Finite Group Databasebeta

← #26109844

Subgroups — #26109844

Order 12032.

498 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 131 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#20243380C211yesmaximal, characteristic
2#20241709C211yesmaximal, characteristic
3#20241291C212yesmaximal, complemented
4#20241246C212yesmaximal, complemented
5C47×#2349C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093941—21no—
9#15688892—21no—
10#15688847—22no—
11C2×C4×D4×C47—21no—
12C2×C4×D4×C47C2211yescharacteristic
13C47×#389—21no—
14C47×#385—22no—
15C47×#382C2211yescharacteristic
16C47×#380C412yes—
17C4×C4⋊C4×C47C2211yescharacteristic
18C2×#95×C47C412yes—
19C2×#10625329—22no—
20C4⋊(C47⋊C8)—22no—
21C4⋊(C47⋊C8)—22no—
22C4⋊(C47⋊C8)—22no—
23#13093941—42no—
24C22⋊(C47⋊C8)—22no—
25C4×C47⋊C8—22no—
26C22×D4×C47D411yescharacteristic
27C23×C4×C47—22no—
28C23×C4×C47C2×C412yes—
29C4×D4×C47—22no—
30C4×D4×C47—22no—
31C4×D4×C47—42no—
32C4×D4×C47—42no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47D411yescharacteristic
37C2×C22⋊C4×C47—22no—
38C2×C22⋊C4×C47D412yes—
39C2×C42×C47—21no—
40C2×C42×C47C2311yescharacteristic
41#95×C47—22no—
42#95×C47—21no—
43#95×C47C2×C411yescharacteristic
44#95×C47C2×C412yes—
45#95×C47C2×C411yescharacteristic
46C24×C47C22⋊C412yes—
47C2×D4×C47—22no—
48C2×D4×C47—22no—
49C2×D4×C47D812yes—
50C2×D4×C47SD1612yes—
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—21no—
55C22×C4×C47—22no—
56C22×C4×C47—44no—
57C22×C4×C47—44no—
58C22×C4×C47—82no—
59C22×C4×C47—82no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C4⋊C4×C47—22no—
65C4⋊C4×C47—42no—
66C4⋊C4×C47—41no—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—82no—
69C22⋊C4×C47—24no—
70C22⋊C4×C47—44no—
71C42×C47—41no—
72C42×C47—41no—
73C42×C47#4112yes—
74C42×C47#4112yes—
75#10625329—42no—
76#10625329—42no—
77C2×C47⋊C8—42no—
78C2×C47⋊C8—42no—
79C23×C47—22no—
80C23×C47—22no—
81C23×C47—42no—
82C23×C47—42no—
83C23×C47#10214yes—
84C23×C47C2×C22⋊C411yescharacteristic
85D4×C47—24no—
86D4×C47—24no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—42no—
100C2×C4×C47—82no—
101C2×C4×C47—82no—
102C2×C4×C47—88no—
103C2×C4×C47#11711yescharacteristic
104C2×C4×C47#11711yescharacteristic
105C2×C4×C47#12312yes—
106C2×C4×C47#12312yes—
107C2×C4×C47C2×D811yescharacteristic
108C2×C4×C47C2×SD1611yescharacteristic
109C2×C4×C47#13712yes—
110C47⋊C8—82no—
111C47⋊C8—82no—
112#8764—471nomaximal, Sylow
113C22×C47—21no—
114C22×C47—21no—
115C22×C47—24no—
116C22×C47—44no—
117C22×C47—44no—
118C22×C47—42no—
119C22×C47—42no—
120C22×C47#38111yescharacteristic
121C22×C47C2×#10211yescharacteristic
122C22×C47#41911yescharacteristic
123C4×C47—22no—
124C4×C47—21no—
125C4×C47—21no—
126C4×C47—42no—
127C4×C47—42no—
128C4×C47—81no—
129C4×C47—82no—
130C4×C47#48012yes—
131C4×C47#48312yes—
132#2349D4711yescharacteristic
133#1800—471no—
134#1460—471no—
135#1303—472no—
136#1286—472no—
137C2×C47—21no—
138C2×C47—21no—
139C2×C47—44no—
140C2×C47#182311yescharacteristic
141C2×C47#182711yescharacteristic
142C2×C47#182911yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4C2×D4711yescharacteristic
145#431—472no—
146#423—472no—
147C2×#100—941no—
148#389—21no—
149#385—22no—
150#382C2×D4711yescharacteristic
151#380Dic4712yes—
152C4×C4⋊C4C2×D4711yescharacteristic
153C2×#95Dic4712yes—
154#343—941no—
155#323—942no—
156C47#876411yescharacteristic, complemented, Sylow
157C22×D4C22⋊D4711yescharacteristic
158C23×C4—22no—
159C23×C4C2×Dic4712yes—
160C2×M16—942no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—42no—
164C4×D4—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊D4711yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4C22⋊D4712yes—
171C2×C42—21no—
172C2×C42C22×D4711yescharacteristic
173C4⋊C8—942no—
174C4⋊C8—942no—
175C4⋊C8—942no—
176#100—1882no—
177C22⋊C8—942no—
178C4×C8—942no—
179#95—22no—
180#95—21no—
181#95C2×Dic4711yescharacteristic
182#95C2×Dic4712yes—
183#95C2×Dic4711yescharacteristic
184C24C22⋊Dic4712yes—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4#1062533412yes—
188C2×D4#1062533512yes—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4C22⋊Dic4712yes—
199C22×C4C22×Dic4711yescharacteristic
200C22×C4C2×C22⋊D4711yescharacteristic
201C22×C4C2×C22⋊D4711yescharacteristic
202M16—1882no—
203M16—1882no—
204C2×C8—1882no—
205C2×C8—1882no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C22⋊C4—24no—
212C22⋊C4—44no—
213C42—41no—
214C42—41no—
215C42#1062535012yes—
216C42#1062535012yes—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23#1309394514yes—
222C23C2×C22⋊Dic4711yescharacteristic
223D4—24no—
224D4—24no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—88no—
241C2×C4#1309398311yescharacteristic
242C2×C4#1309398311yescharacteristic
243C2×C4#1309400712yes—
244C2×C4#1309400712yes—
245C2×C4C2×#1062533411yescharacteristic
246C2×C4C2×#1062533511yescharacteristic
247C2×C4#1309405512yes—
248C8—3762no—
249C8—3762no—
250C22—21no—
251C22—21no—
252C22—24no—
253C22—44no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22#1568898611yescharacteristic
258C22C2×#1309394511yescharacteristic
259C22#1568913211yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—82no—
266C4—81no—
267C4#1568943612yes—
268C4#1568945312yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#2024347411yescharacteristic
273C2#2024349211yescharacteristic
274C2#2024349911yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2349C2
socleC22×C47#381
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2349D47
hypocenterC47#8764
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #20241291C2semidirect
4. #20241246C2semidirect
156. C47#8764semidirect

Maximal subgroups

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

subgroupindex
#202433802
#202417092
#202412912
#202412462
C47×#23492
#876447