Finite Group Databasebeta

← #26127013

Subgroups — #26127013

Order 12032.

434 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15689062C211yesmaximal, characteristic
2#20243468C211yesmaximal, characteristic
3#20241416C214yesmaximal, complemented
4C47×#1463C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic
6#15689087—21no—
7#15689062—21no—
8#15689062—21no—
9#15689062C412yes—
10#15689062C2212yes—
11C2×C47⋊#95—21no—
12C2×C47⋊#95—21no—
13C2×C47⋊#95C411yescharacteristic
14#15688892—21no—
15#15688892C2212yes—
16C22×C4⋊C4×C47C2211yescharacteristic
17C47×#407—21no—
18C47×#324—21no—
19C47×#324C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C2×#10625329—22no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yes—
27C22⋊(C47⋊C8)—41no—
28C22⋊(C47⋊C8)—41no—
29C47⋊#95—22no—
30C47⋊#95—22no—
31C47⋊#95—21no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95C2×C412yes—
35C23×C4×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C2×M16×C47—41no—
39C2×C4⋊C4×C47—41no—
40C2×C4⋊C4×C47—41no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C4⋊C4×C47C2311yescharacteristic
45C22⋊C8×C47—22no—
46C22⋊C8×C47—41no—
47C24×C47C22⋊C411yescharacteristic
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47—41no—
56C22×C4×C47—42no—
57C22×C4×C47—42no—
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×Dic47—22no—
65C22×Dic47—22no—
66C22×Dic47—42no—
67C22×Dic47—42no—
68C22×Dic47—44no—
69C22×Dic47—42no—
70C22×Dic47—41no—
71C22×Dic47—41no—
72M16×C47—82no—
73C2×C8×C47—42no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C4⋊Dic47—42no—
79C4⋊Dic47—42no—
80#10625329—42no—
81#10625329—42no—
82C2×C47⋊C8—42no—
83C23×C47—22no—
84C23×C47—22no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47#9911yescharacteristic
88C23×C47#9911yescharacteristic
89C23×C47C2×C22⋊C411yescharacteristic
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—81no—
109C2×C4×C47#13712yes—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—44no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—84no—
116C2×Dic47—82no—
117C2×Dic47—82no—
118C8×C47—82no—
119C47⋊C8—82no—
120#11814—471nomaximal, Sylow
121C22×C47—24no—
122C22×C47—24no—
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—21no—
129C22×C47#35212yes—
130C22×C47#35512yes—
131C22×C47C2×#9911yescharacteristic
132C22×C47#41911yescharacteristic
133C22×C47#42111yescharacteristic
134C4×C47—22no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47—41no—
138C4×C47—41no—
139C4×C47—41no—
140Dic47—82no—
141Dic47—82no—
142C2×#400—471no—
143#1821—471no—
144#1463D4711yescharacteristic
145#1353—474no—
146C2×C47—21no—
147C2×C47—22no—
148C2×C47—22no—
149C2×C47—21no—
150C2×C47#151111yescharacteristic
151C2×C47C2×#35211yescharacteristic
152C2×C47C2×#35511yescharacteristic
153C22×C4⋊C4—471no—
154C22×C4⋊C4C2×D4711yescharacteristic
155#407—21no—
156#407—941no—
157#400—472no—
158#400—472no—
159#400—941no—
160#400—941no—
161C2×#95—471no—
162C2×#95—941no—
163C2×#95—941no—
164#343—472no—
165#343—941no—
166#324—21no—
167#324C2×D4712yes—
168C47#1181411yescharacteristic, complemented, Sylow
169C23×C4—941no—
170C23×C4—941no—
171C23×C4C4×D4711yescharacteristic
172C23×C4D18811yescharacteristic
173C23×C4C22⋊D4711yescharacteristic
174C2×M16—41no—
175C2×M16—942no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—472no—
179C2×C4⋊C4—941no—
180C2×C4⋊C4—1881no—
181C2×C4⋊C4—1881no—
182C2×C4⋊C4C4×D4711yescharacteristic
183C2×C4⋊C4D18811yescharacteristic
184C2×C4⋊C4C22⋊D4711yescharacteristic
185C2×C4⋊C4C22×D4711yescharacteristic
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—1881no—
189C22⋊C8—1881no—
190#95—472no—
191#95—941no—
192#95—942no—
193#95—942no—
194#95—942no—
195#95—942no—
196C24(C2×D47)⋊C411yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—942no—
208C22×C4—942no—
209C22×C4—1882no—
210C22×C4—1882no—
211C22×C4—1882no—
212C22×C4—1884no—
213C22×C4—1881no—
214C22×C4—1881no—
215C22×C4(C2×D47)⋊C411yescharacteristic
216C22×C4(C2×D47)⋊C411yescharacteristic
217C22×C4(C2×D47)⋊C411yescharacteristic
218C22×C4C2×C4×D4711yescharacteristic
219C22×C4C2×D18811yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221M16—82no—
222M16—1882no—
223M16—1882no—
224C2×C8—42no—
225C2×C8—1882no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—1882no—
231C4⋊C4—1882no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23#1309393811yescharacteristic
237C23#1309393811yescharacteristic
238C23C2×(C2×D47)⋊C411yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—3764no—
259C2×C4—3762no—
260C2×C4—3762no—
261C2×C4—1882no—
262C2×C4—1882no—
263C2×C4—1884no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4#1309405712yes—
267C8—82no—
268C8—3762no—
269C22—24no—
270C22—24no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22#1568891212yes—
278C22#1568891912yes—
279C22C2×#1309393811yescharacteristic
280C22#1568913311yescharacteristic
281C22#1568913911yescharacteristic
282C4—22no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—3762no—
289C4—3762no—
290C2—21no—
291C2—21no—
292C2—22no—
293C2—22no—
294C2#2024192211yescharacteristic
295C2C2×#1568891211yescharacteristic
296C2C2×#1568891911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1463C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1463D47
hypocenterC47#11814
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20241416C2semidirect
168. C47#11814semidirect

Maximal subgroups

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

subgroupindex
C2×#156890622
#202434682
#202414162
C47×#14632
#1181447