Finite Group Databasebeta

← #26222292

Subgroups — #26222292

Order 12032.

1306 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1306 conjugacy classes of subgroups — too many to draw. Only its 251 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#20258100C211yesmaximal, characteristic, complemented
2#20252780C211yesmaximal, characteristic
3#20252094C213yesmaximal, complemented
4#20243804C213yesmaximal, complemented
5#20243673C213yesmaximal, complemented
6#20243191C213yesmaximal, complemented
7C47×#1791C211yesmaximal, characteristic, complemented
8#15689890—26no—
9#15689890C2213yes—
10C2×#13094031—26no—
11C2×#13094031C2213yes—
12C22×#10625324C2211yescharacteristic, complemented
13#15689630C2213yes—
14#15689313C414yes—
15#15689310C414yes—
16#15689198—26no—
17#15689191—26no—
18#15689149C2213yescomplemented
19#15689097C2213yes—
20#15689086C2213yescomplemented
21#15689081C2213yescomplemented
22#15688895C2216yescomplemented
23C47×#585C2211yescharacteristic
24C47×#413C2213yes—
25C47×#408C2213yes—
26C22×C4×D47C2311yescharacteristic
27#13094031—24no—
28#13094031—212no—
29#13094031—26no—
30#13094031—212no—
31#13094031—26no—
32C2×#10625329—212no—
33C2×#10625329—26no—
34C2×#10625329D416yescomplemented
35C2×#10625329C2313yes—
36C2×#10625324—23no—
37C2×#10625324—23no—
38C22×C47⋊C8—24no—
39C22×C47⋊C8D416yescomplemented
40C22×C47⋊C8C2311yescharacteristic
41#13094005C2×C416yes—
42#13094000—23no—
43#13093995—23no—
44#13093992C2×C416yes—
45C4×C22⋊D47—26no—
46#13093983C2313yes—
47Dic47⋊C8—26no—
48#13093939C2×C4112yes—
49(C2×D47)⋊C8—26no—
50#13093934—23no—
51C8×Dic47—23no—
522−1+4×C47—23no—
53H(5,𝔽2)×C47—23no—
54H(5,𝔽2)×C47C2×C414yes—
55C2×#41×C47—26no—
56C2×#41×C47C2313yes—
57C2×M16×C47C2313yes—
58C22×C8×C47C2311yescharacteristic
59#101×C47—23no—
60#100×C47—23no—
61C22⋊C8×C47—26no—
62C23×D47C22×C411yescharacteristic
63#41×C47—24no—
64#41×C47—212no—
65#41×C47—212no—
66#41×C47—26no—
67#41×C47—26no—
68C2×Q8×C47—26no—
69C2×Q8×C47#4113yes—
70C2×D4×C47—26no—
71C2×D4×C47—212no—
72C2×D4×C47C22×C416yes—
73C2×D4×C47#4113yes—
74C22×C4×C47—24no—
75C22×C4×C47C2×D416yes—
76C22×C4×C47C2411yescharacteristic
77C2×C22⋊D47—26no—
78C22×Dic47C22×C411yescharacteristic
79C2×C4×D47—23no—
80C2×C4×D47—23no—
81M16×C47—26no—
82C2×C8×C47—23no—
83C2×C8×C47—23no—
84C22⋊Dic47—26no—
85C22⋊Dic47C22×C416yes—
86(C2×D47)⋊C4—26no—
87C4⋊Dic47—23no—
88Dic47⋊C4—26no—
89C4×Dic47—23no—
90#10625329—212no—
91#10625329—26no—
92#10625329—212no—
93C2×C47⋊C8—212no—
94C2×C47⋊C8—26no—
95C2×C47⋊C8—26no—
96C2×C47⋊C8C2×D416yes—
97C2×C47⋊C8C2×D413yes—
98C2×C47⋊C8C2×D413yes—
99#10625324—41no—
100#10625324—43no—
101C23×C47—24no—
102C23×C47C4×D416yes—
103C23×C47C23×C411yescharacteristic
104C22×D47—23no—
105C22×D47—81no—
106Q8×C47—24no—
107Q8×C47—26no—
108D4×C47—212no—
109D4×C47—212no—
110D4×C47—26no—
111C2×C4×C47—212no—
112C2×C4×C47—26no—
113C2×C4×C47—26no—
114C2×C4×C47C4×D416yes—
115C2×C4×C47C22×D413yes—
116C2×C4×C47C2×#4113yes—
117C22⋊D47—86no—
118C2×Dic47—26no—
119C2×Dic47—23no—
120C4×D47—41no—
121C4×D47—43no—
122C8×C47—41no—
123C8×C47—43no—
124C47⋊C8—26no—
125C47⋊C8#12011yescharacteristic
126C47⋊C8#12013yes—
127#21484—471nomaximal, Sylow
128C22×C47—26no—
129C22×C47—41no—
130C22×C47—46no—
131C22×C47—84no—
132C22×C47C2×C4×D413yes—
133C2×D47—43no—
134C2×D47—43no—
135C2×D47—41no—
136C4×C47—26no—
137C4×C47C2×#12011yescharacteristic
138C4×C47#52513yes—
139Dic47—46no—
140Dic47—41no—
141#3573—471no—
142#3025—471no—
143#2930—473no—
144#1891—473no—
145#1868—473no—
146#1791D4711yescharacteristic, complemented
147#1757—473no—
148C2×C47—23no—
149C2×C47—46no—
150C2×C47C4×#12011yescharacteristic
151D47—81no—
152#585C2×D4711yescharacteristic
153#568—473no—
154#568—946no—
155C2×#131—473no—
156C2×#131—946no—
157C22×M16—471no—
158#525—473no—
159#458—474no—
160B(4,𝔽2)—474no—
161#436—946no—
162#435—946no—
163#424—473no—
164#413C2×D4713yes—
165#410—473no—
166#408C2×D4713yes—
167#407—473no—
168#405—473no—
169#344—476no—
170C47#2148411yescharacteristic, complemented, Sylow
1712−1+4—23no—
172H(5,𝔽2)—23no—
173H(5,𝔽2)C4×D4714yes—
174C2×#41—26no—
175C2×#41C22×D4713yes—
176C23×C4—471no—
177#131—946no—
178#131—946no—
179#131—9412no—
180#131—944no—
181#131—9412no—
182C2×M16—476no—
183C2×M16—473no—
184C2×M16—943no—
185C2×M16—943no—
186C2×M16—9412no—
187C2×M16—946no—
188C2×M16C22×D4713yes—
189C22×C8—471no—
190C22×C8—476no—
191C22×C8—944no—
192C22×C8C22×D4711yescharacteristic
193#123—476no—
194C22⋊Q8—943no—
195#121—943no—
196#120—476no—
197C4×D4—946no—
198#117—473no—
199C4⋊C8—946no—
200#101—23no—
201#100—23no—
202#99—4712no—
203C22⋊C8—26no—
204C22⋊C8—946no—
205#97—943no—
206C4×C8—943no—
207C24—471no—
208#41—24no—
209#41—212no—
210#41—212no—
211#41—26no—
212#41—26no—
213C2×Q8—26no—
214C2×Q8#1062535213yes—
215C2×D4—26no—
216C2×D4—212no—
217C2×D4—946no—
218C2×D4C2×C4×D4716yes—
219C2×D4#1062535213yes—
220C22×C4—24no—
221C22×C4—471no—
222C22×C4—943no—
223C22×C4—943no—
224C22×C4D4×D4716yes—
225C22×C4C23×D4711yescharacteristic
226M16—26no—
227M16—9412no—
228M16—9412no—
229M16—946no—
230M16—1881no—
231M16—1883no—
232C2×C8—23no—
233C2×C8—23no—
234C2×C8—473no—
235C2×C8—473no—
236C2×C8—476no—
237C2×C8—946no—
238C2×C8—946no—
239C2×C8—9412no—
240C4⋊C4—946no—
241C4⋊C4—943no—
242C22⋊C4—476no—
243C22⋊C4—946no—
244C22⋊C4—946no—
245C42—943no—
246C23—24no—
247C23—943no—
248C23—3761no—
249C23(C22⋊D47)⋊C416yes—
250C23C22×C4×D4711yescharacteristic
251Q8—24no—
252Q8—26no—
253D4—212no—
254D4—212no—
255D4—26no—
256D4—3766no—
257C2×C4—212no—
258C2×C4—26no—
259C2×C4—26no—
260C2×C4—946no—
261C2×C4—943no—
262C2×C4—1881no—
263C2×C4—1883no—
264C2×C4(C22⋊D47)⋊C416yes—
265C2×C4C2×D4×D4713yes—
266C2×C4C2×#1062535213yes—
267C8—41no—
268C8—43no—
269C8—471no—
270C8—473no—
271C8—946no—
272C22—26no—
273C22—41no—
274C22—46no—
275C22—84no—
276C22—1883no—
277C22—1883no—
278C22—1881no—
279C22C2×(C22⋊D47)⋊C413yes—
280C4—26no—
281C4—1886no—
282C4—1881no—
283C4#120×D4711yescharacteristic
284C4#1568962913yes—
285C2—23no—
286C2—46no—
287C2—3761no—
288C2#2024588711yescharacteristic
subgroupG/H
centerC4#120×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1791C2
socleC2×C47C4×#120
prime radical#20252780C2
cosocleC22×C4×C47C24
hypercenter#1791D47
hypocenterC47#21484
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1791

Complements

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

NH ≅ G/Ntype
1. #20258100C2semidirect
3. #20252094C2semidirect
4. #20243804C2semidirect
5. #20243673C2semidirect
6. #20243191C2semidirect
7. C47×#1791C2semidirect
12. C22×#10625324C22semidirect
18. #15689149C22semidirect
20. #15689086C22semidirect
21. #15689081C22semidirect
22. #15688895C22semidirect
34. C2×#10625329D4semidirect
39. C22×C47⋊C8D4semidirect
146. #1791D47semidirect
170. C47#21484semidirect

Maximal subgroups

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

subgroupindex
#202581002
#202527802
#202520942
#202438042
#202436732
#202431912
C47×#17912
#2148447