Finite Group Databasebeta

← #26128735

Subgroups — #26128735

Order 12032.

398 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 67 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#20254698C211yesmaximal, characteristic
2#20245305C212yesmaximal, complemented
3#20241396C212yesmaximal, complemented
4#20241328C211yesmaximal, characteristic, complemented
5C47×#2056C211yesmaximal, characteristic, complemented
6#15689673—21no—
7C2×#13093967—22no—
8#15689404—22no—
9#15689359—21no—
10#15689359C2212yes—
11#15689319—22no—
12#15688927—22no—
13C22⋊(C47⋊C16)C2212yes—
14C22×Q16×C47C2211yescharacteristic
15C2×C4⋊C8×C47C2211yescharacteristic
16C2×#103×C47—21no—
17C47×#391—21no—
18C47×#340C2211yescharacteristic
19C2×#10625350—41no—
20C2×D376—21no—
21C2×D376D412yes—
22#13094024—21no—
23#13094021—21no—
24#13094012—41no—
25#13093998—22no—
26#13093998—41no—
27C4×D188—22no—
28C4×D188—41no—
29#13093967—44no—
30C2×C47⋊C16—22no—
31#13093958—21no—
32#13093958D412yes—
33#13093946—22no—
34#13093946—22no—
35C22×Q8×C47—21no—
36C2×Q16×C47—22no—
37C2×Q16×C47—42no—
38C22×C8×C47D411yescharacteristic
39C22×C8×C47C2311yescharacteristic
40C2×C4⋊C4×C47—21no—
41C2×C42×C47D411yescharacteristic
42C4⋊C8×C47—22no—
43#103×C47—42no—
44#95×C47—41no—
45#95×C47—41no—
46C2×Q8×C47—22no—
47C2×Q8×C47—41no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C2×Q8×C47—41no—
52C22×C4×C47—41no—
53C22×C4×C47—41no—
54C22×C4×C47D811yescharacteristic
55C22×C4×C47D811yescharacteristic
56C22×C4×C47C2×D411yescharacteristic
57C2×C22⋊D47—42no—
58#10625350—82no—
59#10625350—82no—
60C2×D188—22no—
61C2×D188—41no—
62C2×D188—41no—
63C2×C4×D47—42no—
64C2×Dic94—41no—
65Q16×C47—24no—
66Q16×C47—42no—
67C2×C8×C47—21no—
68C2×C8×C47—41no—
69C2×C8×C47—41no—
70C2×C8×C47C2×D412yes—
71C4⋊C4×C47—41no—
72C4⋊C4×C47—41no—
73C42×C47—21no—
74C42×C47—21no—
75(C2×D47)⋊C4—42no—
76(C2×D47)⋊C4—42no—
77C4⋊Dic47—22no—
78C4⋊Dic47—41no—
79D376—42no—
80D376—81no—
81C47⋊C16—42no—
82C23×C47C2×D811yescharacteristic
83C22×D47—42no—
84Q8×C47—42no—
85Q8×C47—42no—
86Q8×C47—81no—
87Q8×C47—81no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—41no—
94C2×C4×C47—42no—
95C2×C4×C47—81no—
96C2×C4×C47—81no—
97C2×C4×C47—81no—
98C2×C4×C47—81no—
99C2×C4×C47—81no—
100C2×C4×C47SD3212yescomplemented
101C2×C4×C47#12011yescharacteristic
102C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
103C22⋊D47—82no—
104C22⋊D47—82no—
105C2×Dic47—42no—
106D188—42no—
107D188—42no—
108D188—82no—
109D188—82no—
110C4×D47—84no—
111Dic94—82no—
112C8×C47—22no—
113C8×C47—41no—
114#12161—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47#44711yescharacteristic
118C22×C47C2×SD3211yescharacteristic
119C22×C47#50911yescharacteristic
120C2×D47—82no—
121C2×D47—82no—
122C2×D47—82no—
123C4×C47—21no—
124C4×C47—22no—
125C4×C47—41no—
126C4×C47—82no—
127C4×C47—161no—
128C4×C47#45411yescharacteristic
129C4×C47B(4,𝔽2)11yescharacteristic
130Dic47—82no—
131#3245—471no—
132#2185—472no—
133#2056D4711yescharacteristic, complemented
134#1347—472no—
135#1322—471no—
136C2×C47—21no—
137C2×C47—21no—
138C2×C47#159811yescharacteristic
139C2×C47#218711yescharacteristic
140C2×C47#219111yescharacteristic
141D47—162no—
142C22×Q16C2×D4711yescharacteristic
143#532—941no—
144C2×SD32—942no—
145#474—942no—
146#466—472no—
147#466—941no—
148#459—942no—
149C2×C4⋊C8C2×D4711yescharacteristic
150C2×#103—21no—
151#391—21no—
152#358—942no—
153C22⋊C16—472no—
154#340C2×D4711yescharacteristic
155C47#1216111yescharacteristic, complemented, Sylow
156C2×#41—1881no—
157C22×Q8—21no—
158C2×Q16—22no—
159C2×Q16—42no—
160C2×D8—472no—
161C2×D8—941no—
162C22×C8C22⋊D4711yescharacteristic
163C22×C8C22×D4711yescharacteristic
164C4⋊Q8—941no—
165#127—941no—
166#124—1881no—
167#121—942no—
168#121—1881no—
169C4×D4—942no—
170C4×D4—1881no—
171C2×C4⋊C4—21no—
172C2×C42C22⋊D4711yescharacteristic
173SD32—1884no—
174C2×C16—942no—
175#107—472no—
176#107—941no—
177C4⋊C8—22no—
178#103—42no—
179#102—942no—
180#102—942no—
181#95—41no—
182#95—41no—
183#41—3762no—
184#41—3762no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—1881no—
192C2×D4—942no—
193C2×D4—1881no—
194C2×D4—1881no—
195C2×D4—1882no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—1882no—
199C22×C4#1062533411yescharacteristic
200C22×C4#1062533411yescharacteristic
201C22×C4C2×C22⋊D4711yescharacteristic
202Q16—24no—
203Q16—42no—
204D8—3761no—
205D8—1882no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D4712yes—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—942no—
213C4⋊C4—1881no—
214C22⋊C4—1882no—
215C22⋊C4—1882no—
216C42—21no—
217C42—21no—
218C16—1882no—
219C23—1882no—
220C23C2×#1062533411yescharacteristic
221Q8—42no—
222Q8—42no—
223Q8—81no—
224Q8—81no—
225Q8—3762no—
226D4—3762no—
227D4—3762no—
228D4—3762no—
229D4—3762no—
230D4—1882no—
231D4—1882no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—3764no—
245C2×C4—1882no—
246C2×C4#1309396712yescomplemented
247C2×C4#1309399211yescharacteristic
248C2×C4#1309404911yescharacteristic
249C8—22no—
250C8—41no—
251C22—21no—
252C22—21no—
253C22—3762no—
254C22—3762no—
255C22—3762no—
256C22#1568925611yescharacteristic
257C22C2×#1309396711yescharacteristic
258C22#1568955811yescharacteristic
259C4—22no—
260C4—21no—
261C4—41no—
262C4—82no—
263C4—161no—
264C4—3762no—
265C4#1568929511yescharacteristic
266C4#1568930911yescharacteristic
267C2—21no—
268C2—21no—
269C2—7522no—
270C2#2024232211yescharacteristic
271C2#2024531611yescharacteristic
272C2#2024533811yescharacteristic
subgroupG/H
centerC22#15689256
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#2056C2
socleC22×C47#447
prime radical#20254698C2
hypercenter#2056D47
hypocenterC47#12161
LCSG ▷ C22×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2056

Complements

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

NH ≅ G/Ntype
2. #20245305C2semidirect
3. #20241396C2semidirect
4. #20241328C2semidirect
5. C47×#2056C2semidirect
100. C2×C4×C47SD32semidirect
133. #2056D47semidirect
155. C47#12161semidirect
246. C2×C4#13093967semidirect

Maximal subgroups

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

subgroupindex
#202546982
#202453052
#202413962
#202413282
C47×#20562
#1216147