Finite Group Databasebeta

← #27180489

Subgroups — #27180489

Order 12928.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#20265975C212yesmaximal, complemented
2#20265744C211yesmaximal, characteristic, complemented
3#20265355C214yesmaximal, complemented
4#20265345C214yesmaximal, complemented
5#20265194C212yesmaximal, complemented
6C2×#15692172C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic, complemented
9#15692286—22no—
10#15692285C2214yescomplemented
11#15692281—22no—
12#15692280C2214yescomplemented
13#15692275—24no—
14C2×#13108058C2212yescomplemented
15#15692251C2212yescomplemented
16#15692242C2212yescomplemented
17#15692226—22no—
18#15692224—22no—
19#15692214C2211yescharacteristic, complemented
20#15692182C2218yescomplemented
21#15692172C2214yescomplemented
22H(5,𝔽2)×C101—24no—
23H(5,𝔽2)×C101—22no—
24H(5,𝔽2)×C101C2214yes—
25C2×#41×C101—22no—
26C2×#41×C101C2212yes—
27C22×D4×C101—22no—
28C22×D4×C101—22no—
29C22×D4×C101C2211yescharacteristic
30#13108074—22no—
31#13108074—22no—
32C2×Dic202C2311yescharacteristic, complemented
33C2×D404C2311yescharacteristic, complemented
34C2×C22⋊D101—21no—
35C2×C22⋊D101—24no—
36C2×C4×D101—21no—
37#13108064—24no—
38#13108062—24no—
39#13108061—24no—
40#13108059—24no—
41#13108058—22no—
42#13108058C2314yescomplemented
43C2×#11717579—22no—
44C4⋊Dic101—21no—
45C22⋊Dic101—21no—
46C22⋊Dic101—24no—
47C4×Dic101C2312yescomplemented
48C24×C101—22no—
49C24×C101D412yes—
50#41×C101—28no—
51#41×C101—28no—
52#41×C101—22no—
53#41×C101—22no—
54#41×C101D414yescomplemented
55#41×C101C2314yes—
56C2×Q8×C101D412yescomplemented
57C2×D4×C101—24no—
58C2×D4×C101—28no—
59C2×D4×C101—28no—
60C2×D4×C101—22no—
61C2×D4×C101—28no—
62C2×D4×C101—28no—
63C2×D4×C101—22no—
64C2×D4×C101—22no—
65C2×D4×C101D412yes—
66C2×D4×C101D412yescomplemented
67C2×D4×C101C2312yes—
68C22×C4×C101—22no—
69C22×C4×C101—22no—
70C22×C4×C101C2311yescharacteristic
71C23×C101—28no—
72C23×C101—22no—
73C23×C101—22no—
74C23×C101—22no—
75C23×C101—82no—
76C23×C101—84no—
77C23×C101C2×D414yes—
78C23×C101C2×D411yescharacteristic
79C22×D101—21no—
80Q8×C101—22no—
81Q8×C101C2×D414yes—
82D4×C101—216no—
83D4×C101—24no—
84D4×C101—24no—
85D4×C101—24no—
86D4×C101—22no—
87D4×C101—24no—
88D4×C101C2×D414yes—
89C2×C4×C101—28no—
90C2×C4×C101—22no—
91C2×C4×C101—24no—
92C2×C4×C101—24no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D412yes—
95C2×C4×C101C2×D412yes—
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—41no—
98C22⋊D101—41no—
99C22⋊D101—84no—
100C2×Dic101—22no—
101C2×Dic101—21no—
102D404—22no—
103C4×D101—42no—
104Dic202—22no—
105#11717579—24no—
106C22×C101—24no—
107C22×C101—24no—
108C22×C101—41no—
109C22×C101—44no—
110C22×C101—44no—
111C22×C101—44no—
112C22×C101—82no—
113C22×C101—88no—
114C22×C101#12011yescharacteristic
115C22×C101#12011yescharacteristic
116C22×C101C22×D411yescharacteristic
117C2×D101—42no—
118C2×D101—41no—
119C4×C101—24no—
120C4×C101#12012yes—
121C4×C101C22×D412yes—
122Dic101—42no—
123Dic101—41no—
124C2×C101—21no—
125C2×C101—21no—
126C2×C101—21no—
127C2×C101—44no—
128C2×C101—44no—
129C2×C101C2×#12011yescharacteristic
130D101—81no—
131#3019—1011nomaximal, Sylow
132C101#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)D10111yescharacteristic, complemented
134#575—1012no—
135#535—1011no—
136#455—1014no—
137#453—1014no—
138#421—1012no—
139C2×#100—1011no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D10114yes—
143C2×#41—22no—
144C2×#41—1011no—
145C2×#41C2×D10112yes—
146C22×D4—22no—
147C22×D4—22no—
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#127—1012no—
156#124—1012no—
157#121—2022no—
158#120—2022no—
159#117—1011no—
160#104—1018no—
161#100—1014no—
162C24—22no—
163C24C22⋊D10112yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—2022no—
169#41—2022no—
170#41C22⋊D10114yescomplemented
171#41C22×D10114yes—
172C2×Q8—1011no—
173C2×Q8C22⋊D10112yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—1011no—
183C2×D4—2024no—
184C2×D4—2021no—
185C2×D4C22⋊D10112yes—
186C2×D4C22⋊D10112yescomplemented
187C2×D4C22×D10112yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—2021no—
191C22×C4C22×D10111yescharacteristic
192Q16—2024no—
193SD16—2024no—
194SD16—2024no—
195D8—2024no—
196M16—1014no—
197M16—2022no—
198C2×C8—2022no—
199C4⋊C4—2021no—
200C22⋊C4—2021no—
201C22⋊C4—2024no—
202C42—1012no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—2021no—
210C23C2×C22⋊D10114yes—
211C23C2×C22⋊D10111yescharacteristic
212Q8—22no—
213Q8—2022no—
214Q8C2×C22⋊D10114yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—8084no—
222D4—4041no—
223D4—4041no—
224D4—2022no—
225D4C2×C22⋊D10114yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—4042no—
231C2×C4—2022no—
232C2×C4—2021no—
233C2×C4C2×C22⋊D10112yes—
234C2×C4C2×C22⋊D10112yes—
235C2×C4C2×C22⋊D10111yescharacteristic
236C2×C4C23×D10111yescharacteristic
237C8—2024no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—4042no—
247C22—4041no—
248C22#1569222411yescharacteristic
249C22#1569222411yescharacteristic
250C22C22×C22⋊D10111yescharacteristic
251C4—24no—
252C4—4042no—
253C4—4041no—
254C4#1569222412yes—
255C4C22×C22⋊D10112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—8081no—
262C2C2×#1569222411yescharacteristic
subgroupG/H
centerC2C2×#15692224
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×H(5,𝔽2)×C101C2
socleC2×C101C2×#120
hypercenterC2×H(5,𝔽2)D101
hypocenterC101#3019
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20265975C2semidirect
2. #20265744C2semidirect
3. #20265355C2semidirect
4. #20265345C2semidirect
5. #20265194C2semidirect
6. C2×#15692172C2semidirect
7. C2×H(5,𝔽2)×C101C2semidirect
8. C2×#13108074C22semidirect
10. #15692285C22semidirect
12. #15692280C22semidirect
14. C2×#13108058C22semidirect
15. #15692251C22semidirect
16. #15692242C22semidirect
19. #15692214C22semidirect
20. #15692182C22semidirect
21. #15692172C22semidirect
32. C2×Dic202C23semidirect
33. C2×D404C23semidirect
42. #13108058C23semidirect
47. C4×Dic101C23semidirect
54. #41×C101D4semidirect
56. C2×Q8×C101D4semidirect
66. C2×D4×C101D4semidirect
132. C101#3019semidirect
133. C2×H(5,𝔽2)D101semidirect
170. #41C22⋊D101semidirect
173. C2×Q8C22⋊D101semidirect
186. C2×D4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202659752
#202657442
#202653552
#202653452
#202651942
C2×#156921722
C2×H(5,𝔽2)×C1012
#3019101