Finite Group Databasebeta

← #26129620

Subgroups — #26129620

Order 12032.

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

normal subgroups only

This group has 342 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#20245600C212yesmaximal, complemented
2#20244941C211yesmaximal, characteristic, complemented
3#20241373C212yesmaximal, complemented
4#20241326C211yesmaximal, characteristic, complemented
5C47×#3199C211yesmaximal, characteristic
6#15689172C2211yescharacteristic, complemented
7#15689149—21no—
8#15689140—21no—
9#15688945—22no—
10#15688936—22no—
11#15688905C2212yescomplemented
12#15688880C2211yescharacteristic, complemented
13C47×#535—21no—
14C47×#482—22no—
15C47×#469C2212yes—
16C47×#466—21no—
17C47×#447—22no—
18C47×#408C2211yescharacteristic
19C2×#10625329—21no—
20#13093983—21no—
21#13093964—22no—
22#13093960—22no—
23Dic47⋊C8—42no—
24#13093951—42no—
25C2×#41×C47D411yescharacteristic
26C22×D4×C47—21no—
27C2×D8×C47—21no—
28C2×D8×C47D412yes—
29C2×M16×C47C2311yescharacteristic
30C22×C8×C47D411yescharacteristic
31#127×C47—41no—
32#124×C47—41no—
33#121×C47—22no—
34#121×C47—22no—
35#120×C47—41no—
36#120×C47—41no—
37#117×C47—21no—
38#107×C47—21no—
39#106×C47D412yes—
40#102×C47—22no—
41#102×C47—22no—
42C22⋊C8×C47—22no—
43C24×C47—41no—
44#41×C47—41no—
45#41×C47—41no—
46C2×Q8×C47SD1611yescharacteristic
47C2×D4×C47—21no—
48C2×D4×C47—22no—
49C2×D4×C47—41no—
50C2×D4×C47—42no—
51C2×D4×C47—42no—
52C2×D4×C47—81no—
53C2×D4×C47—81no—
54C2×D4×C47SD1611yescharacteristic
55C22×C4×C47—21no—
56C22×C4×C47C2×D411yescharacteristic
57D8×C47—42no—
58D8×C47—81no—
59M16×C47—41no—
60C2×C8×C47—21no—
61C2×C8×C47—41no—
62C2×C8×C47C2×D412yes—
63C4⋊C4×C47—22no—
64C22⋊C4×C47—42no—
65C22⋊C4×C47—41no—
66C22⋊C4×C47—42no—
67C42×C47—41no—
68C22⋊Dic47—41no—
69C4⋊Dic47—41no—
70C4×Dic47—22no—
71#10625329—41no—
72#10625329—41no—
73C2×C47⋊C8—41no—
74C47⋊C16—42no—
75C23×C47—21no—
76C23×C47—42no—
77C23×C47—82no—
78C23×C47—162no—
79C23×C47AGL(1,ℤ/8)11yescharacteristic
80Q8×C47—41no—
81D4×C47—41no—
82D4×C47—42no—
83D4×C47—42no—
84D4×C47—81no—
85D4×C47—82no—
86D4×C47—162no—
87D4×C47—162no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47#12011yescharacteristic
95C2×C4×C47C2×SD1611yescharacteristic
96C2×Dic47—42no—
97C8×C47—22no—
98C8×C47—41no—
99C47⋊C8—81no—
100#12301—471nomaximal, Sylow
101C22×C47—21no—
102C22×C47—41no—
103C22×C47—42no—
104C22×C47—41no—
105C22×C47—82no—
106C22×C47—82no—
107C22×C47—82no—
108C22×C47—164no—
109C22×C47#45011yescharacteristic
110C4×C47—21no—
111C4×C47—41no—
112C4×C47—82no—
113C4×C47#45311yescharacteristic
114C4×C47#45811yescharacteristic
115Dic47—82no—
116#3199D4711yescharacteristic
117#2238—472no—
118#2098—471no—
119#1339—472no—
120#1321—471no—
121C2×C47—21no—
122C2×C47—41no—
123C2×C47—81no—
124C2×C47—162no—
125C2×C47#161311yescharacteristic
126#535—21no—
127#482—22no—
128#469C2×D4712yes—
129#466—21no—
130#447—22no—
131#430—471no—
132#424—941no—
133#421—941no—
134#408C2×D4711yescharacteristic
135#365—942no—
136#361—942no—
137#349—472no—
138H(3,ℤ/4)—471no—
139C47#1230111yescharacteristic, complemented, Sylow
140C2×#41C22⋊D4711yescharacteristic
141C22×D4—21no—
142C2×D8—21no—
143C2×D8C22⋊D4712yes—
144C2×M16—941no—
145C2×M16C22×D4711yescharacteristic
146C22×C8C22⋊D4711yescharacteristic
147#127—41no—
148#124—41no—
149#121—22no—
150#121—22no—
151#120—41no—
152#120—41no—
153#117—21no—
154#117—941no—
155M32—942no—
156#108—942no—
157#107—21no—
158#106C22⋊D4712yes—
159C4⋊C8—1882no—
160#104—1882no—
161#102—22no—
162#102—22no—
163C22⋊C8—22no—
164C24—41no—
165#41—41no—
166#41—41no—
167C2×Q8#1062533511yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4#1062533511yescharacteristic
176C22×C4—21no—
177C22×C4C2×C22⋊D4711yescharacteristic
178D8—42no—
179D8—81no—
180M16—41no—
181M16—1881no—
182M16—1881no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—1881no—
186C2×C8C2×C22⋊D4712yes—
187C4⋊C4—22no—
188C4⋊C4—1881no—
189C22⋊C4—42no—
190C22⋊C4—41no—
191C22⋊C4—42no—
192C22⋊C4—1881no—
193C42—41no—
194C42—942no—
195C16—1882no—
196C23—21no—
197C23—42no—
198C23—82no—
199C23—162no—
200C23#1309405011yescharacteristic
201Q8—41no—
202D4—41no—
203D4—42no—
204D4—42no—
205D4—81no—
206D4—82no—
207D4—162no—
208D4—162no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—1882no—
216C2×C4#1309399311yescharacteristic
217C2×C4C2×#1062533511yescharacteristic
218C8—22no—
219C8—41no—
220C8—3761no—
221C22—21no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—164no—
229C22#1568927411yescharacteristic
230C4—21no—
231C4—41no—
232C4—82no—
233C4—3762no—
234C4#1568929111yescharacteristic
235C4#1568931311yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—162no—
240C2#2024240811yescharacteristic
subgroupG/H
centerC2#20242408
commutator, cosocleC2×M16×C47C23
FrattiniC2×M16C22×D47
Fitting, prime radicalC47×#3199C2
socleC2×C47#1613
hypercenter#3199D47
hypocenterC47#12301
LCSG ▷ C2×M16×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3199

Complements

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

NH ≅ G/Ntype
1. #20245600C2semidirect
2. #20244941C2semidirect
3. #20241373C2semidirect
4. #20241326C2semidirect
6. #15689172C22semidirect
11. #15688905C22semidirect
12. #15688880C22semidirect
139. C47#12301semidirect

Maximal subgroups

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

subgroupindex
#202456002
#202449412
#202413732
#202413262
C47×#31992
#1230147