Finite Group Databasebeta

← #26102986

Subgroups — #26102986

Order 12032.

248 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 49 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#20244699C211yesmaximal, characteristic
2#20241328C211yesmaximal, characteristic, complemented
3C47×#1292C211yesmaximal, characteristic, complemented
4C22×D376C411yescharacteristic
5C2×#13093946—21no—
6#15689025—21no—
7C22⋊(C47⋊C16)—21no—
8#15688886C411yescharacteristic
9C47×#423—21no—
10C2×C4⋊C8×C47C2211yescharacteristic
11C22×D188—21no—
12C2×D376—41no—
13C2×D376—42no—
14C2×C4⋊Dic47—21no—
15C2×(C2×D47)⋊C4—41no—
16C2×(C2×D47)⋊C4—41no—
17C2×C47⋊C16—41no—
18#13093946—42no—
19C2×M16×C47—41no—
20C22×C8×C47C2×C411yescharacteristic
21C22×C8×C47D411yescharacteristic
22C2×C42×C47D411yescharacteristic
23C4⋊C8×C47—22no—
24C4⋊C8×C47—41no—
25C4⋊C8×C47—41no—
26C23×D47—41no—
27C22×C4×C47C22⋊C411yescharacteristic
28C22×C4×C47D811yescharacteristic
29C22×C4×C47SD1611yescharacteristic
30C22×Dic47—41no—
31C2×D188—22no—
32C2×D188—41no—
33C2×D188—41no—
34C2×D188—41no—
35C2×D188—41no—
36C2×D188—41no—
37M16×C47—81no—
38M16×C47—81no—
39C2×C8×C47—21no—
40C2×C8×C47—21no—
41C2×C8×C47—41no—
42C2×C8×C47—41no—
43C2×C8×C47—42no—
44C42×C47—21no—
45C42×C47—21no—
46(C2×D47)⋊C4—82no—
47(C2×D47)⋊C4—81no—
48(C2×D47)⋊C4—81no—
49C4⋊Dic47—41no—
50C4⋊Dic47—41no—
51D376—42no—
52D376—42no—
53C47⋊C16—81no—
54C23×C47#10211yescharacteristic
55C22×D47—42no—
56C22×D47—81no—
57C22×D47—81no—
58C22×D47—81no—
59C22×D47—81no—
60C22×D47—81no—
61C22×D47—81no—
62C2×C4×C47—21no—
63C2×C4×C47—21no—
64C2×C4×C47—21no—
65C2×C4×C47—21no—
66C2×C4×C47—21no—
67C2×C4×C47—41no—
68C2×C4×C47#9911yescharacteristic
69C2×C4×C47#10411yescharacteristic
70C2×C4×C47D1611yescharacteristic
71C2×C4×C47SD3211yescharacteristic
72C2×Dic47—81no—
73C2×Dic47—81no—
74C2×Dic47—81no—
75D188—42no—
76D188—42no—
77D188—81no—
78D188—81no—
79C8×C47—41no—
80C8×C47—41no—
81C8×C47—82no—
82#7088—471nomaximal, Sylow
83C22×C47—21no—
84C22×C47—21no—
85C22×C47#32711yescharacteristic
86C22×C47#35711yescharacteristic
87C22×C47#36011yescharacteristic
88C2×D47—82no—
89C2×D47—82no—
90C2×D47—82no—
91C2×D47—161no—
92C2×D47—161no—
93C2×D47—161no—
94C2×D47—161no—
95C4×C47—21no—
96C4×C47—21no—
97C4×C47—21no—
98C4×C47—41no—
99C4×C47AΣL(1,𝔽16)11yescharacteristic
100C4×C47#35511yescharacteristic
101Dic47—161no—
102#2054—471no—
103#1322—471no—
104#1292D4711yescharacteristic, complemented
105C2×C47—21no—
106C2×C47—21no—
107C2×C47#134111yescharacteristic
108C2×C47#134511yescharacteristic
109C2×C47#134711yescharacteristic
110D47—162no—
111C22×D8—471no—
112#423—21no—
113C2×C4⋊C8C2×D4711yescharacteristic
114C2×#102—941no—
115#390—941no—
116C22⋊C16—941no—
117#340—471no—
118C47#708811yescharacteristic, complemented, Sylow
119C22×D4—941no—
120C2×D8—1881no—
121C2×D8—1882no—
122C2×M16—41no—
123C22×C8C4×D4711yescharacteristic
124C22×C8D18811yescharacteristic
125C2×C4⋊C4—941no—
126C2×C22⋊C4—1881no—
127C2×C22⋊C4—1881no—
128C2×C42C22⋊D4711yescharacteristic
129C2×C16—1881no—
130C4⋊C8—22no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133#102—1882no—
134C24—1881no—
135C2×D4—942no—
136C2×D4—1881no—
137C2×D4—1881no—
138C2×D4—1881no—
139C2×D4—1881no—
140C2×D4—1881no—
141C22×C4—1881no—
142C22×C4(C2×D47)⋊C411yescharacteristic
143C22×C4#1062533411yescharacteristic
144C22×C4#1062533611yescharacteristic
145D8—1882no—
146D8—1882no—
147M16—81no—
148M16—81no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—42no—
154C4⋊C4—1881no—
155C4⋊C4—1881no—
156C22⋊C4—3762no—
157C22⋊C4—3761no—
158C22⋊C4—3761no—
159C42—21no—
160C42—21no—
161C16—3761no—
162C23—3761no—
163C23—3761no—
164C23—3761no—
165C23—3761no—
166C23—3761no—
167C23—3761no—
168C23—1882no—
169C23#1309394411yescharacteristic
170D4—3761no—
171D4—3761no—
172D4—1882no—
173D4—1882no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—3761no—
181C2×C4—3761no—
182C2×C4—3761no—
183C2×C4#1309393811yescharacteristic
184C2×C4#1309395211yescharacteristic
185C2×C4#1309396511yescharacteristic
186C2×C4#1309396711yescharacteristic
187C8—41no—
188C8—41no—
189C8—82no—
190C22—21no—
191C22—21no—
192C22—3762no—
193C22—3762no—
194C22—3762no—
195C22—7521no—
196C22—7521no—
197C22—7521no—
198C22—7521no—
199C22#1568885611yescharacteristic
200C22#1568892311yescharacteristic
201C22#1568893211yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—7521no—
207C4#1568891111yescharacteristic
208C4#1568891911yescharacteristic
209C2—21no—
210C2—21no—
211C2—7522no—
212C2#2024137811yescharacteristic
213C2#2024139011yescharacteristic
214C2#2024139611yescharacteristic
subgroupG/H
centerC22#15688856
commutatorC22×C8×C47C2×C4
FrattiniC2×C4⋊C8C2×D47
FittingC47×#1292C2
socleC22×C47#327
prime radicalC22×D376C4
cosocleC2×C4⋊C8×C47C22
hypercenter#1292D47
hypocenterC47#7088
LCSG ▷ C22×C8×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #20241328C2semidirect
3. C47×#1292C2semidirect
104. #1292D47semidirect
118. C47#7088semidirect

Maximal subgroups

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

subgroupindex
#202446992
#202413282
C47×#12922
#708847