Finite Group Databasebeta

← #26113681

Subgroups — #26113681

Order 12032.

534 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 115 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#20252429C211yesmaximal, characteristic
2#20248380C211yesmaximal, characteristic
3#20242061C212yesmaximal, complemented
4#20241289C212yesmaximal, complemented
5C47×#1800C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7C2×#13094011—21no—
8#15689201—24no—
9#15689176C2212yes—
10#15689087—22no—
11#15689048C414yes—
12#15689028—21no—
13#15688983—22no—
14C4×C4⋊Dic47C2211yescharacteristic
15#15688870—24no—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#423C2212yes—
18C2×#100×C47—21no—
19C47×#343—21no—
20C22×Dic94—21no—
21C2×#10625329—24no—
22#13094024—22no—
23#13094024—22no—
24#13094011—44no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C412yes—
28C2×C22⋊Dic47—24no—
29C2×C4×Dic47—21no—
30C4⋊(C47⋊C8)D414yes—
31C22⋊(C47⋊C8)—24no—
32C22⋊(C47⋊C8)—22no—
33C4×C47⋊C8—22no—
34C47⋊#95—24no—
35C47⋊#95C2×C412yes—
36C22×D4×C47D411yescharacteristic
37C23×C4×C47D412yes—
38C2×M16×C47—22no—
39C4×D4×C47—22no—
40C4×D4×C47—22no—
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47C2×C412yes—
43C2×C42×C47C2311yescharacteristic
44C4⋊C8×C47—22no—
45C4⋊C8×C47—22no—
46#100×C47—42no—
47C24×C47C22⋊C412yes—
48C2×D4×C47—22no—
49C2×D4×C47—22no—
50C2×D4×C47—21no—
51C2×D4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—44no—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—22no—
60C22×Dic47—22no—
61C2×Dic94—42no—
62C2×Dic94—42no—
63C2×Dic94—41no—
64C2×Dic94—41no—
65M16×C47—42no—
66M16×C47—42no—
67C2×C8×C47—42no—
68C4⋊C4×C47—22no—
69C22⋊C4×C47—24no—
70C42×C47C2×D412yes—
71C42×C47#4112yes—
72C22⋊Dic47—48no—
73C4⋊Dic47—24no—
74C4⋊Dic47—42no—
75C4⋊Dic47—41no—
76C4⋊Dic47—41no—
77C4×Dic47—42no—
78#10625329—84no—
79C2×C47⋊C8—24no—
80C23×C47—22no—
81C23×C47—22no—
82C23×C47—22no—
83C23×C47—42no—
84C23×C47—42no—
85C23×C47C2×C22⋊C411yescharacteristic
86D4×C47—42no—
87D4×C47—42no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—41no—
91C2×C4×C47—44no—
92C2×C4×C47—42no—
93C2×C4×C47—82no—
94C2×C4×C47C4×D412yes—
95C2×C4×C47#12011yescharacteristic
96C2×C4×C47#12112yes—
97C2×C4×C47#12311yescharacteristic
98C2×C4×C47AGL(1,ℤ/8)12yes—
99C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C47#13711yescharacteristic
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104C2×Dic47—42no—
105C2×Dic47—42no—
106C2×Dic47—42no—
107Dic94—82no—
108Dic94—82no—
109C8×C47—82no—
110C47⋊C8—44no—
111#9401—471nomaximal, Sylow
112C22×C47—21no—
113C22×C47—21no—
114C22×C47—44no—
115C22×C47—42no—
116C22×C47—42no—
117C22×C47—42no—
118C22×C47—82no—
119C22×C47#38611yescharacteristic
120C22×C47#41711yescharacteristic
121C22×C47#41811yescharacteristic
122C4×C47—21no—
123C4×C47—22no—
124C4×C47—42no—
125C4×C47#45512yes—
126C4×C47#46812yes—
127C4×C47#46912yes—
128Dic47—82no—
129Dic47—82no—
130#2975—471no—
131#2547—471no—
132#1800D4711yescharacteristic
133#1546—472no—
134#1303—472no—
135C2×C47—21no—
136C2×C47—21no—
137C2×C47—82no—
138C2×C47#188511yescharacteristic
139C2×C47#189511yescharacteristic
140C2×C47#191511yescharacteristic
141C2×C4⋊Q8—471no—
142C2×#124—941no—
143C2×C4×D4C2×D4711yescharacteristic
144#436—944no—
145#431—472no—
146#423C2×D4712yes—
147C2×#100—21no—
148#407—942no—
149#397—474no—
150#391—941no—
151#380—942no—
152C4×C4⋊C4—471no—
153#343—21no—
154#332—944no—
155C47#940111yescharacteristic, complemented, Sylow
156C22×Q8—941no—
157C22×D4C22⋊D4711yescharacteristic
158C23×C4D18812yes—
159C2×M16—22no—
160C2×M16—944no—
161C4⋊Q8—942no—
162C4⋊Q8—942no—
163#124—1884no—
164C4×D4—22no—
165C4×D4—22no—
166C2×C4⋊C4—472no—
167C2×C4⋊C4—941no—
168C2×C4⋊C4—941no—
169C2×C4⋊C4C22⋊D4711yescharacteristic
170C2×C22⋊C4—944no—
171C2×C22⋊C4C4×D4712yes—
172C2×C42—941no—
173C2×C42C22×D4711yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—474no—
177#100—42no—
178C22⋊C8—944no—
179C22⋊C8—942no—
180C4×C8—942no—
181#95—472no—
182#95—944no—
183C24(C2×D47)⋊C412yes—
184C2×Q8—1882no—
185C2×Q8—1882no—
186C2×Q8—1881no—
187C2×Q8—1881no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—942no—
196C22×C4—942no—
197C22×C4(C2×D47)⋊C412yes—
198C22×C4C2×C4×D4711yescharacteristic
199C22×C4C2×D18811yescharacteristic
200C22×C4C2×C22⋊D4711yescharacteristic
201M16—42no—
202M16—42no—
203M16—3764no—
204C2×C8—42no—
205C2×C8—944no—
206C4⋊C4—22no—
207C4⋊C4—944no—
208C4⋊C4—1882no—
209C4⋊C4—1881no—
210C4⋊C4—1881no—
211C22⋊C4—24no—
212C22⋊C4—1888no—
213C42—1882no—
214C42D4×D4712yes—
215C42#1062535212yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23C2×(C2×D47)⋊C411yescharacteristic
222Q8—3762no—
223Q8—3762no—
224D4—42no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—1882no—
233C2×C4—1882no—
234C2×C4—1882no—
235C2×C4—1882no—
236C2×C4—1882no—
237C2×C4—1882no—
238C2×C4(C22⋊D47)⋊C412yes—
239C2×C4#1309399411yescharacteristic
240C2×C4#1309399612yes—
241C2×C4#1309400811yescharacteristic
242C2×C4#1309404912yes—
243C2×C4#1309405011yescharacteristic
244C2×C4#1309405511yescharacteristic
245C8—82no—
246C8—1884no—
247C22—21no—
248C22—21no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22#1568900011yescharacteristic
255C22#1568912211yescharacteristic
256C22#1568912711yescharacteristic
257C4—21no—
258C4—22no—
259C4—42no—
260C4—3762no—
261C4—3762no—
262C4#1568930312yes—
263C4#1568937112yes—
264C4#1568937412yes—
265C2—21no—
266C2—21no—
267C2—82no—
268C2#2024377411yescharacteristic
269C2#2024382411yescharacteristic
270C2#2024393811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1800C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#1800D47
hypocenterC47#9401
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20242061C2semidirect
4. #20241289C2semidirect
155. C47#9401semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202483802
#202420612
#202412892
C47×#18002
#940147