Finite Group Databasebeta

← #26109737

Subgroups — #26109737

Order 12032.

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

normal subgroups only

This group has 622 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
1C2×#15688847C212yesmaximal
2#20241236C214yesmaximal, complemented
3C47×#2273C211yesmaximal, characteristic
4C2×C22⋊(C47⋊C8)C2214yes—
5#15688847—22no—
6#15688847—22no—
7C22×C22⋊C4×C47C411yescharacteristic
8C22×C22⋊C4×C47C411yescharacteristic
9C22×C22⋊C4×C47C2211yescharacteristic
10C4×C22⋊C4×C47—22no—
11C4×C22⋊C4×C47—22no—
12C4×C22⋊C4×C47C412yes—
13C4×C22⋊C4×C47C2212yes—
14C22×C47⋊C8—24no—
15C22⋊(C47⋊C8)—24no—
16C22⋊(C47⋊C8)—24no—
17C25×C47C2×C411yescharacteristic
18C23×C4×C47—21no—
19C23×C4×C47—21no—
20C23×C4×C47C2×C411yescharacteristic
21C23×C4×C47C2311yescharacteristic
22C2×C22⋊C4×C47—21no—
23C2×C22⋊C4×C47—21no—
24C2×C22⋊C4×C47—22no—
25C2×C22⋊C4×C47—24no—
26C2×C22⋊C4×C47—24no—
27C2×C22⋊C4×C47C814yescomplemented
28C2×C22⋊C4×C47C2×C412yes—
29C2×C22⋊C4×C47C2×C412yes—
30C2×C22⋊C4×C47D412yes—
31C2×C22⋊C4×C47D412yes—
32C2×C42×C47—22no—
33C2×C42×C47—22no—
34#95×C47—22no—
35#95×C47—22no—
36C24×C47—22no—
37C24×C47—22no—
38C24×C47—81no—
39C24×C47—81no—
40C24×C47C22⋊C411yescharacteristic
41C24×C47C22⋊C411yescharacteristic
42C24×C47C2×C812yes—
43C24×C47M1612yes—
44C24×C47C22×C411yescharacteristic
45C22×C4×C47—22no—
46C22×C4×C47—22no—
47C22×C4×C47—24no—
48C22×C4×C47—24no—
49C22×C4×C47—22no—
50C22×C4×C47—81no—
51C22×C4×C47—81no—
52C22×C4×C47—81no—
53C22×C4×C47—81no—
54C22×C4×C47—81no—
55C22×C4×C47—81no—
56C22×C4×C47C22⋊C412yes—
57C22×C4×C47C2×D412yes—
58C22×C4×C47#4114yes—
59C22⋊C4×C47—24no—
60C22⋊C4×C47—24no—
61C22⋊C4×C47—22no—
62C22⋊C4×C47—22no—
63C22⋊C4×C47—24no—
64C22⋊C4×C47—24no—
65C22⋊C4×C47C2×C814yes—
66C22⋊C4×C47M1614yes—
67C42×C47—82no—
68C42×C47—82no—
69C2×C47⋊C8—44no—
70C2×C47⋊C8—84no—
71C23×C47—21no—
72C23×C47—21no—
73C23×C47—21no—
74C23×C47—21no—
75C23×C47—24no—
76C23×C47—24no—
77C23×C47—41no—
78C23×C47—41no—
79C23×C47—41no—
80C23×C47—41no—
81C23×C47—41no—
82C23×C47—41no—
83C23×C47—82no—
84C23×C47—82no—
85C23×C47—82no—
86C23×C47—82no—
87C23×C47—84no—
88C23×C47C22⋊C812yes—
89C23×C47C22⋊C812yes—
90C23×C47C2×C22⋊C411yescharacteristic
91C23×C47#11712yes—
92C23×C47C22×C811yescharacteristic
93C23×C47C2×M1611yescharacteristic
94C23×C47C2×M1612yes—
95C2×C4×C47—22no—
96C2×C4×C47—24no—
97C2×C4×C47—24no—
98C2×C4×C47—44no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—82no—
106C2×C4×C47—84no—
107C2×C4×C47—84no—
108C2×C4×C47#12314yes—
109C47⋊C8—84no—
110#8734—471nomaximal, Sylow
111C22×C47—21no—
112C22×C47—22no—
113C22×C47—21no—
114C22×C47—22no—
115C22×C47—22no—
116C22×C47—22no—
117C22×C47—41no—
118C22×C47—41no—
119C22×C47—42no—
120C22×C47—42no—
121C22×C47—42no—
122C22×C47—42no—
123C22×C47—42no—
124C22×C47—41no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—41no—
128C22×C47—84no—
129C22×C47—84no—
130C22×C47#38111yescharacteristic
131C22×C47C2×C22⋊C811yescharacteristic
132C22×C47#40711yescharacteristic
133C22×C47#43112yes—
134C22×C47#43212yes—
135C4×C47—24no—
136C4×C47—44no—
137C4×C47—44no—
138C4×C47—42no—
139#2273D4711yescharacteristic
140C2×#323—472no—
141#1282—474no—
142C2×C47—21no—
143C2×C47—21no—
144C2×C47—22no—
145C2×C47—22no—
146C2×C47—42no—
147C2×C47—42no—
148C2×C47#181711yescharacteristic
149C2×C47#182811yescharacteristic
150C2×C47#182911yescharacteristic
151C22×C22⋊C4Dic4711yescharacteristic
152C22×C22⋊C4Dic4711yescharacteristic
153C22×C22⋊C4C2×D4711yescharacteristic
154C2×C22⋊C8—474no—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—22no—
157C4×C22⋊C4Dic4712yes—
158C4×C22⋊C4C2×D4712yes—
159#323—942no—
160#323—942no—
161C47#873411yescharacteristic, complemented, Sylow
162C25C2×Dic4711yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×Dic4711yescharacteristic
166C23×C4C22×D4711yescharacteristic
167C22×C8—944no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4C47⋊C814yescomplemented
174C2×C22⋊C4C2×Dic4712yes—
175C2×C22⋊C4C2×Dic4712yes—
176C2×C22⋊C4C22⋊D4712yes—
177C2×C22⋊C4C22⋊D4712yes—
178C2×C42—22no—
179C2×C42—22no—
180C22⋊C8—944no—
181C22⋊C8—944no—
182#95—22no—
183#95—22no—
184C24—22no—
185C24—22no—
186C24—81no—
187C24—81no—
188C24C2×C47⋊C812yes—
189C24#1062532912yes—
190C24C22⋊Dic4711yescharacteristic
191C24C22⋊Dic4711yescharacteristic
192C24C22×Dic4711yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4C22⋊Dic4712yes—
205C22×C4#1062535014yes—
206C22×C4C2×C22⋊D4712yes—
207C2×C8—3764no—
208C2×C8—1884no—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4C2×C47⋊C814yes—
216C22⋊C4#1062532914yes—
217C42—82no—
218C42—82no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—24no—
224C23—24no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23—84no—
236C23C22⋊(C47⋊C8)12yes—
237C23C22⋊(C47⋊C8)12yes—
238C23C2×C22⋊Dic4711yescharacteristic
239C23#1309398312yes—
240C23C22×C47⋊C811yescharacteristic
241C23C2×#1062532911yescharacteristic
242C23C2×#1062532912yes—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4#1309400714yes—
257C8—3764no—
258C22—21no—
259C22—22no—
260C22—21no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—84no—
276C22—84no—
277C22#1568898611yescharacteristic
278C22C2×C22⋊(C47⋊C8)11yescharacteristic
279C22#1568908711yescharacteristic
280C22#1568917612yes—
281C22#1568918012yes—
282C4—24no—
283C4—44no—
284C4—44no—
285C4—42no—
286C2—21no—
287C2—21no—
288C2—22no—
289C2—22no—
290C2—42no—
291C2—42no—
292C2#2024345611yescharacteristic
293C2#2024349611yescharacteristic
294C2#2024349911yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC47×#2273C2
socleC22×C47#381
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#2273D47
hypocenterC47#8734
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2273

Complements

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

NH ≅ G/Ntype
2. #20241236C2semidirect
27. C2×C22⋊C4×C47C8semidirect
161. C47#8734semidirect
173. C2×C22⋊C4C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#156888472
#202412362
C47×#22732
#873447