Finite Group Databasebeta

← #26128736

Subgroups — #26128736

Order 12032.

398 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 71 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#20245304C212yesmaximal, complemented
2#20244711C211yesmaximal, characteristic, complemented
3#20241397C212yesmaximal, complemented
4#20241328C211yesmaximal, characteristic, complemented
5C47×#3245C211yesmaximal, characteristic
6C22×Dic188C2211yescharacteristic
7C2×#13093968—22no—
8C2×#13093949—21no—
9#15689029—21no—
10#15688926—22no—
11C22⋊(C47⋊C16)C2212yes—
12#15688886C2211yescharacteristic
13C47×#532—21no—
14C47×#474—22no—
15C47×#466—21no—
16C47×#466C2212yes—
17C47×#459—22no—
18C2×C4⋊C8×C47C2211yescharacteristic
19C22×Dic94—21no—
20C2×Dic188—22no—
21C2×Dic188—42no—
22C2×C4⋊Dic47—21no—
23#13093968—44no—
24C2×C47⋊C16—22no—
25#13093949—42no—
26C47⋊#95—41no—
27C47⋊#95—41no—
28C2×#41×C47—41no—
29C2×D8×C47—21no—
30C2×D8×C47D412yes—
31C22×C8×C47D411yescharacteristic
32C22×C8×C47C2311yescharacteristic
33C4⋊Q8×C47—21no—
34#127×C47—21no—
35#124×C47—41no—
36#121×C47—22no—
37#121×C47—41no—
38C4×D4×C47—22no—
39C4×D4×C47—41no—
40C2×C42×C47D411yescharacteristic
41#107×C47—21no—
42#107×C47D412yes—
43C4⋊C8×C47—22no—
44#102×C47—22no—
45#102×C47—22no—
46#41×C47—82no—
47#41×C47—82no—
48C2×Q8×C47—41no—
49C2×D4×C47—22no—
50C2×D4×C47—41no—
51C2×D4×C47—41no—
52C2×D4×C47—42no—
53C22×C4×C47—42no—
54C22×C4×C47D811yescharacteristic
55C22×C4×C47D811yescharacteristic
56C22×C4×C47C2×D411yescharacteristic
57C22×Dic47—41no—
58C22×Dic47—41no—
59C2×Dic94—22no—
60C2×Dic94—41no—
61C2×Dic94—41no—
62C2×Dic94—41no—
63C2×Dic94—41no—
64C2×Dic94—41no—
65D8×C47—42no—
66D8×C47—81no—
67C2×C8×C47—21no—
68C2×C8×C47—41no—
69C2×C8×C47—41no—
70C2×C8×C47C2×D412yes—
71C4⋊C4×C47—22no—
72C4⋊C4×C47—41no—
73C22⋊C4×C47—42no—
74C22⋊C4×C47—42no—
75C42×C47—21no—
76C42×C47—21no—
77C4⋊Dic47—41no—
78C4⋊Dic47—41no—
79Dic188—24no—
80Dic188—42no—
81C47⋊C16—42no—
82C23×C47—42no—
83C23×C47C2×D811yescharacteristic
84Q8×C47—82no—
85D4×C47—42no—
86D4×C47—42no—
87D4×C47—82no—
88D4×C47—82no—
89D4×C47—82no—
90D4×C47—82no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—41no—
97C2×C4×C47—42no—
98C2×C4×C47—84no—
99C2×C4×C47SD3212yescomplemented
100C2×C4×C47#12011yescharacteristic
101C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
102C2×Dic47—42no—
103C2×Dic47—81no—
104C2×Dic47—81no—
105C2×Dic47—81no—
106C2×Dic47—81no—
107C2×Dic47—81no—
108Dic94—42no—
109Dic94—42no—
110Dic94—81no—
111Dic94—81no—
112C8×C47—22no—
113C8×C47—41no—
114#12161—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47—82no—
118C22×C47—82no—
119C22×C47—82no—
120C22×C47#44711yescharacteristic
121C22×C47C2×SD3211yescharacteristic
122C22×C47#50911yescharacteristic
123C4×C47—21no—
124C4×C47—22no—
125C4×C47—41no—
126C4×C47—82no—
127C4×C47#45411yescharacteristic
128C4×C47B(4,𝔽2)11yescharacteristic
129Dic47—82no—
130Dic47—161no—
131#3245D4711yescharacteristic
132#2185—472no—
133#2056—471no—
134#1347—472no—
135#1322—471no—
136C2×C47—21no—
137C2×C47—21no—
138C2×C47—162no—
139C2×C47#159811yescharacteristic
140C2×C47#218711yescharacteristic
141C2×C47#219111yescharacteristic
142C22×Q16—471no—
143#532—21no—
144C2×SD32—942no—
145#474—22no—
146#466—21no—
147#466C2×D4712yes—
148#459—22no—
149C2×C4⋊C8C2×D4711yescharacteristic
150C2×#103—941no—
151#391—941no—
152#358—942no—
153C22⋊C16—472no—
154#340—471no—
155C47#1216111yescharacteristic, complemented, Sylow
156C2×#41—41no—
157C22×Q8—941no—
158C2×Q16—942no—
159C2×Q16—1882no—
160C2×D8—21no—
161C2×D8C22⋊D4712yes—
162C22×C8C22⋊D4711yescharacteristic
163C22×C8C22×D4711yescharacteristic
164C4⋊Q8—21no—
165#127—21no—
166#124—41no—
167#121—22no—
168#121—41no—
169C4×D4—22no—
170C4×D4—41no—
171C2×C4⋊C4—941no—
172C2×C42C22⋊D4711yescharacteristic
173SD32—1884no—
174C2×C16—942no—
175#107—21no—
176#107C22⋊D4712yes—
177C4⋊C8—22no—
178#103—1882no—
179#102—22no—
180#102—22no—
181#95—1881no—
182#95—1881no—
183#41—82no—
184#41—82no—
185C2×Q8—41no—
186C2×Q8—942no—
187C2×Q8—1881no—
188C2×Q8—1881no—
189C2×Q8—1881no—
190C2×Q8—1881no—
191C2×Q8—1881no—
192C2×D4—22no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—42no—
196C22×C4—42no—
197C22×C4—1881no—
198C22×C4—1881no—
199C22×C4#1062533411yescharacteristic
200C22×C4#1062533411yescharacteristic
201C22×C4C2×C22⋊D4711yescharacteristic
202Q16—944no—
203Q16—1882no—
204D8—42no—
205D8—81no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×C22⋊D4712yes—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—1881no—
213C4⋊C4—1881no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—21no—
217C42—21no—
218C16—1882no—
219C23—42no—
220C23C2×#1062533411yescharacteristic
221Q8—82no—
222Q8—3761no—
223Q8—3761no—
224Q8—1882no—
225Q8—1882no—
226D4—42no—
227D4—42no—
228D4—82no—
229D4—82no—
230D4—82no—
231D4—82no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—84no—
240C2×C4—3761no—
241C2×C4—3761no—
242C2×C4—3761no—
243C2×C4—3761no—
244C2×C4—3761no—
245C2×C4—1882no—
246C2×C4#1309396812yescomplemented
247C2×C4#1309399311yescharacteristic
248C2×C4#1309405011yescharacteristic
249C8—22no—
250C8—41no—
251C22—21no—
252C22—21no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22#1568925711yescharacteristic
257C22C2×#1309396811yescharacteristic
258C22#1568955911yescharacteristic
259C4—22no—
260C4—21no—
261C4—41no—
262C4—82no—
263C4—3762no—
264C4—7521no—
265C4#1568929611yescharacteristic
266C4#1568931011yescharacteristic
267C2—21no—
268C2—21no—
269C2—162no—
270C2#2024232311yescharacteristic
271C2#2024531711yescharacteristic
272C2#2024533911yescharacteristic
subgroupG/H
centerC22#15689257
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
Fitting, prime radicalC47×#3245C2
socleC22×C47#447
hypercenter#3245D47
hypocenterC47#12161
LCSG ▷ C22×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #20245304C2semidirect
2. #20244711C2semidirect
3. #20241397C2semidirect
4. #20241328C2semidirect
99. C2×C4×C47SD32semidirect
155. C47#12161semidirect
246. C2×C4#13093968semidirect

Maximal subgroups

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

subgroupindex
#202453042
#202447112
#202413972
#202413282
C47×#32452
#1216147