Finite Group Databasebeta

← #26113704

Subgroups — #26113704

Order 12032.

706 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 123 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#20243380C211yesmaximal, characteristic, complemented
3#20242056C212yesmaximal, complemented
4#20241291C212yesmaximal, complemented
5C47×#2541C211yesmaximal, characteristic
6#15689201—24no—
7#15689176C2212yes—
8#15689147C2212yes—
9C2×#13093941—21no—
10#15689087—22no—
11#15688892—21no—
12#15688867—24no—
13C2×#127×C47—21no—
14C2×#127×C47C2211yescharacteristic
15C2×#121×C47—22no—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#392C414yes—
18C47×#390—21no—
19C4×C4⋊C4×C47C2211yescharacteristic
20C2×#10625329—24no—
21C2×#10625329—22no—
22C4⋊(C47⋊C8)—22no—
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)D414yescomplemented
25#13093941—42no—
26C22⋊(C47⋊C8)—24no—
27C22⋊(C47⋊C8)—22no—
28C4×C47⋊C8—22no—
29C22×D4×C47—24no—
30C22×D4×C47—21no—
31C22×D4×C47—21no—
32C22×D4×C47C2×C412yes—
33C22×D4×C47D411yescharacteristic
34C23×C4×C47D412yes—
35#127×C47—22no—
36#127×C47—22no—
37#127×C47—44no—
38#121×C47—48no—
39C4×D4×C47—22no—
40C4×D4×C47—22no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47—24no—
44C2×C22⋊C4×C47C2×C412yes—
45C2×C42×C47—21no—
46C2×C42×C47C2311yescharacteristic
47#95×C47C2×C412yes—
48C24×C47—22no—
49C24×C47C22⋊C412yes—
50C2×D4×C47—24no—
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47—21no—
54C2×D4×C47—21no—
55C2×D4×C47—44no—
56C2×D4×C47—48no—
57C2×D4×C47—48no—
58C2×D4×C47—41no—
59C2×D4×C47—42no—
60C2×D4×C47—41no—
61C2×D4×C47—42no—
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47—84no—
65C2×D4×C47—82no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—44no—
70C22×C4×C47C22⋊C412yes—
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C4⋊C4×C47—24no—
75C4⋊C4×C47—22no—
76C22⋊C4×C47—24no—
77C22⋊C4×C47—48no—
78C42×C47—42no—
79C42×C47C2×D412yes—
80C42×C47#4112yes—
81#10625329—42no—
82#10625329—42no—
83#10625329—84no—
84C2×C47⋊C8—24no—
85C2×C47⋊C8—42no—
86C23×C47—22no—
87C23×C47—22no—
88C23×C47—22no—
89C23×C47—42no—
90C23×C47—42no—
91C23×C47—44no—
92C23×C47—44no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47—42no—
96C23×C47C2×C22⋊C411yescharacteristic
97D4×C47—42no—
98D4×C47—42no—
99D4×C47—84no—
100D4×C47—82no—
101D4×C47—816no—
102D4×C47—82no—
103D4×C47—82no—
104D4×C47—82no—
105C2×C4×C47—22no—
106C2×C4×C47—24no—
107C2×C4×C47—22no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—44no—
112C2×C4×C47—42no—
113C2×C4×C47—82no—
114C2×C4×C47C4×D412yes—
115C2×C4×C47#12011yescharacteristic
116C2×C4×C47#12112yes—
117C2×C4×C47#12311yescharacteristic
118C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C47AGL(1,ℤ/8)12yes—
120C2×C4×C47#13711yescharacteristic
121C47⋊C8—44no—
122C47⋊C8—82no—
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—84no—
132C22×C47—84no—
133C22×C47—82no—
134C22×C47—82no—
135C22×C47—82no—
136C22×C47#38611yescharacteristic
137C22×C47#41711yescharacteristic
138C22×C47#41811yescharacteristic
139C4×C47—21no—
140C4×C47—22no—
141C4×C47—44no—
142C4×C47—42no—
143C4×C47#45312yes—
144C4×C47#46812yes—
145C4×C47#46912yes—
146#2975—471no—
147#2541D4711yescharacteristic
148#1800—471no—
149#1545—472no—
150#1303—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47—82no—
154C2×C47—162no—
155C2×C47#188711yescharacteristic
156C2×C47#189511yescharacteristic
157C2×C47#191411yescharacteristic
158C2×#127—21no—
159C2×#127C2×D4711yescharacteristic
160C2×#121—22no—
161C2×C4×D4C2×D4711yescharacteristic
162#436—944no—
163#431—472no—
164#423—472no—
165C2×#100—941no—
166#407—942no—
167#392Dic4714yes—
168#390—21no—
169C4×C4⋊C4C2×D4711yescharacteristic
170#343—941no—
171#331—944no—
172C47#940511yescharacteristic, complemented, Sylow
173C22×D4—24no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C2×Dic4712yes—
177C22×D4C22⋊D4711yescharacteristic
178C23×C4C22⋊D4712yes—
179C2×M16—944no—
180C2×M16—942no—
181#127—22no—
182#127—22no—
183#127—44no—
184#121—48no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C22⋊D4711yescharacteristic
189C2×C22⋊C4—24no—
190C2×C22⋊C4C2×Dic4712yes—
191C2×C42—21no—
192C2×C42C22×D4711yescharacteristic
193C4⋊C8—474no—
194C4⋊C8—942no—
195C4⋊C8—942no—
196#100—1882no—
197C22⋊C8—944no—
198C22⋊C8—942no—
199C4×C8—942no—
200#95C2×Dic4712yes—
201C24—22no—
202C24C22⋊Dic4712yes—
203C2×D4—24no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—44no—
209C2×D4—48no—
210C2×D4—48no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—84no—
218C2×D4—82no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—44no—
223C22×C4C22⋊Dic4712yes—
224C22×C4C22×Dic4711yescharacteristic
225C22×C4C2×C22⋊D4711yescharacteristic
226C22×C4C2×C22⋊D4711yescharacteristic
227M16—3764no—
228M16—1882no—
229M16—1882no—
230C2×C8—944no—
231C2×C8—1882no—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C22⋊C4—24no—
235C22⋊C4—48no—
236C42—42no—
237C42D4×D4712yes—
238C42#1062535212yes—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—44no—
245C23—44no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C2×C22⋊Dic4711yescharacteristic
250D4—42no—
251D4—42no—
252D4—84no—
253D4—82no—
254D4—816no—
255D4—82no—
256D4—82no—
257D4—82no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4D4×Dic4712yes—
268C2×C4#1309399211yescharacteristic
269C2×C4#1309399612yes—
270C2×C4#1309400511yescharacteristic
271C2×C4#1309404911yescharacteristic
272C2×C4#1309405012yes—
273C2×C4#1309405411yescharacteristic
274C8—3762no—
275C8—1884no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—82no—
283C22—84no—
284C22—84no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22#1568900311yescharacteristic
289C22#1568912411yescharacteristic
290C22#1568912811yescharacteristic
291C4—21no—
292C4—22no—
293C4—44no—
294C4—42no—
295C4#1568929012yes—
296C4#1568936712yes—
297C4#1568937412yes—
298C2—21no—
299C2—21no—
300C2—82no—
301C2—162no—
302C2#2024378711yescharacteristic
303C2#2024382711yescharacteristic
304C2#2024393611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#2541C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#2541D47
hypocenterC47#9405
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. #20252429C2semidirect
2. #20243380C2semidirect
3. #20242056C2semidirect
4. #20241291C2semidirect
24. C4⋊(C47⋊C8)D4semidirect
172. C47#9405semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202433802
#202420562
#202412912
C47×#25412
#940547