Finite Group Databasebeta

← #26135963

Subgroups — #26135963

Order 12032.

416 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 79 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#20251381C211yesmaximal, characteristic
2#20243603C212yesmaximal
3#20241515C211yesmaximal, characteristic
4#20241514C212yesmaximal
5C47×#1857C211yesmaximal, characteristic
6#15689166—22no—
7C2×#13093943C2212yes—
8#15689044—22no—
9#15689043—21no—
10#15689030C2212yes—
11#15689029—21no—
12#15689027—22no—
13#15688882—22no—
14C47×#519C2211yescharacteristic
15C47×#428—21no—
16C2×#101×C47C2211yescharacteristic
17C47×#391C2211yescharacteristic
18C47×#339—21no—
19C22×Dic94—21no—
20C22×Dic94—21no—
21C2×#10625329—22no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47D412yes—
24C2×Dic47⋊C4—22no—
25C2×C4×Dic47D412yes—
26#13093957—42no—
27#13093955—42no—
28#13093943—22no—
29#13093943Q814yescomplemented
30C47⋊#95—21no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C22×Q8×C47C2311yescharacteristic
35C2×M16×C47—21no—
36C4×Q8×C47—41no—
37C4×Q8×C47—41no—
38#117×C47—21no—
39#117×C47—22no—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C42×C47D411yescharacteristic
42#107×C47—41no—
43#106×C47—41no—
44#101×C47—21no—
45#101×C47Q812yescomplemented
46#95×C47—22no—
47C2×Q8×C47—41no—
48C2×Q8×C47—41no—
49C2×Q8×C47C2×Q811yescharacteristic
50C2×Q8×C47C2×Q812yes—
51C2×Q8×C47#4111yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47C2×D412yes—
55C22×C4×C47C2×D411yescharacteristic
56C22×Dic47—22no—
57C22×Dic47—22no—
58C2×Dic94—24no—
59C2×Dic94—81no—
60C2×Dic94—81no—
61C2×Dic94—82no—
62C2×Dic94—81no—
63M16×C47—22no—
64M16×C47—41no—
65C2×C8×C47—41no—
66C4⋊C4×C47—21no—
67C4⋊C4×C47—21no—
68C4⋊C4×C47—22no—
69C4⋊C4×C47—22no—
70C4⋊C4×C47—41no—
71C22⋊C4×C47—41no—
72C42×C47—42no—
73C42×C47—41no—
74C42×C47—41no—
75C4⋊Dic47—42no—
76C4⋊Dic47—81no—
77Dic47⋊C4—82no—
78C4×Dic47—42no—
79#10625329—24no—
80#10625329—42no—
81C2×C47⋊C8—42no—
82C23×C47#12011yescharacteristic
83Q8×C47—41no—
84Q8×C47—42no—
85Q8×C47—41no—
86C2×C4×C47—22no—
87C2×C4×C47—21no—
88C2×C4×C47—22no—
89C2×C4×C47—21no—
90C2×C4×C47—22no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47C22⋊Q812yes—
98C2×C4×C47C22⋊Q811yescharacteristic
99C2×C4×C47C4⋊Q811yescharacteristic
100C2×C4×C47C4⋊Q812yes—
101C2×Dic47—24no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104C2×Dic47—42no—
105C2×Dic47—42no—
106C2×Dic47—42no—
107Dic94—84no—
108Dic94—84no—
109C8×C47—42no—
110C47⋊C8—44no—
111#13464—471nomaximal, Sylow
112C22×C47—21no—
113C22×C47—21no—
114C22×C47#39511yescharacteristic
115C22×C47B(4,𝔽2)11yescharacteristic
116C22×C47#45811yescharacteristic
117C4×C47—22no—
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—22no—
121C4×C47—42no—
122C4×C47—81no—
123C4×C47—81no—
124C4×C47—81no—
125Dic47—44no—
126Dic47—82no—
127#2811—471no—
128#1857—472no—
129#1857D4711yescharacteristic
130#1394—472no—
131#1394—471no—
132C2×C47—21no—
133C2×C47—21no—
134C2×C47#206211yescharacteristic
135C2×C47#220211yescharacteristic
136C2×C47#220511yescharacteristic
137#519C2×D4711yescharacteristic
138#428—21no—
139#428—942no—
140C2×#101—472no—
141C2×#101C2×D4711yescharacteristic
142#395—942no—
143#395—941no—
144#391—472no—
145#391—942no—
146#391—941no—
147#391C2×D4711yescharacteristic
148#339—21no—
149#339—942no—
150C47#1346411yescharacteristic, complemented, Sylow
151C22×Q8—941no—
152C22×Q8—941no—
153C22×Q8C22×D4711yescharacteristic
154C2×M16—21no—
155C2×M16—942no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—22no—
159#117—21no—
160C2×C4⋊C4—472no—
161C2×C4⋊C4—942no—
162C2×C4⋊C4—941no—
163C2×C4⋊C4C22⋊D4711yescharacteristic
164C2×C42—472no—
165C2×C42C22⋊D4711yescharacteristic
166#107—41no—
167#107—1882no—
168#106—41no—
169#106—1882no—
170#101—21no—
171#101—474no—
172#101—942no—
173#101Dic9412yescomplemented
174#95—22no—
175#95—942no—
176#95—941no—
177#95—942no—
178#95—942no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—944no—
182C2×Q8—3761no—
183C2×Q8—3761no—
184C2×Q8—3762no—
185C2×Q8—3761no—
186C2×Q8C2×Dic9411yescharacteristic
187C2×Q8#1062535011yescharacteristic
188C2×Q8Q8×D4712yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—942no—
192C22×C4—942no—
193C22×C4D4×D4712yes—
194C22×C4C2×C22⋊D4711yescharacteristic
195M16—22no—
196M16—41no—
197M16—944no—
198M16—1882no—
199C2×C8—41no—
200C2×C8—1882no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—3762no—
207C4⋊C4—3761no—
208C4⋊C4—1882no—
209C22⋊C4—41no—
210C42—41no—
211C42—41no—
212C42—42no—
213C42—1882no—
214C23#1309399211yescharacteristic
215Q8—41no—
216Q8—41no—
217Q8—42no—
218Q8—3764no—
219Q8—3764no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—944no—
232C2×C4—1882no—
233C2×C4—1882no—
234C2×C4—1882no—
235C2×C4—1882no—
236C2×C4—1882no—
237C2×C4#1309400212yes—
238C2×C4#1309400311yescharacteristic
239C2×C4#1309402211yescharacteristic
240C2×C4#1309402312yes—
241C8—42no—
242C8—1884no—
243C22—21no—
244C22—21no—
245C22#1568904311yescharacteristic
246C22#1568930911yescharacteristic
247C22#1568931211yescharacteristic
248C4—21no—
249C4—22no—
250C4—21no—
251C4—22no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4—3762no—
257C4—1884no—
258C2—21no—
259C2—21no—
260C2#2024474111yescharacteristic
261C2#2024540411yescharacteristic
262C2#2024541711yescharacteristic
subgroupG/H
centerC22#15689309
commutator, cosocleC22×Q8×C47C23
FrattiniC22×Q8C22×D47
FittingC47×#1857C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
hypercenter#1857D47
hypocenterC47#13464
LCSG ▷ C22×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
29. #13093943Q8semidirect
45. #101×C47Q8semidirect
150. C47#13464semidirect
173. #101Dic94semidirect

Maximal subgroups

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

subgroupindex
#202513812
#202436032
#202415152
#202415142
C47×#18572
#1346447