Finite Group Databasebeta

← #26105301

Subgroups — #26105301

Order 12032.

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

normal subgroups only

This group has 682 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#20244916C211yesmaximal, characteristic, complemented
2#20244791C212yesmaximal, complemented
3#20241209C211yesmaximal, characteristic, complemented
4#20241209C212yesmaximal, complemented
5C47×#2679C211yesmaximal, characteristic
6C2×#13093957—21no—
7C2×#13093945—22no—
8C2×#13093945—21no—
9#15688993C2211yescharacteristic, complemented
10#15688885—22no—
11#15688878C2212yescomplemented
12#15688878C2211yescharacteristic, complemented
13C2×#120×C47—22no—
14C2×#120×C47—21no—
15C47×#394—22no—
16C47×#394—21no—
17C47×#390C2212yes—
18C47×#390C2211yescharacteristic
19C22×C47⋊C8—21no—
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47—22no—
22C2×C4⋊Dic47—21no—
23#13093957—42no—
24#13093945—42no—
25#13093945—44no—
26C25×C47—21no—
27C22×D4×C47—21no—
28C22×D4×C47—22no—
29C22×D4×C47D412yes—
30C22×D4×C47D411yescharacteristic
31#120×C47—44no—
32#120×C47—42no—
33#120×C47—44no—
34#120×C47—42no—
35C2×C4⋊C4×C47D412yes—
36C2×C4⋊C4×C47D411yescharacteristic
37C2×C22⋊C4×C47—22no—
38C2×C22⋊C4×C47—21no—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47—22no—
42C2×C42×C47C2311yescharacteristic
43#95×C47—21no—
44C24×C47—22no—
45C24×C47—21no—
46C24×C47—44no—
47C24×C47—42no—
48C24×C47—81no—
49C24×C47—82no—
50C24×C47—81no—
51C2×D4×C47—22no—
52C2×D4×C47—21no—
53C2×D4×C47—42no—
54C2×D4×C47—42no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47—44no—
58C2×D4×C47—44no—
59C2×D4×C47—44no—
60C2×D4×C47—81no—
61C2×D4×C47—82no—
62C2×D4×C47D814yes—
63C2×D4×C47SD1612yes—
64C22×C4×C47—21no—
65C22×C4×C47—22no—
66C22×C4×C47C2×D412yes—
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—41no—
69C4⋊C4×C47—42no—
70C4⋊C4×C47—41no—
71C22⋊C4×C47—44no—
72C22⋊C4×C47—42no—
73C22⋊C4×C47—42no—
74C22⋊C4×C47—42no—
75C22⋊C4×C47—42no—
76C22⋊C4×C47—42no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—42no—
79C42×C47—41no—
80C4⋊Dic47—41no—
81C4⋊Dic47—41no—
82C4⋊Dic47—42no—
83C4⋊Dic47—42no—
84C2×C47⋊C8—22no—
85C2×C47⋊C8—41no—
86C2×C47⋊C8—41no—
87C2×C47⋊C8—42no—
88C2×C47⋊C8—42no—
89C2×C47⋊C8—42no—
90C23×C47—24no—
91C23×C47—22no—
92C23×C47—42no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47—41no—
96C23×C47—42no—
97C23×C47—42no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47—42no—
101C23×C47—41no—
102C23×C47—84no—
103C23×C47—88no—
104C23×C47—88no—
105C23×C47—84no—
106C23×C47—84no—
107C23×C47#12011yescharacteristic
108D4×C47—44no—
109D4×C47—42no—
110D4×C47—82no—
111D4×C47—81no—
112D4×C47—84no—
113D4×C47—88no—
114C2×C4×C47—21no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—21no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—41no—
122C2×C4×C47—42no—
123C2×C4×C47C2×D812yes—
124C2×C4×C47C2×SD1611yescharacteristic
125C2×C4×C47AGL(1,ℤ/8)12yes—
126C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
127C2×Dic47—82no—
128C2×Dic47—81no—
129C47⋊C8—42no—
130C47⋊C8—82no—
131#7786—471nomaximal, Sylow
132C22×C47—44no—
133C22×C47—42no—
134C22×C47—44no—
135C22×C47—44no—
136C22×C47—44no—
137C22×C47—816no—
138C22×C47—82no—
139C22×C47—82no—
140C22×C47—82no—
141C22×C47—81no—
142C22×C47—82no—
143C22×C47—81no—
144C22×C47—82no—
145C22×C47#44712yes—
146C22×C47#45011yescharacteristic
147C22×C47#45311yescharacteristic
148C22×C47#45512yes—
149C22×C47#45811yescharacteristic
150C4×C47—22no—
151C4×C47—22no—
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—81no—
155C4×C47—82no—
156Dic47—161no—
157#2679D4711yescharacteristic
158#2093—471no—
159#2071—472no—
160#1270—471no—
161#1270—472no—
162C2×C47—84no—
163C2×C47—82no—
164C2×C47#160712yes—
165C2×C47#161311yescharacteristic
166C2×C47#161911yescharacteristic
167C2×C47#163912yes—
168C2×C47#165911yescharacteristic
169C2×#120—22no—
170C2×#120—21no—
171C2×#107—941no—
172C2×#102—942no—
173C2×#102—941no—
174#394—22no—
175#394—21no—
176#390C2×D4712yes—
177#390C2×D4711yescharacteristic
178#384—471no—
179#340—942no—
180#336—472no—
181#336—471no—
182C47#778611yescharacteristic, complemented, Sylow
183C25—21no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4C22⋊D4712yes—
187C22×D4C22⋊D4711yescharacteristic
188C22×C8—941no—
189C22×C8—942no—
190#120—44no—
191#120—42no—
192#120—44no—
193#120—42no—
194C2×C4⋊C4—942no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4C22⋊D4712yes—
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C42C22×D4711yescharacteristic
204#107—1882no—
205#102—1882no—
206#102—1884no—
207#95—21no—
208C24—22no—
209C24—21no—
210C24—44no—
211C24—42no—
212C24—81no—
213C24—82no—
214C24—81no—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—81no—
225C2×D4—82no—
226C2×D4#1062533414yes—
227C2×D4#1062533512yes—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—1881no—
231C22×C4C2×C22⋊D4712yes—
232C22×C4C2×C22⋊D4711yescharacteristic
233C2×C8—942no—
234C2×C8—1882no—
235C2×C8—1882no—
236C2×C8—1882no—
237C2×C8—1881no—
238C2×C8—1881no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—1881no—
242C4⋊C4—1881no—
243C4⋊C4—1882no—
244C4⋊C4—1882no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—41no—
254C23—24no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—84no—
267C23—88no—
268C23—88no—
269C23—84no—
270C23—84no—
271C23#1309399311yescharacteristic
272D4—44no—
273D4—42no—
274D4—82no—
275D4—81no—
276D4—84no—
277D4—88no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—3762no—
288C2×C4—3761no—
289C2×C4C2×#1062533412yes—
290C2×C4C2×#1062533511yescharacteristic
291C2×C4#1309405012yes—
292C2×C4#1309405011yescharacteristic
293C8—3762no—
294C8—1882no—
295C22—44no—
296C22—44no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—816no—
301C22—81no—
302C22—82no—
303C22—81no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22#1568925712yes—
309C22#1568927411yescharacteristic
310C22#1568929111yescharacteristic
311C22#1568930112yes—
312C22#1568931311yescharacteristic
313C4—22no—
314C4—22no—
315C4—21no—
316C4—21no—
317C4—81no—
318C4—82no—
319C4—7521no—
320C2—84no—
321C2—82no—
322C2#2024237412yes—
323C2#2024240811yescharacteristic
324C2#2024244211yescharacteristic
325C2#2024256712yes—
326C2#2024269411yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#2679C2
socleC23×C47#120
hypercenter#2679D47
hypocenterC47#7786
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2679

Complements

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

NH ≅ G/Ntype
1. #20244916C2semidirect
2. #20244791C2semidirect
3. #20241209C2semidirect
4. #20241209C2semidirect
9. #15688993C22semidirect
11. #15688878C22semidirect
12. #15688878C22semidirect
182. C47#7786semidirect

Maximal subgroups

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

subgroupindex
#202449162
#202447912
#202412092
C47×#26792
#778647