Finite Group Databasebeta

← #26117242

Subgroups — #26117242

Order 12032.

636 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 105 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#20253384C212yesmaximal, complemented
2#20244186C211yesmaximal, characteristic, complemented
3#20241787C212yesmaximal, complemented
4#20241219C211yesmaximal, characteristic, complemented
5C47×#3150C211yesmaximal, characteristic, complemented
6C2×C4×D188C2211yescharacteristic
7#15689434—22no—
8#15689400—22no—
9#15689315—22no—
10#15689256—22no—
11C4×#10625334—24no—
12#15689176C2212yes—
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×(C2×D47)⋊C8—22no—
15D4⋊(C47⋊C8)—24no—
16C2×#135×C47—22no—
17C47×#532C2212yes—
18C47×#497—22no—
19C47×#496—22no—
20C2×#97×C47C2211yescharacteristic
21C22×D188D411yescharacteristic
22C22×C4×D47—21no—
23C2×#10625334—24no—
24C22×C47⋊C8—21no—
25C4×D188—22no—
26C4×D188—22no—
27C2×C4⋊Dic47D411yescharacteristic
28C2×(C2×D47)⋊C4—21no—
29#13093957—24no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)—22no—
32#13093945—24no—
33#13093944—24no—
34C22⋊(C47⋊C8)—22no—
35(C2×D47)⋊C8—44no—
36C4×C47⋊C8D414yes—
37C2×#41×C47—22no—
38#135×C47—44no—
39#135×C47—44no—
40C2×Q16×C47—22no—
41C2×SD16×C47—24no—
42C2×D8×C47—22no—
43C22×C8×C47D412yes—
44C4⋊Q8×C47D412yes—
45#127×C47D412yes—
46#124×C47—22no—
47#121×C47—22no—
48#121×C47—22no—
49C4×D4×C47—22no—
50C4×D4×C47—22no—
51C2×C42×C47C2311yescharacteristic
52#97×C47—22no—
53C23×D47—21no—
54#41×C47—44no—
55#41×C47—44no—
56C2×Q8×C47—22no—
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—22no—
60C2×D4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47#4111yescharacteristic
65C22×Dic47—21no—
66C2×D188—22no—
67C2×D188—42no—
68C2×C4×D47—41no—
69C2×C4×D47—41no—
70C2×C4×D47—42no—
71C2×C4×D47—42no—
72Q16×C47—44no—
73SD16×C47—48no—
74D8×C47—44no—
75C2×C8×C47—22no—
76C2×C8×C47—22no—
77C2×C8×C47D814yescomplemented
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C22⋊C4×C47—42no—
81C22⋊C4×C47—42no—
82C42×C47C2×D412yes—
83C42×C47C2×D412yes—
84#10625334—48no—
85(C2×D47)⋊C4—42no—
86C4⋊Dic47—22no—
87C2×C47⋊C8—24no—
88C2×C47⋊C8—41no—
89C2×C47⋊C8—42no—
90C23×C47—42no—
91C23×C47#12111yescharacteristic
92C22×D47—22no—
93C22×D47—41no—
94C22×D47—42no—
95C22×D47—41no—
96C22×D47—42no—
97Q8×C47—44no—
98D4×C47—44no—
99D4×C47—44no—
100D4×C47—44no—
101D4×C47—84no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—42no—
105C2×C4×C47—82no—
106C2×C4×C47—82no—
107C2×C4×C47#12012yes—
108C2×C4×C47#12011yescharacteristic
109C2×C4×C47#12112yes—
110C2×C4×C47#12711yescharacteristic
111C2×C4×C47C2×D812yes—
112C2×C4×C47AGL(1,ℤ/8)12yes—
113C2×Dic47—41no—
114C2×Dic47—42no—
115D188—24no—
116D188—42no—
117C4×D47—82no—
118C4×D47—81no—
119C4×D47—81no—
120C8×C47—24no—
121C47⋊C8—44no—
122C47⋊C8—81no—
123#10032—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—82no—
127C22×C47—82no—
128C22×C47—82no—
129C22×C47#39211yescharacteristic
130C22×C47#45912yes—
131C2×D47—44no—
132C2×D47—42no—
133C2×D47—82no—
134C2×D47—81no—
135C2×D47—81no—
136C4×C47—22no—
137C4×C47—22no—
138C4×C47—82no—
139C4×C47#44712yes—
140C4×C47#49311yescharacteristic
141C4×C47#49611yescharacteristic
142Dic47—81no—
143#3150D4711yescharacteristic, complemented
144#3100—472no—
145#1955—471no—
146#1480—472no—
147#1275—471no—
148C2×C47—22no—
149C2×C47—162no—
150C2×C47#201011yescharacteristic
151C2×C47#201111yescharacteristic
152C2×C47#201711yescharacteristic
153D47—82no—
154C2×#135—22no—
155#532C2×D4712yes—
156C2×C4×D4—471no—
157#497—22no—
158#496—22no—
159#480—942no—
160#474—942no—
161#459—942no—
162#447—942no—
163C4×D8—944no—
164#431—472no—
165C2×C4⋊C8—471no—
166C2×C22⋊C8—942no—
167C2×#97C2×D4711yescharacteristic
168D4⋊C8—944no—
169C47#1003211yescharacteristic, complemented, Sylow
170C2×#41—22no—
171C22×D4—471no—
172C23×C4—941no—
173#135—44no—
174#135—44no—
175C2×Q16—22no—
176C2×SD16—24no—
177C2×D8—22no—
178C2×D8—944no—
179C22×C8—941no—
180C22×C8C22⋊D4712yes—
181C4⋊Q8C22⋊D4712yes—
182#127C22⋊D4712yes—
183#124—22no—
184#121—22no—
185#121—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—942no—
189C4×D4—942no—
190C2×C4⋊C4—471no—
191C2×C22⋊C4—941no—
192C2×C42C22×D4711yescharacteristic
193#107—944no—
194C4⋊C8—942no—
195C4⋊C8—942no—
196#102—944no—
197#102—944no—
198C22⋊C8—942no—
199C22⋊C8—1884no—
200#97—22no—
201C4×C8—474no—
202C24—941no—
203#41—44no—
204#41—44no—
205C2×Q8—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—42no—
210C2×D4—942no—
211C2×D4—1882no—
212C22×C4—42no—
213C22×C4—941no—
214C22×C4—1881no—
215C22×C4—1881no—
216C22×C4—1882no—
217C22×C4—1882no—
218C22×C4D4×D4711yescharacteristic
219C22×C4#1062535211yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221Q16—44no—
222SD16—48no—
223D8—44no—
224D8—1888no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—944no—
228C2×C8—1881no—
229C2×C8—1882no—
230C2×C8#1062533414yescomplemented
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—942no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—1882no—
237C42D4×D4712yes—
238C42C2×C22⋊D4712yes—
239C23—42no—
240C23—942no—
241C23—1881no—
242C23—1882no—
243C23—1881no—
244C23—1882no—
245C23#1309399511yescharacteristic
246Q8—44no—
247D4—44no—
248D4—44no—
249D4—44no—
250D4—84no—
251D4—944no—
252D4—1882no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—3761no—
259C2×C4—3761no—
260C2×C4—3762no—
261C2×C4—1881no—
262C2×C4—1882no—
263C2×C4#1309399212yes—
264C2×C4#1309399311yescharacteristic
265C2×C4#1309399612yes—
266C2×C4#1309402011yescharacteristic
267C2×C4C2×#1062533412yes—
268C2×C4#1309405012yes—
269C8—24no—
270C8—3761no—
271C8—1884no—
272C22—21no—
273C22—21no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—3762no—
278C22—3761no—
279C22—3761no—
280C22—1884no—
281C22—1882no—
282C22#1568903111yescharacteristic
283C22#1568931512yes—
284C4—22no—
285C4—22no—
286C4—82no—
287C4—3761no—
288C4#1568925712yes—
289C4#1568949911yescharacteristic
290C4#1568951011yescharacteristic
291C2—22no—
292C2—162no—
293C2—3762no—
294C2#2024445211yescharacteristic
295C2#2024445911yescharacteristic
296C2#2024449111yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3150C2
socleC22×C47#392
hypercenter#3150D47
hypocenterC47#10032
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3150

Complements

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

NH ≅ G/Ntype
1. #20253384C2semidirect
2. #20244186C2semidirect
3. #20241787C2semidirect
4. #20241219C2semidirect
5. C47×#3150C2semidirect
77. C2×C8×C47D8semidirect
143. #3150D47semidirect
169. C47#10032semidirect
230. C2×C8#10625334semidirect

Maximal subgroups

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

subgroupindex
#202533842
#202441862
#202417872
#202412192
C47×#31502
#1003247