Finite Group Databasebeta

← #26103660

Subgroups — #26103660

Order 12032.

268 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 43 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#20244193C211yesmaximal, characteristic
2#20241334C211yesmaximal, characteristic, complemented
3C47×#1325C211yesmaximal, characteristic, complemented
4C2×#13094021C411yescharacteristic
5C2×#13093944—21no—
6C2×#13093944—21no—
7C2×C4×C47⋊C8C411yescharacteristic
8#15688849—21no—
9C47×#384C2211yescharacteristic
10C47×#324—21no—
11C22×D188—21no—
12C22×D188—41no—
13C22×C47⋊C8—21no—
14#13094021—21no—
15#13094021—41no—
16#13094021C812yescomplemented
17#13093944—42no—
18#13093944—42no—
19C22⋊(C47⋊C8)—41no—
20C4×C47⋊C8—41no—
21C2×C4⋊C4×C47—41no—
22C2×C4⋊C4×C47D411yescharacteristic
23C2×C4⋊C4×C47D411yescharacteristic
24C2×C42×C47C2×C411yescharacteristic
25C22⋊C8×C47—41no—
26C23×D47—41no—
27C22×C4×C47—21no—
28C22×C4×C47—21no—
29C22×C4×C47—21no—
30C22×C4×C47C22⋊C411yescharacteristic
31C2×D188—22no—
32C2×D188—41no—
33C2×D188—41no—
34C2×D188—41no—
35C2×D188—41no—
36C2×D188—42no—
37C2×D188—42no—
38C2×D188—41no—
39C2×D188—82no—
40C2×D188—82no—
41C2×C8×C47—81no—
42C4⋊C4×C47—41no—
43C4⋊C4×C47—41no—
44C4⋊C4×C47—82no—
45C42×C47—21no—
46C42×C47C2×C811yescharacteristic
47C42×C47M1611yescharacteristic
48C2×C47⋊C8—22no—
49C2×C47⋊C8—41no—
50C2×C47⋊C8—41no—
51C2×C47⋊C8—81no—
52C23×C47#9911yescharacteristic
53C22×D47—42no—
54C22×D47—82no—
55C22×D47—82no—
56C22×D47—81no—
57C22×D47—81no—
58C2×C4×C47—22no—
59C2×C4×C47—22no—
60C2×C4×C47—21no—
61C2×C4×C47—21no—
62C2×C4×C47—21no—
63C2×C4×C47—21no—
64C2×C4×C47—41no—
65C2×C4×C47—41no—
66C2×C4×C47—81no—
67C2×C4×C47—81no—
68C2×C4×C47C22⋊C811yescharacteristic
69C2×C4×C47#10011yescharacteristic
70D188—42no—
71D188—42no—
72D188—81no—
73D188—81no—
74D188—84no—
75D188—84no—
76C8×C47—161no—
77C47⋊C8—42no—
78C47⋊C8—161no—
79#7354—471nomaximal, Sylow
80C22×C47—21no—
81C22×C47—21no—
82C22×C47#32311yescharacteristic
83C22×C47#35311yescharacteristic
84C22×C47#35511yescharacteristic
85C2×D47—84no—
86C2×D47—82no—
87C2×D47—162no—
88C2×D47—161no—
89C2×D47—161no—
90C4×C47—22no—
91C4×C47—22no—
92C4×C47—21no—
93C4×C47—21no—
94C4×C47—81no—
95C4×C47—81no—
96#1957—471no—
97#1325—471no—
98#1325D4711yescharacteristic, complemented
99C2×C47—21no—
100C2×C47—21no—
101C2×C47#131811yescharacteristic
102C2×C47#140811yescharacteristic
103C2×C47#141011yescharacteristic
104D47—162no—
105C2×#127—471no—
106C2×#102—941no—
107C2×#102—941no—
108C2×C4×C8—471no—
109#384C2×D4711yescharacteristic
110#324—21no—
111#324—941no—
112C47#735411yescharacteristic, complemented, Sylow
113C22×D4—941no—
114C22×D4—1881no—
115C22×C8—941no—
116#127—472no—
117#127—941no—
118#127—1881no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4D18811yescharacteristic
121C2×C4⋊C4C22⋊D4711yescharacteristic
122C2×C42C4×D4711yescharacteristic
123#102—1882no—
124#102—1882no—
125C22⋊C8—41no—
126C22⋊C8—1881no—
127C4×C8—1881no—
128C24—1881no—
129C2×D4—942no—
130C2×D4—3762no—
131C2×D4—3762no—
132C2×D4—1882no—
133C2×D4—1882no—
134C2×D4—1881no—
135C2×D4—1881no—
136C2×D4—1881no—
137C2×D4—1881no—
138C2×D4—1881no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4(C2×D47)⋊C411yescharacteristic
143C2×C8—81no—
144C2×C8—942no—
145C2×C8—3761no—
146C2×C8—1881no—
147C2×C8—1881no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—82no—
151C42—21no—
152C42C8×D4711yescharacteristic
153C42#1062532411yescharacteristic
154C23—3762no—
155C23—3762no—
156C23—3761no—
157C23—3761no—
158C23—1882no—
159C23#1309393811yescharacteristic
160D4—3764no—
161D4—3764no—
162D4—3761no—
163D4—3761no—
164D4—1882no—
165D4—1882no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4(C2×D47)⋊C811yescharacteristic
177C2×C4#1309394011yescharacteristic
178C8—161no—
179C8—1882no—
180C8—7521no—
181C22—21no—
182C22—21no—
183C22—3764no—
184C22—3762no—
185C22—7522no—
186C22—7521no—
187C22—7521no—
188C22#1568884611yescharacteristic
189C22#1568891511yescharacteristic
190C22#1568891911yescharacteristic
191C4—22no—
192C4—22no—
193C4—21no—
194C4—21no—
195C4—81no—
196C4—81no—
197C2—21no—
198C2—21no—
199C2—7522no—
200C2#2024131911yescharacteristic
201C2#2024155011yescharacteristic
202C2#2024155411yescharacteristic
subgroupG/H
centerC22#15688915
commutatorC42×C47C2×C8
Frattini#384C2×D47
FittingC47×#1325C2
socleC22×C47#353
prime radicalC2×#13094021C4
cosocleC47×#384C22
hypercenter#1325D47
hypocenterC47#7354
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #20241334C2semidirect
3. C47×#1325C2semidirect
16. #13094021C8semidirect
98. #1325D47semidirect
112. C47#7354semidirect

Maximal subgroups

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

subgroupindex
#202441932
#202413342
C47×#13252
#735447