Finite Group Databasebeta

← #27180488

Subgroups — #27180488

Order 12928.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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×#15692300C211yesmaximal, characteristic, complemented
2#20265975C212yesmaximal, complemented
3#20265354C214yesmaximal, complemented
4#20265344C214yesmaximal, complemented
5#20265195C212yesmaximal, complemented
6C2×#15692172C211yesmaximal, characteristic, complemented
7#535×C101C211yesmaximal, characteristic, complemented
8#15692300—24no—
9#15692300—22no—
10#15692300C2214yes—
11C2×#13108074C2212yes—
12C2×#13108073—22no—
13C22×C22⋊D101—22no—
14C2×D4×D101—22no—
15#15692286C2214yescomplemented
16#15692285—22no—
17#15692281C2214yescomplemented
18#15692280—22no—
19#15692275—24no—
20C2×#13108058C2212yescomplemented
21#15692183C2218yescomplemented
22#15692172C2214yescomplemented
23C2×#41×C101C2211yescharacteristic, complemented
24C22×D4×C101C2211yescharacteristic
25#127×C101C2212yescomplemented
26#124×C101C2212yescomplemented
27#121×C101—22no—
28#120×C101—22no—
29#117×C101C2211yescharacteristic, complemented
30C23×D101—22no—
31#13108074—22no—
32#13108074—22no—
33#13108074D414yescomplemented
34#13108074C2314yes—
35#13108073—28no—
36#13108073—28no—
37C2×Dic202D412yescomplemented
38C2×D404D412yescomplemented
39C2×C22⋊D101—24no—
40C2×C22⋊D101—28no—
41C2×C22⋊D101—22no—
42C2×C22⋊D101—28no—
43D4×D101—28no—
44D4×D101—28no—
45C22×Dic101—22no—
46C2×C4×D101—22no—
47#13108064—24no—
48#13108062—24no—
49#13108061—24no—
50#13108059—24no—
51#13108058—22no—
52#13108058C2314yescomplemented
53C2×#11717579—22no—
54C24×C101D412yes—
55#41×C101—22no—
56#41×C101—22no—
57C2×Q8×C101C2311yescharacteristic, complemented
58C2×D4×C101—21no—
59C2×D4×C101—24no—
60C2×D4×C101—22no—
61C2×D4×C101—22no—
62C2×D4×C101D412yes—
63C2×D4×C101C2312yes—
64C2×D4×C101C2311yescharacteristic, complemented
65C22×C4×C101—21no—
66C22×C4×C101C2311yescharacteristic
67C4⋊C4×C101—21no—
68C22⋊C4×C101—21no—
69C22⋊C4×C101—24no—
70C42×C101C2312yescomplemented
71C23×C101—22no—
72C23×C101—22no—
73C23×C101—21no—
74C23×C101—82no—
75C23×C101C2×D414yes—
76C23×C101C2×D411yescharacteristic
77C22×D101—22no—
78C22×D101—28no—
79C22×D101—84no—
80Q8×C101—22no—
81D4×C101—24no—
82D4×C101—24no—
83D4×C101—22no—
84D4×C101—41no—
85D4×C101—41no—
86D4×C101—84no—
87C2×C4×C101—22no—
88C2×C4×C101—21no—
89C2×C4×C101—42no—
90C2×C4×C101C2×D412yes—
91C2×C4×C101C2×D412yes—
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2411yescharacteristic
94C22⋊D101—216no—
95C22⋊D101—24no—
96C22⋊D101—24no—
97C2×Dic101—28no—
98C2×Dic101—22no—
99D404—22no—
100D404C2×D414yes—
101C4×D101—24no—
102C4×D101—24no—
103Dic202—22no—
104Dic202C2×D414yes—
105#11717579—24no—
106C22×C101—24no—
107C22×C101—41no—
108C22×C101—44no—
109C22×C101—44no—
110C22×C101—42no—
111C22×C101—41no—
112C22×C101—82no—
113C22×C101#12011yescharacteristic
114C22×C101#12011yescharacteristic
115C22×C101C22×D411yescharacteristic
116C2×D101—24no—
117C2×D101—44no—
118C2×D101—88no—
119C4×C101—42no—
120C4×C101—41no—
121C4×C101#12012yes—
122C4×C101C22×D412yes—
123Dic101—24no—
124C2×C101—21no—
125C2×C101—21no—
126C2×C101—21no—
127C2×C101—44no—
128C2×C101—81no—
129C2×C101C2×#12011yescharacteristic
130D101—44no—
131#3019—1011nomaximal, Sylow
132C101#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)—1011no—
134#575—1012no—
135#535D10111yescharacteristic, complemented
136#455—1014no—
137#453—1014no—
138#421—1012no—
139C2×#100—1011no—
140H(5,𝔽2)—1014no—
141H(5,𝔽2)—2024no—
142H(5,𝔽2)—2022no—
143C2×#41—1012no—
144C2×#41—2022no—
145C2×#41C2×D10111yescharacteristic, complemented
146C22×D4—2022no—
147C22×D4—2022no—
148C22×D4C2×D10111yescharacteristic
149#137—1014no—
150#137—2022no—
151AGL(1,ℤ/8)—1014no—
152AGL(1,ℤ/8)—2022no—
153#135—2024no—
154C2×M16—1012no—
155#127C2×D10112yescomplemented
156#124C2×D10112yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D10111yescharacteristic, complemented
160#104—1018no—
161#100—1014no—
162C24—2022no—
163C24C22⋊D10112yes—
164#41—22no—
165#41—22no—
166#41—1014no—
167#41—1014no—
168#41—2028no—
169#41—2028no—
170#41—2022no—
171#41—2022no—
172C2×Q8—1012no—
173C2×Q8C22×D10111yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—1012no—
179C2×D4—2028no—
180C2×D4—2028no—
181C2×D4—2024no—
182C2×D4—2028no—
183C2×D4—2022no—
184C2×D4—2028no—
185C2×D4C22⋊D10112yes—
186C2×D4C22×D10112yes—
187C2×D4C22×D10111yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—2022no—
190C22×C4—2022no—
191C22×C4C22×D10111yescharacteristic
192Q16—2024no—
193SD16—2024no—
194SD16—2024no—
195D8—2024no—
196M16—1014no—
197M16—2022no—
198C2×C8—2022no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D10112yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—8084no—
208C23—2028no—
209C23—2022no—
210C23C2×C22⋊D10114yes—
211C23C2×C22⋊D10111yescharacteristic
212Q8—22no—
213Q8—1014no—
214Q8—2022no—
215D4—24no—
216D4—22no—
217D4—24no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—1014no—
222D4—20216no—
223D4—2024no—
224D4—2024no—
225D4—2022no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—2028no—
230C2×C4—2022no—
231C2×C4—2024no—
232C2×C4—2024no—
233C2×C4D4×D10112yes—
234C2×C4D4×D10112yes—
235C2×C4C2×C22⋊D10111yescharacteristic
236C2×C4C23×D10111yescharacteristic
237C8—2024no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—8088no—
246C22—4044no—
247C22—2024no—
248C22#1569222311yescharacteristic
249C22#1569222311yescharacteristic
250C22C22×C22⋊D10111yescharacteristic
251C4—42no—
252C4—41no—
253C4—2024no—
254C4#1569222312yes—
255C4C2×D4×D10112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—4044no—
262C2C2×#1569222311yescharacteristic
subgroupG/H
centerC2C2×#15692223
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#535×C101C2
socleC2×C101C2×#120
hypercenter#535D101
hypocenterC101#3019
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#15692300C2semidirect
2. #20265975C2semidirect
3. #20265354C2semidirect
4. #20265344C2semidirect
5. #20265195C2semidirect
6. C2×#15692172C2semidirect
7. #535×C101C2semidirect
15. #15692286C22semidirect
17. #15692281C22semidirect
20. C2×#13108058C22semidirect
21. #15692183C22semidirect
22. #15692172C22semidirect
23. C2×#41×C101C22semidirect
25. #127×C101C22semidirect
26. #124×C101C22semidirect
29. #117×C101C22semidirect
33. #13108074D4semidirect
37. C2×Dic202D4semidirect
38. C2×D404D4semidirect
52. #13108058C23semidirect
57. C2×Q8×C101C23semidirect
64. C2×D4×C101C23semidirect
70. C42×C101C23semidirect
132. C101#3019semidirect
135. #535D101semidirect
145. C2×#41C2×D101semidirect
155. #127C2×D101semidirect
156. #124C2×D101semidirect
159. #117C2×D101semidirect
173. C2×Q8C22×D101semidirect
187. C2×D4C22×D101semidirect
202. C42C22×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156923002
#202659752
#202653542
#202653442
#202651952
C2×#156921722
#535×C1012
#3019101