Finite Group Databasebeta

← #27083047

Subgroups — #27083047

Order 12032.

4778 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 807 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#20257043C213yesmaximal, complemented
2#20252790C2112yesmaximal, complemented
3#20252780C2112yesmaximal, complemented
4#20251852C213yesmaximal, complemented
5C47×2−1+6C211yesmaximal, characteristic, complemented
6C22×#10625354C2211yescharacteristic, complemented
7#15689792—212no—
8#15689784—212no—
9#15689687C22118yescomplemented
10#15689630C22118yescomplemented
11#15689601C2213yescomplemented
12#15689313C22116yescomplemented
13#15689310C22148yescomplemented
14#15689097C22136yescomplemented
15C47×#585—212no—
16C47×#585C22112yes—
17C2×2−1+4×C47—212no—
18C2×2−1+4×C47C2213yes—
19C2×#10625354—212no—
20C22×D188C2313yes—
21C22×C4×D47C2313yes—
22#13094020—218no—
23#13094013—218no—
24#13094005C23112yes—
25#13094000—218no—
26#13093998—236no—
27#13093995—218no—
28#13093992C23136yes—
29Q8×Dic47—212no—
30C4×C22⋊D47—236no—
31#13093983C23118yes—
32#13093939C23148yes—
332−1+4×C47—248no—
342−1+4×C47—236no—
352−1+4×C47—224no—
36H(5,𝔽2)×C47—216no—
37H(5,𝔽2)×C47—236no—
38H(5,𝔽2)×C47C23116yes—
39C2×#41×C47—248no—
40C2×#41×C47—272no—
41C2×#41×C47D4112yescomplemented
42C2×#41×C47C23118yes—
43C22×Q8×C47—216no—
44C22×Q8×C47D4112yescomplemented
45C22×Q8×C47C2311yescharacteristic
46C23×D47C2413yes—
47#41×C47—2144no—
48#41×C47—248no—
49#41×C47—248no—
50#41×C47—248no—
51#41×C47—272no—
52C2×Q8×C47—272no—
53C2×Q8×C47—248no—
54C2×Q8×C47C2×D4118yes—
55C2×Q8×C47C2×D4112yes—
56C2×D4×C47—272no—
57C2×D4×C47—248no—
58C2×D4×C47C2×D4118yes—
59C2×D4×C47C24112yes—
60C22×C4×C47—248no—
61C22×C4×C47C2×D4136yes—
62C22×C4×C47C2413yes—
63C2×C22⋊D47—236no—
64C22×Dic47C2411yescharacteristic
65#10625354—416no—
66C2×D188—218no—
67C2×C4×D47—218no—
68C22⋊Dic47—212no—
69C22⋊Dic47C24112yes—
70(C2×D47)⋊C4—236no—
71C4⋊Dic47—218no—
72Dic47⋊C4—212no—
73C4×Dic47—218no—
74C23×C47—216no—
75C23×C47C22×D4112yes—
76C23×C47C2511yescharacteristic
77C22×D47—29no—
78C22×D47—83no—
79Q8×C47—216no—
80Q8×C47—236no—
81Q8×C47#120116yes—
82D4×C47—248no—
83D4×C47—224no—
84D4×C47—236no—
85C2×C4×C47—248no—
86C2×C4×C47—272no—
87C2×C4×C47C22×D4112yes—
88C2×C4×C47C22×D4118yes—
89C22⋊D47—836no—
90C2×Dic47—212no—
91C2×Dic47—23no—
92D188—412no—
93C4×D47—412no—
94#60402—471nomaximal, Sylow
95C22×C47—212no—
96C22×C47—41no—
97C22×C47—412no—
98C22×C47—816no—
99C22×C47C23×D413yes—
100C2×D47—418no—
101C2×D47—43no—
102C4×C47—212no—
103C4×C47C2×#120112yes—
104Dic47—412no—
105Dic47—41no—
1062−1+6D4711yescharacteristic, complemented
107#3487—473no—
108#3027—4712no—
109#3025—4712no—
110#2884—473no—
111C2×C47—23no—
112C2×C47—412no—
113C2×C47C22×#12011yescharacteristic
114D47—83no—
115#585—212no—
116#585C2×D47112yes—
117C2×2−1+4—212no—
118C2×2−1+4C2×D4713yes—
119C22×#41—471no—
120#550—9412no—
121#548—9412no—
122#535—4718no—
123#525—4718no—
124#519—473no—
125#458—4716no—
126B(4,𝔽2)—4748no—
127#410—4736no—
128C47#6040211yescharacteristic, complemented, Sylow
1292−1+4—248no—
1302−1+4—236no—
1312−1+4—224no—
132H(5,𝔽2)—216no—
133H(5,𝔽2)—236no—
134H(5,𝔽2)C22×D47116yes—
135C2×#41—248no—
136C2×#41—272no—
137C2×#41—9412no—
138C2×#41C22⋊D47112yescomplemented
139C2×#41C22×D47118yes—
140C22×Q8—216no—
141C22×Q8C22⋊D47112yescomplemented
142C22×Q8C22×D4711yescharacteristic
143C22×D4—473no—
144C23×C4—473no—
145#127—9418no—
146#124—9418no—
147#123—4712no—
148C22⋊Q8—9418no—
149#121—9418no—
150#121—9436no—
151#120—4736no—
152C4×Q8—9412no—
153C4×D4—9436no—
154#117—4718no—
155#99—4748no—
156C24—473no—
157#41—2144no—
158#41—248no—
159#41—248no—
160#41—248no—
161#41—272no—
162#41—18816no—
163C2×Q8—272no—
164C2×Q8—248no—
165C2×Q8C2×C22⋊D47118yes—
166C2×Q8C2×C22⋊D47112yes—
167C2×D4—272no—
168C2×D4—248no—
169C2×D4—9436no—
170C2×D4—9418no—
171C2×D4C2×C22⋊D47118yes—
172C2×D4C23×D47112yes—
173C22×C4—248no—
174C22×C4—471no—
175C22×C4—9418no—
176C22×C4C2×C22⋊D47136yes—
177C22×C4C23×D4713yes—
178C4⋊C4—9412no—
179C4⋊C4—9418no—
180C22⋊C4—4712no—
181C22⋊C4—9412no—
182C22⋊C4—9436no—
183C42—9418no—
184C23—216no—
185C23—949no—
186C23—3763no—
187C23C22×C22⋊D47112yes—
188C23C24×D4711yescharacteristic
189Q8—216no—
190Q8—236no—
191Q8#13093993116yes—
192D4—248no—
193D4—224no—
194D4—236no—
195D4—37636no—
196D4—18812no—
197C2×C4—248no—
198C2×C4—272no—
199C2×C4—9412no—
200C2×C4—943no—
201C2×C4—18812no—
202C2×C4C22×C22⋊D47112yes—
203C2×C4C22×C22⋊D47118yes—
204C22—212no—
205C22—41no—
206C22—412no—
207C22—816no—
208C22—18818no—
209C22—1883no—
210C22C23×C22⋊D4713yes—
211C4—212no—
212C4—18812no—
213C4—1881no—
214C4C2×#13093993112yes—
215C2—23no—
216C2—412no—
217C2—3763no—
218C2C22×#1309399311yescharacteristic
subgroupG/H
centerC2C22×#13093993
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC47×2−1+6C2
socleC2×C47C22×#120
hypercenter2−1+6D47
hypocenterC47#60402
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×Q8 ◁ 2−1+6

Complements

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

NH ≅ G/Ntype
1. #20257043C2semidirect
2. #20252790C2semidirect
3. #20252780C2semidirect
4. #20251852C2semidirect
5. C47×2−1+6C2semidirect
6. C22×#10625354C22semidirect
9. #15689687C22semidirect
10. #15689630C22semidirect
11. #15689601C22semidirect
12. #15689313C22semidirect
13. #15689310C22semidirect
14. #15689097C22semidirect
41. C2×#41×C47D4semidirect
44. C22×Q8×C47D4semidirect
106. 2−1+6D47semidirect
128. C47#60402semidirect
138. C2×#41C22⋊D47semidirect
141. C22×Q8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202570432
#202527902
#202527802
#202518522
C47×2−1+62
#6040247