Finite Group Databasebeta

← #26104020

Subgroups — #26104020

Order 12032.

286 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 59 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#20245053C211yesmaximal, characteristic
2#20241980C211yesmaximal, characteristic
3C47×#1884C211yesmaximal, characteristic
4#15689097—21no—
5C4×#10625329—21no—
6#15688918—21no—
7#15688916C412yescomplemented
8#15688863—21no—
9#15688863C412yescomplemented
10C47×#568C411yescharacteristic, complemented
11C47×#532C2211yescharacteristic
12C47×#421—21no—
13C47×#410C411yescharacteristic, complemented
14C47×#339—21no—
15C2×#10625329—41no—
16#13093983—41no—
17#13093939—22no—
18#13093939—41no—
19#13093935—22no—
20C4×C47⋊C8—41no—
21C2×#41×C47C2×C411yescharacteristic
22C2×#41×C47D411yescharacteristic
23#131×C47—42no—
24C2×M16×C47—21no—
25C2×M16×C47—41no—
26C4⋊Q8×C47Q811yescharacteristic
27#127×C47C2×C411yescharacteristic
28#124×C47C2×C411yescharacteristic
29#124×C47D411yescharacteristic
30#121×C47—41no—
31C4×D4×C47—41no—
32#117×C47—21no—
33C2×C42×C47D411yescharacteristic
34#104×C47—42no—
35#99×C47—41no—
36#41×C47—41no—
37#41×C47—41no—
38#41×C47—41no—
39#41×C47—41no—
40C2×Q8×C47C4⋊C411yescharacteristic
41C2×Q8×C47C4⋊C411yescharacteristic
42C2×D4×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47C4211yescharacteristic
46C2×D4×C47C22⋊C411yescharacteristic
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47C22⋊C411yescharacteristic
51M16×C47—22no—
52M16×C47—42no—
53M16×C47—41no—
54C2×C8×C47—41no—
55C2×C8×C47—42no—
56C4⋊C4×C47—41no—
57C4⋊C4×C47—41no—
58C22⋊C4×C47—41no—
59C22⋊C4×C47—41no—
60C22⋊C4×C47—41no—
61C42×C47—21no—
62C42×C47—41no—
63C42×C47C22⋊C411yescharacteristic
64C42×C47C4⋊C411yescharacteristic
65C22⋊Dic47—42no—
66C4⋊Dic47—81no—
67C4×Dic47—81no—
68#10625329—44no—
69C2×C47⋊C8—42no—
70C23×C47—21no—
71C23×C47—21no—
72C23×C47#9911yescharacteristic
73Q8×C47—41no—
74Q8×C47—41no—
75D4×C47—41no—
76D4×C47—41no—
77D4×C47—42no—
78D4×C47—82no—
79D4×C47—81no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—42no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—81no—
92C2×C4×C47#9511yescharacteristic
93C2×C4×C47#9911yescharacteristic
94C2×Dic47—82no—
95C8×C47—42no—
96C47⋊C8—44no—
97#7486—471nomaximal, Sylow
98C22×C47—21no—
99C22×C47—41no—
100C22×C47—41no—
101C22×C47—41no—
102C22×C47—81no—
103C22×C47—81no—
104C22×C47#34211yescharacteristic
105C4×C47—22no—
106C4×C47—21no—
107C4×C47—41no—
108C4×C47—41no—
109C4×C47—41no—
110C4×C47—161no—
111C4×C47AΣL(1,𝔽16)11yescharacteristic
112C4×C47#35211yescharacteristic
113Dic47—162no—
114#2126—471no—
115#1884D4711yescharacteristic
116#1525—471no—
117C2×C47—21no—
118C2×C47—41no—
119C2×C47—81no—
120C2×C47—81no—
121C2×C47#139011yescharacteristic
122#568Dic4711yescharacteristic, complemented
123#532C2×D4711yescharacteristic
124#421—21no—
125#410—941no—
126#410Dic4711yescharacteristic, complemented
127C4×M16—941no—
128#354—941no—
129#353—472no—
130#339—21no—
131#329—472no—
132#329—941no—
133C47#748611yescharacteristic, complemented, Sylow
134C2×#41C2×Dic4711yescharacteristic
135C2×#41C22⋊D4711yescharacteristic
136#131—42no—
137C2×M16—21no—
138C2×M16—41no—
139C2×M16—1881no—
140C4⋊Q8Dic9411yescharacteristic
141#127C4×D4711yescharacteristic
142#124C4×D4711yescharacteristic
143#124D18811yescharacteristic
144#121—41no—
145C4×D4—41no—
146#117—21no—
147#117—1881no—
148C2×C42C22⋊D4711yescharacteristic
149#104—42no—
150#99—41no—
151#99—942no—
152#99—1881no—
153#97—942no—
154C4×C8—1881no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8Dic47⋊C411yescharacteristic
160C2×Q8C4⋊Dic4711yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4C4×Dic4711yescharacteristic
165C2×D4(C2×D47)⋊C411yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4C22⋊Dic4711yescharacteristic
170M16—22no—
171M16—42no—
172M16—41no—
173M16—1884no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—1882no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—3761no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—1882no—
184C42—21no—
185C42—41no—
186C42—3761no—
187C42Dic47⋊C411yescharacteristic
188C42(C2×D47)⋊C411yescharacteristic
189C23—21no—
190C23—21no—
191C23#1309393911yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—42no—
197D4—82no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—81no—
211C2×C4—3762no—
212C2×C4C47⋊#9511yescharacteristic
213C2×C4#1309393911yescharacteristic
214C8—42no—
215C8—1884no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#1568889011yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—161no—
229C4—7522no—
230C4#1568891011yescharacteristic
231C4#1568891311yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2024150611yescharacteristic
subgroupG/H
centerC4#15688910
commutatorC2×D4×C47C42
Frattini#532C2×D47
FittingC47×#1884C2
socleC2×C47#1390
prime radical, cosocleC47×#532C22
hypercenter#1884D47
hypocenterC47#7486
LCSG ▷ C2×D4×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1884

Complements

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

NH ≅ G/Ntype
7. #15688916C4semidirect
9. #15688863C4semidirect
10. C47×#568C4semidirect
13. C47×#410C4semidirect
122. #568Dic47semidirect
126. #410Dic47semidirect
133. C47#7486semidirect

Maximal subgroups

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

subgroupindex
#202450532
#202419802
C47×#18842
#748647