Finite Group Databasebeta

← #26105089

Subgroups — #26105089

Order 12032.

536 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 95 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#20244721C212yesmaximal, complemented
2#20244470C211yesmaximal, characteristic
3#20241211C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#2683C211yesmaximal, characteristic
6C22×#10625335—21no—
7C2×#13094024C2211yescharacteristic
8C2×#13093955—22no—
9C2×#13093947—21no—
10#15688885—22no—
11#15688878C2212yes—
12#15688878C2211yescharacteristic
13C2×C22⋊Q8×C47—22no—
14C47×#396—22no—
15C47×#395—21no—
16C47×#390C2211yescharacteristic
17C47×#389C2212yes—
18C47×#386—21no—
19C22×Dic94—21no—
20C2×#10625335—42no—
21C2×#10625335—42no—
22C2×#10625335—42no—
23C22×C47⋊C8—21no—
24C22×C47⋊C8—22no—
25#13094024—42no—
26C2×C4⋊Dic47—22no—
27#13093955—44no—
28#13093947—42no—
29C22×Q8×C47—21no—
30C22×D4×C47D411yescharacteristic
31C23×C4×C47—21no—
32C22⋊Q8×C47—44no—
33C22⋊Q8×C47—44no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47D412yes—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C4⋊C4×C47D412yes—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47—21no—
42C2×C42×C47C2311yescharacteristic
43#95×C47—21no—
44#95×C47—22no—
45C24×C47—21no—
46C2×Q8×C47—44no—
47C2×Q8×C47—81no—
48C2×D4×C47—21no—
49C2×D4×C47—41no—
50C2×D4×C47—41no—
51C2×D4×C47SD1612yes—
52C22×C4×C47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47—44no—
56C22×C4×C47—81no—
57C22×C4×C47C2×D412yes—
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—41no—
60C2×Dic94—22no—
61C2×Dic94—41no—
62C2×Dic94—41no—
63C2×Dic94—41no—
64C2×Dic94—41no—
65C2×Dic94—41no—
66C4⋊C4×C47—22no—
67C4⋊C4×C47—44no—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—42no—
70C4⋊C4×C47—42no—
71C4⋊C4×C47—42no—
72C4⋊C4×C47SD1614yescomplemented
73C22⋊C4×C47—44no—
74C22⋊C4×C47—42no—
75C22⋊C4×C47—42no—
76C22⋊C4×C47—81no—
77C42×C47—41no—
78#10625335—48no—
79C4⋊Dic47—24no—
80C4⋊Dic47—42no—
81C2×C47⋊C8—22no—
82C2×C47⋊C8—24no—
83C2×C47⋊C8—42no—
84C2×C47⋊C8—41no—
85C2×C47⋊C8—41no—
86C2×C47⋊C8—42no—
87C23×C47—22no—
88C23×C47—42no—
89C23×C47—41no—
90C23×C47—42no—
91C23×C47—41no—
92C23×C47#12011yescharacteristic
93Q8×C47—84no—
94D4×C47—42no—
95D4×C47—81no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—24no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—41no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—84no—
109C2×C4×C47C2×SD1611yescharacteristic
110C2×C4×C47C2×SD1612yes—
111C2×C4×C47AGL(1,ℤ/8)12yes—
112C2×C4×C47#13711yescharacteristic
113C2×Dic47—42no—
114C2×Dic47—81no—
115C2×Dic47—81no—
116Dic94—42no—
117Dic94—42no—
118Dic94—81no—
119Dic94—81no—
120C47⋊C8—44no—
121C47⋊C8—42no—
122#7754—471nomaximal, Sylow
123C22×C47—42no—
124C22×C47—44no—
125C22×C47—82no—
126C22×C47—81no—
127C22×C47—81no—
128C22×C47#44811yescharacteristic
129C22×C47#45012yes—
130C22×C47#45311yescharacteristic
131C22×C47#45612yes—
132C22×C47B(4,𝔽2)11yescharacteristic
133C4×C47—21no—
134C4×C47—21no—
135C4×C47—22no—
136C4×C47—22no—
137C4×C47—44no—
138C4×C47—82no—
139C4×C47—81no—
140Dic47—82no—
141#2683D4711yescharacteristic
142#2058—472no—
143#2013—471no—
144#1271—472no—
145#1270—471no—
146C2×C47—82no—
147C2×C47#159611yescharacteristic
148C2×C47#160412yes—
149C2×C47#162911yescharacteristic
150C2×C47#165112yes—
151C2×C47#166011yescharacteristic
152C22×SD16—941no—
153C2×C4⋊Q8—471no—
154C2×C22⋊Q8—22no—
155C2×#106—942no—
156C2×#103—941no—
157#396—22no—
158#395—21no—
159#390C2×D4711yescharacteristic
160#389C2×D4712yes—
161#386—21no—
162#340—942no—
163#336—472no—
164#336—471no—
165C47#775411yescharacteristic, complemented, Sylow
166C22×Q8—21no—
167C22×Q8—941no—
168C22×D4C22⋊D4711yescharacteristic
169C23×C4—21no—
170C2×SD16—1882no—
171C2×SD16—1882no—
172C2×SD16—1882no—
173C22×C8—941no—
174C22×C8—942no—
175C4⋊Q8—1882no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—942no—
181C2×C4⋊C4C22⋊D4712yes—
182C2×C4⋊C4C22⋊D4712yes—
183C2×C4⋊C4C22⋊D4711yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C42C22×D4711yescharacteristic
188#106—1884no—
189#103—1882no—
190#95—21no—
191#95—22no—
192C24—21no—
193C2×Q8—44no—
194C2×Q8—81no—
195C2×Q8—942no—
196C2×Q8—1881no—
197C2×Q8—1881no—
198C2×Q8—1881no—
199C2×Q8—1881no—
200C2×Q8—1881no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4#1062533612yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—44no—
209C22×C4—81no—
210C22×C4—1881no—
211C22×C4C2×C22⋊D4711yescharacteristic
212C22×C4C2×C22⋊D4712yes—
213SD16—1888no—
214C2×C8—942no—
215C2×C8—944no—
216C2×C8—1882no—
217C2×C8—1881no—
218C2×C8—1881no—
219C2×C8—1882no—
220C4⋊C4—22no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—944no—
227C4⋊C4—1882no—
228C4⋊C4#1062533514yescomplemented
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—81no—
233C42—41no—
234C23—22no—
235C23—42no—
236C23—41no—
237C23—42no—
238C23—41no—
239C23#1309399311yescharacteristic
240Q8—84no—
241Q8—3761no—
242Q8—3761no—
243Q8—1882no—
244Q8—1882no—
245D4—42no—
246D4—81no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—84no—
260C2×C4—3761no—
261C2×C4—3761no—
262C2×C4—1882no—
263C2×C4C2×#1062533611yescharacteristic
264C2×C4C2×#1062533512yes—
265C2×C4#1309405012yes—
266C2×C4#1309405511yescharacteristic
267C8—1884no—
268C8—1882no—
269C22—42no—
270C22—44no—
271C22—81no—
272C22—81no—
273C22—82no—
274C22#1568926211yescharacteristic
275C22#1568927412yes—
276C22#1568929111yescharacteristic
277C22#1568930612yes—
278C22#1568931011yescharacteristic
279C4—21no—
280C4—21no—
281C4—22no—
282C4—22no—
283C4—44no—
284C4—81no—
285C4—82no—
286C4—3762no—
287C2—82no—
288C2#2024231111yescharacteristic
289C2#2024235712yes—
290C2#2024250411yescharacteristic
291C2#2024265112yes—
292C2#2024269911yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2683C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2683D47
hypocenterC47#7754
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #20244721C2semidirect
3. #20241211C2semidirect
72. C4⋊C4×C47SD16semidirect
165. C47#7754semidirect
228. C4⋊C4#10625335semidirect

Maximal subgroups

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

subgroupindex
#202447212
#202444702
#202412112
#202412092
C47×#26832
#775447