Finite Group Databasebeta

← #26127471

Subgroups — #26127471

Order 12032.

326 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 71 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#20253763C211yesmaximal, characteristic
2#20245200C211yesmaximal, characteristic, complemented
3#20241420C214yesmaximal, complemented
4C47×#2165C211yesmaximal, characteristic, complemented
5#15689680C411yescharacteristic
6C2×#13094021C411yescharacteristic
7#15689488C412yes—
8#15689488C2212yes—
9#15689470—21no—
10#15689470—21no—
11#15689129—21no—
12#15689129—21no—
13#15688944—21no—
14#15688905—21no—
15#15688901C2212yescomplemented
16C4×M16×C47C2211yescharacteristic
17C47×#364—21no—
18C47×#349—21no—
19C47×#347C2212yescomplemented
20C22×D188—21no—
21C22×D188—41no—
22#13094024—21no—
23#13094021—21no—
24#13094021—41no—
25#13094021C2×C412yes—
26#13094016C2×C412yes—
27#13094003—41no—
28C4×Dic94—41no—
29#13093983—21no—
30#13093964—22no—
31#13093946—22no—
32#13093946—22no—
33#13093946—22no—
34#13093946—22no—
35C2×M16×C47D411yescharacteristic
36C2×M16×C47D411yescharacteristic
37C2×C42×C47C2×C411yescharacteristic
38M32×C47—22no—
39#97×C47C2×C411yescharacteristic
40#97×C47C2311yescharacteristic
41C4×C8×C47D411yescharacteristic
42C4×C8×C47D411yescharacteristic
43C23×D47—41no—
44C22×C4×C47—21no—
45C22×C4×C47C22⋊C411yescharacteristic
46C2×D188—22no—
47C2×D188—42no—
48C2×D188—42no—
49C2×D188—42no—
50C2×D188—42no—
51C2×D188—41no—
52C2×D188—84no—
53C2×Dic94—41no—
54M16×C47—41no—
55M16×C47—41no—
56C2×C8×C47C22⋊C411yescharacteristic
57C2×C8×C47C22⋊C411yescharacteristic
58C2×C8×C47C2×D411yescharacteristic
59C2×C8×C47C2×D411yescharacteristic
60C42×C47—21no—
61C42×C47C22⋊C411yescharacteristic
62C42×C47C22×C411yescharacteristic
63C22⋊Dic47—41no—
64C4⋊Dic47—22no—
65C4⋊Dic47—41no—
66Dic47⋊C4—42no—
67C4×Dic47—41no—
68C16×C47—42no—
69C47⋊C16—42no—
70C23×C47#10011yescharacteristic
71C22×D47—42no—
72C22×D47—82no—
73C22×D47—84no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—42no—
79C2×C4×C47#10011yescharacteristic
80C2×C4×C47C2×C22⋊C411yescharacteristic
81C2×Dic47—42no—
82D188—42no—
83D188—42no—
84D188—82no—
85D188—84no—
86D188—84no—
87Dic94—82no—
88C8×C47—21no—
89C8×C47—21no—
90C8×C47—21no—
91C8×C47—21no—
92#11915—471nomaximal, Sylow
93C22×C47—21no—
94C22×C47—41no—
95C22×C47C2×#10011yescharacteristic
96C2×D47—84no—
97C2×D47—82no—
98C2×D47—164no—
99C4×C47—22no—
100C4×C47—21no—
101C4×C47—41no—
102C4×C47#42111yescharacteristic
103C4×C47#42111yescharacteristic
104Dic47—82no—
105#3138—471no—
106#2165—471no—
107#2165D4711yescharacteristic, complemented
108#1355—474no—
109C2×C47—21no—
110C2×C47—41no—
111C2×C47#153311yescharacteristic
112D47—162no—
113#533—471no—
114C2×#127—471no—
115#490—472no—
116#490—472no—
117#486—941no—
118#486—941no—
119#418—941no—
120#418—941no—
121C4×M16C2×D4711yescharacteristic
122#364—21no—
123#364—941no—
124#349—21no—
125#349—941no—
126#347—472no—
127#347C2×D4712yescomplemented
128C47#1191511yescharacteristic, complemented, Sylow
129C22×D4—941no—
130C22×D4—1881no—
131C2×M16D18811yescharacteristic
132C2×M16C22⋊D4711yescharacteristic
133C4⋊Q8—941no—
134#127—472no—
135#127—941no—
136#127—1881no—
137#125—472no—
138C22⋊Q8—1881no—
139C4×Q8—1881no—
140#117—941no—
141C2×C42C4×D4711yescharacteristic
142M32—22no—
143M32—942no—
144#102—942no—
145#102—942no—
146#102—942no—
147#102—942no—
148#97C4×D4711yescharacteristic
149#97C22×D4711yescharacteristic
150C4×C8D18811yescharacteristic
151C4×C8C22⋊D4711yescharacteristic
152C24—1881no—
153C2×Q8—1881no—
154C2×D4—942no—
155C2×D4—3764no—
156C2×D4—1882no—
157C2×D4—1882no—
158C2×D4—1882no—
159C2×D4—1882no—
160C2×D4—1881no—
161C22×C4—21no—
162C22×C4(C2×D47)⋊C411yescharacteristic
163M16—41no—
164M16—41no—
165C2×C8(C2×D47)⋊C411yescharacteristic
166C2×C8(C2×D47)⋊C411yescharacteristic
167C2×C8C2×D18811yescharacteristic
168C2×C8C2×C22⋊D4711yescharacteristic
169C4⋊C4—942no—
170C4⋊C4—1882no—
171C4⋊C4—1881no—
172C22⋊C4—1881no—
173C42—21no—
174C42—1881no—
175C42(C2×D47)⋊C411yescharacteristic
176C42C2×C4×D4711yescharacteristic
177C16—42no—
178C16—1882no—
179C23—3762no—
180C23—3764no—
181C23—1882no—
182C23#1309394011yescharacteristic
183Q8—3762no—
184D4—3764no—
185D4—3762no—
186D4—3764no—
187D4—1882no—
188D4—1882no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—1882no—
195C2×C4#1309394011yescharacteristic
196C2×C4C2×(C2×D47)⋊C411yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—21no—
201C22—21no—
202C22—41no—
203C22—3764no—
204C22—3762no—
205C22—7524no—
206C22C2×#1309394011yescharacteristic
207C4—22no—
208C4—21no—
209C4—41no—
210C4—3762no—
211C4#1568913911yescharacteristic
212C4#1568914111yescharacteristic
213C2—21no—
214C2—41no—
215C2—7522no—
216C2#2024201111yescharacteristic
subgroupG/H
centerC2#20242011
commutatorC42×C47C22×C4
Frattini#97C22×D47
FittingC47×#2165C2
socleC2×C47#1533
prime radicalC2×#13094021C4
cosocle#97×C47C23
hypercenter#2165D47
hypocenterC47#11915
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #20245200C2semidirect
3. #20241420C2semidirect
4. C47×#2165C2semidirect
15. #15688901C22semidirect
19. C47×#347C22semidirect
107. #2165D47semidirect
127. #347C2×D47semidirect
128. C47#11915semidirect

Maximal subgroups

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

subgroupindex
#202537632
#202452002
#202414202
C47×#21652
#1191547