Finite Group Databasebeta

← #26105027

Subgroups — #26105027

Order 12032.

492 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 83 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#20251376C211yesmaximal, characteristic, complemented
2#20244619C212yesmaximal
3#20241214C211yesmaximal, characteristic, complemented
4#20241210C212yesmaximal, complemented
5C47×#2014C211yesmaximal, characteristic
6C2×#13093945—22no—
7C2×#13093945—22no—
8#15689062—21no—
9#15689055—22no—
10#15689030—21no—
11#15689021C2212yes—
12#15689010—22no—
13#15688878C2212yes—
14C22×SD16×C47—21no—
15C2×#127×C47C2211yescharacteristic
16C2×#102×C47—21no—
17C47×#391C2211yescharacteristic
18C47×#336C2211yescharacteristic
19C22×C47⋊C8—22no—
20C2×C4⋊Dic47—21no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47D412yes—
23C2×C4⋊Dic47D412yes—
24C2×Dic47⋊C4—22no—
25C2×C4×Dic47—21no—
26#13093945—44no—
27#13093945—44no—
28C47⋊#95—21no—
29C47⋊#95—22no—
30C47⋊#95—22no—
31C47⋊#95—22no—
32C22×Q8×C47D411yescharacteristic
33C22×D4×C47—21no—
34C22×D4×C47—22no—
35C2×SD16×C47—42no—
36C2×SD16×C47—42no—
37C2×SD16×C47—42no—
38C22×C8×C47—21no—
39#127×C47—42no—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C42×C47C2311yescharacteristic
42#102×C47—42no—
43#95×C47—22no—
44C24×C47—41no—
45C2×Q8×C47—21no—
46C2×Q8×C47—41no—
47C2×Q8×C47—41no—
48C2×Q8×C47SD1612yes—
49C2×D4×C47—22no—
50C2×D4×C47—24no—
51C2×D4×C47—41no—
52C2×D4×C47—42no—
53C2×D4×C47—42no—
54C2×D4×C47—41no—
55C2×D4×C47—42no—
56C2×D4×C47—41no—
57C2×D4×C47—41no—
58C2×D4×C47—42no—
59C2×D4×C47—42no—
60C2×D4×C47—41no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D412yes—
65C22×Dic47—22no—
66C22×Dic47—22no—
67SD16×C47—48no—
68C2×C8×C47—22no—
69C2×C8×C47—41no—
70C2×C8×C47—41no—
71C4⋊C4×C47—41no—
72C42×C47—41no—
73C4⋊Dic47—22no—
74C4⋊Dic47—22no—
75C4⋊Dic47—42no—
76C4⋊Dic47—81no—
77C4⋊Dic47—81no—
78Dic47⋊C4—42no—
79Dic47⋊C4—42no—
80C4×Dic47—81no—
81C2×C47⋊C8—42no—
82C2×C47⋊C8—42no—
83C2×C47⋊C8—42no—
84C23×C47—42no—
85C23×C47—81no—
86C23×C47—82no—
87C23×C47—81no—
88C23×C47—82no—
89C23×C47#12011yescharacteristic
90Q8×C47—42no—
91Q8×C47—81no—
92D4×C47—42no—
93D4×C47—42no—
94D4×C47—44no—
95D4×C47—44no—
96D4×C47—81no—
97D4×C47—81no—
98D4×C47—82no—
99D4×C47—82no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—42no—
109C2×C4×C47C2×SD1611yescharacteristic
110C2×C4×C47#13512yes—
111C2×C4×C47AGL(1,ℤ/8)12yes—
112C2×C4×C47#13711yescharacteristic
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C8×C47—42no—
120C47⋊C8—82no—
121#7743—471nomaximal, Sylow
122C22×C47—82no—
123C22×C47—84no—
124C22×C47—161no—
125C22×C47—162no—
126C22×C47—161no—
127C22×C47#44811yescharacteristic
128C22×C47#44912yes—
129C22×C47#45311yescharacteristic
130C22×C47#45412yes—
131C22×C47B(4,𝔽2)11yescharacteristic
132C4×C47—21no—
133C4×C47—21no—
134C4×C47—22no—
135C4×C47—22no—
136C4×C47—42no—
137C4×C47—81no—
138Dic47—82no—
139Dic47—82no—
140#2809—471no—
141#2041—472no—
142#2014D4711yescharacteristic
143#1272—471no—
144#1271—472no—
145C2×C47—162no—
146C2×C47#159611yescharacteristic
147C2×C47#159712yes—
148C2×C47#162411yescharacteristic
149C2×C47#162612yes—
150C2×C47#165711yescharacteristic
151C22×SD16—21no—
152C2×#127C2×D4711yescharacteristic
153C2×#102—21no—
154C2×#102—942no—
155C2×#102—942no—
156#400—941no—
157#398—942no—
158#391—941no—
159#391C2×D4711yescharacteristic
160#389—472no—
161#387—942no—
162#336—472no—
163#336C2×D4711yescharacteristic
164C47#774311yescharacteristic, complemented, Sylow
165C22×Q8C22⋊D4711yescharacteristic
166C22×D4—22no—
167C22×D4—21no—
168C2×SD16—42no—
169C2×SD16—42no—
170C2×SD16—42no—
171C22×C8—21no—
172C22×C8—942no—
173#127—42no—
174C2×C4⋊C4—472no—
175C2×C4⋊C4—472no—
176C2×C4⋊C4—941no—
177C2×C4⋊C4—941no—
178C2×C4⋊C4—942no—
179C2×C4⋊C4C22⋊D4711yescharacteristic
180C2×C42—941no—
181C2×C42C22×D4711yescharacteristic
182#102—42no—
183#102—1884no—
184#102—1884no—
185#95—22no—
186#95—941no—
187#95—942no—
188#95—942no—
189#95—942no—
190C24—41no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8#1062533512yes—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—942no—
210C22×C4—942no—
211C22×C4D4×D4712yes—
212C22×C4C2×C22⋊D4711yescharacteristic
213SD16—48no—
214C2×C8—22no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—1882no—
218C2×C8—1882no—
219C2×C8—1882no—
220C4⋊C4—41no—
221C4⋊C4—942no—
222C4⋊C4—942no—
223C4⋊C4—3761no—
224C4⋊C4—3761no—
225C4⋊C4—1882no—
226C4⋊C4—1882no—
227C4⋊C4—1882no—
228C42—41no—
229C42—3761no—
230C23—42no—
231C23—81no—
232C23—82no—
233C23—81no—
234C23—82no—
235C23#1309399211yescharacteristic
236Q8—42no—
237Q8—81no—
238D4—42no—
239D4—42no—
240D4—44no—
241D4—44no—
242D4—81no—
243D4—81no—
244D4—82no—
245D4—82no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4—1882no—
260C2×C4—1882no—
261C2×C4C2×#1062533511yescharacteristic
262C2×C4#1309404512yes—
263C2×C4#1309405112yes—
264C2×C4#1309405411yescharacteristic
265C8—42no—
266C8—3762no—
267C22—82no—
268C22—84no—
269C22—161no—
270C22—162no—
271C22—161no—
272C22#1568926111yescharacteristic
273C22#1568926712yes—
274C22#1568929011yescharacteristic
275C22#1568929412yes—
276C22#1568930911yescharacteristic
277C4—21no—
278C4—21no—
279C4—22no—
280C4—22no—
281C4—42no—
282C4—81no—
283C4—3762no—
284C4—3762no—
285C2—162no—
286C2#2024231011yescharacteristic
287C2#2024231612yes—
288C2#2024246911yescharacteristic
289C2#2024248312yes—
290C2#2024268411yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2014C2
socleC23×C47#120
prime radicalC2×#127×C47C22
hypercenter#2014D47
hypocenterC47#7743
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2014

Complements

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

NH ≅ G/Ntype
1. #20251376C2semidirect
3. #20241214C2semidirect
4. #20241210C2semidirect
164. C47#7743semidirect

Maximal subgroups

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

subgroupindex
#202513762
#202446192
#202412142
#202412102
C47×#20142
#774347