Finite Group Databasebeta

← #26113671

Subgroups — #26113671

Order 12032.

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

normal subgroups only

This group has 514 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#20248561C211yesmaximal, characteristic
3#20242070C212yesmaximal, complemented
4#20241289C212yesmaximal, complemented
5C47×#1800C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7#15689201—24no—
8#15689176C2212yes—
9#15689087—22no—
10#15689035C414yes—
11#15689026—21no—
12#15688988—21no—
13#15688983—22no—
14C4×C4⋊Dic47C2211yescharacteristic
15#15688871—24no—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#423C2212yes—
18C2×#100×C47—21no—
19C47×#343—21no—
20C22×Dic94C2×C412yes—
21C2×#10625329—24no—
22#13094024—22no—
23#13094024—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C22⋊Dic47—24no—
28C2×C4×Dic47—21no—
29C4⋊(C47⋊C8)D414yes—
30C22⋊(C47⋊C8)—24no—
31C22⋊(C47⋊C8)—22no—
32C4×C47⋊C8—22no—
33C47⋊#95—24no—
34C47⋊#95C2×C412yes—
35C22×D4×C47D411yescharacteristic
36C23×C4×C47D412yes—
37C2×M16×C47—22no—
38C4×D4×C47—22no—
39C4×D4×C47—22no—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C22⋊C4×C47C2×C412yes—
42C2×C42×C47C2311yescharacteristic
43C4⋊C8×C47—22no—
44C4⋊C8×C47—22no—
45#100×C47—42no—
46C24×C47C22⋊C412yes—
47C2×D4×C47—22no—
48C2×D4×C47—21no—
49C2×D4×C47—22no—
50C2×D4×C47—21no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—44no—
54C22×C4×C47C22⋊C412yes—
55C22×C4×C47C22×C411yescharacteristic
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×Dic47—22no—
59C22×Dic47—22no—
60C2×Dic94—24no—
61C2×Dic94—82no—
62M16×C47—42no—
63M16×C47—42no—
64C2×C8×C47—42no—
65C4⋊C4×C47—22no—
66C22⋊C4×C47—24no—
67C42×C47C2×D412yes—
68C42×C47#4112yes—
69C22⋊Dic47—84no—
70C4⋊Dic47—42no—
71C4⋊Dic47—41no—
72C4⋊Dic47—42no—
73C4⋊Dic47—41no—
74C4×Dic47—41no—
75C4×Dic47—41no—
76#10625329—84no—
77C2×C47⋊C8—24no—
78C23×C47—22no—
79C23×C47—22no—
80C23×C47—22no—
81C23×C47—42no—
82C23×C47—42no—
83C23×C47C2×C22⋊C411yescharacteristic
84D4×C47—42no—
85D4×C47—42no—
86C2×C4×C47—22no—
87C2×C4×C47—22no—
88C2×C4×C47—41no—
89C2×C4×C47—44no—
90C2×C4×C47—42no—
91C2×C4×C47—82no—
92C2×C4×C47C4×D412yes—
93C2×C4×C47#12011yescharacteristic
94C2×C4×C47#12112yes—
95C2×C4×C47#12311yescharacteristic
96C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
97C2×C4×C47#13712yes—
98C2×C4×C47#13711yescharacteristic
99C2×Dic47—42no—
100C2×Dic47—42no—
101C2×Dic47—44no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104Dic94—84no—
105C8×C47—82no—
106C47⋊C8—44no—
107#9399—471nomaximal, Sylow
108C22×C47—21no—
109C22×C47—21no—
110C22×C47—44no—
111C22×C47—42no—
112C22×C47—42no—
113C22×C47—42no—
114C22×C47—82no—
115C22×C47#38611yescharacteristic
116C22×C47#41711yescharacteristic
117C22×C47#41911yescharacteristic
118C4×C47—21no—
119C4×C47—22no—
120C4×C47—42no—
121C4×C47#45512yes—
122C4×C47#46812yes—
123C4×C47#47012yes—
124Dic47—82no—
125Dic47—82no—
126#2975—471no—
127#2563—471no—
128#1800D4711yescharacteristic
129#1548—472no—
130#1303—472no—
131C2×C47—21no—
132C2×C47—21no—
133C2×C47—82no—
134C2×C47#188511yescharacteristic
135C2×C47#189411yescharacteristic
136C2×C47#191411yescharacteristic
137C2×C4⋊Q8—471no—
138C2×C4×D4C2×D4711yescharacteristic
139#436—944no—
140#431—472no—
141#423C2×D4712yes—
142C2×#100—21no—
143#407—942no—
144#393—474no—
145#390—941no—
146#382—941no—
147#380—942no—
148C4×C4⋊C4—471no—
149#343—21no—
150#333—944no—
151C47#939911yescharacteristic, complemented, Sylow
152C22×Q8—472no—
153C22×D4C22⋊D4711yescharacteristic
154C23×C4D18812yes—
155C2×M16—22no—
156C2×M16—944no—
157C4⋊Q8—942no—
158C4⋊Q8—942no—
159C4×D4—22no—
160C4×D4—22no—
161C2×C4⋊C4—941no—
162C2×C4⋊C4—941no—
163C2×C4⋊C4—941no—
164C2×C4⋊C4C22⋊D4711yescharacteristic
165C2×C22⋊C4—944no—
166C2×C22⋊C4C4×D4712yes—
167C2×C42—941no—
168C2×C42C22×D4711yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—474no—
172#100—42no—
173C22⋊C8—944no—
174C22⋊C8—942no—
175C4×C8—942no—
176#95—472no—
177#95—944no—
178C24(C2×D47)⋊C412yes—
179C2×Q8—944no—
180C2×Q8—3762no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—942no—
189C22×C4—942no—
190C22×C4(C2×D47)⋊C412yes—
191C22×C4C2×C4×D4711yescharacteristic
192C22×C4C2×D18811yescharacteristic
193C22×C4C2×C22⋊D4711yescharacteristic
194M16—42no—
195M16—42no—
196M16—3764no—
197C2×C8—42no—
198C2×C8—944no—
199C4⋊C4—22no—
200C4⋊C4—1882no—
201C4⋊C4—1881no—
202C4⋊C4—1882no—
203C4⋊C4—1881no—
204C22⋊C4—24no—
205C22⋊C4—3764no—
206C42—1881no—
207C42—1881no—
208C42D4×D4712yes—
209C42#1062535212yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×(C2×D47)⋊C411yescharacteristic
216Q8—3764no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—1882no—
226C2×C4—1884no—
227C2×C4—1882no—
228C2×C4—1882no—
229C2×C4—1882no—
230C2×C4(C22⋊D47)⋊C412yes—
231C2×C4#1309399411yescharacteristic
232C2×C4#1309399612yes—
233C2×C4#1309400811yescharacteristic
234C2×C4#1309405011yescharacteristic
235C2×C4#1309405412yes—
236C2×C4#1309405511yescharacteristic
237C8—82no—
238C8—1884no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1568900011yescharacteristic
247C22#1568912211yescharacteristic
248C22#1568913111yescharacteristic
249C4—21no—
250C4—22no—
251C4—42no—
252C4—3762no—
253C4—3762no—
254C4#1568930312yes—
255C4#1568936712yes—
256C4#1568937912yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2024377411yescharacteristic
261C2#2024381911yescharacteristic
262C2#2024393311yescharacteristic
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#9399
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. #20242070C2semidirect
4. #20241289C2semidirect
151. C47#9399semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202485612
#202420702
#202412892
C47×#18002
#939947