Finite Group Databasebeta

← #27181062

Subgroups — #27181062

Order 12928.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20265684C211yesmaximal, characteristic
2C2×#15692213C211yesmaximal, characteristic, complemented
3#20265517C212yesmaximal, complemented
4#20265516C212yesmaximal, complemented
5#20265500C212yesmaximal, complemented
6#20265499C212yesmaximal, complemented
7#20265219C211yesmaximal, characteristic, complemented
8#20265217C211yesmaximal, characteristic, complemented
9#20265178C211yesmaximal, characteristic, complemented
10#20265140C211yesmaximal, characteristic, complemented
11#417×C101C211yesmaximal, characteristic, complemented
12C22×C4×D101C2211yescharacteristic
13C2×#13108058C2211yescharacteristic
14#15692236—21no—
15#15692231C2211yescharacteristic
16#15692231C2211yescharacteristic
17#15692226C2211yescharacteristic
18#15692226C2211yescharacteristic
19#15692223—21no—
20C4×C22⋊D101—21no—
21C4×C22⋊D101—21no—
22#15692214C2211yescharacteristic
23#15692213—21no—
24#15692213—21no—
25#15692213C2212yes—
26#15692213C2212yes—
27C2×C4⋊Dic101C2211yescharacteristic
28C2×#13108051—21no—
29C2×C4×Dic101—21no—
30#15692189C2212yescomplemented
31#15692188C2212yescomplemented
32#15692187C2212yescomplemented
33#15692186C2212yescomplemented
34#15692179C2212yescomplemented
35#15692176C2212yescomplemented
36#15692167C2212yes—
37#15692167C2212yes—
38C2×#41×C101C2211yescharacteristic
39C2×M16×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#103×C101C2212yescomplemented
42#102×C101C2212yescomplemented
43C23×D101D411yescharacteristic
44C2×C22⋊D101—21no—
45C2×C22⋊D101—21no—
46C22×Dic101—21no—
47C22×Dic101D411yescharacteristic
48C2×C4×D101—22no—
49C2×C4×D101—22no—
50C2×C4×D101D411yescharacteristic
51C2×C4×D101D411yescharacteristic
52C2×C4×D101C2311yescharacteristic
53C2×C4×D101C2311yescharacteristic
54#13108058—22no—
55C2×#11717579C2312yes—
56C4⋊Dic101—21no—
57C4⋊Dic101C2312yes—
58C4⋊Dic101C2311yescharacteristic
59C4⋊Dic101C2311yescharacteristic
60#13108053—21no—
61#13108053—21no—
62C22⋊Dic101—21no—
63C22⋊Dic101—21no—
64C22⋊Dic101—21no—
65#13108051—22no—
66#13108051—22no—
67#13108051—21no—
68#13108051—21no—
69C4×Dic101—21no—
70C4×Dic101—22no—
71C4×Dic101—22no—
72#41×C101—22no—
73#41×C101—22no—
74C2×Q8×C101C2311yescharacteristic
75C2×D4×C101—21no—
76C2×D4×C101C2311yescharacteristic
77C22×C4×C101—21no—
78C22×C4×C101—21no—
79C22×C4×C101C2311yescharacteristic
80M16×C101—22no—
81C2×C8×C101C2312yes—
82C4⋊C4×C101—21no—
83C4⋊C4×C101C2312yes—
84C23×C101—21no—
85C23×C101C2×D411yescharacteristic
86C22×D101—22no—
87C22×D101—41no—
88C22×D101—41no—
89C22×D101C2×D411yescharacteristic
90C22×D101C2×D411yescharacteristic
91Q8×C101—21no—
92Q8×C101—21no—
93D4×C101—21no—
94D4×C101—21no—
95D4×C101—42no—
96C2×C4×C101—22no—
97C2×C4×C101—21no—
98C2×C4×C101—42no—
99C2×C4×C101—42no—
100C2×C4×C101C2×D411yescharacteristic
101C2×C4×C101#4112yes—
102C2×C4×C101#4112yes—
103C2×C4×C101C2411yescharacteristic
104C22⋊D101—42no—
105C22⋊D101—42no—
106C2×Dic101—22no—
107C2×Dic101—22no—
108C2×Dic101—21no—
109C2×Dic101—21no—
110C2×Dic101—42no—
111C2×Dic101C2×D411yescharacteristic
112C2×Dic101C2×D411yescharacteristic
113C4×D101—22no—
114C4×D101—22no—
115C4×D101—21no—
116C4×D101—21no—
117C4×D101—21no—
118C4×D101—21no—
119C8×C101—22no—
120#11717579—22no—
121C22×C101—21no—
122C22×C101—21no—
123C22×C101—41no—
124C22×C101—41no—
125C22×C101—41no—
126C22×C101#12312yes—
127C22×C101C22×D411yescharacteristic
128C2×D101—21no—
129C2×D101—21no—
130C2×D101—21no—
131C2×D101—21no—
132C2×D101—21no—
133C2×D101—21no—
134C2×D101—44no—
135C4×C101—41no—
136C4×C101—42no—
137C4×C101#12312yes—
138C4×C101C2×#4111yescharacteristic
139C4×C101C2×#4111yescharacteristic
140Dic101—21no—
141Dic101—21no—
142Dic101—42no—
143Dic101—41no—
144Dic101—41no—
145C2×C101—22no—
146C2×C101—81no—
147C2×C101C2×#12311yescharacteristic
148C2×C101#57511yescharacteristic
149C2×C101#57511yescharacteristic
150D101—41no—
151D101—41no—
152#3093—1011nomaximal, Sylow
153C101#309311yescharacteristic, complemented, Sylow
154#525—1011no—
155C2×#117—1011no—
156#485—1012no—
157#485—1012no—
158#482—1012no—
159#482—1012no—
160#428—1011no—
161#428—1011no—
162#417—1011no—
163#417D10111yescharacteristic, complemented
164#407—1011no—
165C2×#41C2×D10111yescharacteristic
166C23×C4—1011no—
167C2×M16—1011no—
168C2×M16C2×D10111yescharacteristic
169#123—2021no—
170C22⋊Q8—1011no—
171C22⋊Q8—1011no—
172#121—1011no—
173#121—1011no—
174#120—2021no—
175C4×D4—2021no—
176C4×D4—2021no—
177#117—1011no—
178#117—1012no—
179#117—1012no—
180#117—2021no—
181#117—2021no—
182C2×C4⋊C4—1011no—
183C2×C4⋊C4C2×D10111yescharacteristic
184C2×C22⋊C4—2021no—
185C2×C42—2021no—
186#107—1012no—
187#107—1012no—
188#106—1012no—
189#106—1012no—
190#103—1012no—
191#103C2×D10112yescomplemented
192#102—1012no—
193#102C2×D10112yescomplemented
194C22⋊C8—1012no—
195C22⋊C8—1012no—
196C24—1011no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D10111yescharacteristic
200C2×D4—21no—
201C2×D4—2021no—
202C2×D4—2021no—
203C2×D4C22×D10111yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1011no—
207C22×C4—1011no—
208C22×C4—1011no—
209C22×C4—1011no—
210C22×C4—1011no—
211C22×C4—2021no—
212C22×C4—2022no—
213C22×C4—2022no—
214C22×C4C22×D10111yescharacteristic
215M16—22no—
216M16—2022no—
217C2×C8—1012no—
218C2×C8C22×D10112yes—
219C4⋊C4—21no—
220C4⋊C4—1012no—
221C4⋊C4—1011no—
222C4⋊C4—1011no—
223C4⋊C4—2021no—
224C4⋊C4—2021no—
225C4⋊C4—2021no—
226C4⋊C4C22×D10112yes—
227C22⋊C4—2022no—
228C22⋊C4—2022no—
229C22⋊C4—2021no—
230C22⋊C4—2021no—
231C22⋊C4—2021no—
232C22⋊C4—2021no—
233C22⋊C4—2021no—
234C42—2022no—
235C42—2022no—
236C42—2021no—
237C23—21no—
238C23—1011no—
239C23—1011no—
240C23—4041no—
241C23—4041no—
242C23—2022no—
243C23D4×D10111yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—4042no—
250D4—4042no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1011no—
256C2×C4—1011no—
257C2×C4—4042no—
258C2×C4—2022no—
259C2×C4—2021no—
260C2×C4—2021no—
261C2×C4—2022no—
262C2×C4—2021no—
263C2×C4—2021no—
264C2×C4—2021no—
265C2×C4—2021no—
266C2×C4—2022no—
267C2×C4—2022no—
268C2×C4D4×D10111yescharacteristic
269C2×C4#1310807312yes—
270C2×C4#1310807412yes—
271C2×C4C23×D10111yescharacteristic
272C8—22no—
273C8—2022no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—4044no—
280C22—2021no—
281C22—2021no—
282C22—2021no—
283C22—2021no—
284C22—2021no—
285C22—2021no—
286C22#1569223712yes—
287C22C2×D4×D10111yescharacteristic
288C4—41no—
289C4—42no—
290C4—4041no—
291C4—4042no—
292C4—4041no—
293C4—2021no—
294C4—2021no—
295C4#1569223712yes—
296C4C2×#1310807311yescharacteristic
297C4C2×#1310807411yescharacteristic
298C2—22no—
299C2—81no—
300C2—4041no—
301C2—4041no—
302C2C2×#1569223711yescharacteristic
303C2#2026597311yescharacteristic
304C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#417×C101C2
socleC22×C101C22×D4
prime radical#20265684C2
hypercenter#417D101
hypocenterC101#3093
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#15692213C2semidirect
3. #20265517C2semidirect
4. #20265516C2semidirect
5. #20265500C2semidirect
6. #20265499C2semidirect
7. #20265219C2semidirect
8. #20265217C2semidirect
9. #20265178C2semidirect
10. #20265140C2semidirect
11. #417×C101C2semidirect
30. #15692189C22semidirect
31. #15692188C22semidirect
32. #15692187C22semidirect
33. #15692186C22semidirect
34. #15692179C22semidirect
35. #15692176C22semidirect
41. #103×C101C22semidirect
42. #102×C101C22semidirect
153. C101#3093semidirect
163. #417D101semidirect
191. #103C2×D101semidirect
193. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202656842
C2×#156922132
#202655172
#202655162
#202655002
#202654992
#202652192
#202652172
#202651782
#202651402
#417×C1012
#3093101