Finite Group Databasebeta

← #26125719

Subgroups — #26125719

Order 12032.

288 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 69 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#20252028C211yesmaximal, characteristic
2#20245197C212yesmaximal, complemented
3#20241340C212yesmaximal, complemented
4#20241326C211yesmaximal, characteristic
5C47×#1321C211yesmaximal, characteristic
6#41×Dic47C411yescharacteristic
7#15689181—22no—
8#15689173—22no—
9#15689151C2211yescharacteristic
10#15689149C411yescharacteristic
11#15689090C411yescharacteristic
12M16×Dic47C2211yescharacteristic
13C2×C8×Dic47C411yescharacteristic
14#15688943—22no—
15#15688905C2212yes—
16#15688900—22no—
17C47×#408C2211yescharacteristic
18C47×#349C2212yes—
19C2×#10625329C2×C411yescharacteristic
20C22×C47⋊C8C2×C411yescharacteristic
21Q8×Dic47—21no—
22D4×Dic47—21no—
23D4×Dic47—22no—
24#13093983—21no—
25#13093983C2×C411yescharacteristic
26C2×C4×Dic47—21no—
27C2×C4×Dic47C2×C411yescharacteristic
28#13093964—22no—
29Dic47⋊C8—22no—
30Dic47⋊C8—22no—
31C22⋊(C47⋊C8)—22no—
32#13093934D412yes—
33C8×Dic47—21no—
34C8×Dic47—21no—
35C8×Dic47D412yes—
36C2×#41×C47C2×C411yescharacteristic
37C2×M16×C47C2311yescharacteristic
38C22×C8×C47C2×C411yescharacteristic
39M32×C47—22no—
40C22⋊C8×C47—22no—
41#41×C47—41no—
42#41×C47—41no—
43C2×Q8×C47M1611yescharacteristic
44C2×D4×C47—21no—
45C2×D4×C47M1611yescharacteristic
46C22×C4×C47—21no—
47C22×C4×C47C22×C411yescharacteristic
48C22×Dic47—22no—
49C22×Dic47—21no—
50M16×C47—41no—
51C2×C8×C47—21no—
52C2×C8×C47—41no—
53C2×C8×C47C2×D412yes—
54C22⋊Dic47—22no—
55C22⋊Dic47M1612yes—
56C4⋊Dic47—22no—
57C4⋊Dic47—21no—
58C4×Dic47—22no—
59C4×Dic47—21no—
60C4×Dic47—21no—
61C4×Dic47C22⋊C411yescharacteristic
62C4×Dic47C22⋊C411yescharacteristic
63#10625329—41no—
64C2×C47⋊C8—21no—
65C2×C47⋊C8—41no—
66C2×C47⋊C8C22⋊C412yes—
67C16×C47—42no—
68C47⋊C16—42no—
69C23×C47—21no—
70C23×C47C2×M1611yescharacteristic
71Q8×C47—41no—
72D4×C47—41no—
73D4×C47—42no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—41no—
78C2×C4×C47—41no—
79C2×C4×C47C2×C22⋊C411yescharacteristic
80C2×C4×C47C2×M1611yescharacteristic
81C2×Dic47—21no—
82C2×Dic47—22no—
83C2×Dic47—21no—
84C2×Dic47—42no—
85C2×Dic47—44no—
86C8×C47—22no—
87C8×C47—41no—
88C47⋊C8—22no—
89C47⋊C8—41no—
90#11524—471nomaximal, Sylow
91C22×C47—21no—
92C22×C47—41no—
93C22×C47—42no—
94C22×C47—41no—
95C22×C47#40711yescharacteristic
96C4×C47—21no—
97C4×C47—41no—
98C4×C47#41011yescharacteristic
99C4×C47#41311yescharacteristic
100Dic47—22no—
101Dic47—42no—
102Dic47—41no—
103#2922—471no—
104#2165—472no—
105#1328—472no—
106#1321—471no—
107#1321D4711yescharacteristic
108C2×C47—21no—
109C2×C47—41no—
110C2×C47—81no—
111C2×C47#146411yescharacteristic
112C4×#41—471no—
113#433—942no—
114#431—942no—
115#424—471no—
116#424—471no—
117#408—471no—
118#408C2×D4711yescharacteristic
119C4×M16—471no—
120C2×C4×C8—471no—
121#364—942no—
122#349—472no—
123#349C2×D4712yes—
124#347—942no—
125C47#1152411yescharacteristic, complemented, Sylow
126C2×#41C4×D4711yescharacteristic
127C2×M16—471no—
128C2×M16C22×D4711yescharacteristic
129C22×C8—471no—
130C22×C8C4×D4711yescharacteristic
131C4×Q8—941no—
132C4×D4—942no—
133C4×D4—941no—
134#117—471no—
135#117—941no—
136C2×C42—471no—
137C2×C42—941no—
138M32—22no—
139M32—942no—
140C4⋊C8—942no—
141C4⋊C8—942no—
142C22⋊C8—22no—
143C22⋊C8—942no—
144#97—472no—
145C4×C8—472no—
146C4×C8—941no—
147C4×C8—941no—
148#41—41no—
149#41—41no—
150C2×Q8#1062532411yescharacteristic
151C2×D4—21no—
152C2×D4#1062532411yescharacteristic
153C22×C4—21no—
154C22×C4—942no—
155C22×C4—941no—
156C22×C4C2×C4×D4711yescharacteristic
157M16—41no—
158M16—1881no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—472no—
162C2×C8—941no—
163C2×C8—1881no—
164C2×C8D4×D4712yes—
165C4⋊C4—942no—
166C4⋊C4—941no—
167C22⋊C4—472no—
168C22⋊C4—942no—
169C42—471no—
170C42—471no—
171C42—942no—
172C42—941no—
173C42—941no—
174C16—42no—
175C16—1882no—
176C23—21no—
177C23M16×D4711yescharacteristic
178Q8—41no—
179D4—41no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—941no—
187C2×C4—941no—
188C2×C4—942no—
189C2×C4—1882no—
190C2×C4—1884no—
191C2×C4C22⋊C4×D4711yescharacteristic
192C2×C4C2×#1062532411yescharacteristic
193C8—22no—
194C8—41no—
195C8—942no—
196C8—1881no—
197C22—21no—
198C22—41no—
199C22—42no—
200C22—41no—
201C22#1568908511yescharacteristic
202C4—21no—
203C4—41no—
204C4—942no—
205C4—1881no—
206C4—1882no—
207C4#1568909511yescharacteristic
208C4#1568910411yescharacteristic
209C2—21no—
210C2—41no—
211C2—81no—
212C2#2024171911yescharacteristic
subgroupG/H
centerC4#15689095
commutatorC22×C4×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#1321C2
socleC2×C47#1464
prime radicalC2×#41×C47C2×C4
cosocleC2×M16×C47C23
hypercenter#1321D47
hypocenterC47#11524
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #20245197C2semidirect
3. #20241340C2semidirect
125. C47#11524semidirect

Maximal subgroups

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

subgroupindex
#202520282
#202451972
#202413402
#202413262
C47×#13212
#1152447