Finite Group Databasebeta

← #26104355

Subgroups — #26104355

Order 12032.

846 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 155 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#20243456C212yesmaximal, complemented
2#20241209C214yesmaximal, complemented
3C2×C47×#390C211yesmaximal, characteristic
4C2×C22⋊(C47⋊C8)—22no—
5#15688878—22no—
6#15688878C2214yescomplemented
7C23×D4×C47C411yescharacteristic
8C22×C4⋊C4×C47C411yescharacteristic
9C22×C22⋊C4×C47—22no—
10C22×C42×C47C2211yescharacteristic
11C47×#390—22no—
12C47×#390C412yes—
13C47×#390C2212yes—
14C22×C47⋊C8—24no—
15C22⋊(C47⋊C8)—44no—
16C25×C47—21no—
17C22×D4×C47—21no—
18C22×D4×C47—22no—
19C22×D4×C47—42no—
20C22×D4×C47—42no—
21C22×D4×C47C814yescomplemented
22C22×D4×C47C2×C412yes—
23C23×C4×C47—21no—
24C23×C4×C47C2×C411yescharacteristic
25C23×C4×C47D412yes—
26C2×C4⋊C4×C47—21no—
27C2×C4⋊C4×C47—42no—
28C2×C4⋊C4×C47C2×C412yes—
29C2×C22⋊C4×C47—24no—
30C2×C22⋊C4×C47—24no—
31C2×C22⋊C4×C47—48no—
32C2×C42×C47—42no—
33C2×C42×C47C2×C411yescharacteristic
34C2×C42×C47D412yes—
35C2×C42×C47C2311yescharacteristic
36C24×C47—24no—
37C24×C47—22no—
38C24×C47—44no—
39C24×C47—44no—
40C24×C47—42no—
41C24×C47—42no—
42C24×C47C22⋊C411yescharacteristic
43C2×D4×C47—24no—
44C2×D4×C47—22no—
45C2×D4×C47—48no—
46C2×D4×C47—42no—
47C2×D4×C47—42no—
48C2×D4×C47—48no—
49C2×D4×C47—42no—
50C2×D4×C47—42no—
51C2×D4×C47C2×C814yes—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—24no—
55C22×C4×C47—24no—
56C22×C4×C47—22no—
57C22×C4×C47—42no—
58C22×C4×C47—44no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C2×C812yes—
63C22×C4×C47M1612yes—
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D412yes—
66C4⋊C4×C47—42no—
67C4⋊C4×C47—42no—
68C22⋊C4×C47—48no—
69C22⋊C4×C47—48no—
70C42×C47—41no—
71C42×C47—42no—
72C42×C47—41no—
73C2×C47⋊C8—44no—
74C2×C47⋊C8—44no—
75C2×C47⋊C8—44no—
76C23×C47—22no—
77C23×C47—22no—
78C23×C47—24no—
79C23×C47—416no—
80C23×C47—48no—
81C23×C47—42no—
82C23×C47—42no—
83C23×C47—44no—
84C23×C47—44no—
85C23×C47—84no—
86C23×C47—88no—
87C23×C47—84no—
88C23×C47C22⋊C812yes—
89C23×C47#9912yes—
90C23×C47#10012yes—
91C23×C47C2×C22⋊C411yescharacteristic
92D4×C47—44no—
93D4×C47—48no—
94D4×C47—44no—
95C2×C4×C47—24no—
96C2×C4×C47—24no—
97C2×C4×C47—22no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—48no—
107C2×C4×C47—42no—
108C2×C4×C47—44no—
109C2×C4×C47—84no—
110C2×C4×C47C22⋊C812yes—
111C2×C4×C47C22×C811yescharacteristic
112C2×C4×C47C2×M1611yescharacteristic
113C2×C4×C47AGL(1,ℤ/8)14yes—
114C47⋊C8—84no—
115#7592—471nomaximal, Sylow
116C22×C47—22no—
117C22×C47—24no—
118C22×C47—24no—
119C22×C47—22no—
120C22×C47—44no—
121C22×C47—48no—
122C22×C47—816no—
123C22×C47—82no—
124C22×C47—84no—
125C22×C47—82no—
126C22×C47#32314yes—
127C22×C47C2×C22⋊C811yescharacteristic
128C22×C47C2×#9911yescharacteristic
129C22×C47C2×#10011yescharacteristic
130C22×C47#41812yes—
131C22×C47#42112yes—
132C4×C47—22no—
133C4×C47—21no—
134C4×C47—21no—
135C4×C47—24no—
136C4×C47—42no—
137C4×C47—82no—
138C2×#390D4711yescharacteristic
139#1817—472no—
140#1270—474no—
141C2×C47—24no—
142C2×C47—84no—
143C2×C47C2×#32311yescharacteristic
144C2×C47#147812yes—
145C2×C47#151012yes—
146C2×C47#153311yescharacteristic
147C2×C47#155511yescharacteristic
148C23×D4Dic4711yescharacteristic
149C22×C4⋊C4Dic4711yescharacteristic
150C22×C22⋊C4—22no—
151C22×C42C2×D4711yescharacteristic
152C2×C22⋊C8—942no—
153#390—22no—
154#390Dic4712yes—
155#390C2×D4712yes—
156#336—474no—
157#336—942no—
158C47#759211yescharacteristic, complemented, Sylow
159C25—21no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4C47⋊C814yescomplemented
165C22×D4C2×Dic4712yes—
166C23×C4—21no—
167C23×C4C2×Dic4711yescharacteristic
168C23×C4C22⋊D4712yes—
169C22×C8—944no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×Dic4712yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—48no—
176C2×C42—42no—
177C2×C42C2×Dic4711yescharacteristic
178C2×C42C22⋊D4712yes—
179C2×C42C22×D4711yescharacteristic
180C22⋊C8—1884no—
181C24—24no—
182C24—22no—
183C24—44no—
184C24—44no—
185C24—42no—
186C24—42no—
187C24C22⋊Dic4711yescharacteristic
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—48no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—48no—
196C2×D4C2×C47⋊C814yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4C2×C47⋊C812yes—
206C22×C4#1062532912yes—
207C22×C4C22⋊Dic4712yes—
208C22×C4C22⋊Dic4711yescharacteristic
209C22×C4C22×Dic4711yescharacteristic
210C22×C4C2×C22⋊D4712yes—
211C2×C8—1884no—
212C2×C8—1884no—
213C2×C8—1884no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C42—41no—
219C42—42no—
220C42—41no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—416no—
225C23—42no—
226C23—42no—
227C23—48no—
228C23—44no—
229C23—44no—
230C23—84no—
231C23—84no—
232C23—88no—
233C23C22⋊(C47⋊C8)12yes—
234C23#1309393912yes—
235C23#1309394112yes—
236C23C2×C22⋊Dic4711yescharacteristic
237D4—44no—
238D4—48no—
239D4—44no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—48no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—84no—
255C2×C4C22⋊(C47⋊C8)12yes—
256C2×C4C22×C47⋊C811yescharacteristic
257C2×C4C2×#1062532911yescharacteristic
258C2×C4#1309405014yes—
259C8—3764no—
260C22—22no—
261C22—24no—
262C22—22no—
263C22—24no—
264C22—44no—
265C22—48no—
266C22—816no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22#1568884714yes—
271C22C2×C22⋊(C47⋊C8)11yescharacteristic
272C22C2×#1309393911yescharacteristic
273C22C2×#1309394111yescharacteristic
274C22#1568912812yes—
275C22#1568914012yes—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—24no—
280C4—42no—
281C4—82no—
282C2—24no—
283C2—84no—
284C2C2×#1568884711yescharacteristic
285C2#2024177912yes—
286C2#2024191712yes—
287C2#2024201211yescharacteristic
288C2#2024210211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC2×C47×#390C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D4×C47C4
cosocleC2×C42×C47C23
hypercenterC2×#390D47
hypocenterC47#7592
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. #20243456C2semidirect
2. #20241209C2semidirect
6. #15688878C22semidirect
21. C22×D4×C47C8semidirect
158. C47#7592semidirect
164. C22×D4C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202434562
#202412092
C2×C47×#3902
#759247