Finite Group Databasebeta

← #26104019

Subgroups — #26104019

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#20243772C211yesmaximal, characteristic
2#20241980C211yesmaximal, characteristic
3C47×#2126C211yesmaximal, characteristic
4#15689890C411yescharacteristic, complemented
5#15689140—21no—
6#15689097C411yescharacteristic, complemented
7C4×#10625329—21no—
8#15688883—21no—
9#15688863—21no—
10#15688863C412yescomplemented
11C47×#532C2211yescharacteristic
12C47×#410—21no—
13C47×#354—21no—
14C47×#353C412yescomplemented
15#13094031—42no—
16C2×#10625329—21no—
17C2×#10625329—41no—
18C2×#10625329—41no—
19#13093983—21no—
20#13093951—42no—
21#13093939—41no—
22#13093935—22no—
23C4×C47⋊C8—41no—
24C2×#41×C47C2×C411yescharacteristic
25C2×#41×C47D411yescharacteristic
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—41no—
33C2×C42×C47D411yescharacteristic
34#99×C47—22no—
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
51C4⋊C4×C47—41no—
52C4⋊C4×C47—81no—
53C22⋊C4×C47—41no—
54C22⋊C4×C47—41no—
55C22⋊C4×C47—42no—
56C42×C47—21no—
57C42×C47—81no—
58C42×C47C22⋊C411yescharacteristic
59C42×C47C4⋊C411yescharacteristic
60C22⋊Dic47—41no—
61C4⋊Dic47—41no—
62C4×Dic47—41no—
63#10625329—22no—
64#10625329—44no—
65#10625329—42no—
66#10625329—41no—
67C2×C47⋊C8—42no—
68C2×C47⋊C8—41no—
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—82no—
92C2×C4×C47#9511yescharacteristic
93C2×C4×C47#9911yescharacteristic
94C2×Dic47—81no—
95C47⋊C8—44no—
96C47⋊C8—42no—
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—162no—
111C4×C47AΣL(1,𝔽16)11yescharacteristic
112C4×C47#35211yescharacteristic
113Dic47—161no—
114#2126D4711yescharacteristic
115#1884—471no—
116#1525—471no—
117C2×C47—21no—
118C2×C47—41no—
119C2×C47—81no—
120C2×C47—81no—
121C2×C47#139011yescharacteristic
122#568—471no—
123#532C2×D4711yescharacteristic
124#421—941no—
125#410—21no—
126#410—471no—
127C4×M16—941no—
128#354—21no—
129#353Dic4712yescomplemented
130#339—941no—
131#329—472no—
132#329—941no—
133C47#748611yescharacteristic, complemented, Sylow
134C2×#41C4×D4711yescharacteristic
135C2×#41C22⋊D4711yescharacteristic
136#131—1882no—
137C2×M16—941no—
138C2×M16—1881no—
139C2×M16—1881no—
140C4⋊Q8Dic9411yescharacteristic
141#127C2×Dic4711yescharacteristic
142#124C4×D4711yescharacteristic
143#124C22⋊D4711yescharacteristic
144#121—41no—
145C4×D4—41no—
146#117—41no—
147#117—941no—
148C2×C42D18811yescharacteristic
149#104—1882no—
150#99—22no—
151#99—41no—
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×Q8Dic47⋊C411yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4C4×Dic4711yescharacteristic
165C2×D4C22⋊Dic4711yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4(C2×D47)⋊C411yescharacteristic
170M16—942no—
171M16—1884no—
172M16—1882no—
173M16—1881no—
174C2×C8—1882no—
175C2×C8—1881no—
176C2×C8—1882no—
177C4⋊C4—41no—
178C4⋊C4—81no—
179C4⋊C4—1881no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—1881no—
184C42—21no—
185C42—81no—
186C42—1881no—
187C42C4⋊Dic4711yescharacteristic
188C42(C2×D47)⋊C411yescharacteristic
189C23—21no—
190C23—21no—
191C23#1309393811yescharacteristic
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—82no—
211C2×C4—3761no—
212C2×C4C47⋊#9511yescharacteristic
213C2×C4#1309393811yescharacteristic
214C8—1884no—
215C8—1882no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#1568888911yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—162no—
229C4—7521no—
230C4#1568890911yescharacteristic
231C4#1568891211yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2024150511yescharacteristic
subgroupG/H
centerC4#15688909
commutatorC2×D4×C47C42
Frattini#532C2×D47
FittingC47×#2126C2
socleC2×C47#1390
prime radical, cosocleC47×#532C22
hypercenter#2126D47
hypocenterC47#7486
LCSG ▷ C2×D4×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2126

Complements

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

NH ≅ G/Ntype
4. #15689890C4semidirect
6. #15689097C4semidirect
10. #15688863C4semidirect
14. C47×#353C4semidirect
129. #353Dic47semidirect
133. C47#7486semidirect

Maximal subgroups

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

subgroupindex
#202437722
#202419802
C47×#21262
#748647