Finite Group Databasebeta

← #26103806

Subgroups — #26103806

Order 12032.

268 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 43 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#20241515C211yesmaximal, characteristic, complemented
2#20241214C211yesmaximal, characteristic, complemented
3C47×#2007C211yesmaximal, characteristic
4C2×#13093943—21no—
5#15688878—21no—
6C2×#137×C47—21no—
7C47×#537C411yescharacteristic
8C2×#104×C47—21no—
9C47×#391C2211yescharacteristic
10C47×#343C411yescharacteristic
11C2×#10625329—41no—
12C22×C47⋊C8—41no—
13#13093943—22no—
14#13093943—41no—
15C2×#41×C47—21no—
16C22×Q8×C47D411yescharacteristic
17#137×C47—42no—
18#137×C47—41no—
19#137×C47—41no—
20C2×Q16×C47—41no—
21C2×SD16×C47—41no—
22C2×M16×C47—21no—
23C22⋊Q8×C47—41no—
24#121×C47—41no—
25#121×C47—41no—
26C2×C4⋊C4×C47C2×C411yescharacteristic
27C2×C42×C47D411yescharacteristic
28#104×C47—42no—
29#95×C47—41no—
30#41×C47—41no—
31#41×C47—41no—
32#41×C47—41no—
33#41×C47—41no—
34C2×Q8×C47—21no—
35C2×Q8×C47—21no—
36C2×Q8×C47—41no—
37C2×Q8×C47—41no—
38C2×Q8×C47D811yescharacteristic
39C2×Q8×C47SD1611yescharacteristic
40C2×D4×C47—21no—
41C2×D4×C47—41no—
42C2×D4×C47—41no—
43C22×C4×C47—21no—
44C22×C4×C47—21no—
45C22×C4×C47—21no—
46C22×C4×C47—41no—
47C22×C4×C47C22⋊C411yescharacteristic
48Q16×C47—42no—
49Q16×C47—42no—
50SD16×C47—42no—
51SD16×C47—42no—
52M16×C47—22no—
53M16×C47—41no—
54C2×C8×C47—41no—
55C4⋊C4×C47—41no—
56C4⋊C4×C47—41no—
57C22⋊C4×C47—42no—
58C42×C47—41no—
59#10625329—42no—
60#10625329—81no—
61C2×C47⋊C8—42no—
62C2×C47⋊C8—81no—
63C2×C47⋊C8—81no—
64C2×C47⋊C8—81no—
65C23×C47—41no—
66C23×C47#9911yescharacteristic
67Q8×C47—41no—
68Q8×C47—41no—
69Q8×C47—41no—
70Q8×C47—41no—
71Q8×C47—81no—
72D4×C47—41no—
73D4×C47—41no—
74D4×C47—81no—
75D4×C47—81no—
76D4×C47—161no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—41no—
84C2×C4×C47—41no—
85C2×C4×C47—41no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—81no—
90C2×C4×C47—81no—
91C2×C4×C47—81no—
92C2×C4×C47#10211yescharacteristic
93C2×C4×C47#10411yescharacteristic
94C8×C47—42no—
95C47⋊C8—82no—
96C47⋊C8—82no—
97#7409—471nomaximal, Sylow
98C22×C47—21no—
99C22×C47—21no—
100C22×C47—81no—
101C22×C47—81no—
102C22×C47—81no—
103C22×C47#32711yescharacteristic
104C22×C47AΣL(1,𝔽16)11yescharacteristic
105C22×C47#35311yescharacteristic
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—41no—
109C4×C47—41no—
110C4×C47—41no—
111C4×C47—41no—
112C4×C47—81no—
113C4×C47—81no—
114#2007D4711yescharacteristic
115#1394—471no—
116#1272—471no—
117C2×C47—21no—
118C2×C47—21no—
119C2×C47—161no—
120C2×C47#134311yescharacteristic
121C2×C47#140311yescharacteristic
122C2×C47#140511yescharacteristic
123C2×#137—21no—
124#537Dic4711yescharacteristic
125C2×#104—21no—
126C2×#101—941no—
127#391C2×D4711yescharacteristic
128#343Dic4711yescharacteristic
129#336—941no—
130C47#740911yescharacteristic, complemented, Sylow
131C2×#41—21no—
132C22×Q8C22⋊D4711yescharacteristic
133#137—42no—
134#137—41no—
135#137—41no—
136C2×Q16—41no—
137C2×SD16—41no—
138C2×M16—21no—
139C2×M16—1881no—
140C22×C8—1881no—
141C22⋊Q8—41no—
142#121—41no—
143#121—41no—
144C2×C4⋊C4C2×Dic4711yescharacteristic
145C2×C42C22⋊D4711yescharacteristic
146#104—42no—
147#101—942no—
148#101—1881no—
149#95—41no—
150#41—41no—
151#41—41no—
152#41—41no—
153#41—41no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8#1062533411yescharacteristic
159C2×Q8#1062533511yescharacteristic
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4C22⋊Dic4711yescharacteristic
168Q16—42no—
169Q16—42no—
170SD16—42no—
171SD16—42no—
172M16—22no—
173M16—41no—
174M16—3761no—
175M16—1882no—
176C2×C8—41no—
177C2×C8—3761no—
178C2×C8—3761no—
179C2×C8—3761no—
180C2×C8—1882no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C22⋊C4—42no—
184C42—41no—
185C23—41no—
186C23#1309393911yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—81no—
192D4—41no—
193D4—41no—
194D4—81no—
195D4—81no—
196D4—161no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4#1309394511yescharacteristic
213C2×C4#1309395111yescharacteristic
214C8—42no—
215C8—3762no—
216C8—3762no—
217C22—21no—
218C22—21no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22#1568885711yescharacteristic
223C22#1568891011yescharacteristic
224C22#1568891611yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—81no—
233C2—21no—
234C2—21no—
235C2—161no—
236C2#2024138511yescharacteristic
237C2#2024153611yescharacteristic
238C2#2024154211yescharacteristic
subgroupG/H
centerC22#15688910
commutatorC2×C4⋊C4×C47C2×C4
Frattini#391C2×D47
FittingC47×#2007C2
socleC22×C47AΣL(1,𝔽16)
prime radicalC47×#537C4
cosocleC47×#391C22
hypercenter#2007D47
hypocenterC47#7409
LCSG ▷ C2×C4⋊C4×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #2007

Complements

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

NH ≅ G/Ntype
1. #20241515C2semidirect
2. #20241214C2semidirect
130. C47#7409semidirect

Maximal subgroups

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

subgroupindex
#202415152
#202412142
C47×#20072
#740947