Finite Group Databasebeta

← #26125718

Subgroups — #26125718

Order 12032.

280 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 87 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#20245200C212yesmaximal, complemented
2#20241341C212yesmaximal, complemented
3#20241326C212yesmaximal
4C47×#2923C211yesmaximal, characteristic
5#15688944—22no—
6#15688905C2214yes—
7#15688901—22no—
8C4×#41×C47C411yescharacteristic
9C47×#433—22no—
10C47×#432—22no—
11C47×#424C411yescharacteristic
12C47×#424C2211yescharacteristic
13C47×#408C411yescharacteristic
14C47×#408C2211yescharacteristic
15C4×M16×C47C2211yescharacteristic
16C2×#97×C47C411yescharacteristic
17#13093964—24no—
18C2×#41×C47C2×C411yescharacteristic
19C2×M16×C47C2×C411yescharacteristic
20C2×M16×C47C2311yescharacteristic
21C22×C8×C47C2×C411yescharacteristic
22C22×C8×C47C2×C411yescharacteristic
23C4×Q8×C47—21no—
24C4×D4×C47—21no—
25C4×D4×C47—22no—
26#117×C47—21no—
27#117×C47C2×C411yescharacteristic
28C2×C42×C47—21no—
29C2×C42×C47C2×C411yescharacteristic
30C4⋊C8×C47—22no—
31C4⋊C8×C47—22no—
32C22⋊C8×C47—22no—
33C22⋊C8×C47—22no—
34#97×C47—21no—
35#97×C47—21no—
36#97×C47D412yes—
37C4×C8×C47D412yes—
38#41×C47—41no—
39#41×C47—41no—
40C2×Q8×C47M1611yescharacteristic
41C2×D4×C47—21no—
42C2×D4×C47M1611yescharacteristic
43C22×C4×C47—22no—
44C22×C4×C47—21no—
45C22×C4×C47—21no—
46C22×C4×C47C22×C411yescharacteristic
47M16×C47—41no—
48M16×C47—41no—
49C2×C8×C47—21no—
50C2×C8×C47—21no—
51C2×C8×C47—41no—
52C2×C8×C47—41no—
53C2×C8×C47C22⋊C412yes—
54C2×C8×C47C2×D412yes—
55C4⋊C4×C47—22no—
56C4⋊C4×C47—21no—
57C22⋊C4×C47—22no—
58C22⋊C4×C47M1612yes—
59C42×C47—22no—
60C42×C47—21no—
61C42×C47—21no—
62C42×C47C22⋊C411yescharacteristic
63C42×C47C22⋊C411yescharacteristic
64C47⋊C16—44no—
65C23×C47—21no—
66C23×C47C2×M1611yescharacteristic
67Q8×C47—41no—
68D4×C47—41no—
69D4×C47—42no—
70C2×C4×C47—21no—
71C2×C4×C47—21no—
72C2×C4×C47—21no—
73C2×C4×C47—22no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—41no—
77C2×C4×C47—41no—
78C2×C4×C47—81no—
79C2×C4×C47—82no—
80C2×C4×C47C2×C22⋊C411yescharacteristic
81C2×C4×C47C2×M1611yescharacteristic
82C8×C47—22no—
83C8×C47—22no—
84C8×C47—41no—
85C8×C47—41no—
86#11523—471nomaximal, Sylow
87C22×C47—21no—
88C22×C47—41no—
89C22×C47—42no—
90C22×C47—41no—
91C22×C47#40711yescharacteristic
92C4×C47—21no—
93C4×C47—42no—
94C4×C47—41no—
95C4×C47—41no—
96C4×C47—41no—
97C4×C47#41011yescharacteristic
98C4×C47#41311yescharacteristic
99#2923D4711yescharacteristic
100#2165—472no—
101#1328—472no—
102#1321—472no—
103C2×C47—21no—
104C2×C47—41no—
105C2×C47—81no—
106C2×C47#146411yescharacteristic
107C4×#41Dic4711yescharacteristic
108#433—22no—
109#432—22no—
110#424Dic4711yescharacteristic
111#424C2×D4711yescharacteristic
112#408Dic4711yescharacteristic
113#408C2×D4711yescharacteristic
114C4×M16C2×D4711yescharacteristic
115C2×#97Dic4711yescharacteristic
116#364—942no—
117#349—474no—
118#347—942no—
119C47#1152311yescharacteristic, complemented, Sylow
120C2×#41C2×Dic4711yescharacteristic
121C2×M16C2×Dic4711yescharacteristic
122C2×M16C22×D4711yescharacteristic
123C22×C8C2×Dic4711yescharacteristic
124C22×C8C2×Dic4711yescharacteristic
125C4×Q8—21no—
126C4×D4—21no—
127C4×D4—22no—
128#117—21no—
129#117C2×Dic4711yescharacteristic
130C2×C42—21no—
131C2×C42C2×Dic4711yescharacteristic
132M32—944no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#97—21no—
138#97—21no—
139#97C22⋊D4712yes—
140C4×C8C22⋊D4712yes—
141#41—41no—
142#41—41no—
143C2×Q8#1062532911yescharacteristic
144C2×D4—21no—
145C2×D4#1062532911yescharacteristic
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4C22×Dic4711yescharacteristic
150M16—41no—
151M16—41no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8C22⋊Dic4712yes—
157C2×C8C2×C22⋊D4712yes—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C22⋊C4—22no—
161C22⋊C4#1062532912yes—
162C42—22no—
163C42—21no—
164C42—21no—
165C42C22⋊Dic4711yescharacteristic
166C42C22⋊Dic4711yescharacteristic
167C16—1884no—
168C23—21no—
169C23C2×#1062532911yescharacteristic
170Q8—41no—
171D4—41no—
172D4—42no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—81no—
182C2×C4—82no—
183C2×C4C2×C22⋊Dic4711yescharacteristic
184C2×C4C2×#1062532911yescharacteristic
185C8—22no—
186C8—22no—
187C8—41no—
188C8—41no—
189C22—21no—
190C22—41no—
191C22—42no—
192C22—41no—
193C22#1568908711yescharacteristic
194C4—21no—
195C4—41no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4#1568909711yescharacteristic
200C4#1568910711yescharacteristic
201C2—21no—
202C2—41no—
203C2—81no—
204C2#2024172211yescharacteristic
subgroupG/H
centerC4#15689097
commutatorC22×C4×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#2923C2
socleC2×C47#1464
prime radicalC2×#41×C47C2×C4
cosocleC2×M16×C47C23
hypercenter#2923D47
hypocenterC47#11523
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2923

Complements

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

NH ≅ G/Ntype
1. #20245200C2semidirect
2. #20241341C2semidirect
119. C47#11523semidirect

Maximal subgroups

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

subgroupindex
#202452002
#202413412
#202413262
C47×#29232
#1152347