Finite Group Databasebeta

← #27180738

Subgroups — #27180738

Order 12928.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#15692275C211yesmaximal, characteristic, complemented
2C22×#13108058C211yesmaximal, characteristic, complemented
3#20265688C211yesmaximal, characteristic, complemented
4#20265433C212yesmaximal, complemented
5#20265428C212yesmaximal, complemented
6#20265425C212yesmaximal, complemented
7#20265421C212yesmaximal, complemented
8#20265213C211yesmaximal, characteristic, complemented
9#20265173C211yesmaximal, characteristic, complemented
10#20265171C211yesmaximal, characteristic, complemented
11#525×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13#15692275—24no—
14C2×#13108064C2211yescharacteristic, complemented
15C2×#13108062C2211yescharacteristic, complemented
16C2×#13108061C2211yescharacteristic, complemented
17C2×#13108059C2211yescharacteristic, complemented
18C2×#13108058—24no—
19C2×#13108058C2212yes—
20C2×#13108058C2212yes—
21C22×#11717579C2211yescharacteristic
22C22×#11717579C2211yescharacteristic
23#15692238—21no—
24#15692234C2212yes—
25#15692229C2212yes—
26#15692224—21no—
27C4×C22⋊D101—22no—
28#15692214C2211yescharacteristic
29#15692188C2212yescomplemented
30#15692186C2212yescomplemented
31#15692179C2212yescomplemented
32#15692178C2212yescomplemented
33#15692176C2212yescomplemented
34#15692175C2212yescomplemented
35C2×#41×C101C2211yescharacteristic
36C23×C4×C101C2211yescharacteristic
37#123×C101—21no—
38C22⋊Q8×C101C2212yes—
39#121×C101C2212yes—
40#120×C101—21no—
41C4×D4×C101—22no—
42#117×C101C2211yescharacteristic
43#13108074—22no—
44#13108074—22no—
45C2×Dic202C2311yescharacteristic
46C2×D404C2311yescharacteristic
47C2×C22⋊D101—21no—
48C2×C22⋊D101—22no—
49C2×C4×D101—21no—
50#13108064—22no—
51#13108062—22no—
52#13108061—22no—
53#13108059—22no—
54#13108058—24no—
55#13108058—24no—
56C2×#11717579—22no—
57C2×#11717579D414yescomplemented
58C2×#11717579C2312yes—
59C2×#11717579C2311yescharacteristic
60C2×#11717579C2311yescharacteristic
61C4⋊Dic101C2312yes—
62#13108053—22no—
63C22⋊Dic101—21no—
64C22⋊Dic101—22no—
65#13108051—22no—
66C4×Dic101—21no—
67C24×C101D411yescharacteristic
68#41×C101—22no—
69#41×C101—22no—
70C2×Q8×C101C2311yescharacteristic
71C2×D4×C101—22no—
72C2×D4×C101—21no—
73C2×D4×C101C2311yescharacteristic
74C22×C4×C101—22no—
75C22×C4×C101—22no—
76C22×C4×C101—21no—
77C22×C4×C101D412yes—
78C22×C4×C101D411yescharacteristic
79C22×C4×C101C2312yes—
80C22×C4×C101C2311yescharacteristic
81C4⋊C4×C101—22no—
82C4⋊C4×C101C2312yes—
83C22⋊C4×C101—21no—
84C22⋊C4×C101—22no—
85C22⋊C4×C101—22no—
86C42×C101—21no—
87C23×C101—22no—
88C23×C101—21no—
89C23×C101—42no—
90C23×C101C2×D412yes—
91C23×C101C2×D411yescharacteristic
92C22×D101—21no—
93Q8×C101—21no—
94Q8×C101—21no—
95D4×C101—21no—
96D4×C101—21no—
97D4×C101—44no—
98D4×C101—42no—
99C2×C4×C101—22no—
100C2×C4×C101—24no—
101C2×C4×C101—22no—
102C2×C4×C101—22no—
103C2×C4×C101—22no—
104C2×C4×C101—21no—
105C2×C4×C101—42no—
106C2×C4×C101C2×D412yes—
107C2×C4×C101C2×D412yes—
108C2×C4×C101C2×D411yescharacteristic
109C2×C4×C101#4112yes—
110C2×C4×C101C2411yescharacteristic
111C22⋊D101—44no—
112C22⋊D101—42no—
113C2×Dic101—22no—
114C2×Dic101—21no—
115D404—21no—
116D404—21no—
117C4×D101—42no—
118Dic202—21no—
119Dic202—21no—
120#11717579—22no—
121#11717579C2×D412yes—
122#11717579C2×D412yes—
123C22×C101—22no—
124C22×C101—22no—
125C22×C101—22no—
126C22×C101—22no—
127C22×C101—44no—
128C22×C101—41no—
129C22×C101—42no—
130C22×C101#12112yes—
131C22×C101C22×D411yescharacteristic
132C2×D101—41no—
133C2×D101—42no—
134C4×C101—22no—
135C4×C101—41no—
136C4×C101—42no—
137C4×C101#12112yes—
138C4×C101C22×D411yescharacteristic
139C4×C101C2×#4111yescharacteristic
140Dic101—41no—
141Dic101—42no—
142C2×C101—22no—
143C2×C101—42no—
144C2×C101—81no—
145C2×C101C2×#12111yescharacteristic
146C2×C101#57512yes—
147D101—81no—
148#3054—1011nomaximal, Sylow
149C101#305411yescharacteristic, complemented, Sylow
150C2×#135—1011no—
151C22×M16—1011no—
152#525—1011no—
153#525D10111yescharacteristic, complemented
154#470—1012no—
155#469—1012no—
156#468—1012no—
157#468—1012no—
158#427—1011no—
159#416—1011no—
160#416—1011no—
161C2×#41—1011no—
162C2×#41C2×D10111yescharacteristic
163C23×C4C2×D10111yescharacteristic
164#135—2024no—
165C2×Q16—1011no—
166C2×SD16—1011no—
167C2×SD16—1011no—
168C2×D8—1011no—
169C2×M16—1012no—
170C2×M16—1012no—
171C2×M16—2024no—
172C22×C8—1011no—
173C22×C8—1011no—
174#123—21no—
175#123—2021no—
176C22⋊Q8—1012no—
177C22⋊Q8C2×D10112yes—
178#121—1012no—
179#121C2×D10112yes—
180#120—21no—
181#120—2021no—
182C4×D4—22no—
183C4×D4—2022no—
184#117—1011no—
185#117C2×D10111yescharacteristic
186#107—1012no—
187#106—1012no—
188#103—1012no—
189#103—1012no—
190#102—1012no—
191#102—1012no—
192C24C22⋊D10111yescharacteristic
193#41—22no—
194#41—22no—
195#41—2022no—
196#41—2022no—
197C2×Q8—1011no—
198C2×Q8C22×D10111yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—1011no—
202C2×D4—2022no—
203C2×D4—2021no—
204C2×D4C22×D10111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—2021no—
209C22×C4C22⋊D10111yescharacteristic
210C22×C4C22⋊D10112yes—
211C22×C4C22×D10112yes—
212C22×C4C22×D10111yescharacteristic
213Q16—2022no—
214SD16—2022no—
215SD16—2022no—
216D8—2022no—
217M16—2024no—
218M16—2024no—
219C2×C8—1014no—
220C2×C8—1011no—
221C2×C8—1011no—
222C2×C8—1012no—
223C2×C8—2022no—
224C4⋊C4—22no—
225C4⋊C4—1012no—
226C4⋊C4—2022no—
227C4⋊C4C22×D10112yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—2021no—
232C22⋊C4—2022no—
233C22⋊C4—2022no—
234C42—21no—
235C42—2021no—
236C23—22no—
237C23—21no—
238C23—42no—
239C23—2021no—
240C23C2×C22⋊D10112yes—
241C23C2×C22⋊D10111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2021no—
245Q8—2021no—
246D4—21no—
247D4—21no—
248D4—44no—
249D4—42no—
250D4—4044no—
251D4—4042no—
252D4—2021no—
253D4—2021no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—4042no—
262C2×C4—2022no—
263C2×C4—2021no—
264C2×C4D4×D10112yes—
265C2×C4C2×C22⋊D10112yes—
266C2×C4C2×C22⋊D10111yescharacteristic
267C2×C4#1310807312yes—
268C2×C4C23×D10111yescharacteristic
269C8—1012no—
270C8—1012no—
271C8—2022no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—44no—
277C22—41no—
278C22—42no—
279C22—4041no—
280C22—4042no—
281C22#1569222712yes—
282C22C22×C22⋊D10111yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—4041no—
287C4—4042no—
288C4#1569222712yes—
289C4C2×D4×D10111yescharacteristic
290C4C2×#1310807311yescharacteristic
291C2—22no—
292C2—42no—
293C2—81no—
294C2—8081no—
295C2C2×#1569222711yescharacteristic
296C2#2026597512yes—
subgroupG/H
centerC2×C4C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#525×C101C2
socleC22×C101C22×D4
hypercenter#525D101
hypocenterC101#3054
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#15692275C2semidirect
2. C22×#13108058C2semidirect
3. #20265688C2semidirect
4. #20265433C2semidirect
5. #20265428C2semidirect
6. #20265425C2semidirect
7. #20265421C2semidirect
8. #20265213C2semidirect
9. #20265173C2semidirect
10. #20265171C2semidirect
11. #525×C101C2semidirect
14. C2×#13108064C22semidirect
15. C2×#13108062C22semidirect
16. C2×#13108061C22semidirect
17. C2×#13108059C22semidirect
29. #15692188C22semidirect
30. #15692186C22semidirect
31. #15692179C22semidirect
32. #15692178C22semidirect
33. #15692176C22semidirect
34. #15692175C22semidirect
57. C2×#11717579D4semidirect
149. C101#3054semidirect
153. #525D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922752
C22×#131080582
#202656882
#202654332
#202654282
#202654252
#202654212
#202652132
#202651732
#202651712
#525×C1012
#3054101