024no—
272C22—2024no—
273C22—2024no—
274C22—2024no—
275C22—2024no—
276C22#1569222614yescomplemented
277C22C2×D4×D10112yes—
278C22C2×D4×D10111yescharacteristic
279C22C22×C22⋊D10111yescharacteristic
280C22#41×D10112yes—
281C22C2×#1310807311yescharacteristic
282C4—22no—
283C4—42no—
284C4—4044no—
285C4—2024no—
286C4#1569222314yescomplemented
287C2—24no—
288C2—82no—
289C2—2028no—
290C2C2×#1569222311yescharacteristic
291C2#121×D10112yes—
292C2C2×#1569222611yescharacteristic
293C2#2026568411yescharacteristic
294C2#2026568712yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
FittingC2×#121×C101C2
hypercenterC2×#121D101
hypocenterC101#2847
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×C4×D101C2semidirect
2. C2×#15692226C2semidirect
3. C2×#15692223C2semidirect
4. #20265057C2semidirect
5. #20265053C2semidirect
6. #20265037C2semidirect
7. #20265034C2semidirect
8. C2×#121×C101C2semidirect
18. C2×C4⋊Dic101C22semidirect
20. C2×C22⋊Dic101C22semidirect
22. #15692162C22semidirect
23. #15692162C22semidirect
25. C22×D4×C101C22semidirect
28. C2×C4⋊C4×C101C22semidirect
29. C2×C22⋊C4×C101C22semidirect
34. C23×D101D4semidirect
49. C2×C4×D101D4semidirect
62. C22×C4×C101D4semidirect
120. C22×C101#121semidirect
133. C2×D101#121semidirect
136. C4×C101#120semidirect
148. C101#2847semidirect
151. C2×#121D101semidirect
160. C22×D4C2×D101semidirect
171. C2×C4⋊C4C2×D101semidirect
175. C2×C22⋊C4C2×D101semidirect
206. C22×C4C22⋊D101semidirect
276. C22#15692226semidirect
286. C4#15692223semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1012
C2×#156922262
C2×#156922232
#202650572
#202650532
#202650372
#202650342
C2×#121×C1012
#2847101