Finite Group Databasebeta

← #26113701

Subgroups — #26113701

Order 12032.

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

normal subgroups only

This group has 706 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, complemented
2#20248334C211yesmaximal, characteristic
3#20242057C212yesmaximal, complemented
4#20241289C212yesmaximal, complemented
5C47×#1800C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7C2×#13094020—21no—
8C2×#13093998—22no—
9#15689201—24no—
10#15689176C2212yes—
11#15689087—22no—
12#15689032C414yes—
13#15689024—21no—
14C4×C4⋊Dic47C2211yescharacteristic
15#15688866—24no—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#423C2212yes—
18C2×#100×C47—21no—
19C47×#343—21no—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23C22×C22⋊D47—24no—
24C2×#10625329—24no—
25#13094021—22no—
26#13094021—22no—
27#13094020—44no—
28#13093998—48no—
29C2×C4⋊Dic47—21no—
30C2×(C2×D47)⋊C4—24no—
31C2×C4×Dic47—21no—
32C4⋊(C47⋊C8)D414yescomplemented
33C22⋊(C47⋊C8)—24no—
34C22⋊(C47⋊C8)—22no—
35C4×C47⋊C8—22no—
36C47⋊#95C2×C412yes—
37C22×D4×C47D411yescharacteristic
38C23×C4×C47D412yes—
39C2×M16×C47—22no—
40C4×D4×C47—22no—
41C4×D4×C47—22no—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47C2×C412yes—
44C2×C42×C47C2311yescharacteristic
45C4⋊C8×C47—22no—
46C4⋊C8×C47—22no—
47#100×C47—42no—
48C24×C47C22⋊C412yes—
49C23×D47—22no—
50C2×D4×C47—22no—
51C2×D4×C47—22no—
52C2×D4×C47—21no—
53C2×D4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—44no—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C2×C22⋊D47—48no—
62C2×C22⋊D47—48no—
63C2×C22⋊D47—84no—
64C22×Dic47—22no—
65C2×D188—24no—
66C2×D188—44no—
67C2×D188—41no—
68C2×D188—42no—
69C2×D188—41no—
70C2×D188—42no—
71C2×D188—41no—
72C2×D188—41no—
73C2×D188—82no—
74M16×C47—42no—
75M16×C47—42no—
76C2×C8×C47—42no—
77C4⋊C4×C47—22no—
78C22⋊C4×C47—24no—
79C42×C47C2×D412yes—
80C42×C47#4112yes—
81(C2×D47)⋊C4—48no—
82C4⋊Dic47—24no—
83C4×Dic47—42no—
84#10625329—84no—
85C2×C47⋊C8—24no—
86C23×C47—22no—
87C23×C47—22no—
88C23×C47—22no—
89C23×C47—42no—
90C23×C47—42no—
91C23×C47C2×C22⋊C411yescharacteristic
92C22×D47—44no—
93C22×D47—44no—
94C22×D47—42no—
95C22×D47—42no—
96C22×D47—42no—
97D4×C47—42no—
98D4×C47—42no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—41no—
102C2×C4×C47—44no—
103C2×C4×C47—42no—
104C2×C4×C47—82no—
105C2×C4×C47C4×D412yes—
106C2×C4×C47#12011yescharacteristic
107C2×C4×C47#12112yes—
108C2×C4×C47#12311yescharacteristic
109C2×C4×C47AGL(1,ℤ/8)12yes—
110C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C47#13711yescharacteristic
112C22⋊D47—816no—
113C2×Dic47—24no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116D188—84no—
117D188—82no—
118D188—82no—
119D188—82no—
120D188—82no—
121C8×C47—82no—
122C47⋊C8—44no—
123#9405—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—44no—
127C22×C47—42no—
128C22×C47—42no—
129C22×C47—42no—
130C22×C47—82no—
131C22×C47#38611yescharacteristic
132C22×C47#41711yescharacteristic
133C22×C47#41811yescharacteristic
134C2×D47—84no—
135C2×D47—84no—
136C2×D47—82no—
137C2×D47—82no—
138C2×D47—82no—
139C4×C47—21no—
140C4×C47—22no—
141C4×C47—42no—
142C4×C47#45312yes—
143C4×C47#46812yes—
144C4×C47#46912yes—
145Dic47—44no—
146#2975—471no—
147#2541—471no—
148#1800D4711yescharacteristic, complemented
149#1545—472no—
150#1303—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47—82no—
154C2×C47#188711yescharacteristic
155C2×C47#189511yescharacteristic
156C2×C47#191411yescharacteristic
157D47—162no—
158C2×#127—471no—
159C2×#127—941no—
160C2×#121—942no—
161C2×C4×D4C2×D4711yescharacteristic
162#436—944no—
163#431—472no—
164#423C2×D4712yes—
165C2×#100—21no—
166#407—942no—
167#392—474no—
168#390—941no—
169C4×C4⋊C4—471no—
170#343—21no—
171#331—944no—
172C47#940511yescharacteristic, complemented, Sylow
173C22×D4—472no—
174C22×D4—944no—
175C22×D4—941no—
176C22×D4—941no—
177C22×D4C22⋊D4711yescharacteristic
178C23×C4D18812yes—
179C2×M16—22no—
180C2×M16—944no—
181#127—942no—
182#127—942no—
183#127—1884no—
184#121—1888no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—941no—
188C2×C4⋊C4C22⋊D4711yescharacteristic
189C2×C22⋊C4—944no—
190C2×C22⋊C4C4×D4712yes—
191C2×C42—941no—
192C2×C42C22×D4711yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—474no—
196#100—42no—
197C22⋊C8—944no—
198C22⋊C8—942no—
199C4×C8—942no—
200#95—472no—
201C24—942no—
202C24(C2×D47)⋊C412yes—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—944no—
208C2×D4—3762no—
209C2×D4—3764no—
210C2×D4—1884no—
211C2×D4—1888no—
212C2×D4—1888no—
213C2×D4—1881no—
214C2×D4—1882no—
215C2×D4—1881no—
216C2×D4—1882no—
217C2×D4—1881no—
218C2×D4—1881no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—44no—
222C22×C4—942no—
223C22×C4(C2×D47)⋊C412yes—
224C22×C4C2×C4×D4711yescharacteristic
225C22×C4C2×D18811yescharacteristic
226C22×C4C2×C22⋊D4711yescharacteristic
227M16—42no—
228M16—42no—
229M16—3764no—
230C2×C8—42no—
231C2×C8—944no—
232C4⋊C4—22no—
233C4⋊C4—944no—
234C22⋊C4—24no—
235C22⋊C4—1888no—
236C42—1882no—
237C42D4×D4712yes—
238C42#1062535212yes—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—1884no—
245C23—1884no—
246C23—1882no—
247C23—1882no—
248C23—1882no—
249C23C2×(C2×D47)⋊C411yescharacteristic
250D4—42no—
251D4—42no—
252D4—3764no—
253D4—3762no—
254D4—37616no—
255D4—3762no—
256D4—3762no—
257D4—3762no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—944no—
265C2×C4—1882no—
266C2×C4—1882no—
267C2×C4(C22⋊D47)⋊C412yes—
268C2×C4#1309399411yescharacteristic
269C2×C4#1309399612yes—
270C2×C4#1309400811yescharacteristic
271C2×C4#1309404912yes—
272C2×C4#1309405011yescharacteristic
273C2×C4#1309405511yescharacteristic
274C8—82no—
275C8—1884no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—82no—
283C22—3764no—
284C22—3764no—
285C22—3762no—
286C22—3762no—
287C22—3762no—
288C22#1568900011yescharacteristic
289C22#1568912211yescharacteristic
290C22#1568912711yescharacteristic
291C4—21no—
292C4—22no—
293C4—42no—
294C4—1884no—
295C4#1568929312yes—
296C4#1568937112yes—
297C4#1568937412yes—
298C2—21no—
299C2—21no—
300C2—82no—
301C2—7522no—
302C2#2024378411yescharacteristic
303C2#2024382411yescharacteristic
304C2#2024393311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1800C2
socleC22×C47#386
prime radical#20248334C2
cosocleC2×C42×C47C23
hypercenter#1800D47
hypocenterC47#9405
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
1. #20252429C2semidirect
3. #20242057C2semidirect
4. #20241289C2semidirect
5. C47×#1800C2semidirect
32. C4⋊(C47⋊C8)D4semidirect
148. #1800D47semidirect
172. C47#9405semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202483342
#202420572
#202412892
C47×#18002
#940547