Finite Group Databasebeta

← #26136617

Subgroups — #26136617

Order 12032.

450 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 79 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#20245073C212yesmaximal, complemented
2#20241555C214yesmaximal, complemented
3C47×#3151C211yesmaximal, characteristic
4#15689107—22no—
5#15688922C2214yescomplemented
6#15688920—22no—
7C2×AGL(1,ℤ/8)×C47—22no—
8C47×#532C2211yescharacteristic
9C2×#127×C47C411yescharacteristic
10C47×#496—22no—
11C47×#493C412yes—
12C47×#492C2212yes—
13C4×M16×C47C411yescharacteristic
14C2×#10625329—42no—
15#13093943—24no—
16#13093941—42no—
17C2×#41×C47D412yes—
18C22×D4×C47—21no—
19C22×D4×C47—41no—
20AGL(1,ℤ/8)×C47—44no—
21AGL(1,ℤ/8)×C47—44no—
22C2×SD16×C47—24no—
23C2×D8×C47—24no—
24C2×M16×C47—21no—
25C4⋊Q8×C47C2311yescharacteristic
26#127×C47—21no—
27#127×C47—41no—
28#127×C47C2×C412yes—
29#127×C47C2×C411yescharacteristic
30#124×C47D412yes—
31#121×C47—41no—
32C4×D4×C47—41no—
33C2×C42×C47C2×C411yescharacteristic
34#97×C47—21no—
35C4×C8×C47C2×C412yes—
36C24×C47—41no—
37#41×C47—42no—
38#41×C47—42no—
39C2×Q8×C47C2×D412yes—
40C2×D4×C47—22no—
41C2×D4×C47—22no—
42C2×D4×C47—42no—
43C2×D4×C47—42no—
44C2×D4×C47—41no—
45C2×D4×C47—42no—
46C2×D4×C47—42no—
47C2×D4×C47—41no—
48C2×D4×C47—84no—
49C2×D4×C47C22⋊C412yes—
50C22×C4×C47—21no—
51C22×C4×C47—22no—
52C22×C4×C47C22⋊C411yescharacteristic
53SD16×C47—48no—
54D8×C47—48no—
55M16×C47—24no—
56C2×C8×C47—22no—
57C4⋊C4×C47—41no—
58C22⋊C4×C47—42no—
59C42×C47—21no—
60C42×C47C22⋊C411yescharacteristic
61C42×C47C22×C411yescharacteristic
62#10625329—44no—
63#10625329—82no—
64C2×C47⋊C8—82no—
65C23×C47—22no—
66C23×C47—42no—
67C23×C47—81no—
68C23×C47—81no—
69C23×C47—84no—
70C23×C47#9911yescharacteristic
71Q8×C47—42no—
72D4×C47—42no—
73D4×C47—44no—
74D4×C47—82no—
75D4×C47—84no—
76D4×C47—82no—
77D4×C47—84no—
78D4×C47—82no—
79C2×C4×C47—21no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—21no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47#9911yescharacteristic
87C2×C4×C47C2×C22⋊C411yescharacteristic
88C8×C47—24no—
89C47⋊C8—84no—
90#13591—471nomaximal, Sylow
91C22×C47—21no—
92C22×C47—41no—
93C22×C47—42no—
94C22×C47—82no—
95C22×C47—84no—
96C22×C47—82no—
97C22×C47—162no—
98C22×C47—162no—
99C22×C47C2×#9911yescharacteristic
100C4×C47—22no—
101C4×C47—21no—
102C4×C47—41no—
103C4×C47—42no—
104C4×C47#35312yes—
105#3151D4711yescharacteristic
106#2132—472no—
107#1410—474no—
108C2×C47—21no—
109C2×C47—41no—
110C2×C47—82no—
111C2×C47—162no—
112C2×C47C2×#35311yescharacteristic
113C2×AGL(1,ℤ/8)—22no—
114#532C2×D4711yescharacteristic
115C2×#127Dic4711yescharacteristic
116#496—22no—
117#493Dic4712yes—
118#492C2×D4712yes—
119#413—942no—
120C4×M16Dic4711yescharacteristic
121#356—474no—
122#355—942no—
123C47#1359111yescharacteristic, complemented, Sylow
124C2×#41C22⋊D4712yes—
125C22×D4—21no—
126C22×D4—41no—
127AGL(1,ℤ/8)—44no—
128AGL(1,ℤ/8)—44no—
129C2×SD16—24no—
130C2×D8—24no—
131C2×M16—21no—
132C2×M16—1882no—
133C4⋊Q8C22×D4711yescharacteristic
134#127—21no—
135#127—41no—
136#127C2×Dic4712yes—
137#127C2×Dic4711yescharacteristic
138#124C22⋊D4712yes—
139#121—41no—
140C4×D4—41no—
141C2×C42C2×Dic4711yescharacteristic
142#101—944no—
143#100—1882no—
144#97—21no—
145C4×C8C2×Dic4712yes—
146C24—41no—
147#41—42no—
148#41—42no—
149C2×Q8C2×C22⋊D4712yes—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—41no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—41no—
158C2×D4—84no—
159C2×D4C22⋊Dic4712yes—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4C22⋊Dic4711yescharacteristic
163SD16—48no—
164D8—48no—
165M16—24no—
166M16—3762no—
167M16—1884no—
168C2×C8—22no—
169C2×C8—3762no—
170C4⋊C4—41no—
171C22⋊C4—42no—
172C42—21no—
173C42C22⋊Dic4711yescharacteristic
174C42C22×Dic4711yescharacteristic
175C23—22no—
176C23—42no—
177C23—81no—
178C23—81no—
179C23—84no—
180C23#1309393911yescharacteristic
181Q8—42no—
182D4—42no—
183D4—44no—
184D4—82no—
185D4—84no—
186D4—82no—
187D4—84no—
188D4—82no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4#1309393911yescharacteristic
197C2×C4C2×C22⋊Dic4711yescharacteristic
198C8—24no—
199C8—3764no—
200C22—21no—
201C22—41no—
202C22—42no—
203C22—82no—
204C22—84no—
205C22—82no—
206C22—162no—
207C22—162no—
208C22C2×#1309393911yescharacteristic
209C4—22no—
210C4—21no—
211C4—41no—
212C4—42no—
213C4#1568891612yes—
214C2—21no—
215C2—41no—
216C2—82no—
217C2—162no—
218C2C2×#1568891611yescharacteristic
subgroupG/H
centerC2C2×#15688916
commutatorC42×C47C22×C4
FrattiniC4⋊Q8C22×D47
Fitting, prime radicalC47×#3151C2
socleC2×C47C2×#353
cosocleC4⋊Q8×C47C23
hypercenter#3151D47
hypocenterC47#13591
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #20245073C2semidirect
2. #20241555C2semidirect
5. #15688922C22semidirect
123. C47#13591semidirect

Maximal subgroups

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

subgroupindex
#202450732
#202415552
C47×#31512
#1359147