Finite Group Databasebeta

← #26121047

Subgroups — #26121047

Order 12032.

330 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 93 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#20244220C211yesmaximal, characteristic
2#20242038C212yesmaximal
3#20241285C212yesmaximal, complemented
4#20241217C211yesmaximal, characteristic, complemented
5C47×#1843C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689176C2212yes—
8#15689147C2211yescharacteristic
9C2×#13093952—21no—
10C2×#13093952—21no—
11#15688870—22no—
12#15688866—22no—
13C2×C4⋊C8×C47—21no—
14C2×C4⋊C8×C47—21no—
15C2×C4⋊C8×C47C2212yes—
16C47×#376C2211yescharacteristic
17C2×#10625350—21no—
18C2×#10625329—21no—
19#13094024D411yescharacteristic
20#13094021D411yescharacteristic
21#13094012—21no—
22#13093998—22no—
23C4×D188—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26#13093952—42no—
27#13093952—42no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8—22no—
30C22×C8×C47—22no—
31C2×C42×C47D411yescharacteristic
32C2×C42×C47D411yescharacteristic
33C2×C42×C47C2311yescharacteristic
34C4⋊C8×C47—22no—
35C4⋊C8×C47—22no—
36C4⋊C8×C47—42no—
37C4⋊C8×C47—42no—
38#95×C47—22no—
39C22×C4×C47—21no—
40C22×C4×C47—21no—
41C22×C4×C47C2×D411yescharacteristic
42C22×C4×C47#4112yes—
43C2×C22⋊D47—41no—
44#10625350—41no—
45#10625350—41no—
46#10625350—41no—
47#10625350—41no—
48C2×D188—21no—
49C2×D188—22no—
50C2×C4×D47—41no—
51C2×Dic94—21no—
52C2×C8×C47—42no—
53C2×C8×C47—42no—
54C2×C8×C47—42no—
55C42×C47—21no—
56C42×C47—21no—
57C42×C47D812yes—
58C42×C47SD1612yes—
59C42×C47C2×D411yescharacteristic
60C42×C47#4111yescharacteristic
61C42×C47#4112yes—
62(C2×D47)⋊C4—42no—
63C4⋊Dic47—22no—
64#10625329—41no—
65#10625329—41no—
66C2×C47⋊C8—42no—
67C2×C47⋊C8—41no—
68C23×C47#12311yescharacteristic
69C22×D47—41no—
70C2×C4×C47—22no—
71C2×C4×C47—22no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47—42no—
75C2×C4×C47#12111yescharacteristic
76C2×C4×C47#12312yes—
77C2×C4×C47#12411yescharacteristic
78C2×C4×C47#12612yes—
79C2×C4×C47C2×D811yescharacteristic
80C2×C4×C47C2×SD1611yescharacteristic
81C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C47#13711yescharacteristic
83C22⋊D47—81no—
84C22⋊D47—81no—
85C2×Dic47—41no—
86D188—41no—
87D188—41no—
88D188—82no—
89C4×D47—81no—
90C4×D47—81no—
91Dic94—41no—
92Dic94—41no—
93C8×C47—82no—
94C47⋊C8—82no—
95C47⋊C8—81no—
96#10627—471nomaximal, Sylow
97C22×C47—22no—
98C22×C47#39711yescharacteristic
99C22×C47#48011yescharacteristic
100C22×C47#48111yescharacteristic
101C2×D47—81no—
102C2×D47—82no—
103C4×C47—22no—
104C4×C47—22no—
105C4×C47—42no—
106C4×C47—42no—
107C4×C47#45911yescharacteristic
108C4×C47#46111yescharacteristic
109C4×C47#48611yescharacteristic
110C4×C47#48911yescharacteristic
111Dic47—81no—
112#1963—471no—
113#1843D4711yescharacteristic, complemented
114#1540—472no—
115#1301—472no—
116#1274—471no—
117C2×C47—21no—
118C2×C47—21no—
119C2×C47#207111yescharacteristic
120C2×C47#208012yes—
121D47—161no—
122#532—471no—
123#431—472no—
124#423—471no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2×D4712yes—
128C2×#104—941no—
129C2×#104—941no—
130#376C2×D4711yescharacteristic
131#332—942no—
132#331—942no—
133C47#1062711yescharacteristic, complemented, Sylow
134C2×#41—941no—
135C2×M16—941no—
136C22×C8—22no—
137C4⋊Q8—471no—
138#127—471no—
139#124—941no—
140#121—942no—
141C4×D4—942no—
142C2×C42C22⋊D4711yescharacteristic
143C2×C42C22⋊D4711yescharacteristic
144C2×C42C22×D4711yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—42no—
148C4⋊C8—42no—
149C4⋊C8—942no—
150C4⋊C8—942no—
151#104—1882no—
152#104—1882no—
153C22⋊C8—942no—
154C4×C8—942no—
155#95—22no—
156#41—1881no—
157#41—1881no—
158#41—1881no—
159#41—1881no—
160C2×Q8—941no—
161C2×D4—941no—
162C2×D4—942no—
163C2×D4—1881no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—1881no—
167C22×C4#1062535012yes—
168C22×C4C2×C22⋊D4711yescharacteristic
169M16—1881no—
170M16—1881no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—1882no—
175C2×C8—1881no—
176C4⋊C4—942no—
177C22⋊C4—1882no—
178C42—21no—
179C42—21no—
180C42#1062533412yes—
181C42#1062533612yes—
182C42D4×D4711yescharacteristic
183C42#1062535211yescharacteristic
184C42#1062535412yes—
185C23—1881no—
186C23#1309400711yescharacteristic
187Q8—1881no—
188Q8—1881no—
189D4—3762no—
190D4—3761no—
191D4—3761no—
192D4—1881no—
193D4—1881no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—3761no—
200C2×C4—3761no—
201C2×C4—1881no—
202C2×C4#1309399511yescharacteristic
203C2×C4#1309400912yes—
204C2×C4#1309401311yescharacteristic
205C2×C4#1309401912yes—
206C2×C4C2×#1062533411yescharacteristic
207C2×C4C2×#1062533611yescharacteristic
208C2×C4#1309405011yescharacteristic
209C2×C4#1309405511yescharacteristic
210C8—82no—
211C8—3762no—
212C8—3761no—
213C22—22no—
214C22—3761no—
215C22—3762no—
216C22#1568904911yescharacteristic
217C22#1568943611yescharacteristic
218C22#1568944111yescharacteristic
219C4—22no—
220C4—22no—
221C4—42no—
222C4—42no—
223C4—3761no—
224C4#1568931511yescharacteristic
225C4#1568932511yescharacteristic
226C4#1568947111yescharacteristic
227C4#1568948511yescharacteristic
228C2—21no—
229C2—21no—
230C2—7521no—
231C2#2024479211yescharacteristic
232C2#2024483912yes—
subgroupG/H
centerC2×C4#13094013
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1843C2
socleC22×C47#397
prime radical#15689673C22
hypercenter#1843D47
hypocenterC47#10627
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1843

Complements

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

NH ≅ G/Ntype
3. #20241285C2semidirect
4. #20241217C2semidirect
5. C47×#1843C2semidirect
113. #1843D47semidirect
133. C47#10627semidirect

Maximal subgroups

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

subgroupindex
#202442202
#202420382
#202412852
#202412172
C47×#18432
#1062747