Finite Group Databasebeta

← #26127479

Subgroups — #26127479

Order 12032.

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

normal subgroups only

This group has 280 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#20245200C212yesmaximal, complemented
2#20241423C212yesmaximal, complemented
3#20241421C212yesmaximal, complemented
4C47×#3142C211yesmaximal, characteristic
5#15688944—22no—
6#15688905—22no—
7#15688901C2214yes—
8C47×#532C411yescharacteristic
9C2×#125×C47C411yescharacteristic
10C47×#491C411yescharacteristic
11C47×#491C2211yescharacteristic
12C47×#490C411yescharacteristic
13C47×#490C2211yescharacteristic
14C47×#488—22no—
15C47×#417—22no—
16C4×M16×C47C2211yescharacteristic
17#13093964—24no—
18C2×#41×C47—21no—
19C2×M16×C47D412yes—
20C4⋊Q8×C47C2×C411yescharacteristic
21#127×C47C2×C411yescharacteristic
22#125×C47—21no—
23#125×C47—41no—
24#125×C47C2×C411yescharacteristic
25#125×C47C2×C411yescharacteristic
26#124×C47—21no—
27#121×C47—41no—
28C4×D4×C47—41no—
29C2×C4⋊C4×C47—21no—
30C2×C4⋊C4×C47—41no—
31C2×C42×C47C2×C411yescharacteristic
32#103×C47—22no—
33#103×C47—22no—
34#102×C47—22no—
35#102×C47—22no—
36#97×C47C2×C411yescharacteristic
37#97×C47C2311yescharacteristic
38C4×C8×C47D412yes—
39#41×C47—44no—
40C2×Q8×C47—21no—
41C2×D4×C47—21no—
42C2×D4×C47—41no—
43C2×D4×C47—41no—
44C22×C4×C47—21no—
45C22×C4×C47—41no—
46C22×C4×C47—41no—
47C22×C4×C47C22⋊C411yescharacteristic
48M16×C47—42no—
49C2×C8×C47C22⋊C412yes—
50C2×C8×C47C2×D412yes—
51C4⋊C4×C47—21no—
52C4⋊C4×C47—21no—
53C4⋊C4×C47—42no—
54C4⋊C4×C47—41no—
55C4⋊C4×C47—41no—
56C4⋊C4×C47—41no—
57C4⋊C4×C47—41no—
58C22⋊C4×C47—42no—
59C42×C47—21no—
60C42×C47C22⋊C411yescharacteristic
61C42×C47C22×C411yescharacteristic
62C47⋊C16—44no—
63C23×C47—41no—
64C23×C47#10011yescharacteristic
65Q8×C47—42no—
66D4×C47—42no—
67D4×C47—82no—
68D4×C47—82no—
69C2×C4×C47—22no—
70C2×C4×C47—21no—
71C2×C4×C47—21no—
72C2×C4×C47—41no—
73C2×C4×C47—41no—
74C2×C4×C47—41no—
75C2×C4×C47—82no—
76C2×C4×C47—81no—
77C2×C4×C47—82no—
78C2×C4×C47#10011yescharacteristic
79C2×C4×C47C2×C22⋊C411yescharacteristic
80C8×C47—22no—
81C8×C47—22no—
82#11917—471nomaximal, Sylow
83C22×C47—21no—
84C22×C47—41no—
85C22×C47—82no—
86C22×C47—81no—
87C22×C47C2×#10011yescharacteristic
88C4×C47—21no—
89C4×C47—41no—
90C4×C47—41no—
91C4×C47—81no—
92C4×C47—81no—
93C4×C47—81no—
94C4×C47#42112yes—
95#3142D4711yescharacteristic
96#2165—472no—
97#1356—472no—
98#1355—472no—
99C2×C47—21no—
100C2×C47—41no—
101C2×C47—161no—
102C2×C47#153311yescharacteristic
103#532Dic4711yescharacteristic
104C2×#125Dic4711yescharacteristic
105#491Dic4711yescharacteristic
106#491C2×D4711yescharacteristic
107#490Dic4711yescharacteristic
108#490C2×D4711yescharacteristic
109#488—22no—
110#417—22no—
111C4×M16C2×D4711yescharacteristic
112#364—942no—
113#349—942no—
114#347—474no—
115C47#1191711yescharacteristic, complemented, Sylow
116C2×#41—21no—
117C2×M16C22⋊D4712yes—
118C4⋊Q8C2×Dic4711yescharacteristic
119#127C2×Dic4711yescharacteristic
120#125—21no—
121#125—41no—
122#125C2×Dic4711yescharacteristic
123#125C2×Dic4711yescharacteristic
124#124—21no—
125#121—41no—
126C4×D4—41no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—41no—
129C2×C42C2×Dic4711yescharacteristic
130M32—944no—
131#103—22no—
132#103—22no—
133#102—22no—
134#102—22no—
135#97C2×Dic4711yescharacteristic
136#97C22×D4711yescharacteristic
137C4×C8C22⋊D4712yes—
138#41—44no—
139C2×Q8—21no—
140C2×D4—21no—
141C2×D4—41no—
142C2×D4—41no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4C22⋊Dic4711yescharacteristic
147M16—42no—
148C2×C8C22⋊Dic4712yes—
149C2×C8C2×C22⋊D4712yes—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C22⋊C4—42no—
158C42—21no—
159C42C22⋊Dic4711yescharacteristic
160C42C22×Dic4711yescharacteristic
161C16—1884no—
162C23—41no—
163C23#1309394111yescharacteristic
164Q8—42no—
165D4—42no—
166D4—82no—
167D4—82no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—82no—
175C2×C4—81no—
176C2×C4—82no—
177C2×C4#1309394111yescharacteristic
178C2×C4C2×C22⋊Dic4711yescharacteristic
179C8—22no—
180C8—22no—
181C22—21no—
182C22—41no—
183C22—82no—
184C22—81no—
185C22C2×#1309394111yescharacteristic
186C4—21no—
187C4—41no—
188C4—41no—
189C4—81no—
190C4—81no—
191C4—81no—
192C4#1568914012yes—
193C2—21no—
194C2—41no—
195C2—161no—
196C2#2024201211yescharacteristic
subgroupG/H
centerC2#20242012
commutatorC42×C47C22×C4
Frattini#97C22×D47
FittingC47×#3142C2
socleC2×C47#1533
prime radicalC47×#532C4
cosocle#97×C47C23
hypercenter#3142D47
hypocenterC47#11917
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3142

Complements

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

NH ≅ G/Ntype
1. #20245200C2semidirect
2. #20241423C2semidirect
3. #20241421C2semidirect
115. C47#11917semidirect

Maximal subgroups

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

subgroupindex
#202452002
#202414232
#202414212
C47×#31422
#1191747