Finite Group Databasebeta

← #26104049

Subgroups — #26104049

Order 12032.

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

normal subgroups only

This group has 274 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#20242123C211yesmaximal, characteristic
3C47×#2132C211yesmaximal, characteristic
4#15689890C411yescharacteristic
5#15689147—21no—
6#15689140—21no—
7#15689097C411yescharacteristic
8#15688883—21no—
9#15688872—21no—
10#15688867C412yescomplemented
11C47×#532C2211yescharacteristic
12C47×#413—21no—
13C47×#356—21no—
14C47×#355C412yescomplemented
15#13094031—42no—
16C2×#10625329—21no—
17C2×#10625329—41no—
18C2×#10625329—41no—
19#13093983—21no—
20C4⋊(C47⋊C8)—22no—
21C4⋊(C47⋊C8)—41no—
22#13093951—42no—
23#13093939—41no—
24C2×#41×C47C2×C411yescharacteristic
25C2×#41×C47D411yescharacteristic
26C2×M16×C47—41no—
27C4⋊Q8×C47D411yescharacteristic
28#127×C47C2×C411yescharacteristic
29#124×C47C2×C411yescharacteristic
30#124×C47Q811yescharacteristic
31#121×C47—41no—
32C4×D4×C47—41no—
33C2×C42×C47D411yescharacteristic
34#101×C47—22no—
35#100×C47—41no—
36#41×C47—41no—
37#41×C47—41no—
38#41×C47—41no—
39#41×C47—41no—
40C2×Q8×C47C22⋊C411yescharacteristic
41C2×Q8×C47C4⋊C411yescharacteristic
42C2×D4×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47C4211yescharacteristic
46C2×D4×C47C4⋊C411yescharacteristic
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47C22⋊C411yescharacteristic
51M16×C47—42no—
52M16×C47—81no—
53C2×C8×C47—81no—
54C4⋊C4×C47—41no—
55C22⋊C4×C47—41no—
56C22⋊C4×C47—41no—
57C42×C47—21no—
58C42×C47C22⋊C411yescharacteristic
59C42×C47C4⋊C411yescharacteristic
60C22⋊Dic47—41no—
61C4⋊Dic47—41no—
62C4×Dic47—41no—
63#10625329—22no—
64#10625329—41no—
65#10625329—42no—
66#10625329—82no—
67C2×C47⋊C8—41no—
68C2×C47⋊C8—42no—
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—81no—
79D4×C47—81no—
80D4×C47—81no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47#9511yescharacteristic
94C2×C4×C47#9911yescharacteristic
95C2×Dic47—81no—
96C8×C47—82no—
97C47⋊C8—42no—
98C47⋊C8—82no—
99#7496—471nomaximal, Sylow
100C22×C47—21no—
101C22×C47—41no—
102C22×C47—41no—
103C22×C47—41no—
104C22×C47—81no—
105C22×C47—81no—
106C22×C47#34211yescharacteristic
107C4×C47—21no—
108C4×C47—21no—
109C4×C47—21no—
110C4×C47—41no—
111C4×C47—41no—
112C4×C47—41no—
113C4×C47AΣL(1,𝔽16)11yescharacteristic
114C4×C47#35211yescharacteristic
115Dic47—161no—
116#2132D4711yescharacteristic
117#1884—471no—
118#1560—471no—
119C2×C47—21no—
120C2×C47—41no—
121C2×C47—81no—
122C2×C47—81no—
123C2×C47#139011yescharacteristic
124#568—471no—
125#532C2×D4711yescharacteristic
126#423—941no—
127#421—941no—
128#413—21no—
129#410—471no—
130#356—21no—
131#355Dic4712yescomplemented
132#339—941no—
133#333—941no—
134#331—472no—
135C47#749611yescharacteristic, complemented, Sylow
136C2×#41C4×D4711yescharacteristic
137C2×#41C22⋊D4711yescharacteristic
138#131—1882no—
139C2×M16—41no—
140C2×M16—941no—
141C2×M16—1881no—
142C2×M16—1881no—
143C4⋊Q8C22⋊D4711yescharacteristic
144#127C4×D4711yescharacteristic
145#124Dic9411yescharacteristic
146#124C2×Dic4711yescharacteristic
147#121—41no—
148C4×D4—41no—
149#117—941no—
150C2×C42D18811yescharacteristic
151C4⋊C8—942no—
152C4⋊C8—1881no—
153#104—1882no—
154#101—22no—
155#100—41no—
156#99—1881no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161C2×Q8Dic47⋊C411yescharacteristic
162C2×Q8C22⋊Dic4711yescharacteristic
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4C4×Dic4711yescharacteristic
167C2×D4Dic47⋊C411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4(C2×D47)⋊C411yescharacteristic
172M16—42no—
173M16—81no—
174M16—942no—
175M16—3762no—
176M16—1881no—
177M16—1882no—
178C2×C8—81no—
179C2×C8—1882no—
180C2×C8—1881no—
181C2×C8—1882no—
182C4⋊C4—41no—
183C4⋊C4—1881no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—1881no—
187C42—21no—
188C42—1881no—
189C42C4⋊Dic4711yescharacteristic
190C42(C2×D47)⋊C411yescharacteristic
191C23—21no—
192C23—21no—
193C23#1309393811yescharacteristic
194Q8—41no—
195Q8—41no—
196D4—41no—
197D4—41no—
198D4—42no—
199D4—81no—
200D4—81no—
201D4—81no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—3761no—
215C2×C4C47⋊#9511yescharacteristic
216C2×C4#1309393811yescharacteristic
217C8—82no—
218C8—3762no—
219C8—1882no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22#1568888911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—7521no—
234C4#1568890911yescharacteristic
235C4#1568891211yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—81no—
240C2#2024150511yescharacteristic
subgroupG/H
centerC2#20241505
commutatorC2×D4×C47C42
Frattini#532C2×D47
FittingC47×#2132C2
socleC2×C47#1390
prime radical, cosocleC47×#532C22
hypercenter#2132D47
hypocenterC47#7496
LCSG ▷ C2×D4×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
10. #15688867C4semidirect
14. C47×#355C4semidirect
131. #355Dic47semidirect
135. C47#7496semidirect

Maximal subgroups

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

subgroupindex
#202437722
#202421232
C47×#21322
#749647